Classical Hilbert–Lie groups, their extensions and their homotopy groups

Karl-Hermann Neeb

Abstract. Let H be a complex Hilbert space and D a hermitian operator on H with finite spectrum. Then the operators for which the commutator with D is of Schatten class p form a Banach algebra B_{p}(H,D). In the present paper we study groups GL_{p}(H,D) associated to this kind of Lie algebra and also groups GL_{p}(H,I,D) associated to the sub Lie algebras B_{p}(H,I,D):={x∈ B_{p}(H,D):Ix^{∗}I^{−1}=−x}, where I is an antilinear isometry with I^{2}∈{±1}. For p=2 we determine the full second continuous cohomology for these Lie algebras and for the groups we compute all homotopy groups. These results then lead to a direct description of universal central extensions of the groups GL_{2}(H,D),GL_{2}(H,I,D) and some of their real forms. In particular we obtain the infinite-dimensional metaplectic and the metagonal group as special examples. In a last section we discuss associated complex flag manifolds and show that the unitary forms of the complex groups act transitively.

Introduction

An important feature of finite-dimensional reductive Lie algebras 𝔤 is that they always possess a positive definite bilinear form ⟨·, ·⟩ and a so called Cartan involution θ such that

(0.1)  ⟨[x, y], z⟩ = −⟨y, [θ(x), z]⟩ for x, y, z ∈ 𝔤.

Here the case θ = id_{𝔤} corresponds to the case of a compact Lie algebra.

An interesting class of infinite-dimensional Lie algebras generalizing finite-dimensional real reductive Lie algebras, in the sense that they still have the structure provided by (0.1), are the L^{∗} -algebras. More precisely, these are Lie algebras 𝔤 which are real Hilbert spaces endowed with an isometric Lie algebra involution x ↦ x^{∗}, i.e.,

x^{∗∗} = x and [x, y]^{∗} = [y^{∗}, x^{∗}] for x, y ∈ 𝔤,

such that

(0.2)  ⟨[x, y], z⟩ = ⟨y, [x^{∗}, z]⟩ for x, y, z ∈ 𝔤.

Using the Closed Graph Theorem, one can derive the continuity of the Lie bracket on 𝔤 from (0.2), so that this requirement does not have to be put into the axioms of an L^{∗} -algebra. If 𝔤 is finite-dimensional real reductive, we may define x^{∗} := −θ(x) for a Cartan involution θ. A complex L^{∗} -algebra is a real L^{∗} -algebra which is a complex Lie algebra for which the involution ∗ is antilinear. This easily implies that 𝔤 can be turned into a complex Hilbert space by

⟨x, y⟩_{ℂ} := ⟨x, y⟩ − i⟨ix, y⟩, x, y ∈ 𝔤,

such that (0.2) is satisfied for the hermitian scalar product ⟨·, ·⟩_{ℂ}. The structure theory of real and complex L^{∗} -algebras has mostly been developed by Schue, Balachandran, de la Harpe and Unsain, where certain key references are [Sch60], [Sch61], [Ba69], [dlH72] and [Un72]. One finds further references in de la Harpe’s book.

The L^{∗} -algebras of compact type, where x^{∗} = −x for all x ∈ 𝔤, are natural generalizations of compact Lie algebras, so that the corresponding groups G are generalizations of compact groups. This point of view suggests that their representation theory lies at the heart of the representation theory of infinite-dimensional Lie groups. Although we won’t deal with representations in the present paper, it has very much been motivated by [Ne01a], where we approach the representation theory via coadjoint orbits of central extensions, or equivalently affine coadjoint orbits of G. At many points it turns out to be important to have specific geometric and topological information on groups corresponding to simple L^{∗} -algebras of compact type and their complexifications. It is the goal of this paper to provide such information in the concrete setting of groups of operators on Hilbert spaces, without using the structure theory of L^{∗} -algebras.

The simple infinite-dimensional complex L^{∗} -algebras arise in three series which can be described as follows. If H is a complex Hilbert space, we write

𝔤𝔩_{2}(H) := {x ∈ B(H): tr(xx^{∗}) < ∞}

for the Lie algebra of Hilbert-Schmidt operators on H (cf. Definition I.8). If I: H → H is an antilinear isometry with I^{2} ∈ {±1}, we define

𝔤𝔩_{2}(H, I) := 𝔤(H, I) ∩ 𝔤𝔩_{2}(H) and 𝔤𝔩(H, I) := {X ∈ 𝔤𝔩(H): X + IX^{∗}I^{−1} = 0}.

For I^{2} = −1 we also write 𝔰𝔭_{2}(H, I) := 𝔤𝔩_{2}(H, I) and for I^{2} = 1 we write 𝔬_{2}(H, I) := 𝔤𝔩_{2}(H, I). This notation is motivated by the observation that β(x, y) := ⟨x, I.y⟩ defines a complex bilinear form on H which is symmetric for I^{2} = 1 and skew-symmetric for I^{2} = −1 and which satisfies

𝔤𝔩_{2}(H, I) = {x ∈ 𝔤𝔩_{2}(H): (∀v, w ∈ H) β(x.v, w) + β(v, x.w) = 0}.

Each simple infinite-dimensional L^{∗} -algebra 𝔤 is isomorphic to 𝔤𝔩_{2}(H), 𝔰𝔭_{2}(H, I) or 𝔬_{2}(H, I) for some infinite-dimensional Hilbert space H, and all these algebras are pairwise non-isomorphic (see [Sch60] for the separable case and [CGM90], [Neh93] and [St99] for different proofs for the general case). Since we want to treat 𝔰𝔭_{2}(H, I) and 𝔬_{2}(H, I) in a uniform way, this leaves us with the two types 𝔤𝔩_{2}(H) and 𝔤𝔩_{2}(H, I). Separable real simple L^{∗} -algebras have been classified independently by Balachandran ([Ba69]), de la Harpe ([dlH70, 71a]) and Unsain ([Un71, 72]).

Passing from finite-dimensional semisimple complex Lie algebras to infinite-dimensional L^{∗} - algebras, an important new feature is that these algebras have many non-trivial central extensions and outer derivations which lead to a large family of related Banach–Lie algebras, where each one of them has its own specific merits.

It is a general phenomenon in infinite-dimensional Lie theory that non-trivial central extensions play an important role because geometric actions of groups on certain manifolds M do not lift to actions on line bundles over M. One first has to enlarge the group by a central extension. On the representation theoretic side this means that the original symmetry groups only have projective representations in natural function spaces, and that central extensions are required to obtain genuine representations. For a general approach to central extensions of infinite-dimensional Lie groups and criteria for their existences in terms of topological data we refer to [Ne00b].

It is related, but in general of a different nature, that infinite-dimensional Lie algebras have many outer derivations, leading to a different kind of extension of the original Lie algebra. For the classical L^{∗} -algebras, this kind of extension process leads in particular to the corresponding restricted Lie algebras and groups. For 𝔤 = 𝔤𝔩_{2}(H), resp., 𝔤𝔩_{2}(H, I) we put 𝔤_{b} := 𝔤𝔩(H): = B(H), resp., 𝔤𝔩(H, I). Let D ∈ 𝔤_{b} be a hermitian element with finite spectrum. Then the Lie algebra 𝔤(D) := 𝔤 + 𝔷_{𝔤_{b}}(D) is called the restricted Lie algebra associated to 𝔤 and D. For 𝔤 = 𝔤𝔩_{2}(H) the restricted Lie algebras 𝔤𝔩_{2}(H, D) := 𝔤(D) has the form

𝔤𝔩_{2}(H, D) = 𝔤𝔩_{2}(H) + ∑_{j=1}^{k} 𝔤𝔩(H_{j}) = 𝔤 + 𝔷_{𝔤_{b}}(D),

where H = H_{1} ⊕ ⋯ ⊕ H_{k} is the orthogonal eigenspace decomposition for D, and 𝔷_{𝔤_{b}}(D) corresponds to the Lie algebra of all operators preserving all the spaces H_{j}. The terminology “restricted” comes from the fact that operator x = (x_{ij}) ∈ 𝔤𝔩(H), viewed as a k ×k -block matrix with entries x_{ij} ∈ B(H_{j}, H_{i}), is contained in 𝔤𝔩(H, D) if and only if all its off-diagonal blocks x_{ij} ∈ B(H_{j}, H_{i}), i ≠ j, are Hilbert–Schmidt, which we view as a restriction on x.

Now we describe the contents of this paper in some more detail. In Section I we deal with the central extensions of the Lie algebras related to 𝔤𝔩_{2}(H) and 𝔤𝔩_{2}(H, I). In particular we show that 𝔤𝔩(H) and 𝔤𝔩(H, I) have no non-trivial central extension if H is infinite-dimensional. Using this information, we calculate the continuous second Lie algebra cohomology group H_{c}^{2}(𝔤(D), ℂ) for the restricted Lie algebras 𝔤(D) which turns out to be finite-dimensional. This in turn implies the existence of a universal central extension which we describe explicitly in Section IV. For universality of central extensions we use results from [Ne01b], where universal central extensions of infinite-dimensional Lie groups are studied in detail.

Sections II and III are devoted to the homotopy groups of the corresponding groups. Here the groups π_{k}, k = 0, 1, 2, are of particular importance. For k = 0 this is obvious, the group π_{1}(G) is an obstruction to integrate Lie algebra homomorphisms to group homomorphisms, and the groups π_{1}(G) and π_{2}(G) are closely related to obstructions for the existence of central extensions ([Ne00b]).

For the full operator groups GL(H) and

GL(H, I) := {g ∈ GL(H): g^{−1} = Ig^{∗}I^{−1}}

one can use Kuiper’s Theorem saying that the group GL(H) is contractible for a separable Hilbert space H over 𝕂 ∈ {ℝ, ℂ, ℍ}. If H is inseparable, then there seems to be no immediate reference for this result for real and quaternionic Hilbert spaces. As we will see in Section II, the inseparable case has a quite elementary proof which does not depend on 𝕂 . In Section II we also compute the homotopy groups of the congruence groups GL_{p}(H) of the Schatten ideals B_{p}(H) ⊆ B(H), of GL_{p}(H, I) := GL(H, I) ∩ GL_{p}(H) for 1 ≤ p ≤ ∞, and of the corresponding direct limit groups without the restriction that H is separable. Since the homotopy groups of the restricted groups GL_{p}(H, D) and GL_{p}(H, I, D) are somewhat more involved, we deal with them separately in Section III.

Throughout this paper we never have to assume that the Hilbert spaces H under consideration are separable. All statements hold for general Hilbert spaces. To obtain this generality, we frequently have to extend results on homotopy groups of groups of operators which are known for the separable case.

Combining the information on central extensions from Section I and on the homotopy groups from Section III, we show in Section IV that the identity components GL_{2}(H, D)_{e} and GL_{2}(H, I, D)_{e} of the restricted groups have a universal central extension in the category of complex Banach–Lie groups whose central fiber is of the type (ℂ^{×})^{k}. We extend these results to certain real forms of these groups. If D has only two eigenvalues, this construction leads, for GL_{2}(H, D), to the central extension of the restricted general linear group of a polarized Hilbert space, which plays a crucial role in the theory of loop groups (cf. [PS86]). For the real forms Sp_{res}(H, Ω) of Sp_{2}(H_{ℂ} , I, D) and O_{res}(H^{ℝ}) of O_{2}(H_{ℂ} , I, D) (see Section IV for the notation), we obtain as universal central extensions the metagonal and the metaplectic groups discussed systematically by Vershik in [Ve90]. The metaplectic group has been introduced by I. Segal and Shale ([Se59], [Sh62]) and the metagonal group by Shale/Stinespring in [ShSt65] (see also [dlH71b] for a construction of the spin group as a two fold cover of O_{1}(H^{ℝ})^{+} , a real form of the identity component O_{1}(H, I)^{+} of O_{1}(H, I)). These two groups are relatives of the finitedimensional spin and metaplectic group, which are twofold covers of SO(2n, ℝ), resp., Sp(2n, ℝ). The results of Section IV on the universality of certain central extensions form the heart of the paper. It requires essentially all the information on the homotopy groups collected in Sections II and III and also the information on Lie algebra cohomology from Section I. Our structure theoretic approach to central extensions, as opposed to the representation theoretic one, has the advantage that it immediately provides a good deal of structural and topological information on the groups: their Banach–Lie group structure, their Lie algebra cocycles, and their topology.

The representation theoretic approach usually has to face the problem to deal with, a priori, unbounded operators on a Hilbert spaces. A general motivation is to understand the full set of central extensions of groups like O_{res}(H^{ℝ}) and Sp_{res}(H, Ω) is that they act naturally as symmetry groups of geometric objects (mostly symmetric spaces), so that the knowledge of their central extensions is important to understand the implementation of the these symmetry groups in natural Hilbert spaces attached to the geometric objects.

In Section V we finally discuss flag manifolds for the groups GL_{2}(H) and GL_{2}(H, I). For GL_{2}(H) we consider a flag F = (F_{0}, F_{1}, …, F_{k}), where

{0} = F_{0} ⊆ F_{1} ⊆ F_{2} ⊆ … ⊆ F_{k} = H

are closed subspaces of H. For GL_{2}(H, I) we consider flags F of the type

{0} = F_{0} ⊆ F_{1} ⊆ F_{2} ⊆ … ⊆ F_{k} ⊆ F_{k}^{⊥_{β}} ⊆ … ⊆ F_{1}^{⊥_{β}} ⊆ F_{0}^{⊥_{β}} = H,

which means that the spaces F_{j}, j = 1, …, k, are isotropic for the bilinear form β(x, y) = ⟨x, I.y⟩. Let P (F) ⊆ G denote the stabilizer of the flag F. Then the homogeneous space G/P (F) has a natural manifold structure and its elements can be viewed as flags g.F of closed subspaces of H. We will see that in all cases the action of G on G/P (F) extends naturally to an action of the corresponding restricted group and that the unitary real form U = G ∩ U(H) acts transitively on G/P (F). For G = GL_{2}(H) and F_{2} = H this construction leads to the restricted Graßmannians Gr_{res}(F_{1}). For G = GL_{2}(H, I) and k = 2 we obtain for F_{1} ⊆ H maximal isotropic the restricted Graßmannian of maximal isotropic subspaces and for dim F_{1} = 1 the space of isotropic lines in H. Both are hermitian symmetric spaces modeled over Hilbert spaces. Using some structural results from Section III, it is not hard to get basic information on the homotopy groups and the corresponding period maps for the flag manifolds. This kind of information is important for the quantizability of symplectic structures on these manifolds, viewed as affine coadjoint orbits of the unitary real forms U = G ∩ U(H). For more details on this interpretation and on affine coadjoint actions of these groups we refer to [Ne01a].

Several special classes of these flag manifolds show up at various places in the literature. The flag manifolds for GL_{2}(H, D) for separable H have been introduced by A. and G. Helminck in [HH94a] and [HH94b]. They apply the representations of central extensions of the complex group GL_{2}(H, D) in Hilbert spaces of holomorphic sections of line bundles on the flag manifolds to integrable systems. Moreover, they study cell decompositions of the flag manifolds and use them to obtain a Birkhoff decomposition of the group GL_{2}(H, D) ([HH94b, Prop. 2.4.16]).

The restricted Graßmannian Gr_{res}(F) of a polarized Hilbert space plays a central role for the structure of loop groups ([PS86]). The Graßmannians are particular cases of hermitian symmetric spaces, which are dual to symmetric Hilbert domains. These manifolds and their automorphism have been studied in [Ka75] and [DNS89], [DNS90]. A classification of hermitian symmetric Hilbert manifolds has been obtained by W. Kaup in [Ka83].

For separable Hilbert spaces the groups GL_{p}(H) and GL_{p}(H, I), 1 ≤ p ≤ ∞, and their real forms have been studied in detail by de la Harpe in [dlH72], where one finds all kinds of information such as the cohomology, the automorphisms, and the derivations of their Lie algebras, which we use in Section I. De la Harpe’s book also contains a discussion of Riemannian symmetric spaces of the real forms of these groups, where the aforementioned Graßmannians and several other related manifolds show up.

In this paper we do not deal with representations of the groups under consideration, although this paper was motivated by and provides important information useful for the theory of unitary representations of real L^{∗} -groups and their realization in Hilbert spaces of holomorphic sections of holomorphic line bundles over coadjoint orbits which are Kähler manifolds. The geometry of the “elliptic” coadjoint orbits and the corresponding unitary representations will be studied in forthcoming papers (cf. [Ne01a]). This theory includes in particular the spin representation of the metagonal group (fermionic second quantization) and the metaplectic representation (Segal–Shale–Weil representation) of the metaplectic group (bosonic second quantization). For a nice exposition of the construction of these representations in an ad hoc fashion, we refer to Ottesen’s book [Ot95], where it is also explained how embeddings of diffeomorphism groups and loop groups into Sp_{res}(H, Ω) and U_{2}(H, D) lead to interesting unitary representations of their central extensions (see also [PS86], [CR87] and [Mi89]). The mixed cases correspond to the infinite wedge representations of the restricted unitary group U_{res}(H_{+}, H_{−}), which in our terminology is U_{2}(H, D) := U(H) ∩ GL_{2}(H, D), where D has only two eigenvalues (cf. [PS86] and also [Wu98] which contains a lot of information on the physical background). The general L^{∗} -approach to these representations provides in particular direct geometric explanations for their intricate analytic properties such as boundedness properties of the corresponding operators (cf. [Ot95]).

Throughout this paper the letter I will always denote an antilinear isometry on a complex Hilbert space H with I^{2} ∈ {±1}. We call I a conjugation if I^{2} = 1 and an anticonjugation for I^{2} = −1.

I. The second cohomology of classical Lie algebras

In this first section we discuss the second continuous Lie algebra cohomology of the Lie algebras 𝔤𝔩(H) and 𝔤𝔩(H, I) and also for the corresponding restricted Lie algebras 𝔤𝔩_{2}(H, D) and 𝔤𝔩_{2}(H, I, D). In particular we will see that H_{c}^{2}(𝔤(D), ℂ) is always finite-dimensional and that it has a universal central extension ([Ne01b]).

The second cohomology groups of full classical Lie algebras

In this subsection we will show that for every Banach space 𝔷, considered as a trivial module of 𝔤 ∈ {𝔤𝔩(H), 𝔤𝔩(H, I)}, the second continuous cohomology group H_{c}^{2}(𝔤, 𝔷) (cf. Definition I.4) vanishes.

If J is a set we define the set J^{±} as the disjoint union J ˙∪ − J, where −J is a copy of the set J whose elements are denoted −j with the convention that −(−j) = j.

Lemma I.1. Let H be a complex Hilbert space and I: H → H an antilinear isometry with I^{2} = ±1. If H is infinite-dimensional or of finite even-dimension, then there exists an orthonormal basis (e_{j})_{j∈J^{±}} with

I.e_{j} = \{ \begin{matrix}e_{−j} & for j ∈ J \\ ±e_{−j} & for j ∈ −J. \end{matrix}

Proof. (cf. [dlH72, App. I]) First we consider the case I^{2} = −1. Since I and the complex structure on H generate a finite group, H is an orthogonal direct sum of complex subspaces on which I acts irreducibly. Let E be one of these subspaces and v ∈ E a unit vector. The complex bilinear form β(x, y) := ⟨x, I.y⟩ is skew-symmetric because

β(y, x) = ⟨y, I.x⟩ = ⟨x, I^{−1}.y⟩ = −β(x, y).

Therefore {v, I.v} is an orthonormal basis of E with the required properties. Since H is an orthogonal direct sum of copies of E, the assertion follows.

Next we consider the case I^{2} = 1. Then our assumptions imply that the real Hilbert space H_{ℝ} := {v ∈ H: I.v = v} has an orthonormal basis of the form (f_{j})_{j∈J^{±}}. We define

e_{±j} := \frac{1}{\sqrt{2}}(f_{j} ± if_{−j})

and obtain a basis with the required properties.

Remark I.2. (a) To obtain a more explicit description of the Lie algebra 𝔰𝔭(H, I), we use Lemma I.1 to obtain an orthonormal basis (e_{j})_{j∈J^{±}} of H with I.e_{j} = e_{−j} for j ∈ J. Then the closed subspace H_{0} generated by the elements e_{j}, j ∈ J, satisfies H_{0} ≅ l^{2}(J, ℂ), and we obtain a conjugation σ_{0} on this space by σ_{0}((x_{j})_{j∈J}) = (\overline{x_{j}})_{j∈J}. If we identify H = H_{0} ⊕ I.H_{0} with the space H_{0} ⊕ H_{0} , the anticonjugation I is given by I.(a, b) = (−σ_{0}(b), σ_{0}(a)).

For x ∈ B(H_{0}) we define x^{⊤} = σ_{0}x^{∗}σ_{0} . Then the Lie algebra 𝔰𝔭(H, I) ⊆ B(H_{0} ⊕ H_{0}) can be described in terms of (2 × 2)-block matrices as

𝔰𝔭(H, I) ≅ \{ ( \begin{matrix}a & b \\ c & −a^{⊤} \end{matrix} ) ∈ 𝔤𝔩(H): b = b^{⊤}, c = c^{⊤}\}.

To get a similar description on the group level, we write I as a composition of (σ_{0}, σ_{0}) and the operator with the matrix

Ĩ = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) .

Then we have for g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) the relation

Ig^{∗}I^{−1} = Ĩg^{⊤}Ĩ^{−1} = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) ( \begin{matrix}a^{⊤} & c^{⊤} \\ b^{⊤} & d^{⊤} \end{matrix} ) ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) = ( \begin{matrix}d^{⊤} & −b^{⊤} \\ −c^{⊤} & a^{⊤} \end{matrix} ) ,

which shows that

Sp(H, I) ≅ \{ ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL(H): ad^{⊤} − bc^{⊤} = 1, ab^{⊤} = ba^{⊤}, cd^{⊤} = dc^{⊤}\}.

In particular we see that

GL(H_{0}) ≅ \{ ( \begin{matrix}a & 0 \\ 0 & (a^{⊤})^{−1} \end{matrix} ): a ∈ GL(H_{0})\} ⊆ Sp(H, I).

(b) For 𝔬(H, I) and dim H = ∞ or dim H even we write H ≅ H_{0} ⊕ H_{0} with I.(a, b) = (σ_{0}(b), σ_{0}(a)), and keep the other notations from above. Then the Lie algebra 𝔬(H, I) can be described in terms of (2 × 2)-block matrices as

𝔬(H, I) ≅ \{ ( \begin{matrix}a & b \\ c & −a^{⊤} \end{matrix} ) ∈ 𝔤𝔩(H_{0} ⊕ H_{0}): b = −b^{⊤}, c = −c^{⊤}\}.

(c) If dim H = ∞ or dim H is odd, then we have an orthogonal decomposition H ≅ H_{0} ⊕ℂ ⊕H_{0} with I.(a, z, b) = (σ_{0}(b), \overline{z}, σ_{0}(a)) and obtain a similar explicit description as above by (3 × 3)- block matrices.

Lemma I.3. Let H be a complex Hilbert space. The Lie algebra 𝔤𝔩(H) is perfect if and only if dim H = ∞, and 𝔤𝔩(H, I) is perfect if and only if not (dim H = 2 and I^{2} = 1).

Proof. If H is of finite dimension n, then 𝔤𝔩(H) is not perfect because tr: 𝔤𝔩(H) → ℂ is a non-trivial Lie algebra homomorphism. If H is infinite-dimensional, then we use [Ha67, Cor. 2 to Probl. 186] to see that every element in 𝔤𝔩(H) is the sum of two commutators, so that 𝔤𝔩(H) is in particular perfect.

Now we consider 𝔤 := 𝔤𝔩(H, I). If H is finite-dimensional, then this Lie algebra is perfect unless I^{2} = 1 and dim H = 2. Suppose that H is infinite-dimensional. Then there exists a closed subspace H_{0} ⊆ H such that H = H_{0} ⊕ I.H_{0} is an orthogonal direct sum (Lemma I.1). We consider the element

X := ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) ∈ 𝔤𝔩(H, I)

and define 𝔤^{0} := 𝔷_{𝔤}(X). Then 𝔤 = 𝔤^{0} ⊕ [X, 𝔤], and 𝔤^{0} ≅ 𝔤𝔩(H_{0}) (Remark I.2). Since the first part of the proof implies that 𝔤^{0} is perfect, we conclude that 𝔤𝔩(H, I) is perfect.

Definition I.4. Let 𝔤 a topological Lie algebra, i.e., a Lie algebra which is a topological vector space with a continuous Lie bracket, and 𝔷 be a topological vector space, considered as a trivial 𝔤-module. A continuous 𝔷-valued 2-cocycle is a continuous skew-symmetric function ω: 𝔤×𝔤 → 𝔷 with

ω([x, y], z) + ω([y, z], x) + ω([z, x], y) = 0.

It is called a coboundary if there exists a continuous linear map α: 𝔤 → 𝔷 with ω(x, y) = α([x, y]) for all x, y ∈ 𝔤. We write Z_{c}^{2}(𝔤, 𝔷) for the space of continuous 𝔷-valued 2-cocycles and B_{c}^{2}(𝔤, 𝔷) for the subspace of coboundaries. We define the second continuous Lie algebra cohomology space

H_{c}^{2}(𝔤, 𝔷) := Z_{c}^{2}(𝔤, 𝔷)/B_{c}^{2}(𝔤, 𝔷).

See [Ja62, Sect. III.10] for the basic concepts related to Lie algebra cohomology.

In [dlH79] it is shown that the second homology space H_{2}(𝔤𝔩(H)) vanishes (on the algebraic level) and it is also shown that this implies that all Banach Lie algebra extensions of 𝔤𝔩(H) with finite-dimensional centers are trivial (cf. [dlH79, Cor. 4]). The following proposition sharpens this result.

Proposition I.5. If H is a complex Hilbert space, then

H_{c}^{2}(𝔤𝔩(H), 𝔷) = 0

holds for all trivial Banach 𝔤-modules 𝔷.

Proof. Let 𝔤 := 𝔤𝔩(H). If dim H = n is finite, then 𝔤 ≅ 𝔤𝔩(n, ℂ) and therefore H_{c}^{2}(𝔤, 𝔷) = H^{2}(𝔤, 𝔷) = 0. In fact, in view of Levi’s Theorem, each element [ω] ∈ H^{2}(𝔤, 𝔷) can be represented by an 𝔰𝔩(n, ℂ)-invariant 𝔷-valued cocycle. Since there is no non-zero skew-symmetric invariant bilinear form on 𝔰𝔩(n, ℂ), and 𝔷(𝔤𝔩(n, ℂ)) ≅ ℂ is one-dimensional, it follows that H^{2}(𝔤, 𝔷) is trivial.

Now we assume that H is infinite-dimensional. We consider 𝔷 as a trivial 𝔤-module. Let ω ∈ Z_{c}^{2}(𝔤, 𝔷) be a continuous 𝔷-valued cocycle. In view of H^{2}(𝔤, 𝔷) ≅ Lin(H_{2}(𝔤), 𝔷) = 0 ([dlH79]), there exists a linear map λ: 𝔤 → 𝔷 with ω(x, y) = λ([x, y]) for all x, y ∈ 𝔤. It remains to show that λ is continuous.

Let x ∈ 𝔤. According to [Ha67, Cor. 2 to Probl. 186], there exist operators a, b, c, d ∈ 𝔤 with x = [a, b] + [c, d], where ∥a∥, ∥c∥ ≤ 2∥x∥ and ∥b∥, ∥d∥ ≤ 1. We obtain

∥λ(x)∥ = ∥λ([a, b] + [c, d])∥ ≤ ∥ω(a, b)∥ + ∥ω(c, d)∥ ≤ ∥ω∥ ∥a∥ ∥b∥ + ∥ω∥ ∥c∥ ∥d∥ ≤ 4∥ω∥ ∥x∥.

This proves that λ is continuous and therefore that H_{c}^{2}(𝔤, 𝔷) = 0.

Next we show that the second cohomology of the Lie algebras 𝔤𝔩(H, I) vanishes. The proof is based on a modification of the strategy used in [dlH79].

Proposition I.6. If H is a complex Hilbert space, then the Lie algebra 𝔰𝔭(H, I) satisfies

H_{c}^{2}(𝔰𝔭(H, I), 𝔷) = 0

for every Banach space 𝔷, considered as a trivial module.

Proof. If H is finite-dimensional with dim H = 2n, then 𝔤 := 𝔰𝔭(H, I) ≅ 𝔰𝔭(2n, ℂ) is a simple complex Lie algebra, and the Whitehead Lemmas ([Ja62, Lemma III.9.6]) imply that H_{c}^{2}(𝔤, 𝔷) = H^{2}(𝔤, 𝔷) = 0 for every Banach space 𝔷.

Now we assume that H is infinite-dimensional. We consider the element

X := ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) ∈ 𝔤

which defines the 3-grading

𝔤 = 𝔤^{−} ⊕ 𝔤^{0} ⊕ 𝔤^{+},

where 𝔤^{0} = ker ad X ≅ 𝔤𝔩(H_{0}) (Remark I.2) and 𝔤^{±} = ker(ad X ∓2). We know from Proposition I.5 that H_{c}^{2}(𝔤^{0}, 𝔷) = 0. This means that all the assumptions of Corollary III.9 in [Ne01a] are satisfied with 𝔡 = 𝔤^{0} and D_{0} = X, so that it suffices to consider a 𝔤^{0} -invariant cocycle φ ∈ Z_{c}^{2}(𝔤, 𝔷) satisfying φ(𝔤^{0}, 𝔤) = 0. Therefore it suffices to show that every 𝔤^{0} -invariant bilinear form φ: 𝔤^{+} × 𝔤^{−} → 𝔷 vanishes.

For the sake of simpler notation, we identify 𝔤^{0} with 𝔤𝔩(H_{0}) by the map

𝔤𝔩(H_{0}) → 𝔤^{0}, x ↦ ( \begin{matrix}x & 0 \\ 0 & −x^{⊤} \end{matrix} )

(Remark I.2). We further identify 𝔤^{±} in the canonical way with Sym(H_{0}) := {a ∈ B(H_{0}): a^{⊤} = a} and consider the natural action of 𝔤𝔩(H_{0}) on Sym(H_{0}) given by x.a := xa + ax^{⊤}. This corresponds to the action of 𝔤^{0} on 𝔤^{+} , and on 𝔤^{−} the action of 𝔤^{0} corresponds to [x, d] = −x^{⊤}.d = −x^{⊤}d − dx. Therefore φ corresponds to a bilinear form

φ: Sym(H_{0}) × Sym(H_{0}) → 𝔷

satisfying

φ(x.a, d) − φ(a, x^{⊤}.d) = 0 for all x ∈ 𝔤𝔩(H_{0}), a, d ∈ Sym(H_{0}).
For x = x^{⊤} ∈ 𝔤𝔩(H_{0}) we have x.1 = x1 + 1x^{⊤} = 2x, and therefore

φ(a, d) = \frac{1}{2}φ(a.1, d) = \frac{1}{2}φ(1, a.d) = \frac{1}{2}φ(1, da + ad).

It follows in particular that φ is symmetric and that it suffices to show that φ(1, ·) = 0. The 𝔤^{0} -invariance of φ leads for x = −x^{⊤} ∈ 𝔤𝔩(H_{0}) to

0 = φ(x.a, 1) − φ(a, x^{⊤}.1) = φ(xa + ax^{⊤}, 1) + φ(a, \underbrace{x.1}_{=0}) = φ(xa − ax, 1) = φ([x, a], 1).

To see that φ vanishes, it therefore suffices to show that

[Skew(H_{0}), Sym(H_{0})] = Sym(H_{0}).

We know already that the Lie algebra 𝔤𝔩(H_{0}) = Skew(H_{0}) ⊕ Sym(H_{0}) is perfect (Lemma I.3), which implies that

𝔤𝔩(H_{0}) = [𝔤𝔩(H_{0}), 𝔤𝔩(H_{0})]
= ([Skew(H_{0}), Skew(H_{0})] + [Sym(H_{0}), Sym(H_{0})]) + [Skew(H_{0}), Sym(H_{0})],

and this implies in particular that Sym(H_{0}) = [Skew(H_{0}), Sym(H_{0})].

Proposition I.7. If dim H > 2, then

H_{c}^{2}(𝔬(H, I), 𝔷) = 0

for every Banach space 𝔷, considered as a trivial module.

Proof. Assume first that n := dim H is finite. Then n > 2 implies that 𝔬(H, I) ≅ 𝔬(n, ℂ) is a semisimple complex Lie algebra, and the assertion follows from the Whitehead Lemmas.

Now we assume that H is infinite-dimensional. Let 𝔤 := 𝔬(H, I). We consider the element

X := ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) ∈ 𝔤

which defines the 3-grading 𝔤 = 𝔤^{−} ⊕ 𝔤^{0} ⊕ 𝔤^{+}, where 𝔤^{j} = ker(ad X − 2j).

With the same argument as in the proof of Proposition I.6, we see that we may w.l.o.g. assume that φ is 𝔤^{0} -invariant and satisfies φ(𝔤^{0}, 𝔤) = 0. As in Proposition I.6, this leads to a bilinear form

φ: Skew(H_{0}) × Skew(H_{0}) → 𝔷

satisfying

φ(x.a, d) − φ(a, x^{⊤}.d) = 0 for all x ∈ 𝔤𝔩(H_{0}), a, d ∈ Skew(H_{0}),

where x.a = xa + ax^{⊤} for x ∈ 𝔤𝔩(H_{0}) and a ∈ Skew(H_{0}). We have to show that this implies that φ = 0.

We write H_{0} as H_{1} ⊕ H_{1} , where H_{1} is endowed with an antilinear isometric involution σ_{1} such that σ_{0}(v, w) = (σ_{1}.v, σ_{1}.w) for v, w ∈ H_{1} . From now on we write operators in 𝔤𝔩(H_{0}) as (2 × 2)-block matrices according to the decomposition of H_{0} into H_{1} ⊕ H_{1} . Let

S := ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) ∈ Skew(H_{0}).

Then

x.S = ( \begin{matrix}x_{12}^{⊤} − x_{12} & x_{11} + x_{22}^{⊤} \\ −x_{11}^{⊤} − x_{22} & x_{21} − x_{21}^{⊤} \end{matrix} )

shows that for each a ∈ Skew(H_{0}) we have

a = ã.S for ã := ( \begin{matrix}a_{12} & −\frac{1}{2}a_{11} \\ \frac{1}{2}a_{22} & 0 \end{matrix} ) .

Therefore

φ(a, d) = φ(ã.S, d) = φ(S, a^{⊤}.d),

and it suffices to prove that φ(S, ·) = 0.

For x ∈ 𝔤𝔩(H_{0}) with x^{⊤}.S = 0 we have

0 = φ(S, x.a) − φ(x^{⊤}.S, a) = φ(S, x.a).

For x = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ), which satisfies x^{⊤}.S = x^{⊤}S + Sx = 0, we obtain in particular

x.a = xa + ax = ( \begin{matrix}a_{11} & −a_{12} \\ −a_{21} & a_{22} \end{matrix} ) ,

showing that x. Skew(H_{0}) = Skew(H_{0}). This implies that φ(S, ·) = 0, and hence that φ = 0.

The second cohomology groups of restricted Lie algebras

In this subsection we will use the results of the preceding subsection and the general tools developed in [Ne01a] to compute the second cohomology of the restricted versions of the Lie algebras 𝔤𝔩_{2}(H) and 𝔤𝔩_{2}(H, I).

Definition I.8. (a) Let H be a Hilbert space. For 1 ≤ p < ∞ we define

B_{p}(H) := \{X ∈ B(H): tr ((XX^{∗})^{\frac{p}{2}} ) < ∞\}.

For p = ∞ we define B_{∞}(H) := K(H) (cf. [RS78]) as the ideal of compact operators on H. More generally we define for two Hilbert spaces H_{1} , H_{2} :

B_{p}(H_{1}, H_{2}) := \{X ∈ B(H_{1}, H_{2}): tr ((XX^{∗})^{\frac{p}{2}} ) < ∞\} and B_{∞}(H_{1}, H_{2}) := K(H_{1}, H_{2}).

These sets are are invariant under left and right multiplication with bounded operators and they are Banach spaces with respect to the norms

∥X∥_{p} := tr ((XX^{∗})^{\frac{p}{2}} )^{\frac{1}{p}} and ∥X∥_{∞} := ∥X∥,

satisfying

∥XY ∥_{p} ≤ ∥X∥ · ∥Y ∥_{p} and ∥XY ∥_{p} ≤ ∥X∥_{p} · ∥Y ∥.

The spaces B_{p}(H) are called the Schatten ideals of B(H).

The congruence subgroups

GL_{p}(H) := GL(H) ∩ (1 + B_{p}(H))

with respect to the Schatten ideals are Banach–Lie groups with Lie algebra 𝔤𝔩_{p}(H) := B_{p}(H) (cf. [Ne00a, Def. IV.20] and [Mi89]). The group GL_{∞}(H) is called the Fredholm group. It is contained in the monoid

Fred(H) := {A ∈ B(H): dim ker A, dim coker A < ∞}

of Fredholm operators on H. The group

U_{p}(H) := U(H) ∩ (1 + B_{p}(H))

is a Lie group with Lie algebra

𝔲_{p}(H) := 𝔲(H) ∩ B_{p}(H) = {X ∈ B_{p}(H): X^{∗} = −X}.

With Herm_{p}(H) := Herm(H) ∩ B_{p}(H) = i𝔲_{p}(H) we then have

𝔤𝔩_{p}(H) = 𝔲_{p}(H) ⊕ Herm_{p}(H) = 𝔲_{p}(H) ⊕ i𝔲_{p}(H)

and the polar map

U_{p}(H) × Herm_{p}(H) → GL_{p}(H), (u, X) ↦ ue^{X}

is a diffeomorphism ([Ne00a, Prop. A.4]).

(b) The restricted classical Lie algebras are defined as follows. For 𝔤 = 𝔤𝔩_{2}(H) we put 𝔤_{b} := 𝔤𝔩(H), and for 𝔤 = 𝔤𝔩_{2}(H, I) we put 𝔤_{b} := 𝔤𝔩(H, I). Let D ∈ 𝔤_{b} be a hermitian element with finite spectrum, 𝔤_{b}^{0} := 𝔷_{𝔤_{b}}(D) and 𝔤^{0} := 𝔷_{𝔤}(D). Then the Lie algebra

𝔤_{r} := 𝔤(D) := 𝔤 + 𝔤_{b}^{0}

is called the restricted Lie algebra associated to 𝔤 and D. For 𝔤 = 𝔤𝔩_{2}(H) we also write 𝔤𝔩_{2}(H, D) := 𝔤(D) and for 𝔤 = 𝔤𝔩_{2}(H, I) we likewise write 𝔤𝔩_{2}(H, I, D) := 𝔤(D).

Examples I.9. (a) Let 𝔤 = 𝔤𝔩_{2}(H) for an infinite-dimensional complex Hilbert space H and D = D^{∗} ∈ 𝔤𝔩(H) diagonalizable with the eigenvalues d_{1}, …, d_{k} and the corresponding eigenspaces H_{j} := ker(D − d_{j}1). Then H = H_{1} ⊕ … ⊕ H_{k} is an orthogonal decomposition, 𝔤^{0} consists of all elements in 𝔤 preserving this decomposition, and therefore

𝔤^{0} ≅ ⨁_{j=1}^{k} 𝔤𝔩_{2}(H_{j}) and 𝔤_{b}^{0} ≅ ⨁_{j=1}^{k} 𝔤𝔩(H_{j})

lead to

𝔤𝔩_{2}(H, D) = {X = (x_{ij})_{i,j=1,...,k}: (∀i ≠ j) x_{ij} ∈ B_{2}(H_{j}, H_{i})}.

(b) For 𝔤 = 𝔤𝔩_{2}(H, I) and D = D^{∗} ∈ 𝔤𝔩(H, I), we write d_{1}, …, d_{k} for the positive eigenvalues of D, d_{−j} := −d_{j}, and d_{0} := 0. Then Spec(D) ∪ {0} = {d_{j}: j = −k, …, k} (cf. [Ne01a, Lemma III.12]) and for H_{j} := ker(D − d_{j}1) we obtain an orthogonal decomposition

H = H_{k} ⊕ … ⊕ H_{0} ⊕ … ⊕ H_{−k}

with I.H_{j} = H_{−j}, so that H_{0} = ker D is I -invariant, but this space might be trivial. With I_{0} := I |_{H_{0}} and Remark I.2 we now obtain

𝔤^{0} ≅ 𝔤𝔩_{2}(H_{0}, I_{0}) ⊕ ⨁_{j=1}^{k} 𝔤𝔩_{2}(H_{j}) and 𝔤_{b}^{0} ≅ 𝔤𝔩(H_{0}, I_{0}) ⊕ ⨁_{j=1}^{k} 𝔤𝔩(H_{j}).

In the following we will keep the notation of Examples I.9 whenever we discuss specific properties of the Lie algebras 𝔤(D) and the corresponding groups.

Proposition I.10. (a) 𝔤𝔩_{2}(H, D) is perfect if and only if dim H = ∞.

(b) 𝔤𝔩_{2}(H, I, D) is perfect if not (dim H = 2 and I^{2} = 1).

Proof. (a) Let 𝔤 := 𝔤𝔩_{2}(H). If If dim H < ∞, then 𝔤 = 𝔤𝔩(H) is not perfect, and [𝔤, 𝔤] = 𝔰𝔩(H). Suppose that dim H = ∞. Then there exists at least on j for which H_{j} is infinite-dimensional. We consider the direct sum decomposition

𝔤(D) = 𝔤_{b}^{0} ⊕ ⨁_{i≠j} B_{2}(H_{j}, H_{i}).

Since 𝔤(D) = 𝔤_{b}^{0} + [D, 𝔤], it suffices to show that 𝔤_{b}^{0} ⊆ [𝔤(D), 𝔤(D)]. We recall that 𝔤_{b}^{0} ≅ ⨁_{j=1}^{k} 𝔤𝔩(H_{j}) (Examples I.9) and view each 𝔤𝔩(H_{j}) as a subalgebra of 𝔤_{b}^{0} .

The commutator algebra of 𝔤_{b}^{0} contains the full algebra 𝔤𝔩(H_{i}) ⊆ 𝔤_{b}^{0} whenever H_{i} is infinite-dimensional ([Ha67, Cor. 2 to Probl. 186]). If H_{i} is finite-dimensional, then we choose j with H_{j} infinite-dimensional and consider elements of B(H_{i} ⊕ H_{j}) as (2 × 2)-block matrices. For such matrices we have

[ ( \begin{matrix}0 & B \\ 0 & 0 \end{matrix} ) , ( \begin{matrix}0 & 0 \\ C & 0 \end{matrix} ) ] = ( \begin{matrix}BC & 0 \\ 0 & −CB \end{matrix} ) .

Then [𝔤(D), 𝔤(D)] contains 𝔤𝔩(H_{j}) and therefore also B_{2}(H_{i}, H_{j})B_{2}(H_{j}, H_{i}) = 𝔤𝔩(H_{i}). Hence [𝔤(D), 𝔤(D)] contains 𝔤_{b}^{0} , which shows that 𝔤(D) is perfect.

(b) Let 𝔤 := 𝔤𝔩_{2}(H, I). If H is of finite dimension n, then 𝔤(D) = 𝔤 ≅ 𝔰𝔭(n, ℂ) or 𝔬(n, ℂ) is semisimple and therefore perfect unless n = 2 and I^{2} = 1 (Lemma I.3).

Suppose that H is infinite-dimensional. We have 𝔤(D) = 𝔤_{b}^{0} + [D, 𝔤], so that it suffices to show that 𝔤_{b}^{0} ⊆ [𝔤(D), 𝔤(D)]. Using the 3 × 3-block description of 𝔤𝔩(H, I) according to H = H_{+} ⊕ H_{0} ⊕ H_{−} with H_{±} := ∑_{j=1}^{k} H_{±j}, we see that 𝔤(D) is adapted to this decomposition, and we get with D_{+} := D |_{H_{+}} :

𝔤(D) ⊇ 𝔤𝔩_{2}(H_{+}, D_{+}) ≅ \{ ( \begin{matrix}a & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & −a^{⊤} \end{matrix} ): a ∈ 𝔤𝔩_{2}(H_{+}, D_{+})\} ⊇ ∑_{j=1}^{k} 𝔤𝔩(H_{j}).

If H_{+} is finite-dimensional, then the same holds for H_{−} = I.H_{+} , so that 𝔤(D) = 𝔤𝔩(H, I), and the perfectness of 𝔤(D) follows from Lemma I.3.

If H_{+} is infinite-dimensional, then 𝔤𝔩_{2}(H_{+}, D_{+}) is perfect by (a), and each 𝔤𝔩(H_{j}), j = 1, …, k, is contained in this algebra, so that it remains to see that 𝔤𝔩(H_{0}, I_{0}) ⊆ [𝔤(D), 𝔤(D)]. If 𝔤𝔩(H_{0}, I_{0}) itself is perfect, this is trivial. If this is not the case, then dim H_{0} = 2 and I^{2} = 1 (Lemma I.3). Then we extend H_{0} to a four-dimensional I -invariant subspace H̃_{0} of H_{1} + H_{0} + H_{−1} , set Ĩ_{0} := I |_{H_{0}} , and obtain

[𝔤(D), 𝔤(D)] ⊇ [𝔬(H̃_{0}, Ĩ_{0}), 𝔬(H̃_{0}, Ĩ_{0})] = 𝔬(H̃_{0}, Ĩ_{0}) ⊇ 𝔬(H_{0}, I_{0}) ≅ 𝔤𝔩(H_{0}, I_{0}).

This implies that 𝔤(D) is perfect.

Proposition I.11. For k_{∞} := |{j ∈ {1, …, k}: dim H_{j} = ∞}| we have

H_{c}^{2}(𝔤𝔩_{2}(H, D), ℂ) ≅ ℂ^{k_{∞}−1} and H_{c}^{2}(𝔤𝔩_{2}(H, I, D), ℂ) ≅ ℂ^{k_{∞}}.

Each cohomology class contains a cocycle of the form

φ(z)(x + d, x′ + d′) := tr([z, x]x′) = tr(z[x, x′]) for d, d′ ∈ 𝔤(D)^{0}, x, x′ ∈ 𝔤,

where z ∈ 𝔷(𝔤(D)^{0}).

Proof. In [Ne01a, Ex. III.13] we have seen how to describe the space H_{c}^{2}(𝔤(D), ℂ) in all cases. Each continuous 2-cocycle on 𝔤(D) is equivalent to a cocycle φ(z) given as follows. For z ∈ 𝔷(𝔤_{b}^{0}) we define

φ(z)(x + d, x′ + d′) := tr([z, x]x′) = tr(z[x, x′]), d, d′ ∈ 𝔤_{b}^{0}, x, x′ ∈ 𝔤.

This cocycle is trivial if and only if z ∈ 𝔷(𝔤^{0}) + ℂ1, which for 𝔤 = 𝔤𝔩_{2}(H, I) is equivalent to z ∈ 𝔷(𝔤^{0}). We always have 𝔷(𝔤𝔩(H_{0}, I_{0})) = 𝔷(𝔤𝔩_{2}(H_{0}, I_{0})), and for j > 0 we have 𝔷(𝔤𝔩(H_{j}, I_{j})) = ℂ id_{H_{j}} and 𝔷(𝔤𝔩_{2}(H_{j}, I_{j})) = 0 if H_{j} is infinite-dimensional. We conclude that each cohomology class can be represented by φ(z) with z = ∑_{dim H_{j}=∞} z_{j} id_{H_{j}}, and that such a cocycle is trivial if and only if z = 0, for 𝔤 = 𝔤𝔩_{2}(H, I), or z_{1} = … = z_{k}, for 𝔤 = 𝔤𝔩_{2}(H). This implies the assertion.

Universal central extensions

Definition I.12. (a) Let 𝔤 be a topological Lie algebra over 𝕂 ∈ {ℝ, ℂ}, 𝔷 a topological vector space, and ω ∈ Z_{c}^{2}(𝔤, 𝔷) a continuous 𝔷-valued 2-cocycle. Then we write 𝔤 ⊕_{ω} 𝔷 for the topological Lie algebra whose underlying topological vector space is the product space 𝔤 × 𝔷 and whose Lie bracket is defined by

[(x, z), (x′, z′)] = ([x, x′], ω(x, x′)).

Then q: 𝔤⊕_{ω} 𝔷 → 𝔤, (x, z) ↦ x is a central extension and σ: 𝔤 → 𝔤⊕_{ω} 𝔷, x ↦ (x, 0) is a continuous linear section of q.

(b) Let 𝔞 be a topological vector space considered as a trivial 𝔤-module. We call a central extension q: 𝔤̂ = 𝔤 ⊕_{ω} 𝔷 → 𝔤 with 𝔷 = ker q weakly 𝔞-universal if the map

δ_{𝔞}: Lin(𝔷, 𝔞) → H_{c}^{2}(𝔤, 𝔞), γ ↦ [γ ∘ ω]

is bijective.

We call q: 𝔤̂ → 𝔤 universal for 𝔞 if for each central 𝔞-extension q_{1}: 𝔤̂_{1} := 𝔤 ⊕_{f} 𝔞 → 𝔤 of 𝔤 there exists a unique continuous homomorphism φ: 𝔤̂ → 𝔤̂_{1} with q_{1} ∘ φ = q. In view of [Ne01b, Remark I.10(b)], the 𝔞-universality is equivalent to the weak 𝔞-universality plus Hom(𝔤, 𝔞) = 0.

For 𝔞 ≠ {0} this implies in particular that Hom(𝔤, 𝕂) = {0} which for a Banach–Lie algebra 𝔤 implies that 𝔤 is topologically perfect.

Proposition I.13. Let 𝔤 be a perfect 𝕂 -Banach–Lie algebra for which H_{c}^{2}(𝔤, 𝕂) is finite-di-mensional. Then 𝔤 has up to isomorphism a unique 𝕂 -universal central extension

𝔷 ↪ 𝔤̂ := 𝔤 ⊕_{ω} 𝔷 → 𝔤

which, in addition, is universal for all Fréchet spaces. Proof. The uniqueness follows from [Ne01b, Lemma I.13] and the existence from [Ne01b, Cor. II.12].

Remark I.14. The background for Definition I.12(b) is that the central extension q: 𝔤̂ = 𝔤 ⊕_{ω} 𝔷 → 𝔤 defines for each topological vector space 𝔞 an exact sequence containing δ_{𝔞}. To describe this exact sequence, let Z_{c}^{2}(𝔤̂, 𝔷, 𝔞) denotes the set of all continuous 𝔞-valued 2-cocycles ω ∈ Z_{c}^{2}(𝔤̂, 𝔞) with ω(𝔷, 𝔤̂) = {0}. Then B_{c}^{2}(𝔤̂, 𝔞) ⊆ Z_{c}^{2}(𝔤̂, 𝔷, 𝔞) because β([𝔤̂, 𝔷]) = {0} for β ∈ Lin(𝔤̂, 𝔷), and we define

H_{c}^{2}(𝔤̂, 𝔷, 𝔞) := Z_{c}^{2}(𝔤̂, 𝔷, 𝔞)/B_{c}^{2}(𝔤̂, 𝔷).

According to [Ne01a, Th. I.4], we now have the exact sequence

(1.1)  0 → Hom(𝔤, 𝔞)⟶^{q^{∗}} Hom(𝔤̂, 𝔞)⟶^{res} Lin(𝔷, 𝔞)⟶^{δ_{𝔞}}H_{c}^{2}(𝔤, 𝔞)⟶^{q^{∗}}H_{c}^{2}(𝔤̂, 𝔷, 𝔞) → 0.

We will see in Section IV below how to realize the universal central extensions of the restricted Lie algebras 𝔤𝔩_{2}(H, D) and 𝔤𝔩_{2}(H, I, D) explicitly.

II. Homotopy groups of classical groups

In this section we first discuss a quite elementary proof of Kuiper’s Theorem for inseparable Hilbert spaces H over 𝕂 ∈ {ℝ, ℂ, ℍ}, and then we use this result to prove that various classical groups of operators on Hilbert spaces such as GL(H, I) are contractible. Then we turn to the direct limit groups GL(J, 𝕂) of those invertible J × J -matrices g for which g − 1 has only finitely many non-zero entries. We will see that for an infinite set J this group is weakly homotopy equivalent to GL(ℕ, 𝕂), i.e., to the direct limit of the groups GL(n, 𝕂). Combining these insights with general results of Palais, we compute the homotopy groups of the congruence groups GL_{p}(H) of the Schatten ideals B_{p}(H) ⊆ B(H) and GL_{p}(H, I) := GL(H, I) ∩ GL_{p}(H) for 1 ≤ p ≤ ∞. In the next section we deal with groups corresponding to the restricted Lie algebras 𝔤(D).

Kuiper’s Theorem

In this subsection we explain how Kuiper’s Theorem that the group GL(H, 𝕂) of 𝕂 -linear continuous operators on an infinite-dimensional separable 𝕂 -Hilbert space H is contractible ([Ku65]) can be obtained in a quite elementary way for inseparable Hilbert spaces.^{1}

The observation is based on the following lemma, which is a refinement of [vNeu50, Th. 14.10].

Lemma II.1. Let H be a Hilbert space over 𝕂 = ℝ, ℂ or ℍ and M ⊆ B(H, 𝕂) a separable set of operators. Then there exists an orthogonal decomposition H ≅ ˆ⨁_{j∈J}H_{j} into M-invariant subspaces such that each H_{j} is separable.

If, in addition, H is infinite-dimensional, then the spaces H_{j} can be chosen in such a way that they are all infinite-dimensional, hence isomorphic to l^{2}(ℕ, 𝕂). Proof. Since the closed ∗-subalgebra of B(H) generated by M is separable, we may assume that M is ∗-invariant with 1 ∈ M, otherwise we replace it by M∪M^{∗} ∪{1}. Now the assertion follows by a standard application of Zorn’s Lemma. Let H_{j}, j ∈ J, be a maximal set of non-zero closed M-invariant separable subspaces of H such that the sum ∑_{j∈J} H_{j} is orthogonal. Set H_{0} := \overline{∑_{j∈J} H_{j}}. Then H_{0}^{⊥} is M-invariant because M is ∗-invariant. Assume that H_{0} ≠ H. For 0 ≠ v ∈ H_{0}^{⊥} the subspace H_{v} := \overline{span M.v} is a cyclic hence separable subspace orthogonal to all the spaces H_{j}, contradicting the maximality of the family (H_{j})_{j∈J}. This proves the first assertion.

To prove the second part, let us assume that H is infinite-dimensional and consider a decomposition H ≅ ˆ⨁_{j∈J}H_{j} as above. Let

I := {j ∈ J: dim H_{j} < ∞}.

Case 1: If I is finite, then there exists a j_{0} ∈ J \I. Replacing H_{j_{0}} by H_{j_{0}} +∑_{i∈I} H_{i}, we obtain the desired decomposition. Case 2: If I is infinite, then |I × ℕ| = |I| ([La93, App. 2]) implies that I can be partitioned into infinite countable subsets I_{i}, i ∈ I. Then all the subspaces K_{i} := \overline{∑_{j∈I_{i}} H_{j}} are infinitedimensional and separable, and we have the derived orthogonal decomposition of H:

H = ˆ⨁_{j∈J\I}H_{j} ⊕ ˆ⨁_{i∈I}K_{i}.

1 The proof is based on a hint in a footnote in Kuiper’s paper but we don’t know of any direct reference in the literature which provides the result also for real and quaternionic inseparable Hilbert spaces. For complex Hilbert spaces it follows from results of Brüning and Willgerodt on the contractibility of unit groups of von Neumann algebras of infinite type ([BW76]).

Theorem II.2. (Palais) For a metrizable topological manifold modeled over a sequentially complete locally convex space the following are equivalent:

(1) π_{n}(X) = 0 for all n ∈ ℕ_{0} .

(2) X is contractible. Proof. [Pa66, Cor. to Th. 15]

The proof of the following proposition is inspired by the setting in Mityagin’s paper [Mit70].

Proposition II.3. Let Y be a separable topological space and H an inseparable Hilbert space over 𝕂 = ℝ, ℂ or ℍ. Then each continuous map f: Y → GL(H, 𝕂) is homotopic to a constant map. Proof. Since f(Y) is a separable set of operators, Lemma II.1 implies that there exists a set J and an isomorphism H → l^{2}(J, H_{s}) with H_{s} := l^{2}(ℕ, 𝕂) such that the operators in f(Y) are diagonal operators on l^{2}(J, H_{s}).

Step 1: Since H is not separable, the set J is (uncountably) infinite. First we consider a decomposition J = J_{1} ˙∪J_{2} with |J_{1}| = |J_{2}| = |J|. This leads to an orthogonal decomposition H ≅ H ⊕ H, and we consider operators on H accordingly as block 2 × 2-matrices. Let f: Y → GL(H, 𝕂) be as above. Then

f(y) = ( \begin{matrix}g_{1}(y) & 0 \\ 0 & g_{2}(y) \end{matrix} ) ,

where g_{j}: Y → GL(H, 𝕂) are continuous maps. We claim that f is homotopic to the map

(2.1)  f_{1}(y) = ( \begin{matrix}g_{1}(y)g_{2}(y) & 0 \\ 0 & 1 \end{matrix} ) .

It suffices to show that

f(y)^{−1}f_{1}(y) = ( \begin{matrix}g_{2}(y) & 0 \\ 0 & g_{2}(y)^{−1} \end{matrix} )

is homotopic to a constant map. This is implemented by the homotopy

H(t, a) := ( \begin{matrix}1 & 0 \\ t(a^{−1} − 1) & 1 \end{matrix} ) ( \begin{matrix}1 & t \\ 0 & 1 \end{matrix} ) ( \begin{matrix}1 & 0 \\ t(a − 1) & 1 \end{matrix} ) ( \begin{matrix}1 & −ta^{−1} \\ 0 & 1 \end{matrix} )

which satisfies

H(1, a) = ( \begin{matrix}a & 0 \\ 0 & a^{−1} \end{matrix} ) and H(0, a) = ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) .

Step 2: In view of Step 1, we may assume that f_{1}: Y → GL(H, 𝕂) has the form (2.1). Next we observe that H ≅ l^{2}(ℕ, H) because |J| = |ℕ × J| ([La93, App. 2]). Therefore

H ≅ H ⊕ l^{2}(ℕ, H),

and we may assume that

f_{1}(y) = diag(g(y), 1, 1, …).

Partitioning ℕ into odd numbers ℕ_{odd} and even numbers ℕ_{even} , and writing accordingly

l^{2}(ℕ, H) ≅ l^{2}(ℕ_{odd}, H) ⊕ l^{2}(ℕ_{even}, H),

it follows from Step 1 that the constant map Y → GL(l^{2}(ℕ, H), 𝕂) is equivalent to the map

y ↦ diag(g(y)^{−1}, g(y), g(y)^{−1}, g(y), …).

Therefore f_{1} is homotopic to

f_{2}(y) = diag(g(y), g(y)^{−1}, g(y), g(y)^{−1}, …).

Applying the same argument again to the decomposition

H ≅ H ⊕ l^{2}(ℕ, H) ≅ l^{2}({0} ∪ ℕ_{even}, H) ⊕ l^{2}(ℕ_{odd}, H),

we see that f_{2} is homotopic to a constant map.

Theorem II.4. (Kuiper’s Theorem for general Hilbert spaces) If H is an infinite-dimensional Hilbert space over 𝕂 = ℝ, ℂ or ℍ, then the group GL(H, 𝕂) is contractible.

Proof. In view of Theorem II.2, it suffices to show that all homotopy groups of GL(H, 𝕂) vanish. In [Ku65], this is proved for infinite-dimensional separable Hilbert spaces, and for inseparable Hilbert spaces, this follows from Proposition II.3 because the spheres 𝕊^{k}, k ∈ ℕ_{0} , are separable.

Consequences of Kuiper’s Theorem

Definition II.5. (a) If H is a Hilbert space over 𝕂 ∈ {ℝ, ℂ, ℍ}, then we define

U(H, 𝕂) := {g ∈ GL(H, 𝕂): g^{∗}g = gg^{∗} = 1}

as the unitary part of this group. We also write

O(H) := U(H, ℝ), U(H) := U(H, ℂ) and Sp(H) := U(H, ℍ).

(b) Let H be a complex Hilbert space and I be an antilinear isometry with I^{2} ∈ {±1}. For I^{2} = 1 we then have

U(H, I) := U(H) ∩ GL(H, I) ≅ O(H_{ℝ}) with H_{ℝ} := {x ∈ H: I.x = x},

and for I^{2} = −1 we have

U(H, I) ≅ U(H, ℍ) ≅ Sp(H),

where the quaternionic structure on H is given by the subalgebra ℂ1 + ℂI ≅ ℍ of B(H, ℝ), the real linear endomorphisms of H.

(c) (Hermitian groups) Let H be a complex Hilbert space and H = H_{+} ⊕ H_{−} be an orthogonal decomposition. Further let T = T^{∗} ∈ B(H) with H_{±} = ker(T ∓1). We define the corresponding pseudo-unitary group

U(H_{+}, H_{−}) := {g ∈ GL(H): T g^{∗}T^{−1} = g^{−1}}.

We define Ω(x, y) := Im⟨x, y⟩ and write H^{ℝ} for the real Hilbert space underlying H. Then

Sp(H, Ω) := {g ∈ GL(H^{ℝ}, ℝ): (∀v, w ∈ H^{ℝ}) Ω(g.v, g.w) = Ω(v, w)}

is called the symplectic group of H. If we start with the real Hilbert space H^{ℝ} and consider an isometric complex structure I on H^{ℝ}, then we can define

Ω(x, y) := −⟨I.x, y⟩ = ⟨x, I.y⟩

and put

Sp(H^{ℝ}, I) := {g ∈ GL(H^{ℝ}, ℝ): (∀v, w ∈ H^{ℝ}) Ω(g.v, g.w) = Ω(v, w)}.

It is easy to see that both constructions lead to isomorphic groups Sp(H^{ℝ}, I) ≅ Sp(H, Ω).

Now let I be a conjugation on the complex Hilbert space H and H_{+} ⊆ H a subspace for which we get an orthogonal decomposition H = H_{+} ⊕ H_{−} with H_{−} := I.H_{+} . Then we define

O^{∗}(H, I) := U(H, I) ∩ U(H_{+}, H_{−}).

Theorem II.6. If H is an infinite-dimensional Hilbert space over 𝕂 ∈ {ℝ, ℂ, ℍ}, then the following groups are contractible:

(i) the group of 𝕂 -linear automorphisms GL(H, 𝕂).

(ii) the group of isometric 𝕂 -linear automorphisms U(H, 𝕂), and in particular the groups

O(H) = U(H, ℝ), U(H) = U(H, ℂ) and Sp(H) = U(H, ℍ). (iii) the group GL(H, I) if H is complex and I an antilinear isometry with I^{2} ∈ {±1}. More-

over, GL(H, I) has a smooth polar decomposition. (iv) the hermitian groups U(H_{+}, H_{−}), H = H_{+} ⊕ H_{−} an orthogonal decomposition with two

infinite-dimensional summands, Sp(H, Ω), and O^{∗}(H, I). Proof. (i) is Theorem II.4. (ii) follows from (i) and the polar decomposition GL(H, 𝕂) ≅ U(H, 𝕂) × Herm(H, 𝕂) of the group GL(H, 𝕂) with the unitary part U(H, 𝕂). (iii) In view of Definition II.5(b), the group U(H, I) is contractible, because it is one of the groups in (ii). Hence the assertion follows from the polar decomposition of GL(H, I) which can be obtained as follows. We consider the automorphism τ(g) := I(g^{∗})^{−1}I^{−1} of GL(H) and write τ_{𝔤}(x) := −Ix^{∗}I^{−1} for the corresponding antilinear automorphism of its Lie algebra 𝔤𝔩(H). Then

GL(H, I) = GL(H)^{τ} := {g ∈ GL(H): τ(g) = g}.

Let g = ue^{x} be the polar decomposition of g ∈ GL(H). Then τ(g) = τ(u)e^{τ_{𝔤}(x)} is the polar decomposition of τ(g), so that the uniqueness of this decomposition implies that τ(g) = g, is equivalent to τ(u) = u and τ_{𝔤}(x) = x, i.e., u ∈ U(H, I) and x ∈ Herm(H, I). (iv) For the hermitian groups we will see below that they have polar decompositions with

U(H_{+}, H_{−}) ∩ U(H) ≅ U(H_{+}) × U(H_{−}), Sp(H, Ω) ∩ O(H^{ℝ}) ≅ U(H)

and

O^{∗}(H, I) ∩ U(H) ≅ U(H_{+}),

where H ≅ H_{+} ⊕ I.H_{+} as in Definition II.5(c). Therefore (ii) implies that all these groups are contractible.

To prove the polar decomposition of U(H_{+}, H_{−}), let g ∈ GL(H) with polar decomposition g = ue^{x}, u ∈ U(H) and x = x^{∗}. For T as in Definition II.5(c) we consider the automorphism τ(g) := T (g^{∗})^{−1}T^{−1} of GL(H) and write τ_{𝔤}(x) := −T x^{∗}T^{−1} for the corresponding antilinear automorphism of its Lie algebra 𝔤𝔩(H). Then τ(g) = τ(u)e^{τ_{𝔤}(x)} is the polar decomposition of τ(g), so that the uniqueness of this decomposition implies that τ(g) = g is equivalent to τ(u) = u and τ_{𝔤}(x) = x. Therefore g ∈ U(H_{+}, H_{−}) if and only if

u ∈ U(H_{+}, H_{−}) ∩ U(H) ≅ U(H_{+}) × U(H_{−}) and x ∈ 𝔲(H_{+}, H_{−}).

To see that Sp(H, Ω) is adapted to the polar decomposition, we observe that

Ω(x, y) = Im⟨x, y⟩ = Re⟨x, iy⟩ = (x, Jy),

where (·, ·) := Re⟨·, ·⟩ denotes the real scalar product on H^{ℝ}. Therefore g ∈ Sp(H, Ω) is equivalent to g^{⊤}Jg = J, i.e., g = τ(g) := J(g^{⊤})^{−1}J^{−1} . Then τ is an involutive automorphism of GL(H^{ℝ}) and τ_{𝔤}(x) := −Jx^{⊤}J^{−1} is the corresponding Lie algebra automorphism. Let g = ue^{x} be the polar decomposition of g ∈ GL(H^{ℝ}), where u ∈ O(H^{ℝ}) and x^{⊤} = x. Then τ(g) = τ(u)e^{τ_{𝔤}(x)} is the polar decomposition of τ(g) because ue^{−x} is the polar decomposition of g^{−⊤}. Therefore g ∈ Sp(H, Ω) is equivalent to τ(u) = u, i.e., u ∈ U(H), and to Jx = −xJ, i.e., x is antilinear.

The argument for the group O^{∗}(H, I) is similar.

Homotopy groups of direct limit groups

Definition II.7. Let (X, x_{0}) and (Y, y_{0}) be pointed topological spaces. A map f ∈ C_{∗}(X, Y) := {h ∈ C(X, Y): h(x_{0}) = y_{0}} is called a weak homotopy equivalence if all induced maps π_{k}(f): π_{k}(X, x_{0}) → π_{k}(Y, y_{0}) are bijections.

A map f ∈ C_{∗}(X, Y) is called a homotopy equivalence if there exists g ∈ C_{∗}(Y, X) such that fg, resp., gf are homotopic to id_{Y} , resp., id_{X} in C_{∗}(Y, Y), resp., C_{∗}(X, X).

Theorem II.8. (a) Let X be a locally convex topological vector space and E ⊆ X a dense subspace. We endow E with the direct limit topology with respect to the finite-dimensional subspaces. For each open subset U ⊆ X the continuous map U ∩ E → U is a weak homotopy equivalence if U ∩ E is considered as a topological subspace of E.

(b) Let X and Y be metrizable locally convex topological vector spaces and f: X → Y a continuous linear map with dense range. For each open subset U ⊆ Y let V := f^{−1}(U) and f_{V} := f |_{V}: V → U. Then f_{V} is a homotopy equivalence. Proof. These are Theorems 12 and 16 in [Pa66].

Lemma II.9. Let E be a real vector space endowed with the direct limit topology with respect to its finite-dimensional subspaces. Then the following assertions hold:

(i) Each linearly independent subset of E is closed and discrete. (ii) Each compact subset of E is contained a finite-dimensional subspace. (iii) For each subset U ⊆ E and u_{0} ∈ U we have

π_{k}(U, u_{0}) ≅ lim_{F ∈F} π_{k}(U ∩ F, u_{0}),

where F denotes the directed set of all finite-dimensional subspaces F ⊆ E containing u_{0} . (iv) If U ⊆ E is a subset for which the intersection with all finite-dimensional subspaces are

open, then the subspace topology on U coincides with the direct limit topology with respect to the sets U ∩ F, F ⊆ E a finite-dimensional subspace.

Proof. (i) (cf. [Pa66, Lemma 5.2]) Let S ⊆ E be a linearly independent subset. Then for each finite-dimensional subspace F ⊆ E the intersection S ∩ F is closed, and therefore S is closed in E. The same argument implies that each subset of S is also closed in E. It follows in particular that S is a discrete topological space. (ii) (cf. [Pa66, Lemma 5.3]) Let K ⊆ E be a compact subset and S ⊆ K a maximal linearly independent subset. Then K ⊆ span S. In view of (i), S is closed, hence compact. On the other hand, S is discrete and therefore finite. (iii) Let Y be a compact space with base point y_{0} and f: Y → U a continuous map with f(y_{0}) = u_{0} . Then f(Y) is a compact subset of E, hence contained in a finite-dimensional subspace F, and we clearly have u_{0} = f(y_{0}) ∈ F.

For Y = 𝕊^{k}, it follows that the natural homomorphism

η: lim_{F ∈F} π_{k}(U ∩ F, u_{0}) → π_{k}(U, u_{0})

is surjective. To see that it is also injective, suppose that f: 𝕊^{k} → U ∩F is a continuous map which in U is homotopic to the constant map 𝕊^{k} → {u_{0}}. Let H: [0, 1] × 𝕊^{k} → U be a homotopy with H(0, x) = f(x) and H(1, x) = u_{0} . Then im(H) is contained in a finite-dimensional subspace F ⊆ E, and therefore the homotopy class of f in π_{k}(U ∩ F, u_{0}) is trivial. This implies that η is injective. (iv) This follows from the observation that a subset V ⊆ U is open in the subspace topology if and only if all intersections V ∩ F, F ⊆ E a finite-dimensional subspace, are open, because this already implies that V is open in E.

Definition II.10. Let J be an infinite set. We view a function m: J × J → 𝕂 as a matrix with entries m(i, j). In this sense we write 𝕄(J, 𝕂) for the set of all J × J -matrices with at most finitely many non-zero entries in 𝕂 . Then 𝕄(J, 𝕂) is a real algebra with respect to matrix multiplication. It has a unit if and only if J is finite. We write 1 = (δ_{ij})_{i,j∈J} for the identity matrix. Then 1 + 𝕄(J, 𝕂) is a multiplicative monoid, and we define GL(J, 𝕂) to be its group of units. We endow GL(J, 𝕂) with the direct limit topology with respect to the subgroups GL(F, 𝕂) := GL(J, 𝕂) ∩ (1 + 𝕄(F, 𝕂)), where F ⊆ J is a finite subset. It follows directly from the constructions that the left and right multiplications in the group GL(J, 𝕂) are continuous, but if J is uncountable, then the multiplication is not jointly continuous ([Gl99, Th. 7.1]). Here we identify 𝕄(F, 𝕂) in a natural way with a subset of 𝕄(J, 𝕂) and likewise GL(F, 𝕂) with a subset of GL(J, 𝕂).

Proposition II.11. Let J be an infinite set. Then for each injective map ℕ ↪ J the corresponding map GL(ℕ, 𝕂) → GL(J, 𝕂) is a weak homotopy equivalence.

Proof. We may w.l.o.g. assume that ℕ ⊆ J. Let η: GL(ℕ, 𝕂) ↪ GL(J, 𝕂) be the corresponding embedding of groups.

Let Y be a compact space and f: Y → GL(J, 𝕂) be a continuous map. Then there exists a finite subset F ⊆ J with f(Y) ⊆ GL(F, 𝕂) (Lemma II.9(ii)). If F′ ⊆ J is finite with F ∩F′ = Ø and |F | = |F′| = n, then GL(F ∪ F′, 𝕂) ≅ GL(2n, 𝕂), where we identify F with {1, …, n} and F′ with {n + 1, …, 2n}. Then f is a map of the form

f(y) = ( \begin{matrix}g(y) & 0 \\ 0 & 1 \end{matrix} ) ,

where we write the elements of GL(2n, 𝕂) as block 2 × 2-matrices with entries in 𝕄(n, 𝕂), and g: Y → GL(n, 𝕂) is a continuous map.

We consider the map H: [0, 1] × GL(n, 𝕂) → GL(2n, 𝕂) given by

H(t, a) := ( \begin{matrix}1 & 0 \\ t(a^{−1} − 1) & 1 \end{matrix} ) ( \begin{matrix}1 & t \\ 0 & 1 \end{matrix} ) ( \begin{matrix}1 & 0 \\ t(a − 1) & 1 \end{matrix} ) ( \begin{matrix}1 & −ta^{−1} \\ 0 & 1 \end{matrix} )

which satisfies

H(1, a) = ( \begin{matrix}a & 0 \\ 0 & a^{−1} \end{matrix} ) and H(0, a) = ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) .

Then

H̃: [0, 1] × Y → GL(2n, 𝕂), (t, y) ↦ f(y)H(t, g(y))^{−1}

is continuous with H̃(0, y) = f(y) and

H̃(1, y) = ( \begin{matrix}1 & 0 \\ 0 & g(y)^{−1} \end{matrix} ) .

This construction shows that every continuous map f: Y → GL(F, 𝕂) is homotopic in GL(J, 𝕂) to a continuous map f′: Y → GL(F′, 𝕂).

In particular we see that for each continuous map f: Y → GL(J, 𝕂) there exists a finite subset E ⊆ ℕ such that f is homotopic to a continuous map f̃: Y → GL(E, 𝕂). In fact, with F as above, we simply choose E ⊆ ℕ such that |E| = |F | and E ∩ F = Ø. This argument shows that the natural homomorphism π_{k}(η): π_{k}(GL(ℕ, 𝕂)) → π_{k}(GL(J, 𝕂)) is surjective.

To see that π_{k}(η) is injective, suppose that f: Y → GL(n, 𝕂) ⊆ GL(ℕ, 𝕂) is in GL(J, 𝕂) homotopic to a constant map. Let H: [0, 1]×Y → GL(J, 𝕂) be a homotopy with H(0, y) = 1 and H(1, y) = f(y) for all y ∈ Y . Then there exists a finite subset F ⊆ J with im(H) ⊆ GL(F, 𝕂). Then we may assume that F ⊇ {1, …, n}, and since GL(|F |, 𝕂) ≅ GL(F, 𝕂), we see that the homotopy class of f vanishes in GL(|F |, 𝕂) ⊆ GL(ℕ, 𝕂). In particular f is homotopic to a constant map in GL(ℕ, 𝕂).

The homotopy groups for GL(ℕ, 𝕂) and hence for all groups GL(J, 𝕂), where J is an infinite set, are given by the following theorem.

Theorem II.12. (Bott Periodicity Theorem) Let 𝕂 ∈ {ℝ, ℂ, ℍ} and d := dim_{ℝ} 𝕂 . Then for k ≤ d(n + 1) − 3 and q ∈ ℕ the maps

π_{k}(GL(n, 𝕂)) → π_{k}(GL(n + q, 𝕂))

are isomorphisms, so that

π_{k}(GL(ℕ, 𝕂)) ≅ π_{k}(GL(n, 𝕂)).

Moreover, we have the periodicity relations

π_{n+2}(GL(ℕ, ℂ)) ≅ π_{n}(GL(ℕ, ℂ)), π_{n+4}(GL(ℕ, ℝ)) ≅ π_{n}(GL(ℕ, ℍ)),

π_{n+4}(GL(ℕ, ℍ)) ≅ π_{n}(GL(ℕ, ℝ)), π_{n}(GL(ℕ, ℍ)/ GL(ℕ, ℂ)) ≅ π_{n+1}(GL(ℕ, ℍ)), π_{n}(GL(ℕ, ℂ)/ GL(ℕ, ℝ)) := lim_{m→∞} π_{n}(GL(m, ℂ)/ GL(m, ℝ)) ≅ π_{n+2}(GL(ℕ, ℍ)),

π_{n}(GL(ℕ, ℝ)/ GL(ℕ, ℂ)) := lim_{m→∞} π_{n}(GL(2m, ℝ)/ GL(m, ℂ)) ≅ π_{n+1}(GL(ℕ, ℝ)),

π_{n}(GL(ℕ, ℂ)/ GL(ℕ, ℍ)) := lim_{m→∞} π_{n}(GL(2m, ℂ)/ GL(m, ℍ)) ≅ π_{n+2}(GL(ℕ, ℝ)).

In particular the homotopy groups of GL(ℕ, 𝕂) are determined by the following table:

GL(ℕ, ℝ)  GL(ℕ, ℂ)  GL(ℕ, ℍ)
π_{0}  ℤ_{2}  0  0
π_{1}  ℤ_{2}  ℤ  0
π_{2}  0  0  0
π_{3}  ℤ  ℤ  ℤ

Proof. The first easy part is [Hu94, Th. 8.4.1] and the remaining assertions can be found in [Bo59, pp.314ff] (cf. also [Hu94, Cor. 9.5.2]).

For the sake of completeness, we include a proof of the first part. Using the polar decomposition, we may consider the corresponding maps of the unitary groups U(n, 𝕂). To understand the effect of the inclusion maps U(n, 𝕂) → U(n+1, 𝕂) for the homotopy groups, we consider the transitive action of U(n + 1, 𝕂) on the sphere 𝕊^{d(n+1)−1} which leads to a locally trivial principal bundle

U(n, 𝕂) ↪ U(n + 1, 𝕂) → 𝕊^{d(n+1)−1}.

The exact homotopy sequence of this bundles contains the piece

… → π_{k+1}(𝕊^{d(n+1)−1}) → π_{k}(U(n, 𝕂))⟶^{π_{k}(η_{n})}π_{k}(U(n + 1, 𝕂)) → π_{k}(𝕊^{d(n+1)−1}) → ….

For k < d(n + 1) − 1, i.e., k ≤ d(n + 1) − 2 the group π_{k}(𝕊^{d(n+1)−1}) vanishes (this follows by smoothing and Sard’s Theorem), so that π_{k}(η_{n}) is surjective. If, in addition, k ≤ d(n + 1) − 3, then k + 1 < d(n + 1) − 1 implies that also π_{k+1}(𝕊^{d(n+1)−1}) vanishes, so that the injectivity of π_{k}(η_{n}) follows.

Homotopy groups of congruence subgroups for Schatten ideals

Let H be a Hilbert space over 𝕂 ∈ {ℝ, ℂ, ℍ}. For x, y ∈ H we define P_{x,y}(v) := ⟨v, y⟩x and put P_{x} := P_{x,x}. Note that P_{x,y} ∈ B_{1}(H).

Lemma II.13. Let H be a 𝕂 -Hilbert space and (e_{j})_{j∈J} an orthonormal basis. Then B_{0}(H) := span{P_{e_{j},e_{k}}: j, k ∈ J} is dense in each of the spaces B_{p}(H), 1 ≤ p ≤ ∞. Proof. For each p ∈ [1, ∞] we have ∥x∥_{∞} ≤ ∥x∥_{p} ≤ ∥x∥_{1} and accordingly B_{1}(H) ⊆ B_{p}(H) ⊆ K(H) = B_{∞}(H).

Since B_{0}(H) and B_{p}(H) are ∗-invariant, it suffices to see that each hermitian operator in B_{p}(H) is contained in the closure of B_{0}(H). The Spectral Theorem for Compact Hermitian Operators directly implies that the ideal B_{fin}(H) of continuous maps with finite-dimensional image is dense in B_{p}(H), and hence that B_{1}(H) is dense in B_{p}(H). Therefore it suffices to see that B_{0}(H) is dense in B_{1}(H) because ∥x∥_{p} ≤ ∥x∥_{1} .

In view of the Hahn–Banach Theorem, we have to show that each continuous linear functional f ∈ B_{1}(H)′ vanishing on B_{0}(H) is zero. As B_{1}(H)′ ≅ B(H) ([Ne99, Prop. A.I.10(vi)]), the functional f can be written as f(X) = tr(AX) for some A ∈ B(H). Hence

f(P_{e_{j},e_{k}}) = tr(AP_{e_{j},e_{k}}) = tr(P_{A.e_{j},e_{k}}) = ⟨A.e_{j}, e_{k}⟩.

If f vanishes on B_{0}(H), then the matrix of A with respect to the orthonormal basis (e_{j})_{j∈J} vanishes, and this means that A = 0.

The following theorem is well known for the case of separable Hilbert spaces (cf. [Pa65] and [dlH72, p.II.29]). The results on direct limit groups obtained in the preceding subsection easily permit us to extend it to general Hilbert spaces.

Theorem II.14. Let H be an infinite-dimensional Hilbert space over 𝕂 ∈ {ℝ, ℂ, ℍ} and p ∈ [1, ∞]. Then the following assertions hold:

(i) For every k ∈ ℕ_{0} we have π_{k}(GL_{p}(H)) ≅ π_{k}(GL(ℕ, 𝕂)) ≅ lim_{⟶} π_{k}(GL(n, 𝕂)).

(ii) If H_{s} ⊆ H is an infinite-dimensional separable subspace, then the inclusion map GL_{p}(H_{s}) ↪

GL_{p}(H) is a weak homotopy equivalence. (iii) For 1 ≤ p ≤ q ≤ ∞ the inclusion map GL_{p}(H) ↪ GL_{q}(H) is a homotopy equivalence.

Proof. (i) Let e_{j}, j ∈ J, be an orthonormal basis of H. Then Lemma II.13 above shows that B_{0}(H) = span{P_{e_{j},e_{k}}: j, k ∈ J} is dense B_{p}(H). We endow B_{0}(H) with the direct limit topology with respect to the directed set of finite-dimensional subspaces of B_{0}(H).

For the open subset U := GL_{p}(H) − 1 ⊆ B_{p}(H) we have

U ∩ B_{0}(H) = GL(J, 𝕂) − 1,

where GL(J, 𝕂) is the set of those elements g ∈ GL(H) for which the matrix of g − 1 with respect to (e_{j})_{j∈J} has only finitely many entries, i.e., g and g^{∗} fix all but finitely many e_{j}. It easily follows from (g^{∗})^{−1} = (g^{−1})^{∗} that g^{−1} has the same property. Therefore the natural identification of B_{0}(H) with the matrix algebra 𝕄(J, 𝕂) leads to an identification of the group 1 + (U ∩ B_{0}(H)) with GL(J, 𝕂) as in Definition II.10.

Theorem II.8 implies that if we endow GL(J, 𝕂) with the final topology with respect to the subgroups GL(F, 𝕂), F ⊆ J a finite subset, the inclusion map GL(J, 𝕂) → GL_{p}(H) is a weak homotopy equivalence. Further Proposition II.11 shows that we have a weak homotopy equivalence GL(ℕ, 𝕂) ↪ GL(J, 𝕂). Composition of these two weak homotopy equivalences yields a weak homotopy equivalence, and, in view of Lemma II.9(iii), this proves (i). (ii) We first choose an orthonormal basis (e_{n})_{n∈ℕ} of H_{s} and then complete it to an orthonormal basis (e_{j})_{j∈J} of H. We consider the corresponding maps

φ_{1}: GL(ℕ, 𝕂) → GL_{p}(H_{s}), φ_{2}: GL(J, 𝕂) → GL_{p}(H), φ_{3}: GL(ℕ, 𝕂) → GL(J, 𝕂)

and

φ_{4}: GL_{p}(H_{s}) → GL_{p}(H)

with φ_{4} ∘ φ_{1} = φ_{2} ∘ φ_{3} . Since φ_{1} and φ_{2} are weak homotopy equivalences by the first part of the proof, and φ_{3} is a weak homotopy equivalence by Proposition II.11, it follows that φ_{4} also is a weak homotopy equivalence. (iii) From the elementary inclusion l^{p}(ℕ, ℝ) ⊆ l^{q}(ℕ, ℝ) we derive that B_{p}(H) ⊆ B_{q}(H), and Lemma II.13 implies that B_{p}(H) is a dense subspace. Therefore (iii) follows by applying Theorem II.8(b) to the open subset U := GL_{q}(H)−1 ⊆ B_{q}(H) which satisfies U ∩B_{p}(H) = GL_{p}(H)−1.

Corollary II.15. If H is an infinite-dimensional complex Hilbert space, 1 ≤ p ≤ ∞, and

GL_{p}(H, I) := GL(H, I) ∩ GL_{p}(H),

then the following assertions hold:

(i) π_{k}(GL_{p}(H, I)) ≅ \{ \begin{matrix}π_{k}(GL(ℕ, ℝ)) & for I^{2} = 1 \\ π_{k}(GL(ℕ, ℍ)) & for I^{2} = −1. \end{matrix}

(ii) If H_{s} ⊆ H is an infinite-dimensional separable I -invariant subspace, then the inclusion map

GL_{p}(H_{s}, I |_{H_{s}}) ↪ GL_{p}(H, I) is a weak homotopy equivalence. (iii) For 1 ≤ p ≤ q ≤ ∞ the inclusion map GL_{p}(H, I) ↪ GL_{q}(H, I) is a homotopy equivalence.

Proof. We first observe that the polar decomposition of GL_{p}(H) ([Ne00a, Prop. A.4]) implies that its intersection with GL_{p}(H, I) also has a polar decomposition (see the proof of Theorem II.6), hence is homotopy equivalent to U_{p}(H, I) := U_{p}(H) ∩ GL(H, I). For I^{2} = −1 we have U_{p}(H, I) ≅ U_{p}(H, ℍ), and for I^{2} = 1 we get U_{p}(H, I) ≅ U_{p}(H_{ℝ}, ℝ), where H_{ℝ} = {x ∈ H: I.x = x}. Since the group U_{p}(H, 𝕂) is homotopy equivalent to GL_{p}(H, 𝕂) (Theorem II.6), the assertions on the groups GL_{p}(H, I) follow Theorem II.14 and the existence of polar decompositions.

III. Homotopy groups of restricted groups

In this section we turn to the homotopy groups of the restricted groups GL_{p}(H, D) and GL_{p}(H, I, D) defined below for a complex Hilbert space H, an antilinear isometry I with I^{2} = ±1, and a hermitian operator D with finite spectrum.

The main motivation for the study of the restricted groups defined by a hermitian element D = D^{∗} ∈ 𝔤𝔩(H) comes from the results in [Ne01a]. There it is shown that each continuous Lie algebra cocycle on 𝔤 ∈ {𝔤𝔩_{2}(H), 𝔤𝔩_{2}(H, I)} can be written in the form ω_{D}(x, y) := tr([D, x]y) for an element D ∈ 𝔤_{b} ∈ {𝔤𝔩(H), 𝔤𝔩(H, I)}. If one considers the unitary real form 𝔤_{ℝ} := {x ∈ 𝔤: x^{∗} = −x}, then the corresponding real cocycles are of the form ω_{D} for D^{∗} = −D. Each of these cocycles determines an affine action of G_{ℝ} := G ∩ U(H) on 𝔤_{ℝ} ≅ 𝔤′_{ℝ} (a twisted coadjoint action) given by

g.x = gxg^{−1} + D − gDg^{−1} = g(x − D)g^{−1} + D

whose orbits carry natural weak symplectic structures generalizing the Kirillov–Kostant–Souriau structure on coadjoint orbits of compact groups. The orbit O_{D} of 0 ∈ 𝔤_{ℝ} is most naturally attached to D, and we show in [Ne01a] that it is a submanifold of 𝔤_{ℝ} if and only if its symplectic structure is strongly symplectic if and only if D has finite spectrum, which of course is equivalent to the hermitian operator iD having finite spectrum. In this sense the condition of having finite spectrum shows up as a natural condition in regard of the geometry of coadjoint orbits. Now the restricted real group G_{ℝ}(D) = G_{ℝ}Z_{G_{ℝ,b}}(D) acts naturally on these orbits and one obtains unitary representations of a central 𝕋-extension on Hilbert spaces of holomorphic sections of holomorphic line bundles on the orbits. For more details we refer to [Ne01a] and [Ne00a].

Restricted classical groups

Definition III.1. (a) Let H be a 𝕂 -Hilbert space with 𝕂 ∈ {ℝ, ℂ, ℍ}. If H = H_{1}⊕…⊕H_{k} is the eigenspace decomposition of D = D^{∗} ∈ B(H), then we write operators on H accordingly as (k × k)-block matrices and consider for 1 ≤ p ≤ ∞ the space

B_{p}(H, D) := {A ∈ B(H): ∥[D, A]∥_{p} < ∞} = {A = (a_{ij}) ∈ B(H): (∀i ≠ j)a_{ij} ∈ B_{p}(H_{j}, H_{i})}.

This space carries the structure of a Banach algebra given by the natural composition of operators and the norm

∥X∥ := max{∥a_{jj}∥, j = 1, …, k; ∥a_{jl}∥_{p}, j ≠ l}.

It is clear that A is complete with respect to this norm. That the norm on A satisfies ∥XY ∥ ≤ k∥X∥ ∥Y ∥ can be seen as follows. We have

∥(XY)_{jj}∥ = ∥ ∑_{l} X_{jl}Y_{lj}∥ ≤ ∥X_{jj}∥ ∥Y_{jj}∥ + ∑_{l≠j} ∥X_{jl}∥_{p}∥Y_{lj}∥_{p} ≤ k∥X∥ ∥Y ∥

because ∥A∥ ≤ ∥A∥_{p}. We likewise obtain for j ≠ l the estimate

\begin{aligned}∥(XY)_{jl}∥_{p} & ≤ ∥X_{jj}Y_{jl}∥_{p} + ∥X_{jl}Y_{ll}∥_{p} + ∑_{i≠j,l} ∥X_{ji}Y_{il}∥_{p} \\ & ≤ ∥X_{jj}∥ ∥Y_{jl}∥_{p} + ∥X_{jl}∥_{p}∥Y_{ll}∥ + ∑_{i≠j,l} ∥X_{ji}∥_{p} ∥Y_{il}∥_{p} ≤ k∥X∥ ∥Y ∥. \end{aligned}

Since B_{p}(H_{i}, H_{j}) = B(H_{i}, H_{j}) if and only if one of the spaces H_{i} and H_{j} is finitedimensional, the algebra B_{p}(H, D) coincides with B(H) if and only if at most one of the spaces H_{j} is infinite-dimensional.

(b) Next we modify the construction in (a) slightly. The space

B_{1,2}(H, D) := {A = (a_{ij}) ∈ B_{2}(H, D): (∀j) a_{jj} ∈ B_{1}(H_{j})}

is a Banach algebra with respect to operator composition and the norm

∥X∥ := max{∥a_{jj}∥_{1}; ∥a_{jl}∥_{2}, j ≠ l}.

It is clear that A is complete with respect to this norm. We recall that

∥XY ∥_{1} ≤ ∥X∥_{2}∥Y ∥_{2} ≤ ∥X∥_{1}∥Y ∥_{1} for X, Y ∈ B_{2}(H).

In view of this fact, we can show that the norm on B_{1,2}(H) satisfies ∥XY ∥ ≤ k∥X∥ ∥Y ∥: For each j we have

∥(XY)_{jj}∥_{1} = ∥ ∑_{l} X_{jl}Y_{lj}∥_{1} ≤ ∥X_{jj}∥_{1} ∥Y_{jj}∥_{1} + ∑_{l≠j} ∥X_{jl}∥_{2}∥Y_{lj}∥_{2} ≤ k∥X∥ ∥Y ∥.

We likewise obtain for j ≠ l the estimate

\begin{aligned}∥(XY)_{jl}∥_{2} & = ∥X_{jj}Y_{jl}∥_{2} + ∥X_{jl}Y_{ll}∥_{2} + ∑_{i≠j,l} ∥X_{ji}Y_{il}∥_{2} \\ & ≤ ∥X_{jj}∥ ∥Y_{jl}∥_{2} + ∥X_{jl}∥_{2}∥Y_{ll}∥ + ∑_{i≠j,l} ∥X_{ji}∥_{2} ∥Y_{il}∥_{2} ≤ k∥X∥ ∥Y ∥ \end{aligned}

because ∥A∥ ≤ ∥A∥_{2} ≤ ∥A∥_{1} .

Proposition III.2. In the setting of Definition III.1 we have:

(a) For each p ∈ [1, ∞] the set GL_{p}(H, D) := GL(H) ∩ B_{p}(H, D) is a group.

(b) GL_{1,2}(H, D) := GL(H) ∩ (1 + B_{1,2}(H, D)) is a group.

(c) The inclusion maps GL_{1}(H) ↪ GL_{1,2}(H, D) ↪ GL_{2}(H) are homotopy equivalences.

(d) For each p ∈ [1, ∞] the inclusion GL_{p}(H, D) ↪ GL_{∞}(H, D) is a homotopy equivalence.

(e) For each p ∈ [1, ∞] the the polar map

U_{p}(H, D) × Herm_{p}(H, D) → GL_{p}(H, D), (u, x) ↦ ue^{x}

is a diffeomorphism and the inclusion U_{p}(H, D) ↪ GL_{p}(H, D) is a homotopy equivalence. Proof. (a) Let g ∈ GL_{p}(H, D). We only have to show that (g^{−1})_{il} ∈ B_{p}(H_{l}, H_{i}) holds for i ≠ l. First we observe that

1 = (g^{−1})_{ii}g_{ii} + ∑_{j≠i}(g^{−1})_{ij}g_{ji} ∈ (g^{−1})_{ii}g_{ii} + B_{p}(H_{i}).

We also have

g_{ii}(g^{−1})_{il} = − ∑_{j≠i} g_{ij}(g^{−1})_{jl} ∈ B_{p}(H_{l}, H_{i}).

Multiplying this equation with (g^{−1})_{ii}, we obtain

(g^{−1})_{ii}g_{ii}(g^{−1})_{il} ∈ B_{p}(H_{l}, H_{i}) ∩ ((g^{−1})_{il} + B_{p}(H_{l}, H_{i})).

The fact that the intersection of these two sets is not empty shows that (g^{−1})_{il} ∈ B_{p}(H_{l}, H_{i}).

(b) Let g ∈ GL(H) ∩ (1 + B_{1,2}(H, D)) ⊆ B_{2}(H, D). Then (a) implies that for j ≠ l we have (g^{−1})_{jl} ∈ B_{2}(H_{l}, H_{j}). We further have for i =, 1 …, k:

1 = g_{ii}(g^{−1})_{ii} + ∑_{j≠i} g_{ij}(g^{−1})_{ji} ∈ (1 + B_{1}(H_{i}))(g^{−1})_{ii} + B_{1}(H_{i}) ⊆ (g^{−1})_{ii} + B_{1}(H_{i}),

so that g^{−1} ∈ 1 + B_{1,2}(H, D).

(c) First we observe that the space B_{1}(H) is dense in B_{1,2}(H, D), so that

GL_{1}(H) = GL(H) ∩ (1 + B_{1}(H))

further yields

GL_{1}(H) = GL_{1,2}(H, D) ∩ (1 + B_{1}(H)),

and now Theorem II.8(b) applies.

Next we note that B_{1,2}(H, D) ⊆ B_{2}(H) is a dense subspace and that (b) implies that GL_{1,2}(H, D) = GL_{2}(H) ∩ (1 + B_{1,2}(H, D)). Therefore Theorem II.8(b) applies again and shows that the inclusion GL_{1,2}(H, D) → GL_{2}(H) is a homotopy equivalence.

(d) follows as in (c) from Theorem II.8(b) because B_{p}(H, D) is dense in B_{∞}(H, D) with

GL_{p}(H, D) = GL(H) ∩ B_{p}(H, D) = GL(H) ∩ B_{∞}(H, D) ∩ B_{p}(H, D) = GL_{∞}(H, D) ∩ B_{p}(H, D).

(e) (See also [HH94b, Prop. 2.1.14] for the existence of a polar decomposition). In view of (a), GL_{p}(H, D) = GL(H) ∩ B_{p}(H, D) is the unit group of the Banach algebra B_{p}(H, D). Hence the spectrum of an element of the Banach algebra B_{p}(H, D) is the same as the spectrum as an element of B(H).

Let g ∈ GL_{p}(H, D). Then g^{∗}g ∈ B_{p}(H, D), and Spec_{B_{p}(H,D)}(g^{∗}g) = Spec(g^{∗}g) is contained in ]0, ∞[. Therefore [Ne00a, Lemma A.1] implies that log(g^{∗}g) ∈ B_{p}(H, D) and that the map GL_{p}(H, D) → B_{p}(H, D), g ↦ log(g^{∗}g), is smooth. If g = ue^{x} is the polar decomposition of g, then we conclude that x = log(g^{∗}g) ∈ B_{p}(H, D), hence that e^{x} ∈ GL_{p}(H, D) and therefore u = ge^{−x} ∈ U_{p}(H, D). Moreover, the polar map

U_{p}(H, D) × Herm_{p}(H, D) → GL_{p}(H, D), (u, x) ↦ ue^{x}

is a diffeomorphism since its inverse is also smooth. This means that GL_{p}(H, D) has a smooth polar decomposition, and (e) follows.

Definition III.3. (Restricted groups) Let 𝔤 ∈ {𝔤𝔩_{2}(H), 𝔤𝔩_{2}(H, I)} and accordingly 𝔤_{b} ∈ {𝔤𝔩(H), 𝔤𝔩(H, I)}. We fix a hermitian element D ∈ 𝔤_{b} with finite spectrum.

For 𝔤 = 𝔤𝔩_{2}(H) we define

G_{b} := GL(H) and G := GL_{2}(H).

For 𝔤 = 𝔤𝔩_{2}(H, I) we likewise put G_{b} := GL(H, I) and G := GL_{2}(H, I).

In both cases we define the restricted groups associated to 𝔤 and D = D^{∗} ∈ 𝔤_{b} by GL_{p}(H, D) (cf. Proposition III.2) for 𝔤 = 𝔤𝔩_{2}(H) and

GL_{p}(H, I, D) := GL(H, I) ∩ GL_{p}(H, D), 1 ≤ p ≤ ∞,

for 𝔤 = 𝔤𝔩_{2}(H, I). We likewise define

GL_{1,2}(H, I, D) := GL(H, I) ∩ GL_{1,2}(H, D).

We also put

GL(H)^{0} := GL(H, D)^{0} := Z_{GL(H)}(D) ≅ ∏_{j=1}^{k} GL(H_{j})

and

GL(H, I)^{0} := GL(H, I, D)^{0} := Z_{GL(H,I)}(D) ≅ GL(H_{0}, I_{0}) × ∏_{j=1}^{k} GL(H_{j}),

where the last isomorphism follows easily from Remark I.2.

Remark III.4. Suppose that H is infinite-dimensional. The following remark shows that we may often reduce considerations about the groups GL_{p}(H, D) or GL_{p}(H, I, D) to the case where all spaces H_{j} are infinite-dimensional.

Let 𝔤_{b} ∈ {𝔤𝔩(H), 𝔤𝔩(H, I)} for 𝔤 ∈ {𝔤𝔩_{2}(H), 𝔤𝔩_{2}(H, I)}. Let D, D′ be hermitian elements of 𝔤_{b} with finite spectrum for which D − D′ has finite rank. Then for x ∈ B(H) the conditions [D, x] ∈ B_{p}(H) and [D′, x] ∈ B_{p}(H) are equivalent because their difference has finite rank, whence B_{p}(H, D) = B_{p}(H, D′) and therefore B_{p}(H, I, D) = B_{p}(H, I, D′).

We explain the construction for 𝔤𝔩_{p}(H, I, D), the other case is even simpler. If D is given, then we construct D′ as follows. If H_{j} is infinite-dimensional, then we define D′ |_{H_{j}} as D |_{H_{j}}. If H_{0} is finite-dimensional, then it is even-dimensional and there exists a subspace H_{0}^{+} for which H_{0} = H_{0}^{+} ⊕ I.H_{0}^{+} is an orthogonal direct sum (cf. Remark I.2). Pick j_{0} > 0 such that H_{j_{0}} is infinite-dimensional. Then we define D′ on all finite-dimensional spaces H_{j}, j > 0, and H_{0}^{+} in such a way that it has the same eigenvalue d_{j_{0}} as D on H_{j_{0}} . Likewise we define it by −d_{j_{0}} on H_{0}^{−} and the finite-dimensional spaces H_{−j}. Then D′ ∈ 𝔤_{b} and D − D′ has finite rank, so that GL_{p}(H, I, D) = GL_{p}(H, I, D′).

In the remainder of this section we discuss the homotopy groups of the two types of restricted groups GL_{p}(H, D) and GL_{p}(H, I, D), where D = D^{∗} ∈ 𝔤_{b} is a hermitian element with finite spectrum.

The homotopy groups of GL_{p}(H, D)

In Proposition III.2(d) we have seen that for p ∈ [1, ∞] the natural inclusion maps GL_{p}(H, D) ↪ GL_{∞}(H, D) are homotopy equivalences. Therefore it suffices to determine the homotopy groups of GL_{2}(H, D) to know them for all the groups GL_{p}(H, D).

In the following we keep the setting of Definition III.1, resp., Examples I.9 and set

k_{∞} := |{j ∈ {1, …, k}: dim H_{j} = ∞}|.

The determination of the connected components of GL_{2}(H, D) described in the following proposition can be found in [HH94b, Prop. 2.3.1].

Proposition III.5. For each g = (g_{ij}) ∈ GL_{2}(H, D) the diagonal operators g_{jj} are Fredholm operators, and the connected components of the group GL_{2}(H, D) coincide with the fibers of the continuous homomorphism

Ind: GL_{2}(H, D) → ℤ^{k_{∞}}, g ↦ (ind(g_{jj}))_{dim H_{j}=∞}

whose image is the set of those tuples (n_{j}) ∈ ℤ^{k_{∞}} with ∑_{j} n_{j} = 0. Moreover, the identity component GL_{2}(H, D)_{e} of GL_{2}(H, D) is given by

ker(Ind) = GL_{2}(H, D)_{e} = GL_{2}(H) GL(H, D)^{0} = GL_{1,2}(H, D) GL(H, D)_{∞}^{0},

where GL(H, D)_{∞}^{0} ⊆ GL(H, D)^{0} is the subgroup corresponding to the infinite-dimensional ones among the spaces H_{j}, j = 1, …, k, and

π_{0}(GL_{2}(H, D)) ≅ \{ \begin{matrix}ℤ^{k_{∞}−1} & for k_{∞} ≥ 1 \\ 0 & for k_{∞} = 0. \end{matrix}

Proof. For each p ∈ [1, ∞] we have GL_{p}(H, D) ⊆ GL_{∞}(H, D), and each element g = (g_{ij}) of this group, written as a k×k -matrix with g_{ij} ∈ B(H_{j}, H_{i}), is a diagonal matrix modulo compact operators and invertible as such. Therefore all diagonal entries g_{jj} are invertible modulo compact operators, hence contained in the monoid Fred(H_{j}) of Fredholm operators on H_{j}. This means that ind(g_{jj}) := dim ker g_{jj} − dim coker g_{jj} is well defined. That Ind is a group homomorphism follows from the observation that modulo compact operators we can view elements of GL_{2}(H, D) as diagonal operators, so that the assertion follows from ind(ab) = ind(a) + ind(b) for Fredholm operators on each space H_{j}.

The inclusions

(3.1)  GL_{1,2}(H, D) GL(H, D)_{∞}^{0} ⊆ GL_{2}(H) GL(H, D)^{0} ⊆ GL_{2}(H, D)_{e} ⊆ ker(Ind)

follow from the connectedness of the groups GL_{1,2}(H) and GL_{2}(H) (Theorem II.14, Proposition III.2(c)) and GL(H_{j}) and the continuity of the index function.

For the converse, let g ∈ ker(Ind). Since each g_{jj} is a Fredholm operator of index 0, we conclude that whenever dim H_{j} = ∞, there exists a finite rank operator b_{j} ∈ B(H_{j}) with ker b_{j} = (ker g_{jj})^{⊥} mapping ker(g_{jj}) bijectively onto im(g_{jj})^{⊥}. Then d_{j} := g_{jj} + b_{j} ∈ GL(H_{j}) satisfies

g_{jj} = g_{jj} + b_{j} − b_{j} ∈ (g_{jj} + b_{j})(1 + B_{1}(H_{j})).

For dim H_{j} < ∞ we put d_{j} := 1. Then d := diag(d_{1}, …, d_{k}) ∈ GL(H, D)_{∞}^{0} satisfies d^{−1}g ∈ GL_{1,2}(H, D). We thus obtain ker(Ind) ⊆ GL_{1,2}(H, D) GL(H, D)_{∞}^{0} and hence equality in (3.1).

Since the off-diagonal entries of g are compact, the invertibility of g implies that 0 = ind(g) = ∑_{j} ind(g_{jj}), hence the corresponding restriction on the image of Ind. If, conversely, (n_{j}) ∈ ℤ^{k_{∞}} satisfies ∑_{j} n_{j} = 0, then we can write H_{j} as l^{2}(J_{j}, ℂ) and accordingly H as l^{2}(J, ℂ) with J = ˙∪_{j}J_{j}. Now there exists a permutation σ of J for which

n_{j} = |J_{j} ∩ σ^{−1}(J \ J_{j})| − |(J_{j} ∩ σ(J \ J_{j}))|.

We leave the easy proof to the reader. Then the isometry σ̃ of H defined by σ is contained in GL_{2}(H, D) and satisfies Ind(σ̃) = (n_{j})_{dim H_{j}=∞}.

Definition III.6. Let G be a Banach–Lie group with Lie algebra 𝔤 = L(G) and 𝔥 ⊆ 𝔤 a closed subalgebra. We call the subgroup H := ⟨exp 𝔥⟩ generated by the exponential image of 𝔥 the corresponding analytic subgroup of G. According to [Ma62], this group has a natural Lie group structure such that the map H ↪ G is a morphism of Lie groups.

For a closed subgroup H ⊆ G we consider the closed Lie subalgebra

𝔥 := L(H) := {X ∈ 𝔤: exp(ℝX) ⊆ H}

of 𝔤 ([Ne00a, Cor. IV.3]) and say that H is a Lie subgroup if there exists an open 0-neighborhood V ⊆ 𝔤 such that exp |_{V} is a diffeomorphism onto an open subset exp(V) and exp(V ∩ 𝔥) = (exp V) ∩ H. Then H carries a natural Lie group structure such that the map H ↪ G is a homomorphism of Lie groups which is a homeomorphism onto its image (cf. [Ne00a, Prop. IV.5]).

We call a Lie subgroup H ⊆ G complemented or split if 𝔤 contains a closed subspace E complementing the closed subalgebra 𝔥. If this condition is satisfied, then H is a submanifold in the sense of Bourbaki, and in particular the homogeneous space G/H carries a natural manifold structure such that the canonical map π: G → G/H is a submersion (cf. [Ne00a, Prop. IV.5]; see also [Bou90, Ch. 3, §1.6, Prop. 11]).

The next step is the determination of all homotopy groups of the restricted group GL_{p}(H, D).

Theorem III.7. If H is infinite-dimensional, then

π_{m}(GL_{2}(H, D)) ≅ π_{m−1}(GL(ℕ, ℂ))^{k_{∞}−1} ≅ \{ \begin{matrix}ℤ^{k_{∞}−1} & for m even \\ 0 & for m odd. \end{matrix}

Proof. We consider the short exact sequence of groups

(3.2) A := GL_{2}(H) ∩ GL(H, D)_{∞}^{0} ↪ B := GL_{2}(H) ⋊ GL(H, D)_{∞}^{0} → → C := GL_{2}(H, D)_{e},

where the surjectivity of the multiplication map B → C follows from Proposition III.5. Moreover, the assumptions of Proposition A.6 are satisfied because GL_{2}(H) is connected and a normal subgroup of GL_{2}(H, D), since B_{2}(H) is an algebra ideal of B(H). Furthermore, it is clear that

A ≅ ∏_{dim H_{j}=∞} GL_{2}(H_{j})

is a complemented Lie subgroup of GL_{2}(H). It follows that B has a natural Banach–Lie group structure and that the map B → C is a locally trivial A-principal bundle.

The homotopy groups of A are given by

π_{m}(A) ≅ π_{m}(GL_{2}(H)^{k_{∞}}) ≅ π_{m}(GL_{2}(H))^{k_{∞}}

(Theorem II.14). Since the group

GL(H, D)_{∞}^{0} ≅ ∏_{dim H_{j}=∞} GL(H_{j})

is contractible (Theorem II.4), the homomorphisms

χ_{m}: π_{m}(A) ≅ π_{m}(GL_{2}(H))^{k_{∞}} → π_{m}(GL_{2}(H)) ≅ π_{m}(B)

can be viewed as the k_{∞} -fold summation maps in the abelian group π_{m}(GL_{2}(H)). This map is surjective with

ker χ_{m} ≅ π_{m}(GL_{2}(H))^{k_{∞}−1}.

Therefore the exact homotopy sequence of (3.2) yields for each m ∈ ℕ_{0} a short exact sequence

⋯ ⟶^{0}π_{m+1}(C) ↪ π_{m}(A)⟶^{χ_{m}}π_{m}(B)⟶^{0}π_{m}(C) ⋯,

whence

π_{m+1}(C) ≅ ker χ_{m} ≅ π_{m}(GL_{2}(H))^{k_{∞}−1}.

The remaining assertions follow from Theorem II.12 and II.14.

The groups GL_{p}(H, I, D)

Our next step it to determine the homotopy groups of the groups GL_{p}(H, I, D). For that we need some preparation because the case H_{0} ≠ 0 is more complicated than the case of GL_{p}(H, D) discussed above.

Up to a discussion of the connected components, Propositions III.9 below reduces the general case p ∈ [1, ∞] to the special case p = ∞, which for GL_{p}(H, I, D) seems to be better accessible than the case p = 2 to determine the connected components.

Proposition III.8. For each 1 ≤ p ≤ ∞ the groups GL_{p}(H, D) and GL_{p}(H, I, D) have smooth polar decompositions. In particular, the inclusion maps

U_{p}(H, I, D) → GL_{p}(H, I, D)

are homotopy equivalences.

Proof. In Proposition II.2(e) we have seen that GL_{p}(H, D) has a smooth polar decomposition. If g = ue^{x} is the polar decomposition of g ∈ GL_{p}(H, I, D), then it follows from the polar decompositions of GL(H, I) (Theorem II.6) and of GL_{p}(H, D) that u ∈ U_{p}(H, D) and x ∈ B_{p}(H, D). That the polar map of GL_{p}(H, I, D) is a diffeomorphism follows by restriction from the corresponding result for GL_{p}(H, D).

Proposition III.9. For 1 ≤ p ≤ q ≤ ∞ the inclusion map of the identity components

GL_{p}(H, I, D)_{e} → GL_{q}(H, I, D)_{e}

is a weak homotopy equivalence.

Proof. In view of Remark III.4, we may assume that all spaces H_{j} are infinite-dimensional. Then the group

GL(H, I, D)^{0} ≅ GL(H_{0}, I_{0}) × ∏_{j=1}^{k} GL(H_{j})

is contractible (Theorem II.6). Since GL(H) acts smoothly by conjugation on the normal subgroup GL_{p}(H), the group GL(H, I, D)^{0} acts smoothly on GL_{p}(H, I), so that we can form the connected Banach–Lie group G_{p} := GL_{p}(H, I)_{e} ⋊ GL(H, I, D)^{0} . From 𝔤𝔩_{p}(H, I, D) = 𝔤𝔩_{p}(H, I) + 𝔤𝔩(H, I, D)^{0} and Lemma A.5 we derive that the multiplication map m: G_{p} → M_{p} := GL_{p}(H, I, D)_{e} is surjective with kernel

H_{p} := GL_{p}(H, I, D)^{0} ≅ GL_{p}(H_{0}, I_{0}) × ∏_{j=1}^{k} GL_{p}(H_{j}).

The assumptions of Proposition A.6 are satisfied, so that the map G_{p} → M_{p} defines a locally trivial H_{p} -bundle. Since all the groups GL(H_{j}) and GL(H_{0}, I_{0}) are contractible, Corollary II.15(iii) implies that the inclusion maps H_{p} ↪ H_{q} and G_{p} ↪ G_{q} are homotopy equivalences. Now Proposition A.8 implies that the inclusion map M_{p} → M_{q} is a weak homotopy equivalence.

Now we prepare the discussion of the case p = ∞.

Lemma III.10. Let A be a unital C^{∗} -algebra and τ a linear antiautomorphism of A commuting with the ∗-map. Then

G := {g ∈ GL(A): τ(g) = g^{−1}}

is adapted to the polar decomposition G(A) = U(A) exp(Herm(A)) of G(A).

Proof. First we note that G is an algebraic subgroup of G(A), hence a Lie group with Lie algebra 𝔤 = {x ∈ A: τ(x) = −x} ([Ne00a, Prop. IV.14]). Since τ commutes with ∗, the group G is ∗-invariant.

Now we consider the automorphism of G(A) given by σ(g) := τ(g)^{−1} . The fact that τ commutes with ∗ implies that this automorphism preserves the subgroup U(A) of unitary elements and the subset exp(Herm(A)). Let g ∈ G(A), and let g = ue^{x} denote its polar decomposition. Then

σ(g) = σ(u)σ(e^{x}) = τ(u)^{−1}e^{−τ(x)}

is the polar decomposition of σ(g). Therefore g is fixed by σ if and only if σ fixes u and e^{x} separately. This means that u ∈ G and τ(x) = −x, i.e., x ∈ 𝔤 ∩ Herm(A). We conclude that G = (G ∩ U(A)) exp(𝔤 ∩ Herm(A)).

Proposition III.11. Let I: H → H be an antilinear isometry with I^{2} ∈ {±1}. We define

Fred(H, I) := {g ∈ Fred(H): gIg^{∗}I^{−1} ∈ 1 + K(H)}.

Then

{g ∈ Fred(H, I): ind(g) = 0} = GL(H, I)(1 + K(H)) = GL(H, I) + K(H).

Proof. It is obvious that GL(H, I) + K(H) ⊆ {g ∈ Fred(H, I): ind(g) = 0}. The proof of the converse is more involved.

Let A := Cal(H) := B(H)/K(H), write q: B(H) → A for the quotient map, and observe that the antiautomorphism a ↦ Ia^{∗}I^{−1} of B(H) preserves K(H), hence induces an antiautomorphism τ on A with

τ(q(a)) := q(Ia^{∗}I^{−1}).

We consider the group

G(A)_{τ} := {g ∈ G(A): τ(g) = g^{−1}}.

Since Fred(H) = q^{−1}(G(A)), G(A)_{e} = {q(g): ind(g) = 0}, and q(g) ∈ G(A)_{τ} is equivalent to q(g) ∈ G(A) and q(gIg^{∗}I^{−1}) = q(g)τ(q(g)) = 1, we see that

Fred(H, I) = q^{−1}(G(A)_{τ}) and q(Fred(H, I)) = G(A)_{τ} .

The assertion of the theorem means that G(A)_{τ} ∩ G(A)_{e} = q(GL(H, I)).

Since G(A)_{τ} is adapted to the polar decomposition of G(A) (Lemma III.10), we have G(A)_{τ} = (G(A)_{τ} ∩ U(A)) exp(L(G(A)_{τ}) ∩ Herm(A)) with

exp(L(G(A)_{τ}) ∩ Herm(A)) = q(exp(𝔤𝔩(H, I) ∩ Herm(H)) ⊆ q(GL(H, I)).

Therefore it suffices to show that G(A)_{τ} ∩ U(A)_{e} ⊆ q(GL(H, I)) which in turn will follow from

Fred(H, I) ∩ U(H) ⊆ GL(H, I) + K(H)

because q(U(H)) = U(A)_{e} follows from the connectedness of U(H) (Theorem II.4) and q(𝔲(H)) = {a ∈ A: a^{∗} = −a} = 𝔲(A).

First we consider an element u ∈ U_{∞}(H) = U(H) ∩ GL_{∞}(H) with Iu^{−1}I^{−1} = u and the eigenspaces H_{μ} = ker(u − μ1). Then we have for v ∈ H_{μ} the relation

uI.v = Iu^{−1}.v = I.\overline{μ}v = μI.v,

showing that I.H_{μ} = H_{μ}. This implies that u = exp x for some x ∈ 𝔲_{∞}(H) = 𝔲(H) ∩ B_{∞}(H) and IxI^{−1} = −x because we can choose x in such a way that on H_{μ} it is given by iλ id_{H_{μ}}, where e^{iλ} = μ for some λ ∈ [−π, π], and u ∈ U_{∞}(H) implies x ∈ 𝔲_{∞}(H). Let 𝔲_{∞}(H)_{−} := {x ∈ 𝔲_{∞}(H): IxI^{−1} = −x}. Then

{u ∈ U_{∞}(H): Iu^{−1}I^{−1} = u} = exp(𝔲_{∞}(H)_{−}).
Now let g ∈ Fred(H, I) ∩ U(H) and define u := gIg^{−1}I^{−1} . Then
Iu^{−1}I^{−1} = I^{2}gI^{−1}g^{−1}I^{−1} = gI^{2}I^{−1}g^{−1}I^{−1} = gIg^{−1}I^{−1} = u,

so that the preceding paragraph shows that u = exp x with x ∈ 𝔲_{∞}(H)_{−}. We put

g̃:= exp(−\frac{1}{2}x)g

and obtain

g̃Ig^{−1}I^{−1} = exp(−\frac{1}{2}x)gIg^{−1} exp(\frac{1}{2}x)I^{−1}
= exp(−\frac{1}{2}x)uI exp(\frac{1}{2}x)I^{−1} = exp(\frac{1}{2}x) exp(−\frac{1}{2}x) = 1.

This means that g̃ ∈ U(H, I). We conclude that

g = exp (\frac{1}{2}x)g̃ ∈ g̃ + K(H) ⊆ GL(H, I) + K(H).

This completes the proof.

Lemma III.12. For g ∈ GL_{∞}(H, I, D) and dim H = ∞ the following assertions hold:

(i) For each j we have g_{jj} ∈ Fred(H_{j}) with ind(g_{jj}) = ind(g_{−j,−j}). (ii) We consider the map

Ind: GL_{∞}(H, I, D) → ℤ × ℤ^{k_{∞}}, Ind(g) = ( ind(g_{00}), ind(g_{jj})_{1≤j≤k,dim H_{j}=∞}).

Then Ind is a continuous group homomorphism,

GL_{∞}(H, I, D)_{Ind} := ker(Ind) = GL_{∞}(H, I) GL(H, I, D)^{0} = GL_{∞}(H, I) GL(H, I, D)_{∞}^{0},

where GL(H, I, D)_{∞}^{0} ⊆ GL(H, I, D)^{0} is the subgroup corresponding to the infinite-dimen-sional ones among the spaces H_{j}, j = 0, …, k. Moreover

im(Ind) = \{ \begin{matrix}\{(n_{j}): n_{0} + 2 ∑_{0<j} n_{j} = 0\} ≅ ℤ^{k_{∞}} & for dim H_{0} = ∞ \\ \{(n_{j}): n_{0} = 0 = ∑_{0<j} n_{j}\} ≅ ℤ^{k_{∞}−1} & for dim H_{0} < ∞. \end{matrix}

Proof. (i) The operator g has an inverse g^{−1} in B_{∞}(H, I, D), which means that all offdiagonal blocks of g^{−1} are compact. Therefore the diagonal block g_{jj} ∈ B(H_{j}) is invertible modulo K(H_{j}), and this means that g_{jj} ∈ Fred(H_{j}).

In view of Ig^{∗}I^{−1} = g^{−1} and I.H_{j} = H_{−j}, we have (g^{−1})_{jj} = Ig_{−j,−j}^{∗}I^{−1}, and therefore

− ind(g_{jj}) = ind(g_{−j,−j}^{∗}) = − ind(g_{−j,−j}).

(ii) That Ind is a group homomorphism follows from Proposition III.5. We may w.l.o.g. assume that the spaces H_{j} are infinite-dimensional for 1 ≤ j ≤ k_{∞} and finite-dimensional for j > k_{∞}.

In view of (i), for each g ∈ GL_{∞}(H, I, D) we have

0 = ind g = ind(g_{00}) + 2 ∑_{j=1}^{k} ind(g_{jj}).

Therefore a necessary condition for n = (n_{j}) ∈ im(Ind) is n_{0} + 2 ∑_{j=1}^{k_{∞}} n_{j} = 0.

For tuples with n_{0} = 0 this leads to the requirement ∑_{j} n_{j} = 0, and Proposition III.5 shows that all these tuples can be obtained from the subgroup

GL_{∞}(H_{+} ⊕ I.H_{+}, I, D) ≅ GL_{∞}(H_{+}, D_{+}),

where H_{+} = ∑_{j=1}^{k} H_{j} (Remark I.2). Therefore we may assume that n_{2} = … = n_{k_{∞}} = 0. Considering only those operators which act non-trivially on the subspaces H_{±1} and H_{0} , we may even assume that k = k_{∞} = 1, and that H_{1} and H_{0} are infinite-dimensional and separable. Then we identify H with l^{2}(ℤ^{±}, ℂ) (in the notation of Lemma I.1), where

H_{±1} = l^{2}(±ℤ_{>0}, ℂ) and H_{0} = l^{2}(±ℤ_{≤0}, ℂ)

and I.e_{j} = e_{−j} for j ∈ ℤ. We consider the operator S ∈ U(H) given by S.e_{±j} := e_{±(j+1)}. Since this is a unitary operator commuting with I, it is an element of U(H, I). Moreover, its off-diagonal terms in the (3 × 3)-block decomposition are of finite rank. This implies that S ∈ GL_{p}(H, I, D) for p ∈ [1, ∞]. The operator S_{11} is a unilateral right shift operator, so that ind(S_{11}) = −1, and S_{00} is a (2 × 2)-block diagonal operator, where both components are unilateral left shift operators, so that ind(S_{00}) = 2. Therefore (2, −1) ∈ im(Ind), and from that the description of im(Ind) follows.

It is clear that GL_{∞}(H, I) GL(H, I, D)^{0} ⊆ ker(Ind). For the converse, assume that Ind(g) = 0. Then each g_{jj} is a Fredholm operator of index 0, so that we find d_{j} ∈ GL(H_{j}), j = 1, …, k, such that d_{j}^{−1}g_{jj} ∈ 1 + B_{1}(H_{j}) ⊆ 1 + K(H_{j}). For j = 0 the relation gIg^{∗}I^{−1} = 1 implies that g_{00} ∈ Fred(H_{0}, I_{0}), so that Proposition III.11 yields an element d_{0} ∈ GL(H_{0}, I_{0}) with d_{0}^{−1}g_{00} ∈ 1 + K(H_{0}). Then

diag(d_{k}^{−1}, …, d_{1}^{−1}, d_{0}, Id_{1}^{∗}I^{−1}, …, Id_{k}^{∗}I^{−1}) ∈ GL(H, I, D)^{0},

so that we may w.l.o.g. assume that g_{jj} ∈ 1+B_{1}(H_{j}) holds for j = 1, …, k and g_{00} ∈ 1+K(H_{0}). Then for j > 0 the relation

(g^{−1})_{jj}g_{jj} ∈ 1 + B_{1}(H_{j})

implies that (g^{−1})_{jj} ∈ 1 + B_{1}(H_{j}), and therefore g^{−1} = Ig^{∗}I^{−1} leads to g_{−j,−j} ∈ 1 + B_{1}(H_{−j}). Therefore g ∈ GL_{∞}(H, I).

If dim H_{j} < ∞, we may put d_{j} = 1, so that we get the sharper assertion that

ker(Ind) = GL_{∞}(H, I) GL(H, I, D)_{∞}^{0}.

We will see below that the group GL_{∞}(H, I, D)_{Ind} is not always connected.

Corollary III.13. For 1 ≤ p ≤ q ≤ ∞ the inclusion map GL_{p}(H, I, D) → GL_{q}(H, I, D) is a weak homotopy equivalence. Proof. In view of Proposition III.9, it remains to show that the induced homomorphism

α_{p,q}: π_{0}( GL_{p}(H, I, D)) → π_{0}( GL_{q}(H, I, D))

is bijective. For this we may assume that q = ∞ because if we can show the assertion in this case, the corollary follows from α_{p,∞} = α_{q,∞}α_{p,q}.

The proof of Lemma III.12 shows in particular that Ind|_{GL_{p}(H,I,D)} has the same range for each p ∈ [1, ∞]. Moreover, GL(H, I, D)^{0} ⊆ GL_{p}(H, I, D) implies that

GL_{p}(H, I, D)_{Ind} := ker Ind ∩ GL_{p}(H, I, D) = GL_{p}(H, I) GL(H, I, D)_{∞}^{0}.

Modulo connected components, the inclusion

GL_{p}(H, I, D)_{Ind} ↪ GL_{∞}(H, I, D)_{Ind}

therefore corresponds to the inclusion map GL_{p}(H, I) ↪ GL_{∞}(H, I) which is a homotopy equivalence (Corollary II.15). This completes the proof.

Theorem III.14. (Homotopy groups of GL_{p}(H, I, D)) If H is an infinite-dimensional complex Hilbert space and k_{∞} := |{j ∈ {1, …, k} : dim H_{j} = ∞}|, then

π_{0}(GL_{p}(H, I, D)_{Ind}) ≅ \{ \begin{matrix}ℤ_{2} & for dim H_{0} < ∞ and I^{2} = 1 \\ 0 & otherwise, \end{matrix}

π_{0}(GL_{p}(H, I, D)) ≅ \{ \begin{matrix}ℤ_{2} × ℤ^{k_{∞}−1} & for dim H_{0} < ∞ and I^{2} = 1 \\ ℤ^{k_{∞}−1} & for dim H_{0} < ∞ and I^{2} = −1 \\ ℤ^{k_{∞}} & for dim H_{0} = ∞, \end{matrix}

π_{1}(GL_{p}(H, I, D)) = 0, and π_{2}(GL_{p}(H, I, D)) ≅ ℤ^{k_{∞}}.

The higher homotopy groups are 8-periodic, i.e., π_{n+8}(GL_{p}(H, I, D)) ≅ π_{n}(GL_{p}(H, I, D)), n ∈ ℕ_{0} , and therefore determined by the following table.

dim H_{0} = ∞  dim H_{0} < ∞, I^{2} = 1  dim H_{0} < ∞, I^{2} = −1
π_{1}  0  0  0
π_{2}  ℤ^{k_{∞}}  ℤ^{k_{∞}}  ℤ^{k_{∞}}
π_{3}  0  0  ℤ_{2}
π_{4}  ℤ^{k_{∞}}  ℤ^{k_{∞}−1}  ℤ_{2} ⊕ ℤ^{k_{∞}}
π_{5}  0  0  0
π_{6}  ℤ^{k_{∞}}  ℤ^{k_{∞}}  ℤ^{k_{∞}}
π_{7}  0  ℤ_{2}  0
π_{8}  ℤ^{k_{∞}}  ℤ_{2} ⊕ ℤ^{k_{∞}−1}  ℤ^{k_{∞}−1}

Proof. First we use Remark III.4 to see that we may assume that all spaces H_{j} are infinitedimensional and Corollary III.13 to see that we may assume that p = ∞.

In view of Lemma III.12(b), we have a surjective homomorphism

φ: GL_{∞}(H, I) ⋊ GL(H, I, D)^{0} → GL_{∞}(H, I, D)_{Ind}

whose kernel is isomorphic to

A := GL_{∞}(H, I) ∩ GL(H, I, D)^{0} ≅ \{ \begin{matrix}∏_{j=1}^{k} GL_{∞}(H_{j}) & for H_{0} = 0 \\ GL_{∞}(H_{0}, I_{0}) × ∏_{j=1}^{k} GL_{∞}(H_{j}) & for H_{0} ≠ 0. \end{matrix}

Let B := GL_{∞}(H, I) ⋊ GL(H, I, D)^{0} and C := GL_{∞}(H, I, D)_{Ind} . To see that the map B → → C defines a locally trivial A-principal bundle, we first observe that, although the group GL_{∞}(H, I) need not be connected, the group GL(H, I < D)^{0} acts smoothly on it since it acts smoothly on the Banach algebra K(H), hence on GL_{∞}(H), and GL_{∞}(H, I) is a complemented Lie subgroup invariant under this action. In view of Remark A.7, and since all assumptions of Proposition A.6 are easily verified, the multiplication map B → C is a locally trivial A-principal bundle.

The exact homotopy sequence of the principal bundle A ↪ B → → C yields a long exact sequence of homotopy groups

(3.3)  ⋯ → π_{k+1}(C) → π_{k}(A) → π_{k}(B) → π_{k}(C) → π_{k−1}(A) → …

ending as an exact sequence in

⋯ → π_{1}(C) → π_{0}(A) → π_{0}(B) → π_{0}(C) → 0.

Since the groups GL_{∞}(H_{j}) are all connected, we have

π_{0}(A) ≅ \{ \begin{matrix}π_{0}(GL_{∞}(H_{0}, I_{0})) ≅ ℤ_{2} & for H_{0} ≠ 0 and I^{2} = 1 \\ 0 & otherwise \end{matrix}

(Theorem II.14). The contractibility of the group GL(H, I, D)^{0} (Theorem II.6) further leads to

π_{0}(B) ≅ π_{0}(GL_{∞}(H, I)) ≅ \{ \begin{matrix}ℤ_{2} & for I^{2} = 1 \\ 0 & for I^{2} = −1. \end{matrix}

Here the homomorphism π_{0}(A) → π_{0}(B) is the identity if π_{0}(A) is non-trivial because the inclusions

O(ℕ, ℂ) ↪ GL_{∞}(H_{0}, I_{0}) ↪ GL_{∞}(H, I)

are weak homotopy equivalences (cf. Corollary II.15). This leads directly to

π_{0}(C) ≅ \{ \begin{matrix}ℤ_{2} & for H_{0} = 0 and I^{2} = 1 \\ 0 & otherwise. \end{matrix}

Now we turn to the higher homotopy groups π_{m}, m ≥ 1. We observe that in all cases the contractibility of GL(H, I, D)^{0} shows that B is homotopy equivalent to GL_{∞}(H, I). First we deal with the simpler case H_{0} ≠ 0. Then the inclusion map GL_{∞}(H_{0}, I_{0}) → GL_{∞}(H, I) is a weak homotopy equivalence (cf. Corollary II.15). Hence for each m ∈ ℕ_{0} the homomorphism χ_{m}: π_{m}(A) → π_{m}(B) is surjective. Therefore the exact homotopy sequence of A ↪ B → → C leads to for each m ∈ ℕ to

π_{m}(C) ≅ ker χ_{m−1} ≅ ∏_{j=1}^{k} π_{m−1}(GL_{∞}(H_{j}))

≅ π_{m−1}(GL(ℕ, ℂ))^{k_{∞}} ≅ \{ \begin{matrix}ℤ^{k_{∞}} & for m even \\ 0 & for m odd. \end{matrix}

Now we turn to the case where H_{0} = 0. Here we will need results from Bott’s paper [Bo59]. Since the first two homotopy groups will be particularly important in the following, it is instructive to determine them directly. For that we note that in all cases the homomorphism π_{0}(A) → π_{0}(B) is injective, so that the homomorphism π_{1}(B) → π_{1}(C) is surjective by the exactness of (3.3). Furthermore π_{2}(A) = 0 and π_{2}(B) = 0 (Theorems II.6, II.14, Corollary II.15), so that π_{2}(C) = ker χ_{1} and π_{1}(C) ≅ coker χ_{1} . To determine these groups, we observe that

π_{1}(A) ≅ \{ \begin{matrix}ℤ_{2} × ℤ^{k} & for dim H_{0} = ∞ and I^{2} = 1 \\ ℤ^{k_{∞}} & otherwise \end{matrix}

and

π_{1}(B) ≅ π_{1}( GL_{∞}(H, I)) ≅ \{ \begin{matrix}ℤ_{2} & for I^{2} = 1 \\ 0 & for I^{2} = −1 \end{matrix} \}.

For I^{2} = −1 the homomorphism χ_{1} is trivial, so that π_{2}(C) ≅ π_{1}(A) ≅ ℤ^{k} and π_{1}(C) ≅ π_{1}(B) = 0.

For the case I^{2} = 1 we first observe that the block diagonal map

GL(n, ℂ) → O(2n, ℂ), g ↦ ( \begin{matrix}g & 0 \\ 0 & (g^{⊤})^{−1} \end{matrix} )

induces a surjective map

π_{1}(GL(n, ℂ)) ≅ ℤ → π_{1}(O(2n, ℂ)) ≅ \{ \begin{matrix}ℤ & for n = 1 \\ ℤ_{2} & for n > 1, \end{matrix}

because the generator of π_{1}(O(2n, ℂ)) can be obtained with the natural embedding of SO(2, ℝ) ≅ 𝕋. The homomorphism χ_{1} is therefore given by

χ_{1}: π_{1}(A) ≅ ℤ^{k} → π_{1}(B) ≅ ℤ_{2}, χ_{1}((n_{j})) = ∑_{j}[n_{j}],

where [n] ∈ ℤ_{2} ≅ ℤ/2ℤ denotes the congruence class modulo 2 of n ∈ ℤ. Since H is infinitedimensional, we have k > 0, showing that χ_{1} is surjective, so that π_{1}(C) = 0. Since ker χ_{1} is a subgroup of ℤ^{k} of index 2, we obtain π_{2}(C) ≅ ker χ_{1} ≅ ℤ^{k}.

We finally turn to the higher homotopy groups. The group A ≅ ∏_{j=1}^{k} GL_{∞}(H_{j}) is homotopy equivalent to GL(ℕ, ℂ)^{k}. In particular we have π_{2n−1}(A) ≅ ℤ^{k} and π_{2n}(A) = 0 for all n ∈ ℕ (Theorem II.12). Therefore the exact homotopy sequence of A ↪ B → → C contains the exact pieces

π_{2n}(B) ↪ π_{2n}(C) → π_{2n−1}(A)⟶^{χ_{2n−1}}π_{2n−1}(B) → → π_{2n−1}(C).

Since every subgroup of π_{2n−1}(A) ≅ ℤ^{k} is free, hence projective, we can apply this to ker χ_{2n−1} ⊆ π_{2n−1}(A) to obtain

(3.4)  π_{2n}(C) ≅ π_{2n}(B) ⊕ ker χ_{2n−1} and π_{2n−1}(C) ≅ coker χ_{2n−1}.

We are thus left with the determination of kernel and cokernel of χ_{2n−1} . Let

η_{m}: GL(ℂ^{m}) → GL(ℂ^{2m}, I), g ↦ ( \begin{matrix}g & 0 \\ 0 & (g^{⊤})^{−1} \end{matrix} )

and η: GL(ℕ, ℂ) → GL(2ℕ, ℂ, I) := lim_{⟶} GL(ℂ^{2m}, I) be the corresponding limit map. Then the homomorphism χ_{2n−1} is equivalent to

(3.5)  \begin{matrix}π_{2n−1}(A) ≅ π_{2n−1}(GL(ℕ, ℂ))^{k} & → π_{2n−1}(B) ≅ π_{2n−1}(GL(2ℕ, ℂ, I)), \\ (x_{1}, …, x_{k}) & ↦ ∑_{j=1}^{k} π_{2n−1}(η)(x_{j}). \end{matrix}

We conclude in particular that

coker χ_{2n−1} = coker π_{2n−1}(η).

Next we use polar decompositions to see that for I^{2} = 1 we have a homotopy equivalence

GL(2ℕ, ℂ, I) ∼ lim_{⟶} O(2m, ℂ) ∼ lim_{⟶} O(2m, ℝ) ∼ GL(ℕ, ℝ)

and for I^{2} = −1 we get

GL(2ℕ, ℂ, I) ∼ lim_{⟶} Sp(2m, ℂ) ∼ lim_{⟶} U(ℍ^{m}, ℍ) ∼ GL(ℕ, ℍ).

The natural embedding GL(ℕ, ℂ) ↪ GL(2ℕ, ℂ, I) correspond then to the natural inclusions

GL(ℕ, ℂ) ↪ GL(2ℕ, ℝ) ≅ GL(ℕ, ℝ) and GL(ℕ, ℂ) ↪ GL(ℕ, ℍ).

Information on the effect of these maps on the level of the homotopy groups comes from Theorem II.12, where we find (3.6) π_{n}(GL(ℕ, ℝ)/ GL(ℕ, ℂ)) ≅ π_{n+1}(GL(ℕ, ℝ)), π_{n}(GL(ℕ, ℍ)/ GL(ℕ, ℂ)) ≅ π_{n+1}(GL(ℕ, ℍ)).

We first consider the case I^{2} = 1 in detail. For each m ∈ ℕ we obtain with (3.6) an exact sequence

π_{2m}(GL(ℕ, ℂ)) = 0 → π_{2m}(GL(ℕ, ℝ)) ↪ π_{2m+1}(GL(ℕ, ℝ))

→ π_{2m−1}(GL(ℕ, ℂ))⟶^{π_{2m−1}(η)}π_{2n−1}(GL(ℕ, ℝ)) → → π_{2m}(GL(ℕ, ℝ))

→ π_{2m−2}(GL(ℕ, ℂ)) = 0.

In view of the Bott periodicity, it suffices to consider m = 1, 2, 3, 4, which lead to

\begin{matrix}m = 1 : & 0 ↪ ℤ → ℤ⟶^{π_{1}(η)}ℤ_{2} → → 0 \\ m = 2 : & 0 ↪ 0 → ℤ⟶^{π_{3}(η)}ℤ → → 0 \\ m = 3 : & 0 ↪ ℤ → ℤ⟶^{π_{5}(η)}0 → → 0 \\ m = 4 : & ℤ_{2} ↪ ℤ_{2} → ℤ⟶^{π_{7}(η)}ℤ → → ℤ_{2}. \end{matrix}

Using (3.4) and (3.5), this information leads to the following table:

m  π_{n}(B)  ker χ_{n−1}  coker χ_{n}  π_{n}(C)
1  ℤ_{2}  —  0  0
2  0  ℤ^{k}  —  ℤ^{k}
3  ℤ  —  0  0
4  0  ℤ^{k−1}  —  ℤ^{k−1}
5  0  —  0  0
6  0  ℤ^{k}  —  ℤ^{k}
7  ℤ  —  ℤ_{2}  ℤ_{2}
8  ℤ_{2}  ℤ^{k−1}  —  ℤ_{2} ⊕ ℤ^{k−1}

From (3.4) and Bott periodicity we further derive that the homotopy groups of C are 8-periodic, so that the table above contains all the information.

For I^{2} = −1, Bott periodicity implies that we obtain a similar picture shifted by 4, therefore the entries for I^{2} = −1 in the table can simply be obtained from them for I^{2} = 1 by a 4-shift.

Proposition III.15. The inclusion maps

GL_{1}(H, I) ↪ GL_{1,2}(H, I, D) ↪ GL_{2}(H, I)

are weak homotopy equivalences.

Proof. Since the subgroup G^{0} := GL_{2}(H, I, D)^{0} ⊆ G := GL_{2}(H, I) acts smoothly on G_{1,2}; = GL_{1,2}(H, I, D) by conjugation, the semidirect product group B := G_{1,2}⋊G^{0} has a natural Banach–Lie group structure. Moreover, the fact hat 𝔤𝔩_{2}(H, I, D) = 𝔤𝔩_{1,2}(H, I, D) + 𝔤𝔩_{2}(H, I)^{0} implies that the multiplication map m: B → G has an open image which therefore is a union of connected components. Since the inclusion map G_{1} := GL_{1}(H, I) ↪ G is a weak homotopy equivalence (Corollary III.13), the map m is surjective.

Using Proposition A.6, we see that G ≅ B/A, where

A ≅ G_{1,2} ∩ G^{0} = G_{1}^{0} ≅ GL_{1}(H_{0}, I_{0}) × ∏_{j=1}^{k} GL_{1}(H_{j}).

We consider the exact homotopy sequence of the A-principal bundle B:

… → π_{k}(A) → π_{k}(B) ≅ π_{k}(G_{1,2}) × π_{k}(G_{2}^{0}) → π_{k}(G) → π_{k−1}(A) → …

Since the inclusion map A ≅ G_{1}^{0} ↪ G_{2}^{0} is a weak homotopy equivalence, the homomorphism π_{k}(A) → π_{k}(G_{1,2}) × π_{k}(G_{2}^{0}) corresponds to the inclusion of the second factor. In particular, it is injective, so that the exactness of the sequence implies that the maps

π_{k}(G_{1,2}) → π_{k}(G), k ∈ ℕ,

are isomorphisms.

Problems III. It is natural to ask for the range of the index map ind on the monoid Fred(H, I). If I^{2} = 1, then Fred(H, I) contains in particular all operators g on the real space H_{ℝ} := {x ∈ H: I.x = x} for which gg^{∗} − 1 is compact. Since this includes unilateral shift operators on separable subspaces, it follows that im(ind) = ℤ in this case.

For I^{2} = −1, we may consider H as a quaternionic Hilbert space and obtain, with similar arguments as above, that im(ind) contains all even numbers, because an unilateral shift operator on l^{2}(ℕ, ℍ) has index 2. It is an interesting question if for I^{2} = −1 there exists an element g ∈ Fred(H, I) with ind(g) = 1. It is clear that this cannot be realized as an ℍ -linear operator, because all these operator have even index.

In both cases we see that im(ind) ≅ ℤ, so that we get π_{0}(G(A)_{τ}) ≅ ℤ with the notation from the proof of Proposition III.11.

IV. Universal central extensions of restricted groups

In this section we will draw the results from the preceding section together to describe universal central extensions Ĝ_{r} of the identity components G_{r} of the restricted groups GL_{2}(H, D) and GL_{2}(H, I, D) in the category of complex Banach–Lie groups. Extending the results to real forms of these groups leads in particular to the metagonal and the metaplectic groups, of which we show that they are universal central extensions of restricted versions of the real groups O(H^{ℝ}) and Sp(H, Ω) in the category of real Banach–Lie groups. This section is the heart of the paper because the proof of the universality of the central extensions requires the results on Lie algebra cohomology from Section I and also the detailed knowledge on the homotopy groups from Section III, which in turn uses Section II. All statements concerning the universality of the considered group extensions are new.

Throughout this section H is assumed to be an infinite-dimensional complex Hilbert space.

Central extensions of complex restricted groups

First we deal with the groups G_{r} := GL_{2}(H, D)_{e}. We start with a few preparations.

Lemma IV.1. For z ∈ 𝔷(𝔤𝔩(H, D)^{0}) the functional x ↦ tr(zx) on 𝔤_{1,2} := 𝔤𝔩_{1,2}(H, D) vanishes on the commutator algebra if and only if all z_{j} are equal. Proof. We have [𝔤_{1,2}, 𝔤_{1,2}] ⊆ [𝔤𝔩_{2}(H), 𝔤𝔩_{2}(H)] = 𝔰𝔩(H), so that the condition that all z_{j} are equal is sufficient.

On the other hand we have for i ≠ j and A ∈ B_{2}(H_{i}, H_{j}), B ∈ B_{2}(H_{j}, H_{i}):

[A, B] = ABE_{ii} − BAE_{jj}

in terms of (k × k)-block matrices. Therefore

tr(z[A, B]) = z_{i} tr(AB) − z_{j} tr(BA) = (z_{i} − z_{j}) tr(AB).

Now B_{2}(H_{i}, H_{j})B_{2}(H_{j}, H_{i}) = B_{1}(H_{i}) implies that z_{i} = z_{j} is also necessary for tr(z[𝔤_{1,2}, 𝔤_{1,2}]) to vanish.

Lemma IV.2. The continuous Lie algebra homomorphism tr: 𝔤𝔩_{1,2}(H, D) → ℂ integrates to a continuous group homomorphism det: GL_{1,2}(H, D) → ℂ^{×}. Proof. (cf. [HH94b] for a similar construction) Since the inclusion map

GL_{1}(H) → GL_{1,2}(H, D)

is a homotopy equivalence (Proposition III.2(c)), we obtain a homomorphism ι: G̃L_{1}(H) → G̃L_{1,2}(D) which induces a surjective homomorphism

π_{1}(GL_{1}(H)) ≅ ℤ → π_{1}(GL_{1,2}(H, D)) ≅ ℤ.

If we compose the unique homomorphism d̃et: G̃L_{1,2}(D) → ℂ^{×} satisfying L(d̃et) = tr with ι, then we obtain a lift of the determinant map det: GL_{1}(H) → ℂ^{×}. We conclude that

π_{1}(GL_{1,2}(H, D)) ⊆ ker d̃et,

and this implies the assertion.

Definition IV.3. We define the group

SL(H, D) := ker(det: GL_{1,2}(H, D) → ℂ^{×}).

Let v ∈ H_{j} for some infinite-dimensional space H_{j} be a unit vector and define the holomorphic homomorphism γ: ℂ^{×} → GL_{1}(H_{1}) ⊆ GL_{1,2}(H, D) by γ(z).v = zv and γ(z).w = w for w⊥v. Then det ∘γ = id_{ℂ}^{×}, and we conclude that the map

GL_{1,2}(H, D) ↦ SL(H, D) ⋊ ℂ^{×}, g ↦ (gγ(det g)^{−1}, det(g))

is a diffeomorphism. On the Lie algebra level we have a corresponding semidirect decomposition

𝔤𝔩_{1,2}(H, D) ≅ 𝔰𝔩(H, D) ⋊ ℂ.

The idea for the direct construction of the central extension in Definition IV.4 is a slight modification of the construction used in [HH94a,b]. Different constructions for the special case k = 2 can be found in [PS86] and [Mi89]. These central extensions could also be obtained more indirectly with the general methods described in [Ne00b], which requires to calculate the corresponding period maps (cf. Proposition IV.9), but in any case it is more convenient to have a concrete realization of the central extension.

Definition IV.4. Let G = GL_{2}(H) and G_{r} := GL_{2}(H, D)_{e}. Since G_{b,∞}^{0} := GL(H, D)_{∞}^{0} (cf. Lemma III.1) acts smoothly on the Banach algebra B_{1,2}(H, D), it acts smoothly by conjugation on G_{1,2} := GL_{1,2}(H, D), so that we can form the semidirect product Banach–Lie group G_{1,2} ⋊ G_{b,∞}^{0}, and the multiplication map G_{1,2} ⋊ G_{b,∞}^{0} → → G_{r} induces an isomorphism

(G_{1,2} ⋊ G_{b,∞}^{0})/N → G_{r}, (a, d)N ↦ ad, where N ≅ G_{1,2} ∩ G_{b,∞}^{0} ≅ ∏_{dim H_{j}=∞} GL_{1}(H_{j})

(cf. Proposition A.6 and Remark A.7). Here we use that G_{1,2} and G_{b,∞}^{0} are connected (Proposition III.2(c) and Theorem III.6).

In view of Definition IV.3, S := SL(H, D) ⊆ G_{1,2} is a Lie subgroup which also satisfies G_{r} = SG_{b,∞}^{0} , so that

G_{r} ≅ (S ⋊ G_{b,∞}^{0})/N_{S} with N_{S} := N ∩ (S ⋊ G_{b,∞}^{0}).

The group N_{S} has a natural holomorphic homomorphism

Δ: N_{S} → (ℂ^{×})^{k_{∞}}, (g, g^{−1}) ↦ (det(g_{j}))_{dim H_{j}=∞}.

With

Z := Δ(N_{S}) = {(z_{j}) ∈ (ℂ^{×})^{k_{∞}}: ∏_{j} z_{j} = 1}

we then have N_{S} ≅ ker Δ ⋊ Z. Since Δ is invariant under conjugation with elements of G_{b,∞}^{0} , the subgroup ker Δ is a normal Lie subgroup in S ⋊ G_{b,∞}^{0} , and it is complemented because N_{S} is complemented in S. Therefore we can form the quotient group

Ĝ_{r} := (S ⋊ G_{b,∞}^{0})/ ker Δ

whose elements we write as [(a, d)] := (a, d) ker Δ (cf. Definition III.6). This group has a natural homomorphism

q: Ĝ_{r} → G_{r}, q([(a, d)]) := ad with ker q ≅ N_{S}/ ker Δ ≅ Z.

We thus obtain a central extension Z ↪ Ĝ_{r}⟶^{q}G_{r}. On the subgroup G_{1,2} = Sγ(ℂ^{×}) ⊆ G_{r} (in the notation of Definition IV.3), this central extension has a natural splitting given by the homomorphism

σ_{1}: G_{1,2} = Sγ(ℂ^{×}) → G_{r}^{♯}, σ_{1}(gγ(z)) := [(g, γ(z))].

Remark IV.5. On the Lie algebra level the construction in Definition IV.4 leads, for 𝔤 = 𝔤𝔩_{2}(H) and 𝔤_{r} := 𝔤𝔩_{2}(H, D), to a surjective homomorphism

𝔰 ⋊ 𝔤_{b,∞}^{0} := 𝔰𝔩(H, D) ⋊ 𝔤𝔩(H, D)_{∞}^{0} → → 𝔤_{r} := 𝔤𝔩_{2}(H, D), (x, y) ↦ x + y,

(cf. Definition III.1(b)), and 𝔤_{b,∞}^{0} = 𝔤𝔩(H, D)_{∞}^{0} is a closed Lie subalgebra of 𝔤_{r}. We have the central extension

𝔷 ≅ ℂ^{k_{∞}−1} ↪ 𝔤̂_{r} → → 𝔤_{r}.

To describe this central extension by a continuous cocycle, we need a continuous splitting map σ: 𝔤_{r} → 𝔤̂_{r}. This can be obtained by from the decomposition 𝔤_{r} = 𝔤_{r}^{0} ⊕ [D, 𝔤], where [D, 𝔤] ⊆ 𝔰 denotes the closed subspace corresponding to the off-diagonal blocks. Writing elements x ∈ 𝔤_{r} as x = x_{0} + x_{1} according to this decomposition, we define

σ(x_{0} + x_{1}) := [(x_{1}, x_{0}, 0)].

It is clear that this is a continuous linear splitting map. The corresponding cocycle is given by

ω(x_{0} + x_{1}, y_{0} + y_{1}) = [σ(x_{0} + x_{1}), σ(y_{0} + y_{1})] − σ([x_{0} + x_{1}, y_{0} + y_{1}])
= [([x_{0}, y_{1}] + [x_{1}, y_{0}] + [x_{1}, y_{1}], [x_{0}, y_{0}], 0)]
− σ([x_{0}, y_{0}] + [x_{1}, y_{1}]_{0}, [x_{1}, y_{1}]_{1} + [x_{0}, y_{1}] + [x_{1}, y_{0}])
= [([x_{0}, y_{1}] + [x_{1}, y_{0}] + [x_{1}, y_{1}], [x_{0}, y_{0}], 0)]
− [([x_{1}, y_{1}]_{1} + [x_{0}, y_{1}] + [x_{1}, y_{0}], [x_{0}, y_{0}] + [x_{1}, y_{1}]_{0}, 0)]
= [([x_{1}, y_{1}]_{0}, −[x_{1}, y_{1}]_{0}, 0)]
= [(0, 0, dΔ([x_{1}, y_{1}]_{0}))] ≅ dΔ([x_{1}, y_{1}]_{0}).

If E := [x_{1}, y_{1}]_{0} denotes the block diagonal part of [x_{1}, y_{1}], then E is of trace class with trace 0 because [B_{2}(H), B_{2}(H)] ⊆ B_{1}(H) consists of matrices with vanishing trace. Now

dΔ(E) = (tr(E_{j}))_{dim H_{j}=∞} ∈ 𝔷 = {(z_{j}) ∈ ℂ^{k_{∞}}: ∑_{j} z_{j} = 0}.

Remark IV.6. It is interesting to compare the group Ĝ_{r} with the group constructed in [HH94b]. As in Definition IV.4, we write G_{r} as a quotient (G_{1,2} ⋊ G_{b,∞}^{0})/N, and consider the homomorphism Δ_{N}: N → (ℂ^{×})^{k_{∞}} given by the same formula as in Definition IV.4. We now obtain a central extension

G_{r}^{♯} := (G_{1,2} ⋊ G_{b,∞}^{0})/ ker Δ_{N}

by the same arguments. It is clear that we may view Ĝ_{r} as a subgroup of G_{r}^{♯} which we now describe as a kernel of a homomorphism to ℂ^{×}.

Since the homomorphism det: G_{1,2} := GL_{1,2}(H, D) → ℂ^{×} is invariant under the action of the group G_{b,∞}^{0} = GL(H, D)_{∞}^{0} , it extends to a holomorphic homomorphism

G_{1,2} ⋊ G_{b,∞}^{0} → ℂ^{×}

which obviously vanishes on the normal subgroup ker Δ_{N}, so that we obtain a holomorphic homomorphism

D: G_{r}^{♯} = (G_{1,2} ⋊ G_{b,∞}^{0})/ ker Δ_{N} → ℂ^{×}

which on Z^{♯} := N/ ker Δ_{N} ≅ (ℂ^{×})^{k_{∞}} restricts to the multiplication map (z_{j}) ↦ ∏_{j} z_{j}. From that we conclude that

ker D = Ĝ_{r} and G_{r}^{♯} ≅ Ĝ_{r} × ℂ^{×},

where the complementary factor can be chosen as the first factor in Z ≅ (ℂ^{×})^{k_{∞}}. Our construction further implies that the section σ: G_{1,2} → G_{r}^{♯}, g ↦ [(g, 1)] satisfies D ∘ σ = det.

Definition IV.7. For 𝔤 = 𝔤𝔩_{2}(H, I) and 𝔤_{r} = 𝔤𝔩_{2}(H, I, D) we have a surjective homomorphism

𝔤𝔩_{1,2}(H, I, D) ⋊ 𝔤𝔩(H, I, D)_{∞}^{0} → → 𝔤_{r} := 𝔤𝔩_{2}(H, I, D),

where 𝔤𝔩_{1,2}(H, I, D) := 𝔤𝔩_{2}(H, I) ∩ B_{1,2}(H, D) (cf. Definition III.1(b)). On the group level, we obtain for G = GL_{2}(H, I) as in Definition IV.4 an isomorphism

(G_{1,2} ⋊ G_{b,∞}^{0})/N → G_{r}, (a, d)N ↦ ad,

where G_{r} := GL_{2}(H, I, D)_{e}, G_{b,∞}^{0} := GL(H, I, D)_{∞}^{0} , G_{1,2} := GL_{1,2}(H, I, D)_{e}, and

N ≅ ∏_{dim H_{j}=∞,j>0} GL_{1}(H_{j}) × \{ \begin{matrix}GL_{1}(H_{0}, I_{0}) & for dim H_{0} = ∞ \\ 0 & for dim H_{0} < ∞ \end{matrix} \}

(cf. Definition III.3). The group N has a natural holomorphic homomorphism

Δ: N → Z^{♯} := (ℂ^{×})^{k_{∞}}, g ↦ (det(g_{j}))_{dim H_{j}=∞,j>0}

with N ≅ ker Δ ⋊ Z^{♯}. As in Definition IV.4, we now obtain a central extension

Z^{♯} ↪ G_{r}^{♯} := (G_{1,2} ⋊ G_{b,∞}^{0})/ ker Δ → → G_{r}.

which splits on the subgroup G_{1,2} (cf. Remark IV.6). If 𝔤̂_{r} denotes the Lie algebra of G_{r}^{♯} and 𝔷 the Lie algebra of Z^{♯}, then we obtain a central Lie algebra extension 𝔷 ↪ 𝔤̂_{r} → → 𝔤_{r}.

Proposition IV.8. For 𝔤_{r} = 𝔤𝔩_{2}(H, D) let 𝔤̂_{r} be as in Definition IV.4 and for 𝔤_{r} = 𝔤𝔩_{2}(H, I, D) as in Definition IV.7. Then the following assertions hold:

(i) The central extension 𝔷 ↪ 𝔤̂_{r} → → 𝔤_{r} is Banach universal over ℂ. (ii) Every real form 𝔷_{ℝ} ↪ (𝔤̂_{r})_{ℝ} → → 𝔤_{r,ℝ} of this central extension is a Banach universal central

extension of the real form 𝔤_{r,ℝ} of 𝔤_{r}.

Proof. (i) First we assume that 𝔤_{r} = 𝔤𝔩_{2}(H, D). Let ω_{j}, dim H_{j} = ∞, denote the components of the cocycle ω from Remark IV.5. Then ∑_{j} ω_{j} = 0, and Proposition I.11 shows that this is the only non-trivial relation between the cohomology classes [ω_{j}] ∈ H_{c}^{2}(𝔤_{r}, ℂ). Hence we obtain an isomorphism

δ_{ℂ}: 𝔷′ = Lin(𝔷, ℂ) → H_{c}^{2}(𝔤_{r}, ℂ), α ↦ [α ∘ ω]

so that the central extension 𝔤̂_{r} → 𝔤_{r} with kernel 𝔷 is ℂ -universal by Proposition I.13 because 𝔤_{r} is perfect (Proposition I.10). Moreover, Proposition I.13 implies that 𝔤̂_{r} is a Banach universal central extension of 𝔤_{r}.

For 𝔤_{r} = 𝔤𝔩_{2}(H, I, D) Proposition I.11 shows that the components ω_{j} with dim H_{j} = ∞ of the corresponding Lie algebra cocycle ω (Remark IV.5) yield a basis [ω_{j}] of H_{c}^{2}(𝔤_{r}, ℂ), so that 𝔤̂_{r} is a universal central extension of 𝔤_{r} by Proposition I.13. (ii) is an immediate consequence of (i) and Remark I.10(c) in [Ne01b].

Before we can turn to the universality assertions on the group level, we have to compute some homotopy groups of Ĝ_{r} and G_{r}^{♯}. For any central extension Z ↪ Ĝ → → G of a connected Banach–Lie group G with an abelian Banach–Lie group Z, which is a locally trivial Z -bundle, the vanishing of π_{m}(Z), m ≥ 2, in view of the exact homotopy sequence of the Z -bundle implies that the maps π_{m}(G) → π_{m}(Ĝ) are isomorphisms for m ≥ 3, and we have an exact sequence

0 → π_{2}(Ĝ) ↪ π_{2}(G)⟶^{δ}π_{1}(Z) → π_{1}(Ĝ) → π_{1}(G) → π_{0}(Z) → 0

which describes the relations between π_{1} and π_{2} of G and Ĝ. Here the period map δ: π_{2}(G) → π_{1}(Z) plays a key role because it determines π_{2}(Ĝ) ≅ ker δ and im δ is the kernel of the map π_{1}(Ĝ) → π_{1}(G). If G is simply connected, we obtain in particular π_{1}(Ĝ) ≅ coker δ.

Next we will analyze the period map for the central extensions Ĝ_{r} → G_{r} and G_{r}^{♯} → G_{r} explicitly in terms of the description of the group π_{2}(G_{r}) given in Theorems III.6 and III.14. We will see that in both cases δ is injective, but that it is not surjective for G_{r}^{♯}. Since G_{r} is simply connected, this leads to

π_{1}(G_{r}^{♯}) ≅ coker δ.

Proposition IV.9.

(i) For G_{r} = GL_{2}(H, D)_{e} the period map δ: π_{2}(G_{r}) → π_{1}(Z) is an isomorphism. The group Ĝ_{r} is simply connected, and π_{2}(Ĝ_{r}) = 0.

(ii) For G_{r} = GL_{2}(H, I, D)_{e} the period map δ is always injective. If I^{2} = −1, then it is also surjective, and for I^{2} = 1 we have coker δ ≅ ℤ_{2} . The group G_{r}^{♯} satisfies π_{1}(G_{r}^{♯}) ≅ ℤ_{2} and π_{2}(G_{r}^{♯}) = 0.

Proof. (i) In our direct construction of Ĝ_{r} in Definition IV.4, we used the description of G_{r} as (S ⋊ G_{b,∞}^{0})/N_{S} and a homomorphism Δ: N_{S} → Z with

Ĝ_{r} ≅ (S ⋊ G_{b,∞}^{0})/ ker Δ and N_{S} ≅ ker Δ ⋊ Z.

The exact homotopy sequence of the N_{S} -principal bundle S ⋊ G_{b,∞}^{0} contains a piece

⋯ π_{2}(G_{1,2} ⋊ G_{b,∞}^{0}) → π_{2}(G_{r})⟶^{δ_{r}}π_{1}(N_{S}) → π_{1}(G_{1,2} ⋊ G_{b,∞}^{0}) → ⋯,

and since π_{2}(G_{1,2} ⋊ G_{b,∞}^{0}) = 0 (Theorems II.6, II.14, and Proposition III.2(c)), the connecting map δ_{r}: π_{2}(G_{r}) → π_{1}(N_{S}) is injective. The Z -bundle Ĝ_{r} → G_{r} is associated to the N_{S} -bundle G_{1,2} ⋊ G_{b,∞}^{0} → G_{r} via the homomorphism Δ: N_{S} → Z, so that the corresponding connecting map δ satisfies δ = π_{1}(Δ) ∘ δ_{r}. Therefore we simply have to identify the image of δ_{r} in π_{1}(N_{S}) and the homomorphism π_{1}(Δ): π_{1}(N_{S}) → π_{1}(Z), restricted to this subgroup.

We have

N_{S} = {d = diag(d_{j}) ∈ ∏_{dim H_{j}=∞} GL_{1}(H_{j}): det d = ∏_{j} det(d_{j}) = 1},

ker Δ ≅ ∏_{dim H_{j}=∞} SL(H_{j}), and Z ≅ (ℂ^{×})^{k_{∞}−1} . From the proof of Theorem III.7 we derive that

im(δ_{r}) ≅ {(n_{j}) ∈ ℤ^{k_{∞}}: ∑ n_{j} = 0} = π_{1}(N) ≅ ℤ^{k_{∞}−1},

and that the map π_{1}(Δ): π_{1}(N) → π_{1}(Z) ≅ ℤ^{k_{∞}−1} is an isomorphism. Therefore δ: π_{2}(G_{r}) → π_{1}(Z) is an isomorphism. The remaining assertions follows from the exact homotopy sequence of the Z -principal bundle Z ↪ Ĝ_{r} → → G_{r} because π_{1}(G_{r}) = 0 (Theorem III.7) implies that π_{1}(Ĝ_{r}) ≅ coker δ = 0 and π_{2}(Ĝ_{r}) ≅ ker δ = 0. (ii) For q: G_{r}^{♯} → G_{r} = GL_{2}(H, I, D)_{e} the situation is slightly different. Here we have N ≅ ∏_{dim H_{j}=∞} GL_{1}(H_{j}) with N ≅ ker Δ ⋊ Z, where ker Δ ≅ ∏_{dim H_{j}=∞} SL(H_{j}) is simply connected. Therefore the map π_{1}(Δ): π_{1}(N) → π_{1}(Z) is an isomorphism, and again we have a factorization δ = π_{1}(Δ) ∘ δ_{r} with δ_{r}: π_{2}(G_{r}) → π_{1}(N). Here the vanishing of π_{2}(G_{1,2} ⋊ G_{b,∞}^{0}) follows from the contractibility of G_{b,∞}^{0} (Theorem II.6), Proposition III.15, and Corollary II.15. We conclude that δ_{r} is injective.

In view of π_{1}(G_{r}) = 0 (Theorem II.14), the exactness of

⋯ π_{2}(G_{r})⟶^{δ_{r}}π_{1}(N) → π_{1}(G_{1,2} ⋊ G_{b,∞}^{0}) → π_{1}(G_{r}) = 0

implies that

coker δ_{r} ≅ π_{1}(G_{1,2} ⋊ G_{b,∞}^{0}) ≅ π_{1}(G_{1,2}).

Using again Proposition III.15 and Corollary II.15, we obtain

π_{1}(G_{1,2}) ≅ \{ \begin{matrix}ℤ_{2} & for I^{2} = 1 \\ 0 & for I^{2} = −1. \end{matrix}

As in (i), we now obtain π_{2}(G_{r}^{♯}) = 0 and π_{1}(G_{r}^{♯}) ≅ coker δ ≅ coker δ_{r}.

From the topological information contained in Proposition IV.9 and the universality of the Lie algebra extensions 𝔤̂_{r} → 𝔤_{r}, we will now derive the description of a universal central group extension Ĝ_{r} → G_{r} in the category of complex Banach–Lie groups. For I^{2} = 1 it turns out that we will have to a twofold covering group Ĝ_{r} of G_{r}^{♯}, which corresponds to the usual passage from orthogonal groups to spin groups.

Theorem IV.10. If H is infinite-dimensional, then for G_{r} = GL_{2}(H, D)_{e} the central extension

Z ≅ (ℂ^{×})^{k_{∞}−1} ↪ Ĝ_{r}⟶^{q}G_{r}

and for G_{r} = GL_{2}(H, I, D)_{e} the universal covering group

Z ↪ Ĝ_{r} := G̃_{r}^{♯}⟶^{q}G_{r}

is universal for all abelian complex Banach–Lie groups in the following sense: For each central extension q_{H}: H → G_{r} which is a locally trivial A-principal bundle for an abelian Banach–Lie group A, there exists a unique morphism φ: Ĝ_{r} → H with q_{H} ∘ φ = q.

Proof. We have seen in Proposition IV.9 that for G_{r} = GL_{2}(H, I, D)_{e} the period map is not always surjective onto π_{1}(Z). Let q: Ĝ_{r} → G_{r}^{♯} denote the universal covering map for G_{r} = O_{2}(H_{ℂ} , I, D)^{+} := GL_{2}(H_{ℂ} , I, D)_{e}. For Z := q^{−1}(Z^{♯}) ⊆ Ĝ_{r} we then have G_{r} ≅ Ĝ_{r}/Z, and since G_{r} is simply connected, the group Z is connected; otherwise Ĝ_{r}/Z_{e} would be a nontrivial connected covering group of G_{r}. Therefore Z = exp_{G_{r}} 𝔷 is central in Ĝ_{r}, and we see

that q: Ĝ_{r} → G_{r} is in fact a central extension of 𝔤_{r}. Furthermore, Proposition IV.9 implies that π_{2}(Ĝ_{r}) ≅ π_{2}(G_{r}^{♯}) = 0.

For G_{r} = GL_{2}(H, D)_{e} we directly get from Proposition IV.9 that π_{2}(Ĝ_{r}) = π_{1}(Ĝ_{r}) = 0. Therefore in both cases G_{r} is simply connected (Theorem III.7, Theorem III.14), Ĝ_{r} is

simply connected, and 𝔤̂_{r} is Banach universal (Proposition IV.8). Now the assertion follows from [Ne01a, Th. IV.14].

Remark IV.11. If ω ∈ Z_{c}^{2}(𝔤_{r}, ℂ) is represented by diag(λ_{j}) ∈ ⨁_{dim H_{j}=∞} ℂ id_{H_{j}} (Proposition I.11), then Proposition IV.9 implies that the corresponding period map per_{ω}: π_{2}(G_{r}) → ℂ factors through

π_{2}(G_{r}) → π_{1}(Z) ↪ 𝔷⟶^{α}ℂ,

where the linear functional α: 𝔷 → ℂ is given by α((z_{j})) = ∑_{j} λ_{j}z_{j}. Then

per_{ω}((n_{j})) = ∑_{j} λ_{j}n_{j}.

For 𝔤 = 𝔤𝔩_{2}(H, I) this leads to

im(per_{ω}) = ∑_{j} ℤλ_{j} ⊆ ℂ

and for 𝔤 = 𝔤𝔩_{2}(H) to

im(per_{ω}) = { ∑_{j} n_{j}λ_{j}: ∑_{j} n_{j} = 0} = ∑_{j≠k} ℤ(λ_{j} − λ_{k}) ⊆ ℂ.

The metaplectic and the metagonal group

In this subsection H denotes a complex infinite-dimensional Hilbert space and J its complex structure given by J.v = iv, v ∈ H. We write H^{ℝ} for the underlying real Hilbert space. The complexification H_{ℂ} := (H^{ℝ})_{ℂ} decomposes into the ±i-eigenspaces H_{ℂ}^{±} for the complex extension of J which we also denote by J. In the following we will consider H_{ℂ} as a space with this decomposition, so that J.(x, y) = (ix, −iy) in these “coordinates.” We write I for the complex conjugation on H_{ℂ} with H^{ℝ} = {x ∈ H_{ℂ}: I.x = x} and note that IJ = JI on H_{ℂ} because J preserves the subspace H^{ℝ}. The ±i-eigenvectors of J can be written v ∓ iJv, v ∈ H^{ℝ}, and the antilinearity of I implies that I(v ∓ iJv) = v ± iJv. To obtain a convenient setting, we identify the i-eigenspace H_{ℂ}^{+} of J with H via the mapping v ↦ \frac{1}{\sqrt{2}}(v − iJv) which is a complex linear isometry. From IJ = IJ and the antilinearity of I, we get I.H_{ℂ}^{+} = H_{ℂ}^{−}. Since each orthonormal basis of H_{ℂ}^{+} is mapped by I into an orthonormal basis of H_{ℂ}^{−}, we see that we may identify H_{ℂ}^{−} with H in such a way that there exists an antilinear involution σ of H (fixing the elements of a given orthonormal basis) such that I is given by product coordinates on H_{ℂ} ≅ H ⊕ H by the formula I.(x, y) = (σ(y), σ(x)) (cf. Remark I.2).

Definition IV.12. (a) We define the restricted real linear group of H by

GL_{res}(H^{ℝ}) := {g ∈ GL(H^{ℝ}): gJg^{−1} − J ∈ B_{2}(H^{ℝ})}.

The condition gJg^{−1} − J ∈ B_{2}(H^{ℝ}) is equivalent to [g, J] ∈ B_{2}(H^{ℝ}). The elements of the group GL_{res}(H^{ℝ}) are called almost linear automorphisms of the complex Hilbert space H.

(b) It is clear that GL(H) is a subgroup of GL_{res}(H^{ℝ}). We also define the corresponding restricted orthogonal and symplectic group

O_{res}(H^{ℝ}) := O(H^{ℝ}) ∩ GL_{res}(H^{ℝ}) and Sp_{res}(H, Ω) := Sp(H, Ω) ∩ GL_{res}(H^{ℝ}).

Lemma IV.13. The operator D := −iJ is a hermitian involution on H_{ℂ}, and the following assertions hold:

(i) GL_{res}(H^{ℝ}) = {g ∈ GL_{2}(H_{ℂ} , D): Ig = gI} = GL_{2}(H_{ℂ} , D) ∩ GL(H^{ℝ}) and the group GL_{res}(H^{ℝ}) has a smooth polar decomposition with GL_{res}(H_{ℝ}) ∩ U(H_{ℂ}) = O_{res}(H^{ℝ}).

(ii) O(H_{ℂ} , I)∩U(H_{ℂ}) = O(H_{ℂ} , I)∩GL(H^{ℝ}) = O(H^{ℝ}) and O(H_{ℂ} , I)∩U_{2}(H_{ℂ} , D) = O_{res}(H^{ℝ}). (iii) The operator Ĩ:= iJI is an antilinear isometry with Ĩ^{2} = −1 and

Sp(H, Ω) = {g ∈ Sp(H_{ℂ} , Ĩ): Ig = gI} and Sp_{res}(H, Ω) = {g ∈ Sp_{2}(H_{ℂ} , Ĩ, D): Ig = gI}.

In particular Sp(H, Ω) is a real form of Sp_{2}(H_{ℂ} , I) and Sp_{res}(H, Ω) is a real form of Sp(H_{ℂ} , I, D). The group Sp_{res}(H, Ω) has a polar decomposition with Sp_{res}(H^{ℝ}) ∩ U(H^{ℝ}) = U(H), and it is contractible.

Proof. First we note that D is a hermitian operator on H_{ℂ} with spectrum {±1}, because in product coordinates on H_{ℂ} it is given by D(x, y) = (−x, y).

(i) The condition g ∈ GL_{res}(H^{ℝ}) means that if we write its complex linear extension, also denoted g, to H_{ℂ} according to the decomposition of H_{ℂ} as a matrix

g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ,

then

[J, g] = [ ( \begin{matrix}i & 0 \\ 0 & −i \end{matrix} ) , ( \begin{matrix}a & b \\ c & d \end{matrix} ) ] = i ( \begin{matrix}0 & 2b \\ −2c & 0 \end{matrix} ) ,

so that g ∈ GL_{res}(H^{ℝ}) is equivalent to g ∈ GL_{2}(H_{ℂ} , D) and gI = Ig.

That the group GL_{2}(H_{ℂ} , D) has a polar decomposition has been shown in Proposition III.7. Let g ∈ GL_{2}(H_{ℂ} , D) and g = ue^{x} be its polar decomposition, where u ∈ U(H_{ℂ}) and x = x^{∗} ∈ Herm(H_{ℂ}). We then have IgI^{−1} = IuI^{−1}e^{IxI^{−1}} , which is the polar decomposition of IgI^{−1} . Therefore the uniqueness of the polar decomposition and the first part of the proof imply that g ∈ GL_{res}(H_{ℝ}) if and only if u ∈ GL_{res}(H_{ℝ}) ∩ U(H_{ℂ}) ≅ O_{res}(H^{ℝ}) and IxI^{−1} = x, i.e., x ∈ Herm(H_{ℂ}) ∩ 𝔤𝔩_{res}(H^{ℝ}). (ii) It is clear that the group O(H^{ℝ}) acts unitarily on H_{ℂ}. For g ∈ O(H^{ℝ}) we therefore have Ig^{∗}I^{−1} = g^{∗}II^{−1} = g^{∗} = g^{−1} which implies that g ∈ O(H_{ℂ} , I). If, conversely, g ∈ O(H_{ℂ} , I), then gIg^{∗} = I, so that g ∈ O(H^{ℝ}) holds if and only if g commutes with I if and only if g is unitary. This is the first assertion. With (i) and the first part we get

O(H_{ℂ} , I) ∩ U_{2}(H_{ℂ} , D) = O(H_{ℂ} , I) ∩ U(H_{ℂ}) ∩ GL_{2}(H_{ℂ} , D) = O(H^{ℝ}) ∩ GL_{2}(H_{ℂ} , D)

= O(H^{ℝ}) ∩ GL(H^{ℝ}) ∩ GL_{2}(H_{ℂ} , D) = O(H^{ℝ}) ∩ GL_{res}(H^{ℝ}) = O_{res}(H^{ℝ}).

(iii) Its definition implies that Ĩ = iJI is isometric and antilinear. Further Ĩ^{2} = iJIiJI = −i^{2}JIJI = J^{2}I^{2} = −1. For g ∈ Sp(H, Ω) ⊆ GL(H^{ℝ}) we have Ĩg^{∗}Ĩ^{−1} = Jg^{∗}J^{−1} = g^{−1} , showing that g ∈ Sp(H_{ℂ} , Ĩ). If, conversely, g ∈ Sp(H_{ℂ} , Ĩ), then IgI^{−1} = J(g^{∗})^{−1}J^{−1} , so that Ig = gI is equivalent to g = J(g^{∗})^{−1}J^{−1} . We further get with (i):

Sp_{res}(H, Ω) = Sp(H, Ω) ∩ GL_{res}(H^{ℝ}) = Sp(H, Ω) ∩ GL_{2}(H_{ℂ} , D)
= GL(H^{ℝ}) ∩ Sp_{2}(H_{ℂ} , Ĩ, D) = {g ∈ Sp_{2}(H_{ℂ} , Ĩ, D): Ig = gI}.

To obtain the polar decomposition of Sp_{res}(H, Ω), we can argue as in (i), and we get

Sp_{res}(H, Ω) ∩ O(H^{ℝ}) = {g ∈ O(H^{ℝ}): Jg^{−1}J = Jg^{∗}J^{−1} = g^{−1}} = U(H).

The contractibility of U(H) (Theorem II.6) implies that Sp_{res}(H, Ω) is contractible.

Remark IV.14. From Lemma IV.13(ii) we get O(H^{ℝ}) = O(H_{ℂ} , I) ∩ GL(H^{ℝ}) and therefore

O_{res}(H^{ℝ}) = O(H_{ℂ} , I) ∩ GL_{res}(H^{ℝ}) = O_{2}(H_{ℂ} , I, D) ∩ GL(H^{ℝ}),

showing that O_{res}(H^{ℝ}) is a real form of O_{2}(H_{ℂ} , I, D). Moreover

O_{res}(H^{ℝ}) = O_{2}(H_{ℂ} , I, D) ∩ U(H_{ℂ})

by Lemma IV.13(ii), so that the polar decomposition of O_{2}(H_{ℂ} , I, D) implies that the inclusion map

O_{res}(H^{ℝ}) ↪ O_{2}(H_{ℂ} , I, D)

is a homotopy equivalence. Therefore Theorem III.14 leads to

π_{k}(O_{res}(H^{ℝ})) ≅ \{ \begin{matrix}ℤ_{2} & for k = 0 \\ 0 & for k = 1 \\ ℤ & for k = 2. \end{matrix}

To see elements of O_{res}(H^{ℝ}) \ O_{res}(H^{ℝ})^{+} , we recall from Lemma III.12 that
O_{∞}(H_{ℂ} , I, D) = O_{∞}(H_{ℂ} , I) O(H_{ℂ} , I, D)_{∞}^{0},

where the group O(H_{ℂ} , I, D)_{∞}^{0} ≅ O(H) is contractible. Moreover, the inclusion map O_{1}(H_{ℂ} , I) ↪ O_{∞}(H_{ℂ} , I) is a homotopy equivalence by Corollary III.9. Therefore O_{1}(H_{ℂ} , I)^{−} := {g ∈ O_{1}(H_{ℂ} , I): det g = −1} is not contained in the identity component of O_{∞}(H_{ℂ} , I, D), and hence

O_{res}(H^{ℝ})^{−} ⊇ O_{1}(H_{ℂ} , I)^{−}.

The terminology for the groups defined below is taken from Vershik [Ve90], where these groups are also discussed. Here the main new point is that we can show their universality as central extensions of the corresponding restricted groups.

Definition IV.15. (Metaplectic and metagonal group) (a) Let D := −iJ as above. Then ω(x + d, x′ + d′) := tr(D[x, x′]) is the universal cocycle for 𝔤_{r} := 𝔰𝔭_{2}(H_{ℂ} , Ĩ, D) (Proposition I.11, Remark IV.5). For the antilinear involution θ(x) = IxI^{−1} we have θ(D) = −IiJI = iIJI = iJ = −D. Therefore

(θ.ω)(x + d, x′ + d′) = \overline{tr(D[θ.x, θ.x′])} = \overline{tr(Dθ.[x, x′])}
= tr((θ.D)[x, x′]) = − tr(D[x, x′]) = −ω(x + d, x′ + d′).

We conclude that

θ̂(x, z) := (θ(x), −\overline{z})

defines a complex conjugation of 𝔰̂𝔭_{2}(H_{ℂ} , Ĩ, D) := 𝔤̂_{r} whose real form is the metaplectic Lie algebra

𝔪𝔭(H, Ω) := {(x, z) ∈ 𝔰̂𝔭_{2}(H_{ℂ} , Ĩ, D): x ∈ 𝔰𝔭_{res}(H, Ω), z ∈ iℝ}.

It is a universal central extension of the real Banach–Lie algebra 𝔰𝔭_{res}(H, Ω) (Proposition IV.3(iii)).

The involution θ̂ integrates to an antiholomorphic involution θ̂_{G} of the simply connected complex group Ŝp_{2}(H_{ℂ} , Ĩ, D) := Ĝ_{r} (Proposition IV.9), and we define the metaplectic group

Mp(H, Ω) := {g ∈ Ŝp_{2}(H_{ℂ} , Ĩ, D): θ̂_{G}(g) = g}.

Then Mp(H, Ω) is a real Lie subgroup of Ŝp_{2}(H_{ℂ} , Ĩ, D) with Lie algebra 𝔪𝔭(H, Ω). This group is a central extension of Sp_{res}(H, Ω) by 𝕋 = \{z ∈ ℂ^{×}: \overline{z} = z^{−1}\}.

(b) The universal cocycle ω(x+d, x′+d′) := tr(D[x, x′]) of 𝔤_{r} := 𝔬_{2}(H_{ℂ} , I, D) satisfies θ.ω = −ω for θ(x) = σxσ^{−1} . Therefore

θ̂(x, z) := (θ(x), −\overline{z})

defines a complex conjugation of 𝔬̂_{2}(H_{ℂ} , D) := 𝔤̂_{r} whose real form is the metagonal Lie algebra

𝔪𝔬(H^{ℝ}) := {(x, z) ∈ 𝔬̂_{2}(H_{ℂ} , D): x ∈ 𝔬_{res}(H^{ℝ}), z ∈ iℝ}

which is a universal central extension of the real Banach–Lie algebra 𝔬_{res}(H^{ℝ}) (Proposition IV.8(iii)).

The involution θ̂ integrates to an antiholomorphic involution θ̂_{G} of the simply connected complex group Ô_{2}(H_{ℂ} , I, D)^{+} := Ĝ_{r} (Theorem IV.10), and we define the connected metagonal group

MO(H^{ℝ})^{+} := {g ∈ Ô_{2}(H_{ℂ} , I, D)^{+}: θ̂_{G}(g) = g}.

Then MO(H^{ℝ})^{+} is a real Lie subgroup of Ô_{2}(H_{ℂ} , I, D)^{+} with Lie algebra 𝔪𝔬(H^{ℝ}) which is a central extension of the identity component O_{res}(H^{ℝ})^{+} by Z_{ℝ} ≅ 𝕋. Its connectedness now follows from the connectedness of Z_{ℝ} and of O_{res}(H^{ℝ})^{+} .

Let r ∈ O(H^{ℝ})^{−} be a simple reflection, i.e., r − 1 has one-dimensional range, which is J -antilinear (cf. Remark IV.14). Then

O_{res}(H^{ℝ}) ≅ O_{res}(H^{ℝ})^{+} ⋊ {1, r}.

The relation rDr^{−1} = −D implies that ω(rxr^{−1}, ryr^{−1}) = −ω(x, y), so that r acts as an involutive automorphism on the Lie algebra 𝔪𝔬(H^{ℝ}) by τ_{𝔤}.(x, z) := (rxr^{−1}, −z).

Anticipating the result that MO(H^{ℝ})^{+} is simply connected (Theorem IV.18), it follows that τ_{𝔤} integrates to an involution τ_{G} on this group. We let the group ℤ_{4} := ℤ/4ℤ act on MO(H^{ℝ})^{+} in such a way that [n] := n + 4ℤ acts as τ_{G}^{n}. Then we define the full metagonal group

MO(H^{ℝ}) := (MO(H^{ℝ})^{+} ⋊ ℤ_{4})/{1, (−1, [2])},

where −1 denotes the unique non-trivial involution in the central circle Z_{ℝ} ⊆ MO(H^{ℝ})^{+} . Using reflections s in H_{ℝ} with dim(im(s − 1)) = 2, we obtain, as in the finite-dimensional case, elements s̃ ∈ MO(H^{ℝ})^{+} with s^{2} = −1, and we may even assume that r and s commute. Then (s̃, [1])^{2} = (−1, [2]) implies that the group MO(H^{ℝ}) can also be written as a semidirect product MO(H^{ℝ})^{+} ⋊ ℤ_{2} . The full metagonal group is a natural analog of the groups Pin(2n, ℝ).

Theorem IV.16. The metaplectic group Mp(H, Ω) satisfies

π_{m}(Mp(H, Ω)) ≅ π_{m}(𝕋) ≅ \{ \begin{matrix}0 & for m ≠ 1 \\ ℤ & for m = 1. \end{matrix}

Its universal covering group M̃p(H, Ω) is contractible, the identity component of its center is isomorphic to ℝ, and it is a universal central extension of the real Banach–Lie group Sp_{res}(H, Ω).

Proof. Since Mp(H, Ω) is a central 𝕋-extension of the contractible group Sp_{res}(H, Ω), it is connected, and the exact homotopy sequence of the locally trivial principal bundle

𝕋 ↪ Mp(H, Ω) → → Sp_{res}(H, Ω)

yields the assertion on the homotopy groups of Mp(H, Ω). Therefore all homotopy groups of the universal covering group M̃p(H, Ω) are trivial, which implies that it is contractible (Theorem II.2). It is easy to see that the center of Sp_{res}(H, Ω) consists of {±1}, so that the identity component of the center of M̃p(H, Ω) is the universal covering group of 𝕋 = Z(Mp(H, Ω))_{e}, hence isomorphic to ℝ. Finally the fact that 𝔪𝔭(H, Ω) is a universal central extension of the Banach– Lie algebra 𝔰𝔭_{res}(H, Ω) (Proposition IV.8(ii)), and the simple connectedness of M̃p(H, Ω) and Sp_{res}(H, Ω), imply the universality of M̃p(H, Ω) as a central extension of Sp_{res}(H, Ω) ([Ne01a, Th. IV.14]).

Remark IV.17. (a) Since the group Sp_{res}(H, Ω) is contractible, the central extension Mp(H, Ω) is a trivial principal bundle, hence has a continuous global section. There is no general reason for this to imply that there is a smooth global section because we do not know whether there exists a contraction which is a smooth map. For a discussion of the existence of smooth global sections of central extensions we refer to Section V in [Ne00b]. The polar decomposition of Sp_{res}(H, Ω) implies that it is diffeomorphic to a product of U(H) and a Hilbert space. Since the space C([0, 1], ℝ) embeds isometrically into 𝔲(H) and has no smooth functions with arbitrarily small support (cf. [KM97]), the same holds for U(H), showing that U(H) and therefore also Sp_{res}(H, Ω) is not smoothly paracompact.

Nevertheless, one can also show directly as follows that Mp(H, Ω) has a smooth global section. First we observe that the the invariance of 𝔰𝔭_{res}(H, Ω) under the involution x ↦ x^{∗} leads to the decomposition

𝔤 := 𝔰𝔭_{res}(H, Ω) = 𝔨 ⊕ 𝔭 with 𝔨 = 𝔲(H) = {x ∈ 𝔤: x^{∗} = −x}, 𝔭 = {x ∈ 𝔤: x^{∗} = x}.

Writing elements x ∈ 𝔤 accordingly as x = x_{𝔨} + x_{𝔭}, the universal cocycle satisfies

ω(x, x′) = tr(D[x, x′]) = tr(D[x_{𝔭}, x′_{𝔭}])

because [D, x_{𝔨}] = 0. Therefore the inverse image

𝔨̂ = 𝔨 ⊕ iℝ ⊆ 𝔤̂ = 𝔤 ⊕_{ω} iℝ

of 𝔨 is a direct Lie algebra sum 𝔨 ⊕ iℝ. Moreover, 𝔤̂ = 𝔨̂ ⊕ 𝔭 has the structure of a symmetric Lie algebra because [𝔨̂, 𝔭] ⊆ 𝔭 and [𝔭, 𝔭] ⊆ 𝔨̂. Now the use the methods from [Ne00c] to see that Mp(H, Ω) has a polar decomposition K̂ exp(𝔭). That the assumptions of [Ne00c, Th. IV.1] are satisfied follows from the fact that for each x ∈ 𝔭 the operator (ad x)^{2} |_{𝔭}: 𝔭 → 𝔭 is positive hermitian ([Ne00c, Prop. III.16]). We then conclude that that simply connected covering group M̃p(H, Ω) has a polar decomposition ( U(H) × iℝ) exp 𝔭 because U(H) × iℝ is the simply connected group with Lie algebra 𝔨̂. From that it follows directly that Mp(H, Ω) has a polar decomposition K̂ exp 𝔭 with K̂ ≅ U(H) × 𝕋. Now we get a smooth section

σ: Sp_{res}(H, Ω) = U(H) exp 𝔭 → Mp(H, Ω) = (U(H) × 𝕋) exp 𝔭, u exp x ↦ (u, 1) exp x

for u ∈ U(H) and x ∈ 𝔭.

(b) In the finite-dimensional case H ≅ ℂ^{n} we have π_{1}(Sp(H, Ω)) ≅ π_{1}(Sp(2n, ℝ)) ≅ ℤ, and the Lie algebra 𝔰𝔭(H, Ω) is centrally closed. Hence the central 𝕋-extensions of Sp(H, Ω) are classified by

Hom(π_{1}(Sp(Ω, H)), 𝕋) ≅ Hom(ℤ, 𝕋) ≅ 𝕋,

and the metaplectic group (which in this case is also called Mp_{c}(H, Ω)) is defined by the homomorphism ℤ → 𝕋 mapping 1 to −1. The corresponding Lie algebra extension is trivial, but the commutator group of Mp_{c}(H, Ω) is a twofold covering of Sp(H, Ω) (which is also frequently called metaplectic group).

Theorem IV.18. The inclusion map

MO(H^{ℝ})^{+} ↪ Ô_{2}(H_{ℂ} , I, D)^{+}

is a weak homotopy equivalence. In particular the group MO(H^{ℝ})^{+} is simply connected. Moreover, it is a universal central extension of the real Banach–Lie group O_{res}(H^{ℝ})^{+} . Proof. Since O_{2}(H_{ℂ} , I, D)^{+} has a polar decomposition with unitary part O_{res}(H_{ℝ})^{+} (Remark IV.14), the inclusion map O_{res}(H_{ℝ})^{+} → O_{2}(H_{ℂ} , I, D)^{+} is a homotopy equivalence. Further the inclusion map Z_{ℝ} ≅ 𝕋 ↪ Z ≅ ℂ^{×} is a homotopy equivalence. Therefore Proposition A.8 implies that the inclusion

MO(H^{ℝ})^{+} ↪ Ô_{2}(H_{ℂ} , I, D)^{+}

is a weak homotopy equivalence. We conclude in particular that MO(H^{ℝ})^{+} is simply connected. Therefore the fact that 𝔪𝔬(H^{ℝ}) is a universal central extension of the real Banach–Lie algebra 𝔬_{res}(H^{ℝ}) implies the universality of MO(H^{ℝ})^{+} as a central extension of the simply connected group O_{res}(H^{ℝ})^{+} ([Ne01a, Th. IV.14]).

Remark IV.19. If H ≅ ℂ^{n} is finite-dimensional and n > 1, then π_{1}(O(2n, ℝ)) ≅ ℤ_{2} , and there exists a natural 𝕋-extension of O(2n, ℝ)^{+} = SO(2n, ℝ) corresponding to the inclusion homomorphism π_{1}(O(2n, ℝ)) ≅ ℤ_{2} ↪ 𝕋. This central extension would be a natural analog of MO(H^{ℝ}). Its commutator subgroup is the universal covering group Spin(2n, ℝ) of SO(2n, ℝ).

V. Infinite-dimensional flag manifolds

In this section we discuss analogs of complex flag manifolds for the groups

G ∈ {GL_{2}(H), GL_{2}(H, I)}.

We define these manifolds as the orbits of certain flags F = (F_{0}, F_{1}, …, F_{k}) in H under G. Let P (F) ⊆ G denote the stabilizer of such a flag. Then the homogeneous space G/P (F) is a complex manifold, called a flag manifold. We also show that the unitary real form U := G∩U(H) acts transitively on G/P (F). Similar results hold for the restricted groups G_{r}.

Definition V.1. (a) We consider a flag F = (F_{0}, F_{1}, …, F_{k}), where

{0} = F_{0} ⊆ F_{1} ⊆ F_{2} ⊆ … ⊆ F_{k} = H

are closed subspaces of H. Let

P_{b}(F) := {g ∈ GL(H): (∀j)g.F_{j} = F_{j}}

denote the stabilizer of this flag. To get a better description of this group, we define closed subspaces H_{j} := F_{j} ∩ F_{j−1}^{⊥} for j = 1, …, k and thus obtain an orthogonal decomposition H = H_{1} ⊕ … ⊕ H_{k}. Accordingly we view operators on H as matrices (x_{ij})_{i,j=1,...,k} with x_{ij} ∈ B(H_{j}, H_{i}). Then

P_{b}(F) = {g ∈ GL(H): (∀i > j) g_{ij} = 0} ≅ N(F) ⋊ M(F),

where

M(F) = {g ∈ GL(H): (∀j)g.H_{j} = H_{j}} = {g ∈ GL(H): (∀i ≠ j)g_{ij} = 0} ≅ ∏_{j=1}^{k} GL(H_{j})

and

N(F) = {g ∈ GL(H): (∀j)(g − 1).H_{j} ⊆ H_{j−1}} = {g ∈ P_{b}(F): (∀j) g_{jj} = 1}.

(b) For G = GL_{2}(H) we now define P := P (F) := P_{b}(F) ∩ G. In this case the description in (a) implies immediately that P (F) is a complemented Lie subgroup of G, so that the homogeneous space G/P (F) has a natural structure of a complex Banach manifold (cf. Definition III.6) which we call a flag manifold associated to G.

(c) For G = GL_{2}(H, I) we now consider a chain of closed subspaces

{0} = F_{0} ⊆ F_{1} ⊆ … ⊆ F_{k}

which are isotropic, i.e., that all spaces F_{j} are isotropic for the bilinear form β(x, y) = ⟨x, I.y⟩, which in turn is equivalent to I.F_{j}⊥F_{j}. We extend this chain of subspaces to the flag F defined by

{0} = F_{0} ⊆ F_{1} ⊆ F_{2} ⊆ … ⊆ F_{k} ⊆ F_{k}^{⊥_{β}} ⊆ … ⊆ F_{1}^{⊥_{β}} ⊆ F_{0}^{⊥_{β}} = H.

For g ∈ GL(H) and τ(g) := I(g^{∗})^{−1}I^{−1} the condition g ∈ GL(H, I) is equivalent to g = τ(g). Moreover, the condition g.E = E for a closed subspace E ⊆ H is equivalent to g^{∗}.E^{⊥} = E^{⊥}, hence to τ(g).E^{⊥_{β}} = E^{⊥_{β}} because E^{⊥_{β}} = I.E^{⊥}. Therefore the subgroup P_{b}(F) ⊆ GL(H) is invariant under the involution τ, and its intersection with GL(H, I) coincides with the set of all elements preserving the subspaces F_{1}, …, F_{k}.

To fix the notation in such a way that it is compatible with Examples I.9, we define for the flag F the spaces H_{1}, …, H_{k} as above, H_{0} := F_{k}^{⊥} ∩ F_{k}^{⊥_{β}}, and H_{−j} := I.H_{j} for j = 1, …, k. Then

F_{j}^{⊥_{β}} = H_{1} ⊕ … ⊕ H_{k} ⊕ H_{0} ⊕ H_{−k} ⊕ … ⊕ H_{−j−1}.

Note that H_{0} is zero if and only if F_{k} is maximal isotropic for β.

It is clear that the group M(F) is invariant under τ. For g ∈ P_{b}(F) we have τ(g)_{jj} = I(g_{−j,−j}^{∗})^{−1}I^{−1} , showing that also N(F) is τ -invariant. Therefore the semidirect decomposition of P_{b}(F) leads with

M_{b} := M(F)^{τ} ≅ GL(H_{0}, I_{0}) × ∏_{j=1}^{k} GL(H_{j})

(cf. Remark I.2) and N_{b} := N(F)^{τ} to the semidirect decomposition P_{b} = N_{b} ⋊ M_{b}. On the Lie algebra level the strictly lower triangular matrices in 𝔤𝔩(H) provide a complement invariant under τ_{𝔤}(x) := −Ix^{∗}I^{−1} , so that passing to τ_{𝔤} -fixed points yields a closed complement to the Lie algebra L(P_{b}) of P_{b} in 𝔤𝔩(H, I). Therefore P_{b} is a complemented Lie subgroup of G_{b} = GL(H, I).

Similar results hold for the group P := P_{b} ∩ GL_{2}(H, I), so that we obtain a complex manifold structure on the homogeneous spaces G/P and G_{b}/P_{b} (cf. Definition III.6).

Remark V.2. (a) The equation

N(F) − 1 = {x ∈ B(H): (∀i ≤ j) x_{ij} = 0}

shows that this is a closed subspace of B(H), hence that N(F) is contractible. Similar assertions hold for the intersection with GL_{2}(H).

(b) To obtain the corresponding result for N(F)∩GL(H, I), we note that the exponential function exp: N(F) − 1 → N(F) is a polynomial diffeomorphism inverted by the logarithm function given by

log(1 + x) = ∑_{j=1}^{∞} \frac{(−1)^{n+1}}{n} x^{n}.

This property is inherited by the subgroup N(F) ∩ GL(H, I), so that it is also contractible, and the same holds for N(F) ∩ GL_{2}(H, I).

Now we turn to the homotopy groups of the flag manifolds G/P (F).

Proposition V.3. (a) For G = GL_{2}(H) and P = P_{b}(F) ∩ G the manifold G/P satisfies

π_{0}(G/P) = 0, π_{1}(G/P) = 0 and π_{2}(G/P) ≅ ℤ^{k−1}.

(b) For G = GL_{2}(H, I) and P = P_{b}(F) ∩ G the flag manifold G/P satisfies

π_{0}(G/P) ≅ \{ \begin{matrix}0 & for I^{2} = −1 \\ ℤ_{2} & for I^{2} = 1 and H_{0} ≠ 0 \end{matrix} \}, π_{1}(G/P) = 0 and π_{2}(G/P) ≅ ℤ^{k}.

Proof. (a) We have P ≅ N ⋊ M, where N is diffeomorphic to a Banach space. Therefore P is homotopy equivalent to M. We conclude that

π_{0}(P) = π_{0}(G) = 0, π_{1}(P) ≅ ℤ^{k}, π_{1}(G) ≅ ℤ and π_{2}(P) ≅ π_{2}(G) = 0.

Hence G/P is connected, and if χ: π_{1}(P) → π_{1}(G) is the homomorphism induced by the inclusion map, the exact homotopy sequence of the principal P -bundle G → G/P implies that

π_{1}(G/P) ≅ coker χ and π_{2}(G/P) ≅ ker χ.

In view of χ(n_{1}, …, n_{k}) = ∑_{j} n_{j}, we get

π_{1}(G/P) ≅ 0 and π_{2}(G/P) ≅ ℤ^{k−1}.

(b) For G = GL_{2}(H, I) we also have P ≅ N ⋊ M, which is homotopy equivalent to M (Remark V.2(b)). For H_{0} = 0 we have

π_{0}(P) = 0, π_{1}(P) ≅ ℤ^{k} and π_{2}(P) = 0,

and

π_{0}(G) ≅ π_{1}(G) ≅ \{ \begin{matrix}0 & for I^{2} = −1 \\ ℤ_{2} & for I^{2} = 1 \end{matrix} \} and π_{2}(G) = 0.

Therefore the exact homotopy sequence of the bundle P ↪ G → G/P yields

π_{0}(G/P) ≅ \{ \begin{matrix}0 & for I^{2} = −1 \\ ℤ_{2} & for I^{2} = 1 \end{matrix} \}, π_{1}(G/P) = 0 and π_{2}(G/P) ≅ ℤ^{k}

because for I^{2} = 1 the homomorphism π_{1}(P) → π_{1}(G) is given by

ℤ^{k} → ℤ_{2} = ℤ/2ℤ, (n_{j}) ↦ ∑_{j}[n_{j}].

For H_{0} ≠ 0 we have

π_{0}(P) ≅ \{ \begin{matrix}0 & for I^{2} = −1 \\ ℤ_{2} & for I^{2} = 1 \end{matrix} \}, π_{1}(P) ≅ \{ \begin{matrix}ℤ^{k} & for I^{2} = −1 \\ ℤ_{2} × ℤ^{k} & for I^{2} = 1 \end{matrix} \}, and π_{2}(P) = 0

and

π_{0}(G) ≅ π_{1}(G) ≅ \{ \begin{matrix}0 & for I^{2} = −1 \\ ℤ_{2} & for I^{2} = 1 \end{matrix} \} and π_{2}(G) = 0.

Therefore the exact homotopy sequence yields

π_{0}(G/P) = 0, π_{1}(G/P) = 0 and π_{2}(G/P) ≅ ℤ^{k}

because for I^{2} = 1 the homomorphism π_{0}(P) → π_{0}(G) is surjective.

Remark V.4. (a) In Proposition III.5 we have seen that for G = GL_{2}(H) we have G_{r} = GG_{b}^{0} which implies that with P_{r} := G_{r} ∩ P (F) we have G_{r} = GP_{r} with P_{r} ∩ G = P, so that G acts transitively on G_{r}/P_{r}, and we obtain G/P ≅ G_{r}/P_{r}.

For G = GL_{2}(H, I) we have on the Lie algebra level 𝔤_{r} = 𝔤 + 𝔤_{b}^{0} , which implies that G_{r} = G_{e}(G_{b}^{0})_{e} (Lemma A.5), and hence that the identity component G_{e} of G acts transitively on the connected manifold G_{r}/P_{r} for P_{r} := G_{r} ∩ P (F). From P_{r} ∩ G = P we derive that G_{r}/P_{r} ≅ G_{e}/(G_{e} ∩ P) is the connected component of G/P containing the base point 1P.

(b) Suppose that all spaces H_{j} in Definition V.1 are infinite-dimensional and that G_{r} = GL(H, I, D)_{e}, where D is compatible with the flag F in the sense of Examples I.9. Then the group P_{r} ≅ N_{r} ⋊ M_{r} is contractible because N_{r} is contractible and all factors in M_{r} are contractible (Theorem II.6, Remark V.2). Therefore the exact homotopy sequence of the P_{r} - principal bundle G_{r} → G_{r}/P_{r} ≅ G_{e}/(P ∩G_{e}) implies that the quotient map is a weak homotopy equivalence. In particular we obtain for each m ∈ ℕ_{0} the relation

π_{m}(G/P) ≅ π_{m}(G_{r}).

Transitivity of the action of the unitary real form on the flag manifolds

In this subsection we show that the unitary real forms of the restricted groups G_{r} also act transitively on the corresponding flag manifolds G_{r}/P_{r}.

Lemma V.5. If φ: H_{1} → H_{2} is a topological isomorphism of Hilbert spaces, then there exists a unitary isomorphism ψ: H_{1} → H_{2} .

Proof. The map φ^{∗}φ ∈ GL(H_{1}) is an invertible positive operator, so that γ := \sqrt{φ^{∗}φ} ∈ GL(H_{1}) is uniquely defined. Now ψ := φ ∘ γ^{−1}: H_{1} → H_{2} is unitary because it is invertible and

ψ^{∗}ψ = γ^{−1}φ^{∗}φγ^{−1} = γ^{−1}γ^{2}γ^{−1} = 1.

Lemma V.6. Let H_{1} and H_{2} be complex Hilbert spaces and I_{j}: H_{j} → H_{j} antilinear isometries with I_{j}^{2} = ±1 (same signs). If φ: (H_{1}, I_{1}) → (H_{2}, I_{2}) is a topological isomorphism with φI_{1}φ^{∗} = I_{2} , then there exists a unitary isomorphism ψ: H_{1} → H_{2} with ψI_{1}ψ^{∗} = I_{2} .

Proof. First we observe that the condition φI_{1}φ^{∗} = I_{2} means that φ is an isometry between the spaces (H_{j}, β_{j}), where β_{j}(x, y) = ⟨x, I_{j}.y⟩ is a non-degenerate complex bilinear form. Indeed,

β_{2}(φ(x), φ(y)) := ⟨φ(x), I_{2}.φ(y)⟩ = ⟨x, I_{1}.y⟩ =: β_{1}(x, y), x, y ∈ H_{1},

is equivalent to φ^{∗}I_{2}φ = I_{1} , i.e., to φ^{−1}I_{2}(φ^{∗})^{−1} = I_{1} which in turn means that φI_{1}φ^{∗} = I_{2} .

We define ψ as in the proof of Lemma V.5. The remark above implies that φ^{∗}φ is a β_{1} -isometry, hence in GL(H_{1}, I_{1}). The polar decomposition of this group (Theorem II.6(iii)) implies that γ ∈ GL(H_{1}, I_{1}), so that ψ = φγ^{−1}: H_{1} → H_{2} satisfies

ψI_{1}ψ^{∗} = φγ^{−1}I_{1}(γ^{∗})^{−1}φ^{∗} = φI_{1}φ^{∗} = I_{2}.

Proposition V.7. Let H = H_{1} ⊕ … ⊕ H_{k} be the orthogonal eigenspace decomposition of D = D^{∗} ∈ B(H), F_{j} := H_{1} + … + H_{j}, P_{b} := P_{b}(F), P_{r} := P_{b} ∩ GL_{2}(H, D) and P := P_{b} ∩ GL_{2}(H). Then

GL(H) = U(H)P_{b}, GL_{2}(H, D) = U_{2}(H, D)P_{r} = U_{2}(H)P_{r}, and GL_{2}(H) = U_{2}(H)P,

i.e., U(H) acts transitively on GL(H)/P_{b} and U_{2}(H) acts transitively on GL_{2}(H, D)/P_{r} and GL_{2}(H)/P. Moreover, if u ∈ U(H) and g ∈ GL_{2}(H, D) satisfy u^{−1}g ∈ P_{b}, then u ∈ U_{2}(H, D).

Proof. (see [PS86, Prop. 7.13] for the case k = 2) Let F′_{j} := g.F_{j}. Then g maps F_{1} isomorphically onto the Hilbert space F′_{1} . Hence Lemma V.5 implies that there exists a unitary isomorphism u_{1}: F_{1} → F′_{1} . Moreover, g induces a topological isomorphism

H_{2} ≅ F_{2}/F_{1} → F′_{2}/F′_{1} ≅ H′_{2} := (F′_{1})^{⊥} ∩ F′_{2}.

Applying Lemma V.5 again, we find a unitary isomorphism u_{2}: H_{2} → H′_{2} . Continuing this way, we obtain unitary isomorphisms

u_{j}: H_{j} → H′_{j} := (F′_{j−1})^{⊥} ∩ F′_{j}, j = 1, …, k.

Putting these maps together, we obtain a unitary map u ∈ U(H) with u(H_{j}) = H′_{j} for all j and therefore in particular with

u(F_{j}) = ∑_{m=1}^{j} H′_{m} = F′_{j}.

This means that u^{−1}g preserves all spaces F_{j}. We conclude that GL(H) = U(H)P_{b}.

Suppose that g ∈ GL_{2}(H, D). Then for each j the orthogonal projection

p_{j}: F′_{j} → F_{j}^{⊥}

is Hilbert–Schmidt because its composition with g is Hilbert–Schmidt as an operator F_{j} → F_{j}^{⊥}. We conclude that p_{j} ∘ u: F_{j} → F_{j}^{⊥} is Hilbert–Schmidt, which implies that for i > j the operator u_{ij} ∈ B(H_{j}, H_{i}) is Hilbert–Schmidt. Moreover, for j > 1 the orthogonal projection

q_{j}: H′_{j} → F_{j−1}

is Hilbert–Schmidt because its composition with g corresponds to the operators g_{1j}, …, g_{j−1,j}, hence is Hilbert–Schmidt. Therefore q_{j} ∘ u |_{H_{j}}: H_{j} → F_{j−1} is Hilbert–Schmidt, which means that u_{1j}, …, u_{j−1,j} are Hilbert–Schmidt. We conclude that also for i < j we have u_{ij} ∈ B_{2}(H_{j}, H_{i}), and hence that u ∈ U_{2}(H, D). Thus u^{−1}g ∈ P_{b} ∩ GL_{2}(H, D) = P_{r}, and this proves that GL_{2}(H, D) = U_{2}(H, D)P_{r}. From the connectedness of the groups U_{2}(H, D) we derive U_{2}(H, D) = U_{2}(H) U(H)^{0} ⊆ U_{2}(H)P_{r} (Lemma A.5), whence GL_{2}(H, D) = U_{2}(H)P_{r}.

Finally GL_{2}(H) ⊆ GL_{2}(H, D) = U_{2}(H)P_{r} leads to GL_{2}(H) = U_{2}(H)(P_{r} ∩ GL_{2}(H)) = U_{2}(H)P.

Proposition V.8. Let H = H_{−k} ⊕ … ⊕ H_{k} be the orthogonal eigenspace decomposition of D = D^{∗} ∈ B(H, I) with IH_{j} = H_{−j}, F_{j} := H_{1} + … + H_{j} for j = 1, …, k, and define P_{b} := P_{b}(F) and P_{e} := P_{b} ∩ GL_{2}(H, I)_{e}. Then

GL(H, I) = U(H, I)P_{b} and GL_{2}(H, I, D) ⊆ U_{2}(H, I, D)P_{b}.

Moreover, with G_{r} := GL_{2}(H, I, D)_{e}, U_{r} := G_{r} ∩ U(H) and P_{r} := P_{b} ∩ G_{r} we get

G_{r} = U_{r}P_{r} = U_{2}(H, I)P_{r} and GL_{2}(H, I)_{e} = U_{2}(H, I)_{e}P_{e}.

In particular U(H, I) acts transitively on GL(H, I)/P_{b} and U_{2}(H, I) acts transitively on G_{r}/P_{r}.

Proof. Let g ∈ GL(H, I). As in the proof of Proposition V.7, we put F′_{j} := g.F_{j} and obtain unitary operators

u_{j}: H_{j} → H′_{j} := (F′_{j−1})^{⊥} ∩ F′_{j}, j = 1, …, k.

Putting these maps together, we obtain a unitary map u_{+}: H_{+} := F_{k} → H′_{+} := F′_{k} mapping each H_{j}, j = 1, …, k, unitarily onto H′_{j}. From g ∈ GL(H, I) we derive that F′_{k} is isotropic for the bilinear form β(x, y) := ⟨x, I.y⟩, which means that H′_{−} := I.H′_{+} ⊆ H_{+}^{⊥}. Therefore H′_{+} + I.H′_{+} ⊆ H is an orthogonal direct sum, hence a closed subspace of H. We define a unitary map

u_{−}: H_{−} := I.H_{+} → H′_{−}, v ↦ I(u_{+}^{∗})^{−1}I^{−1}.v.
Let H′_{0} := g.(H_{0} + H_{+}) ∩ (H′_{+})^{⊥}. Then

(H′_{0} + H′_{+})^{⊥} = (g.(H_{0} + H_{+}))^{⊥} = (g^{∗})^{−1}.((H_{0} + H_{+})^{⊥}) = I^{−1}gI.H_{−} = I.(g.H_{+}) = I.H′_{+} = H′_{−}

implies that H = H′_{+} ⊕ H′_{0} ⊕ H′_{−} is an orthogonal direct sum.

Since (H_{0}, β |_{H_{0}×H_{0}}) is, as a space with bilinear form, isomorphic to

(H_{+} + H_{0})/(H_{+} + H_{0})^{⊥_{β}} = (H_{+} + H_{0})/H_{+} ≅ H_{0},

the map g ∈ GL(H, I) induces an isomorphism

H_{0} → g(H_{0} + H_{+})/(g(H_{0} + H_{+}))^{⊥_{β}} ≅ (H′_{0} + H′_{+})/H′_{+} ≅ H′_{0}

with respect to the restriction of β to both spaces. Therefore we obtain with with Lemma V.6 a unitary β -isometric map u_{0}: H_{0} → H′_{0} .

Combining u_{0} with u_{±}, we now obtain with H = H_{+}⊕H_{0}⊕H_{−} a unitary map u: H → H. To see that u ∈ U(H, I), we first recall that u_{0} is β -isometric. Moreover, u maps (H_{0})^{⊥_{β}} = H_{+} + H_{−} to the closed subspace H′_{+} + H′_{−} = H′_{+} + I.H′_{+} with

(H′_{+} + H′_{−})^{⊥_{β}} = (H′_{+})^{⊥_{β}} ∩ (H′_{−})^{⊥_{β}} = (H′_{+} + H′_{0}) ∩ (I.H′_{−})^{⊥} = (H′_{+} + H′_{0}) ∩ (H′_{+})^{⊥} = H′_{0}.

Therefore it remains to show that u |_{H_{+}+H_{−}} is β -isometric. The subspaces H_{±} and H′_{±} are β -isotropic, so that the assertion follows from

β(u.v_{+}, u.v_{−}) = β(u_{+}.v_{+}, I(u_{+}^{∗})^{−1}I^{−1}.v_{−}) = −⟨u_{+}.v_{+}, (u_{+}^{∗})^{−1}I^{−1}.v_{−}⟩ = ⟨v_{+}, I.v_{−}⟩ = β(v_{+}, v_{−})

for v_{±} ∈ H_{±}. We conclude that u ∈ U(H, I), and that u^{−1}g preserves the spaces F_{1}, …, F_{k}, hence is contained in P_{b}.

Suppose that g ∈ GL_{2}(H, I, D). Then g maps the flag

F_{1} ⊆ F_{2} ⊆ … ⊆ F_{k} ⊆ F_{k}^{⊥_{β}} ⊆ … ⊆ F_{1}^{⊥_{β}}

to

F′_{1} ⊆ F′_{2} ⊆ … ⊆ F′_{k} ⊆ (F′_{k})^{⊥_{β}} ⊆ … ⊆ (F′_{1})^{⊥_{β}}

and u does the same. Therefore the last assertion in Proposition V.7 entails that u ∈ U_{2}(H, D) and hence that u ∈ U_{2}(H, I, D). Finally u^{−1}g ∈ P_{b} implies that GL_{2}(H, I, D) ⊆ U_{2}(H, I, D)P_{b}.

We conclude in particular that the group U_{r} = U_{2}(H, I, D)_{e} ⊆ G_{r} acts on G_{r}/P_{r} with open orbits, and therefore transitively because G_{r}/P_{r} is connected, whence G_{r} = U_{r}P_{r}. With Lemma A.5 we now obtain from 𝔲_{r} = 𝔲_{2}(H, I) + 𝔲(H, I, D)^{0} that U_{r} = U_{2}(H, I)_{e} U(H, I, D)^{0} ⊆ U_{2}(H, I)_{e}P_{r}, so that G_{r} = U_{2}(H, I)_{e}P_{r}, and therefore GL_{2}(H, I)_{e} ⊆ G_{r} = U_{2}(H, I)_{e}P_{r} yields GL_{2}(H, I)_{e} = U_{2}(H, I)_{e}P_{e}.

Remark V.9. (a) The decompositions of type G = UP obtained in Propositions V.7/8 are analogs of the Iwasawa decomposition of finite-dimensional complex reductive Lie groups. It is an interesting question whether such decompositions could be obtained for infinite flags.

(b) It follows from the proof of Theorem II.14 that the manifolds G/P contain a dense subset which is the directed union of orbits of finite-dimensional groups G_{F} which are compact complex flag manifolds. Since the orbits G_{F} P/P ⊆ G/P have the property that all holomorphic functions on them are constant, it follows easily that all holomorphic functions on G/P are constant (cf. [HH94b, Cor. 3.2.2] for the case G = GL_{2}(H)).

For refined information on holomorphic sections of complex line bundles on the manifolds G/P we refer to [HH94a,b], [Ne00a] and [Ne01a]. For an extension of the Bott–Borel–Weil Theorem to direct limit groups, which is closely related to our setting, we refer to [NRW00].

Remark V.10. (a) First let G = GL_{2}(H). For k = 2 and n := dim F_{1} < ∞ the orbit G.F_{1} consists of all n-dimensional subspaces of H. Therefore Gr_{n}(H) := G/P is the Graßmannian of all n-dimensional subspaces of H. For n = 1 we obtain in particular the projective space ℙ(H) = Gr_{1}(H).

For k = 2, H separable, and F := F_{1} of infinite dimension and codimension, the manifold Gr_{res}(F) := G/P is the restricted Graßmannian of the separable Hilbert space H based in F. This manifold plays a crucial role in the structure theory of loop groups and in theoretical physics (cf. [PS86], [Wu98]).

(b) The manifolds G/P for arbitrary length of the flag and G = GL_{2}(H) have been introduced in two papers of A. and G. Helminck (cf. [HH94a] and [HH94b]).

(c) For G = GL_{2}(H) and k = 2 the manifolds G/P ≅ U/(U ∩ P) (Proposition V.7) are symmetric spaces because the group U ∩ P can be written as the fixed point set of an involution on U defined as τ(u) = pup^{−1} , where p^{2} = 1 and ker(p − 1) = F_{1} .

Other symmetric spaces are obtained for G = GL_{2}(H, I) and k = 1 and either dim F_{1} = 1 or F_{1} ⊆ H maximal isotropic. In the first case we obtain the space of all isotropic lines in ℙ(H) for the bilinear form β(x, y) = ⟨x, I.y⟩, and in the second case we obtain a subset of the restricted Graßmannian associated to F_{1} , consisting of all those subspaces which are isotropic for β. In the first case the involution on U_{2}(H, I) can be obtained from p^{2} = p with ker(p + 1) = H_{0} and ker(p − 1) = F_{1} + I.F_{1} . Then each element in U_{2}(H, I) commuting with p either preserves F_{1} or maps it to I.F_{1} , showing that U_{2}(H, I) ∩ P has index 2 in the fixed point group of τ(u) = pup^{−1} , hence is an open subgroup. In the second case, where F_{1} is maximal isotropic, we define p ∈ GL(H, I) by i on F_{1} and −i on F_{1}^{⊥}. Then τ(u) := pup^{−1} defines an involution on U_{2}(H, I) with the required properties.

The important role of the flag manifolds G/P ≅ U/(U ∩ P) stems from the fact that these are precisely those coadjoint orbits of central extensions of the real group U which are strong Kähler manifolds, hence have the “best” geometric structure. We refer to [Ne01a] for this characterization which was one of the main motivations for writing the present paper.

Appendix

Lemma A.1. Let X and Y be Banach spaces and A: X → Y a continuous linear map. Suppose that X_{1}, Y_{1} are Banach spaces with continuous injective linear maps η_{X}: X_{1} → X and η_{Y}: Y_{1} → Y . If A(η_{X}(X_{1})) ⊆ η_{Y} (Y_{1}), then the induced map

A_{1}: X_{1} → Y_{1} with η_{Y} ∘ A_{1} = A ∘ η_{X}

is continuous. Proof. We argue with the Closed Graph Theorem. Assume that (x_{n}, A_{1}.x_{n}) → (x, y) ∈ X_{1} × Y_{1} . Then η_{Y} (A_{1}.x_{n}) = A.η_{X}(x_{n}) → A.η_{X}(x) implies that η_{Y} (y) = A.η_{X}(x) = η_{Y} (A_{1}.x), and therefore A_{1}.x = y. Therefore A_{1} is continuous.

Lemma A.2. Let X, Y, Z be Banach spaces and A: X × Y → Z a continuous bilinear map. Suppose that X_{1}, Y_{1}, Z_{1} are Banach spaces with continuous injective linear maps η_{X}: X_{1} → X, η_{Y}: Y_{1} → Y and η_{Z}: Z_{1} → Z. If A(η_{X}(X_{1}) × η_{Y} (Y_{1})) ⊆ η_{Z}(Z_{1}), then the induced bilinear map

A_{1}: X_{1} × Y_{1} → Z_{1} with η_{Z} ∘ A_{1} = A ∘ (η_{X} × η_{Y})

is continuous. Proof. In view of [Ru73, Th. 2.17], it suffices to show that A_{1} is separately continuous. Fix y_{1} ∈ Y_{1} . Then the map A(·, η_{Y} (y_{1})) maps η_{X}(X_{1}) to η_{Z}(Z_{1}), so that the continuity of the map A_{1}(·, y_{1}): X_{1} → Z_{1} follows from Lemma A.1. We likewise obtain the continuity of the maps A_{1}(x_{1}, ·). Therefore A_{1} is separately continuous and therefore continuous.

Lemma A.3. If 𝔤 is a Banach–Lie algebra and 𝔞, 𝔟 are Banach-Lie algebras with continuous injective homomorphisms η_{𝔞}: 𝔞 → 𝔤 and η_{𝔟}: 𝔟 → 𝔤 such that η_{𝔟}(𝔟) normalizes η_{𝔞}(𝔞), then the induced action of 𝔟 on 𝔞 is continuous. Proof. We apply Lemma A.2 with the continuous bilinear map [·, ·]: 𝔤 × 𝔤 → 𝔤 which maps η_{𝔞}(𝔞) × η_{𝔟}(𝔟) to η_{𝔞}(𝔞).

Lemma A.4. If M and N are Banach manifolds, M_{1} ⊆ M is a submanifold, and f: M_{1} → N is a smooth map, then the graph Γ(f) := {(x, f(x)): x ∈ M_{1}} is a submanifold of M × N. Proof. Passing to local charts, we may assume that M = M_{0} × M_{1} holds for a Banach manifold M_{0} . Let x = (x_{0}, x_{1}) ∈ M. Then we may further assume that N is an open subset of a Banach space X and that f(x_{1}) = 0. We then consider the smooth function

F: M_{0} × M_{1} × N → M_{0} × M_{1} × X, F (y_{0}, y_{1}, y_{2}) := (y_{0}, y_{1}, y_{2} − f(y_{1}))

which is a local diffeomorphism around (x_{0}, x_{1}, 0) ∈ Γ(f) with

F^{−1}(M × {0}) = M_{0} × Γ(f).

This implies that Γ(f) is a submanifold of M × N.

Lemma A.5. Let A, B, C be Banach–Lie groups with morphisms η_{A}: A → C and η_{B}: B → C. Assume that

(1) C is connected,

(2) im L(η_{A}) + im L(η_{B}) = L(C), and

(3) η_{B}(B) normalizes η_{A}(A). Then C = η_{A}(A)η_{B}(B).

Proof. The multiplication map m: A × B → C, (a, b) ↦ η_{A}(a)η_{B}(b) is a smooth map whose differential in (e, e) is given by (x, y) ↦ L(η_{A})(x) + L(η_{B})(y), and hence is surjective. Therefore the Implicit Function Theorem implies that im(m) has inner points. Further (3) implies that im(m) is a subgroup of C and therefore an open subgroup. Now (1) shows that m is surjective.

Proposition A.6. Let A, B, C be Banach–Lie groups and assume that A is connected and that there exist injective morphisms η_{A}: A → C and η_{B}: B → C. Let 𝔞, 𝔟 and 𝔠 denote the corresponding Lie algebras and identify 𝔞 and 𝔟 with their image under L(η_{A}), resp., L(η_{B}). We assume that

(1) 𝔟 is a closed subalgebra of 𝔠,

(2) 𝔠 = 𝔞 + 𝔟,

(3) B normalizes A, and

(4) 𝔞 ∩ 𝔟 is complemented in 𝔞.

Then the conjugation action of B on A is smooth, AB is an open subgroup of C, and the multiplication map m: A ⋊ B → AB ⊆ C is a locally trivial A ∩ B -principal bundle.

Proof. Since 𝔟 normalizes 𝔞, Lemma A.3 implies that the bracket map 𝔞×𝔟 → 𝔞 is continuous with respect to the Banach space structure on 𝔞 which might be finer than that inherited from 𝔠. Therefore 𝔞 ⋊ 𝔟 carries the structure of a Banach–Lie algebra. Lemma A.1 also implies that for each b ∈ B the map Ad_{𝔞}(b) := Ad(b) |_{𝔞}: 𝔞 → 𝔞 is a continuous automorphism. The action of B on 𝔞 is obtained by integrating a continuous representation of its Lie algebra 𝔟, hence is a continuous homomorphism Ad_{𝔞}: B → Aut(𝔞).

For each b ∈ B the conjugation map c_{A}(b): A → A, a ↦ bab^{−1} is a group automorphism with c_{A}(b)(exp_{A}(x)) = exp_{A}(Ad_{𝔞}(x)) for x ∈ 𝔞, because both sides have the same image in C. Therefore each c_{A}(b) is smooth in an identity neighborhood, thus a smooth automorphism of A. Moreover, the smoothness of the action of B on 𝔞 implies that the B -orbit maps of elements of A are smooth for regular values of the exponential map exp_{A}: 𝔞 → A, hence for all points in an identity neighborhood of A. Since B acts by automorphisms of A, the set of all points with smooth orbit map is a subgroup, so that the connectedness of A entails that all orbit maps of elements in A are smooth. Now let b_{0} ∈ B and a_{0} ∈ A_{e}. For b ∈ B and a ∈ A we then have

b_{0}baa_{0}(b_{0}b)^{−1} = c_{A}(b_{0})((bab^{−1}c_{A}(b)(a_{0})).

Since B acts by smooth automorphisms and with smooth orbits maps, the smoothness of the action of B on A follows from the smoothness close to the identity which in turn follows from the fact that the exponential function is a local diffeomorphism and the action of B on 𝔞 is smooth. Thus the group A ⋊ B carries a natural structure of a Banach–Lie group.

As in the proof of Lemma A.5, we see that the multiplication map m: A⋊B → C, (a, b) ↦ ab and an open map. Therefore AB is an open subgroup of C. The kernel of m is the Lie subgroup

N = {(a, a^{−1}) ∈ A ⋊ B: a ∈ A ∩ B}

([Ne00a, Lemma IV.11]). The Lie algebra of N is the subalgebra {(x, −x) ∈ 𝔞 ⋊ 𝔟: x ∈ 𝔞 ∩ 𝔟} which is complemented in 𝔞⋊𝔟 because, in view of (4), 𝔞∩𝔟 is complemented as a subspace of 𝔞. Therefore we see with Definition III.6 that the quotient group (A ⋊ B)/N is a Lie group, which then is isomorphic to AB. Moreover, the quotient map A ⋊ B → (A ⋊ B)/N is a submersion, hence defines a locally trivial A ∩ B -principal bundle.

Remark A.7. If, under the assumption of Proposition A.6, A is not connected, then we first obtain with Proposition A.6 that A_{e} ⋊ B → A_{e}B is an A_{e} ∩ B -principal bundle over the open subgroup A_{e}B ⊆ C. Moreover, A_{e}B_{e} is an open connected subgroup of C, hence the identity component C_{e} of C.

Assume that AB = C. Then the multiplication map A ⋊ B → C → C/C_{e} = π_{0}(C) factors through a surjective homomorphism

(A/A ∩ C_{e}) ⋊ (B/B ∩ C_{e}) → π_{0}(C).

If we know, in addition, that B acts continuously on A, then A ⋊ B is a topological group and the open subgroup A_{e} ⋊ B is a Banach–Lie group. This implies that A ⋊ B is a Banach– Lie group because because conjugating by elements of A induces continuous, hence smooth, isomorphisms of A_{e} ⋊ B.

Proposition A.8. For j = 1, 2 let G_{j} be a topological group and H_{j} ⊆ G_{j} a closed subgroup. We further assume that q_{j}: G_{j} → M_{j} := G_{j}/H_{j} defines a locally trivial principal bundle and that we have a continuous homomorphism φ: G_{1} → G_{2} with φ(H_{1}) ⊆ H_{2} . Let φ_{M}: M_{1} → M_{2}, g_{1}H_{1} ↦ g_{2}H_{2} denote the map induced by φ. If two of the three maps φ, φ_{M} and φ_{H} := φ |_{H_{1}}: H_{1} → H_{2} are weak homotopy equivalences, then the same holds for the third one.

Proof. Since the map φ is compatible with the subgroups H_{j} ⊆ G_{j}, it induces maps between the exact homotopy sequences

⋯ → π_{k}(H_{j}) → π_{k}(G_{j}) → π_{k}(M_{j}) → π_{k−1}(H_{j}) → π_{k−1}(G_{j}) → π_{k−1}(M_{j})

of G_{j} → → M_{j}:

⋯ → π_{k}(H_{1}) → π_{k}(G_{1}) → π_{k}(M_{1}) → π_{k−1}(H_{1}) → π_{k−1}(G_{1}) → …
↓^{π_{k}(φ_{H})} ↓^{π_{k}(φ)} ↓^{π_{k}(φ_{M})} ↓^{π_{k−1}(φ_{H})} ↓^{π_{k−1}(φ)}
⋯ → π_{k}(H_{2}) → π_{k}(G_{2}) → π_{k}(M_{2}) → π_{k−1}(H_{2}) → π_{k−1}(G_{2}) → ….

We assume that φ and φ_{H} are weak homotopy equivalences; the other cases are similar. Then the maps π_{k}(φ) and φ_{k}(φ_{H}) are isomorphisms, and the rows in the above diagram are exact, so that the 5-Lemma ([CE56, Prop. I.1.1]) implies that all homomorphisms π_{k}(φ_{M}), k ∈ ℕ, are isomorphisms. To obtain this also for k = 0 we may extend the exact homotopy sequence by zeros on the right hand side because the maps π_{0}(G_{j}) → π_{0}(M_{j}) are trivially surjective.

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