Banach algebras generated by two idempotents and one flip

Tilo Finck  Steffen Roch^{∗} Bernd Silbermann

Abstract

The paper is concerned with the invertibility of elements in a Banach algebra generated by two idempotents and one flip element. A symbol in form of a 2×2 matrix function which is defined on some Hausdorff compact M will be constructed and a complete description of M will be given.

0 Introduction

The goal of the present paper is to study invertibility of elements in Banach algebras which are generated by two idempotent and one flip element. More precise: throughout this paper, we let B be a complex Banach algebra with identity element e which is not degenerated in the sense that e ≠ 0, and we let p, q and j be elements of B satisfying

p^{2} = p, q^{2} = q, j^{2} = e .  (1)

The smallest closed subalgebra of B which contains the idempotents p and q as well as the flip j will be denoted by A. Clearly, this definition involves that e ∈ A. What we are interested in is criteria for invertibility of elements of A. Moreover, the wanted criteria should be well verifiable which leads to the following precision of our task: we ask whether or not it is possible to associate with every element a of A a certain matrix–valued function, called the symbol of a, having the property that a is invertible if and only if the symbol of a is invertible. As the examples in [2] indicate, this is impossible in general. What we shall point out is that it is enough to impose two more axioms, namely

∗Supported by a DFG-Heisenberg grant.

jpj = e − p, jqj = e − q  (2)

in order to ensure the positive solvability of our task. Let us emphasize that (2) are the only additional conditions we need to verify the existence of a symbol as well as to derive its explicite form (but notice also that (2) are \underline{not} the only additional conditions which fit to our purposes; see e.g.[12], where jqj = e − q is replaced by jqj = q, which leads to a quite different algebra).

The same algebra A – but with additional conditions besides (2) – and the same task were subject of the paper [15] by two of the authors. Unfortunately, in this paper, we did not only impose more (and, as we know now, redundant) conditions on p, q and j; we even did not mention all conditions in needed clearness which might lead to some confusion. We are grateful to Helena Mascarenhas from the IST Lisboa whose questions gave raise to take up the subject again.

The paper is organized as follows: In Section 1, we shall prove the existence of the wanted symbol calculus (Theorem 1). Particularly, we shall see that one can choose the symbol of an element of A in form of a 2 × 2–matrix function which is defined on some Hausdorff compact M and which is even continuous. The compact M depends on two distinguished elements of the algebra A. In Section 2, we shall derive some properties of M which characterize the possibly occuring M’s completely (Theorem 2). The explicit description of M will be obtained in Section 3 where we shall also formulate and prove our main result (Theorem 3). Another approach to determine M will be proposed in Section 4 (Theorem 5). Some final remarks concerning the invertibility of elements of A in the (larger) algebra B as well as the case of non–scalar coefficient algebra will be mentioned in the concluding fifth section.

Let us note finally that Banach algebras generated by two idempotents and the identity element were investigated in [15], [7], [17] and [8], [9]. A generalization of these results to the case of N > 2 idempotents (but with additional conditions) was obtained in [2]. For Banach algebras generated by two idempotents and one flip (satisfying, possibly, other axioms than (2)) we refer to [15], [3] and [12]. Applications of these result to concrete Banach algebras occuring in operator theory or numerical analysis are discussed, e.g. in [15], [7], [3], [4], [8], [9] and others. Recall also that the related problem for C^{∗}–algebras generated by two projections (i.e. selfadjoint idempotents) was subject of a whole series of publications, mainly initiated by Halmos’ paper [10]. For a more detailed account on this special topic we refer to [2].

1 Existence of the symbol calculus

Throughout this paper, let A be a closed subalgebra of a Banach algebra B with identity e ≠ 0 which is generated by two idempotents p and q and by a flip j satisfying (2). Evidently, the distinction between idempotents and flips is not essential since e−2p is a flip for every idempotent p and (e+j)/2 is an idempotent for every flip j. So we could also think of A as an algebra which is generated by three idempotents (or three flips). But it is the concrete form of the axioms (2) and the concrete applications in operator theory which let us prefer the chosen terminology. We start with a trivial observation.

Proposition 1 The subalgebra of A which is generated by p and j is isomorphic to C^{2×2}, and the isomorphism can be chosen in such a way that it takes the elements

e_{11} = p, e_{12} = pj(e − p), e_{21} = (e − p)jp, e_{22} = e − p

into the matrices

E_{11} = ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) , E_{12} = ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ) , E_{21} = ( \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} ) , E_{22} = ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ) ,

respectively.

Proof: Let L ⊂ A stand for the linear space spanned by e_{11}, e_{12}, e_{21} and e_{22}. We claim that the {e_{ij}}_{i,j=1}^{2} are linearly independent. Indeed, suppose

αp + βpj(e − p) + γ(e − p)jp + δ(e − p) = 0  (3)

with some α, β, γ, δ ∈ C. Multiplying (3) by p or e − p from the left or right hand sides yields

αp = 0, βpj(e − p) = 0,
γ(e − p)jp = 0, δ(e − p) = 0.

The assumption p = 0 together with the axiom jpj = e − p would imply 0 = e which was excluded. Thus, α = 0 and, analogously, δ = 0. Similarly, if pj(e − p) were 0, then pj(e − p)j = p would be 0, too, which is impossible as we have just seen. Thus, β = 0 and, analogously, γ = 0. So, L is a four–dimensional subspace of A which contains the elements p = e_{11}, e = e_{11} + e_{22} and j = e_{12} + e_{21}. Moreover, L is an algebra, since

e_{ij} · e_{kl} = δ_{jk}e_{il}  (4)

(with δ_{jk} referring to the Kronecker delta), as one easily checks. Therefore, L is the smallest algebra, which contains p and j, and the algebra is already closed due to its finite dimensionality. The isomorphy of L to C^{2×2} (with e_{ij} corresponding to E_{ij}) is an immediate consequence of (4) and of the analogous relations holding for the matrices E_{ij}.

For every i, j ∈ {1, 2}, we define mappings

w_{ij} : B → B, a ↦ e_{1i}ae_{j1} + e_{2i}ae_{j2}

and a mapping

w : B → B^{2×2}, a ↦ ( \begin{matrix}w_{11}(a) & w_{12}(a) \\ w_{21}(a) & w_{22}(a) \end{matrix} ) .  (5)

Evidently,

w(e) = ( \begin{matrix}e & 0 \\ 0 & e \end{matrix} ) , w(p) = ( \begin{matrix}e & 0 \\ 0 & 0 \end{matrix} ) , w(j) = ( \begin{matrix}0 & e \\ e & 0 \end{matrix} ) .  (6)

The following proposition is a special (2 × 2) case of a more general and well known result holding for all algebras with C^{N×N} as one of its subalgebras. For a proof see [6], Proposition 3.3 in Chapter 4.3., or [16].

Proposition 2 (a) The ranges of w_{11}, w_{12}, w_{21} and w_{22} coincide.

(b) The set K := w_{11}(B) is a closed subalgebra of B which contains e.

(c) The mapping w is a continuous isomorphism from B onto K^{2×2}.

(d) ke_{ij} = e_{ij}k for all k ∈ K and i, j ∈ {1, 2}.

(e) The centers of B and K coincide.

Below we shall apply this proposition to the algebra A. As a preparation we mention the following result.

Proposition 3 (a) Let a ∈ B be an element satisfying jaj = e − a. Then

w(a) = ( \begin{matrix}b & c \\ −c & e − b \end{matrix} ) with b = pap + (e − p)(e − a)(e − p) and c = (pa − ap)j.

(b) The elements b and c commute if and only if p(a^{2} − a) = (a^{2} − a)p.

Proof: (a) The axiom jaj = e−a and (6) involve that w(a) = ( \begin{matrix}b & c \\ −c & e − b \end{matrix} ) with some elements b, c ∈ B. Further,

b = w_{11}(a) = pap + (e − p)jaj(e − p) = pap + (e − p)(e − a)(e − p),
c = w_{12}(a) = pa(e − p)j + (e − p)ja(e − p)
= (pa(e − p) + (e − p)(e − a)p) j = (pa − ap)j.

(b) It is easy to check that

bc − cb = p(a^{2} − a)jp + (e − p)(a^{2} − a)j(e − p).

Thus, bc − cb = 0 if and only if

p(a^{2} − a)jp = 0 and (e − p)(a^{2} − a)j(e − p) = 0.

Since jpj = e − p and j is invertible, this is equivalent to

p(a^{2} − a)(e − p) = (e − p)(a^{2} − a)p = 0

which, on its hand, is equivalent to

p(a^{2} − a) = (a^{2} − a)p.

Particularly, Proposition 3 applies to the element a = q; so in what follows let b, c ∈ A be defined by

b = pqp + (e − p)(e − q)(e − p), c = (pq − qp)j.  (7)

In this special setting, we have bc = cb due to Proposition 3(b) and, moreover,

c^{2} = b(b − e),  (8)

the latter fact being a consequence of

q^{2} = q respective ( \begin{matrix}b & c \\ −c & e − b \end{matrix} )^{2} = ( \begin{matrix}b & c \\ −c & e − b \end{matrix} ) = w(q).

Let C denote the smallest closed subalgebra of A which contains b, c and e. The following result is a summary of Propositions 1–3.

Proposition 4 Let A be a Banach algebra with identity e ≠ 0 which is generated by elements p, q, j satisfying (1) and (2), and let C be the smallest closed subalgebra of A generated by elements b, c and e defined by (7). Then C is the center of A, and the mapping w defined by (5) is a continuous isomorphism from A onto C^{2×2}. The action of w on the generating elements of A is given by (6) and Proposition 3(a).

Thus, an element a ∈ A is invertible in A if and only if the matrix w(a) is invertible in C^{2×2}. Since C is commutative, an element of C^{2×2} is invertible if and only if its determinant (which can be defined as usually) is invertible in C, and the latter invertibility is subject to the Gelfand theory. If we let M refer to the maximal ideal space of C and G : C → C(M) to the Gelfand transform, then we arrive at the following result.

Theorem 1 An element a ∈ A is invertible in A if and only if the function

M → C^{2×2}, m ↦ ( \begin{matrix}G (w_{11}(a)) (m) & G (w_{12}(a)) (m) \\ G (w_{21}(a)) (m) & G (w_{22}(a)) (m) \end{matrix} )

is invertible at every point m ∈ M.

This theorem provides us with the desired matrix–valued calculus for the algebra A. The symbol can be realized as a 2 × 2 matrix function on the maximal ideal space M of C and is – if M is endowed with its Gelfand topology – even a continuous function. The information bc = cb and c^{2} = b(b − e) about b and c is all what can be derived from the axioms (1) and (2). Indeed, if b̂ and ĉ are elements of a Banach algebra B̂ with identity ê with b̂ĉ = ĉb̂ and ĉ^{2} = b̂(b̂ − e) then

e = ( \begin{matrix}ê & 0 \\ 0 & ê \end{matrix} ) , p = ( \begin{matrix}ê & 0 \\ 0 & 0 \end{matrix} ) , j = ( \begin{matrix}0 & ê \\ ê & 0 \end{matrix} ) , q = ( \begin{matrix}b̂ & ĉ \\ −ĉ & e − b̂ \end{matrix} )

are elements of B̂^{2×2} which satisfy (1) and (2). Thus, our problem of establishing a matrix–valued symbol calculus for an algebra A which is generated by elements p, q, j satisfying (1) and (2) is completely equivalent to the determination of the maximal ideal space of a commutative Banach algebra which is generated by e and by elements b, c satisfying c^{2} = b(b − e), and to the computation of the Gelfand transforms, at least for the generating elements b and c. This will be done in the forthcomming sections. Before, let us remark that the characterization of the center of A was observed in [13] for the first time. The first proof of the existence of an (at most) 2 × 2 matrix–valued symbol calculus for the algebra A appeared in [15]. There it is shown that A is a F_{4}–algebra, i.e. that

F_{4}(a_{1}, a_{2}, a_{3}, a_{4}) = ∑_{s∈S_{4}} sgn s a_{s(1)}a_{s(2)}a_{s(3)}a_{s(4)} = 0

for all choices of a_{1}, a_{2}, a_{3}, a_{4} ∈ A where S_{4} refers to the permutation group of {1, 2, 3, 4}. Then Krupnik’s theory of Banach algebras with polynomial identities (see [11], Theorem 21.1) applies to prove the existence of a 2 × 2 symbol. For a further proof, we refer to Section 4 below.

2 Properties of the maximal ideal space M

We are going to establish some characterizing properties of the maximal ideal space M of C. For every algebra D with identity let σ_{D}(d) refer to the spectrum of d ∈ D in D. Since C is the center of A, we have σ_{C}(d) = σ_{A}(d) for every d ∈ C.

Since C is both a commutative algebra and generated by two of its elements, we infer from the general theory of commutative Banach algebras that M is homeomorphic to a compact subset of σ_{C}(b) × σ_{C}(c) ⊆ C × C, and that this subset can be characterized as the joint spectrum σ_{C}(b, c) of b and c in C (see, e.g. [5], Chapter I, §3.5, Proposition 9(i)). Recall that the joint spectrum of elements b, c can be defined as the set of all pairs (x, y) ∈ C × C such that the smallest closed ideal of C which contains b − xe and c − ye is proper. Moreover, since b and c together with e generate the algebra C, their joint spectrum is a polynomial convex set (see again [5], Chapter I, §3.5, Corollary 1). The latter fact will not be used explicitely in what follows.

We shall thouroughly identify M and σ_{C}(b, c). If we do this, then we have

(G P (b, c)) (x, y) = P (x, y)  (9)

for every polynomial P in two variables and (x, y) ∈ M. We have already remarked that, if (x, y) ∈ M, then x ∈ σ_{C}(b). Conversely, given an x ∈ σ_{C}(b), there is (at least) one y ∈ σ_{C}(c) such that (x, y) ∈ M. This is a simple consequence of (8) and of the fact that the range of the Gelfand transform of an element coincides with the spectrum of that element. Hence, M = σ_{C}(b, c) can be decomposed into fibers over σ_{C}(b): Given x ∈ σ_{C}(b), set M_{x} := {(x, y) ∈ M}. Then none of the fibers M_{x} is empty, and ∪_{x∈σ_{C}(b)}M_{x} = M.

Proposition 5 Let x ∈ σ_{C}(b). Then the fiber M_{x} is either a singleton {(x, y)} or a doubleton {(x, y), (x, −y)} where y ∈ C is a number such that y^{2} = x(x − 1).

Proof: From (9) we conclude that

G (c^{2} − b(b − e)) (x, y) = y^{2} − x(x − 1)

and, since c^{2} − b(b − e) = 0, that

y^{2} = x(x − 1) for every (x, y) ∈ M.  (10)

Thus, given x ∈ σ_{C}(b), there are at most two numbers y such that (x, y) ∈ M.

We denote the sets

{x ∈ σ_{C}(b) : M_{x} is a singleton}

and

{x ∈ σ_{C}(b) : M_{x} is a doubleton}

by Σ_{1} and Σ_{2}, respectively.

There is an alternative way to derive this fibration via Allan’s local principle which we quote here for the reader’s convenience. For a proof see [1] or [4], Theorem 1.34.

Allan’s local principle Let A be a Banach algebra with identity, and let B be a closed subalgebra of the center of A which contains the identity. For every maximal ideal x of B, let I_{x} denote the smallest closed two–sided ideal of A which contains x. Then an element a ∈ A is invertible in A if and only if the cosets a + I_{x} are invertible in the quotient algebra A/I_{x} for all x with I_{x} ≠ A.

We shall apply this theorem with C and alg(e, b) (= the smallest closed subalgebra of C which contains e and b) in place of A and B, respectively. It is well– known that the maximal ideal space of the (singly generated) Banach algebra alg(e, b) is homeomorphic to the spectrum of b in this algebra. Moreover, the only maximal ideals x ∈ σ_{alg(e,b)}(b) which generate proper ideals I_{x} in C are those in σ_{C}(b)(⊆ σ_{alg(e,b)}(b)); see [7]. Thus, Allans local principle offers the possibility of localizing C over σ_{C}(b). Given x ∈ σ_{C}(b), we denote the related ”‘local algebra”’ L/I_{x} by C_{x} and the canonical homomorphism from C onto C_{x} by ϕ_{x}.

Proposition 6 Let (x, y) ∈ σ_{C}(b) × σ_{C}(c). Then (x, y) ∈ M if and only if y ∈ σ_{C_{x}} (ϕ_{x}(c)).

Proof: If (x, y) ∉ M, then the ideal of C which is generated by b − xe and c − ye is not proper; that is, there are elements f, g ∈ C such that

f(b − xe) + g(c − ye) = e.

Applying the local homomorphism ϕ_{x} to this equality and taking into account that ϕ_{x}(b) = xϕ_{x}(e), we find

ϕ_{x}(g) (ϕ_{x}(c) − yϕ_{x}(e)) = ϕ_{x}(e),

i.e. y ∉ σ_{C_{x}} (ϕ_{x}(c)). Let, conversely, y ∉ σ_{C_{x}} (ϕ_{x}(c)). Then there are elements f ∈ C and j ∈ I_{x} such that

f(c − ye) = e + j.

The set C(b − xe) is dense in I_{x}; hence, there are elements g, h ∈ C with ∥h∥ < 1 and j = −g(b − xe) + h. Consequently,

(e + h)^{−1}g(b − xe) + (e + h)^{−1}f(c − ye) = e

which implies that (x, y) ∉ M.

Particularly, if x ∈ Σ_{1} resp. x ∈ Σ_{2}, then σ_{C_{x}} (ϕ_{x}(c)) consists of one resp. two points, and the maximal ideal space of C_{x} is a singleton resp. a doubleton.

For further conclusions, we still need a completion of Allan’s local principle. We define the limiting set of a set–valued function M on C,

M : x ↦ M_{x} ⊆ C,

at a point x_{0} ∈ C as the set of all limits of sequences (m_{n}) with m_{n} ∈ M_{x_{n}} and x_{n} → x_{0}. We denote this set by lim_{x→x_{0}} M_{x}.

Proposition 7 Let the notations be as in Allan’s local principle; further let M be the set of all maximal ideals x of B such that I_{x} ≠ A and, given x ∈ M and a ∈ A, set ψ_{x}(a) := a + I_{x}.

(a) The mapping

M → R^{+}, x ↦ ∥ψ_{x}(a)∥

is upper semi–continuous for every a ∈ A.

(b) If ψ_{x_{0}}(a) is invertible for some a ∈ A then there is an ϵ and a neighbourhood U of x_{0} in M such that ψ_{x}(d) is invertible for all d ∈ A with ∥a − d∥ < ϵ and for all x ∈ U.

(c) lim_{x→x_{0}} σ_{C_{x}} (ψ_{x}(a)) ⊆ σ_{C_{x}} (ψ_{x_{0}}(a)) for every a ∈ A.

Property (c) can be rephrased as upper semi–continuity of local spectra. Proof: (a) See, e.g. [1] or[4], Theorem 1.34.

(b) Let ψ_{x_{0}}(a) be invertible in A/I_{x_{0}}, and let f ∈ A be an element such that ψ_{x_{0}}(af − e) = ψ_{x_{0}}(fa − e) = 0. Choose an ϵ < 1/(4∥f∥) as well as an open neighborhood U of x_{0} such that

∥ψ_{x}(af − e)∥ < \frac{1}{4} and ∥ψ_{x}(fa − e)∥ < \frac{1}{4} for all x ∈ U

(the latter being possible due to part (a)). Then we have, for all d ∈ A with ∥a − d∥ < ϵ and for all x ∈ U:

∥ψ_{x}(df − e)∥ ≤ ∥ψ_{x}(af − e)∥ + ∥a − d∥∥f∥ < \frac{1}{2} < 1

and, analogously, ∥ψ_{x}(fd − e)∥ < 1. Hence, the cosets ψ_{x}(df) and ψ_{x}(fd) are invertible, but this involves the invertibility of ψ_{x}(d).

(c) Let s ∈ lim_{x→x_{0}} σ_{C_{x}} (ψ_{x_{0}}(a)). By definition, then there are points x_{n} ∈ M with x_{n} → x_{0} and numbers s_{n} ∈ σ_{C_{x}} (ψ_{x_{n}}(a)) such that s_{n} → s. We consider the elements a − s_{n}e which converge to a − se. Assume the coset ψ_{x_{0}}(a − se) to be invertible. Then, by part (b), the cosets ψ_{x_{n}}(a − s_{n}e) are invertible for all sufficiently large n, which contradicts our hypothesis. Consequently, s ∈ σ_{C_{x}} (ψ_{x_{0}}(a)).

Let us now return to our concrete algebra. We set Σ_{0} := σ_{C}(b)∩{0, 1}. Evidently, Σ_{0} ⊆ Σ_{1}.

Theorem 2 (a) The set Σ_{2} ∪ Σ_{0} is closed in σ_{C}(b) (and , hence, compact).

(b) The set Σ_{1} \ Σ_{0} is open in σ_{C}(b), and the mapping which assigns to every x ∈ Σ_{1} \ Σ_{0} the second component of the (uniquely determined) pair (x, y) ∈ M is continuous on Σ_{1} \ Σ_{0}.

Proof: (a) Let x_{0} belong to the closure of Σ_{2} ∪ Σ_{0}. What we have to show is that x_{0} ∈ Σ_{2} ∪ Σ_{0}. If x_{0} ∈ Σ_{0}, then we are done. So let x_{0} ∉ Σ_{0}. Then there is a sequence (x_{n}) ⊆ Σ_{2} which converges to x_{0}. Our assumption x_{0} ∉ Σ_{0} further guarantees that x_{0}^{2} − x_{0} ≠ 0, thus, we can find an open neighbourhood U of x_{0}^{2} − x_{0} in C such that there exists a continuous branch, sqrt say, of the square root function on U. Hence,

σ (ϕ_{x_{n}}(c)) = {sqrt(x_{n}^{2} − x_{n}), −sqrt(x_{n}^{2} − x_{n})}

for all sufficiently large n. From part (c) of the preceding proposition we know that

lim_{n→∞} σ (ϕ_{x_{n}}(c)) = {sqrt(x_{0}^{2} − x_{0}), −sqrt(x_{0}^{2} − x_{0})} ⊆ σ (ϕ_{x_{0}}(c)) .

But sqrt(x_{0}^{2} − x_{0}) ≠ −sqrt(x_{0}^{2} − x_{0}) because of x_{0} ∉ Σ_{0}, hence, σ (ϕ_{x_{0}}(c)) consists of two points, i.e. x_{0} ∈ Σ_{2}.

(b) It is immediate from part (a) that Σ_{1} \Σ_{0} is an open subset of σ_{C}(b). So what remains to show is the continuity of the mapping

Σ_{1} \ Σ_{0} → σ_{C}(c), x ↦ (x, y) ↦ y.

If x_{0} ∈ Σ_{1} \ Σ_{0}, then x_{0}^{2} − x_{0} ≠ 0, and we can again choose a neighbourhood U of x_{0}^{2} − x_{0} in C as well as a continuous branch sqrt of the square root function on U such that

y_{0} = y(x_{0}) = sqrt(x_{0}^{2} − x_{0}).

Let V be an open neighbourhood of x_{0} in Σ_{1}\Σ_{0} such that x^{2}−x ∈ U for all x ∈ V. Then, for every x ∈ V, either y(x) =sqrt(x^{2}−x) or y(x) = −sqrt(x^{2}−x). Assume there exists a sequence (x_{n}) ⊆ V tending to x_{0} such that y(x_{n}) = −sqrt(x_{n}^{2} − x_{n}) for every n. Since

y(x_{n}) = −sqrt(x_{n}^{2} − x_{n}) → −sqrt(x_{0}^{2} − x_{0})

we conclude, again via Proposition 7(c), that

−y_{0} = −sqrt(x_{0}^{2} − x_{0}) ∈ σ (ϕ_{x_{0}}(c)) .

Since y_{0} ≠ −y_{0} (a consequence of x_{0} ∉ Σ_{0}) and y_{0} ∈ σ (ϕ_{x_{0}}(c)) (by assumption), this contradicts the hypothesis x_{0} ∈ Σ_{1}. Therefore,

y = y(x) = sqrt(x^{2} − x) for all x ∈ V

which involves continuity of the function x ↦ y at every point of Σ_{1} \ Σ_{0}.

The preceding theorem characterizes possible symbol algebras completely. Indeed, let Σ̂ be a compact subset of C, set Σ̂_{0} = Σ̂ ∩ {0, 1}, and let Σ̂_{1}, Σ̂_{2} be subsets of Σ̂ such that

Σ̂_{1} ∩ Σ̂_{2} = ∅, Σ̂_{1} ∪ Σ̂_{2} = Σ̂, Σ̂_{0} ⊆ Σ̂_{1}, Σ̂_{2} ∪ Σ̂_{0} is compact

and such that there is a continuous function ŷ = ŷ(x) on Σ̂_{1} \ Σ̂_{0} satisfying

ŷ(x)^{2} = x^{2} − x for all x ∈ Σ̂_{1} \ Σ̂_{0}.

We define a set M̂ by

M̂ := {(x, y) : x ∈ Σ̂_{1} \ Σ̂_{0}, y = ŷ(x)} ∪ {(x, y) : x ∈ Σ̂_{2} ∪ Σ̂_{0}, y^{2} = x^{2} − x}

and functions ê, p̂, ĵ, q̂ from M̂ into C^{2×2} by

ê(x, y) = ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) , p̂(x, y) = ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) , ĵ(x, y) = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} )

and

q̂(x, y) = ( \begin{matrix}x & y \\ −y & 1 − x \end{matrix} ) .

Then ê, p̂, ĵ and q̂ satisfy the axioms (1) and (2) in place of e, p, j and q, respectively, and starting with ê, p̂, ĵ and q̂ and computing b, c, Σ_{1} and Σ_{2} as above, just yields Σ_{1} = Σ̂_{1}, Σ_{2} = Σ̂_{2} and M = M̂. Indeed, for b and c we find

b(x, y) = ( \begin{matrix}x & 0 \\ 0 & x \end{matrix} ) and c(x, y) = ( \begin{matrix}y & 0 \\ 0 & y \end{matrix} )

which implies the assertions as one easily checks.

3 Determination of the maximal ideal space M

Now we are going to point out one way of identifying the components Σ_{0}, Σ_{1} and Σ_{2} of σ_{C}(b) as well as the correct branch of the square root function for every connected component of Σ_{1} \ Σ_{0}. Particularly, we shall explain that the knowledge of the (global) spectrum of only one element of C is sufficient for the desired identifications. This element is b + c. To see this, we need a trivial observation.

Proposition 8 Let x_{1}, x_{2}, y_{1}, y_{2} ∈ C satisfy the conditions

y_{1}^{2} = x_{1}^{2} − x_{1}, y_{2}^{2} = x_{2}^{2} − x_{2}, and x_{1} + y_{1} = x_{2} + y_{2}.

Then x_{1} = x_{2} and y_{1} = y_{2}.

Proof: Squaring the equality x_{1} − x_{2} = y_{2} − y_{1} we get

x_{1}^{2} − 2x_{1}x_{2} + x_{2}^{2} = y_{1}^{2} − 2y_{1}y_{2} + y_{2}^{2} = x_{1}^{2} − x_{1} − 2y_{1}y_{2} + x_{2}^{2} − x_{2}

resp.

2x_{1}x_{2} − x_{1} − x_{2} = 2y_{1}y_{2}.

Squaring once more, we obtain

x_{1}^{2} + x_{2}^{2} + 4x_{1}^{2}x_{2}^{2} − 4x_{1}^{2}x_{2} − 4x_{1}x_{2}^{2} + 2x_{1}x_{2} = 4y_{1}^{2}y_{2}^{2}
= 4(x_{1}^{2} − x_{1})(x_{2}^{2} − x_{2}) = 4x_{1}^{2}x_{2}^{2} − 4x_{1}^{2}x_{2} − 4x_{1}x_{2}^{2} + 4x_{1}x_{2}

resp.

x_{1}^{2} − 2x_{1}x_{2} + x_{2}^{2} = (x_{1} − x_{2})^{2} = 0,

whence the assertion follows.

Now we can characterize the points in M.

Proposition 9 The mapping

M → C, (x, y) ↦ x + y  (11)

is a bijection between M and σ_{C}(b + c).

Proof: From (9) we infer that

G(b + c)(x, y) = x + y.  (12)

Since the range of the Gelfand transform of an element is nothing but the spectrum of the element, (12) involves that (11) is a mapping from M onto σ_{C}(b + c). It remains to verify the injectivity of (11). Let z ∈ σ_{C}(b + c). Then, as we have just seen, there is (at least) one point (x, y) ∈ M such that z = x + y. But y^{2} = x^{2} − x for every pair (x, y) ∈ M by (10), hence, Proposition 8 yields the uniqueness of (x, y).

Actually, Proposition 8 yields a little bit more than we have employed in the proof of Proposition 9. Namely, every point z ∈ σ_{C}(b + c) can be uniquely written as x + y where y^{2} = x^{2} − x. We see from Proposition 9 that then necessarily x ∈ σ_{C}(b), y ∈ σ_{C}(c) and (x, y) ∈ M. Moreover, if we have the summands x and y at our disposal we can look whether or not z′ = x − y also belongs to σ_{C}(b + c). If no, then x ∈ Σ_{1}, if yes, then x ∈ Σ_{2}. Finally, if x ∈ Σ_{1} \ Σ_{0} then the associated y indicates which branch of the square root function has to be choosen in an open neighbourhood of x. These observations allow to determine M, Σ_{0}, Σ_{1}, Σ_{2} as well as the explicit form of the symbol calculus in practice.

We summarize these facts in our main result.

Theorem 3 Let A be a Banach algebra generated by elements p, q and j satisfying (1) and (2), and let C be the smallest closed subalgebra of A generated by e and by elements b and c defined by (7). Further, let M consist of all pairs (x, y) ∈ C × C where x^{2} − x = y^{2} and x + y ∈ σ_{C}(b + c).

(a) The set M (provided with the topology inherited by that one of C × C) is homeomorphic to the maximal ideal space of C.

(b) The mapping smb which assignes with e, p, j and q a matrix–valued function on M by

(smb e)(x, y) = ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) ,  (smb p)(x, y) = ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} )
(smb j)(x, y) = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) , (smb q)(x, y) = ( \begin{matrix}x & y \\ −y & 1 − x \end{matrix} )

extends to a continuous homomorphism from A into C(M, C^{2×2}).

(c) An element a ∈ A is invertible if and only if (smb a)(x, y) is invertible for every (x, y) ∈ M.

\underline{Example 1} We’d like to discuss now Power’s theorem describing C^{∗}–algebras generated by two projections and one flip [14]. Let B be a C^{∗}–algebra with identity e, and let p, q, j be self–adjoint elements of B satisfying

p^{2} = p, q^{2} = q, j^{2} = e, jpj = e − p jqj = e − q.  (13)

Further we suppose, as Power did,

σ(pqp) = [0, 1],  (14)
ipjqp ≥ 0,  (15)

where i denotes the imaginary unit. Define b_{0} := pqp and c_{0} :=ipjqp. It is elementary to check that

b = b_{0} + jb_{o}j and − ic = c_{0} + jc_{0}j.  (16)

Further, if a is an element of the algebra pAp (with identity element p) then jaj belongs to the algebra (e−p)A(e−p) (with identity e−p), and a−λp is invertible in pAp if and only if jaj − λ(e − p) is invertible in (e − p)A(e − p). From this it is easy to conclude that

σ_{A}(a) ⊆ σ_{A}(a + jaj) ∪ {0} for every a ∈ pAp.

Taking into account (16) and this observation we obtain that (14) and (15) are equivalent to

σ(b) = [0, 1], −ic ≥ 0.

Since (−ic)^{2} = b − b^{2}, this involves that −ic is the (non–negative) square root of the non–negative element b − b^{2}, whence follows c ∈alg(e, b). Consequently, σ(b) = [0, 1] coincides with its component Σ_{1}, and if √ refers to the branch of the square root function on [0, 1] with \sqrt{1} = 1, then Theorem 3 specifies to Power’s context as follows:

Theorem 4 Let A be a C^{∗}–algebra generated by self–adjoint elements e, p, j, q satisfying (13–15). Then A is isometrically isomorphic to C ([0, 1], C^{2×2}) and the isomorphism sends e, p, j, q into the functions ê, p̂, ĵ, q̂ given by

ê(x) = ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) ,  p̂(x) = ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} )

ĵ(x) = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) , q̂(x) = ( \begin{matrix}x & i\sqrt{x(1 − x)} \\ −i\sqrt{x(1 − x)} & 1 − x \end{matrix} )

For a proof observe that M can be identified with Σ_{1} (because Σ_{2} = ∅), so only functions depending on one variable x appear. That A is isomorphic to the whole algebra C ([0, 1], C^{2×2}) is a consequence of the Gelfand–Naimark theorem for commutative C^{∗}–algebras which involves that

C = alg(e, b) ≅ C (σ(b)) = C[0, 1].

Notice that the form of the symbol in Theorem 4 differs slightly from that one used by Power ([14], Lemma 7.1), but Power’s choice follows from our’s by multi-

plying ê, p̂, ĵ, q̂ by ( \begin{matrix}1 & 0 \\ 0 & i \end{matrix} ) and ( \begin{matrix}1 & 0 \\ 0 & −i \end{matrix} ) from the left and right hand side,

respectively.

\underline{Example 2} Now we weaken the assumptions (14), (15) in the previous example and require merely that

σ_{C}(b) \ Σ_{0} is connected,  (17)
Σ_{2} = ∅,  (18)

and we also do not longer consider C^{∗}–algebras but the general case again. These are the conditions which we imposed (more or less clearly) to p, q and j in [15]. In this case we conclude from Theorem 2 that there is a \underline{continuous} function y = y(x) on σ_{C}(b) \ Σ_{0} satisfying y^{2} = x^{2} − x and such that

(smb e)(x) = (smb e) (x, y(x)) = ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) ,

(smb p)(x) = (smb p) (x, y(x)) = ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) ,

(smb j)(x) = (smb j) (x, y(x)) = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) ,

and (smb q)(x) = (smb q) (x, y(x)) is either ( \begin{matrix}x & y(x) \\ −y(x) & 1 − x \end{matrix} ) or ( \begin{matrix}x & −y(x) \\ y(x) & 1 − x \end{matrix} )

for every x ∈ σ_{C}(b) \ Σ_{0} (and, thus, for every x ∈ σ_{C}(b), since y(0) = y(1) = 0). The first consequence is that we can consider the symbol function as a function depending on only one complex variable, and a second one is that it is enough to determine the sign of y in the symbol of q for only \underline{one} point x ∈ σ_{C}(b) \ Σ_{0}; then it is the same for all points x ∈ σ_{C}(b).

4 An alternative description of M

For another local description of the maximal ideal space M we introduce the algebra A_{0} which is the smallest (not necessarily closed) subalgebra of B that contains p, q and j, and we define C_{0} and alg_{0}(e, b) analogously. As in Section 1 we conclude that alg_{0}(e, b) is in the center of A_{0}, and that C_{0} coincides with the center of A_{0}.

Proposition 10 The algebra A_{0} is a module of dimension at most 4 over C_{0} and a module of dimension at most 8 over alg_{0}(e, b).

The first assertion of this proposition offers a further way of deriving Theorem 1 by invoking Corollary 1.2 of [9]. Proof: The first assertion is an immediate consequence of Theorem 1 (choose p, pj, jp and e − p as a basis of the module), but it can also be verified directly without effort. For the second assertion, define ”‘basic”’ elements e_{1}, … e_{8} by

\begin{matrix}e_{1} = pq, & e_{5} = e_{1}j = je_{4} \\ e_{2} = (e − p)q, & e_{6} = e_{2}j = je_{3} \\ e_{3} = p(e − q), & e_{7} = e_{3}j = je_{2} \\ e_{4} = (e − p)(e − q), & e_{8} = e_{4}j = je_{1}, \end{matrix}

and let D stand for the set of all elements ∑ r_{i}(b)e_{i} where the r_{i} are polynomials. Evidently, D is a linear space. For a proof that D is an algebra it is sufficient to check the multiplication table

\begin{matrix}· & e_{1} & e_{2} & e_{3} & e_{4} \\ e_{1} & be_{1} & (1 − b)e_{1} & be_{3} & −be_{3} \\ e_{2} & be_{2} & (1 − b)e_{2} & (b − 1)e_{4} & (1 − b)e_{4} \\ e_{3} & (1 − b)e_{1} & (b − 1)e_{1} & (1 − b)e_{3} & be_{3} \\ e_{4} & −be_{2} & be_{2} & (1 − b)e_{4} & be_{4} \end{matrix}

as well as the identities (where m, n ∈ {1, 2, 3, 4}):

e_{m}e_{4+n} = e_{m}e_{n}j,
e_{4+m}e_{n} = e_{m}je_{n} = e_{m}e_{5−n}j
e_{4+m}e_{4+n} = e_{m}je_{n}j = e_{m}e_{5−n}.

Thus, D is an algebra, and the elements p, q, j belong to D because

p = e_{1} + e_{3}, q = e_{1} + e_{2} and j = e_{5} + e_{6} + e_{7} + e_{8}.

Conversely, every algebra which contains p, q and j also includes D, and this proves our claim.

Employing Allan’s local principle, we can localize the algebra A over its central subalgebra alg(e, b), the maximal ideal space of which is homeomorphic to σ_{A}(b). We denote the local coset of a ∈ A at x ∈ σ_{A}(b) by ϕ_{x}(a). Since D is dense in A, the image ϕ_{x}(D) of D under ϕ_{x} must be dense in ϕ_{x}(A) = A/I_{x}. But

ϕ_{x} (∑ r_{i}(b)e_{i}) = ∑ r_{i}(x)ϕ_{x}(e_{i}),

thus, ϕ_{x}(D) is a complex linear space of dimension at most eight. Particularly, this involves the closedness of ϕ_{x}(D) in A/I_{x}. Hence, A/I_{x} = ϕ_{x}(D), and A/I_{x} is a linear space of dimension at most eight. For the dimension of A/I_{x} to be equal to eight it is necessary and sufficient that the local cosets ϕ_{x}(e_{i}), i = 1, …, 8, are linearly independent.

Proposition 11 The cosets ϕ_{x}(e_{1}), …, ϕ_{x}(e_{8}) are linearly independent if and only if the cosets ϕ_{x}(e_{1}) and ϕ_{x}(e_{7}) are linearly independent.

Proof: Suppose there are numbers α_{i} ∈ C with ∑_{i=1}^{8} |α_{i}|^{2} ≠ 0 such that

α_{1}ϕ_{x}(e_{1}) + … + α_{8}ϕ_{x}(e_{8}) = 0.

Multiplying this equality by ϕ_{x}(p) resp. ϕ_{x}(e − p) from the left hand side and by ϕ_{x}(q) resp. ϕ_{x}(e − q) from the right hand side, we obtain

α_{1}ϕ_{x}(e_{1}) + α_{7}ϕ_{x}(e_{7}) = α_{1}ϕ_{x}(pq) + α_{7}ϕ_{x}(pjq) = 0,  (19)
α_{2}ϕ_{x}(e_{2}) + α_{2}ϕ_{x}(e_{2}) = α_{2}ϕ_{x} ((e − p)q) + α_{8}ϕ_{x} ((e − p)jq) = 0,
α_{3}ϕ_{x}(e_{3}) + α_{5}ϕ_{x}(e_{5}) = α_{3}ϕ_{x} (p(e − q)) + α_{5}ϕ_{x} (pj(e − q)) = 0,

α_{4}ϕ_{x}(e_{4}) + α_{6}ϕ_{x}(e_{6}) = α_{4}ϕ_{x} ((e − p)(e − q)) + α_{6}ϕ_{x} ((e − p)j(e − q)) = 0.

Thus, at least one of the pairs

(ϕ_{x}(e_{1}), ϕ_{x}(e_{7})) , (ϕ_{x}(e_{2}), ϕ_{x}(e_{8})) , (ϕ_{x}(e_{3}), ϕ_{x}(e_{5})) , (ϕ_{x}(e_{4}), ϕ_{x}(e_{6}))  (20)

is linearly dependent. Let for definiteness, (ϕ_{x}(e_{1}), ϕ_{x}(e_{7})) be one of these pairs. Multiplying the equality (19) from the left by ϕ_{x}(j) we get

α_{1}ϕ_{x}(jpq) + α_{7}ϕ_{x}(jpjq) = α_{1}ϕ_{x} ((e − p)jq) + α_{7}ϕ_{x} ((e − p)q) = 0

resp.

α_{7}ϕ_{x}(e_{2}) + α_{1}ϕ_{x}(e_{8}) = 0.

Analogously, (19) implies

α_{7}ϕ_{x}(e_{3}) + α_{1}ϕ_{x}(e_{5}) = 0, α_{7}ϕ_{x}(e_{4}) + α_{1}ϕ_{x}(e_{6}) = 0,

hence, all pairs of (20) are linearly dependent. Similarly, if any pair of (20) is linearly dependent, then all pairs of (20) are so. Particularly, if the ϕ_{x}(e_{i}), i = 1, …, 8, are linearly dependent, then so are ϕ_{x}(e_{1}) and ϕ_{x}(e_{7}) as we claimed. The reverse conclusion is obvious.

Proposition 12 The dimension of A/I_{x} is either four or eight.

Proof: We have already remarked that the dimension of A/I_{x} cannot be greater than eight. If it is less than eight, then the preceding proposition and its proof involve that the dimension is not greater than four. Suppose the dimension of A/I_{x} to be less than four. Then there are complex numbers α_{1}, α_{2}, α_{3}, α_{4} with ∑ |α_{i}|^{2} ≠ 0 such that

α_{1}ϕ_{x}(pq) + α_{2}ϕ_{x} ((e − p)q) + α_{3}ϕ_{x} (p(e − q)) + α_{4}ϕ_{x} ((e − p)(e − q)) = 0.

Multiplication by ϕ_{x}(p) resp. ϕ_{x}(e − p) from the left and by ϕ_{x}(q) resp. ϕ_{x}(e − q) from the right hand side yields

α_{1}ϕ_{x}(pq) = α_{2}ϕ_{x} ((e − p)q) = α_{3}ϕ_{x} (p(e − q)) = α_{4}ϕ_{x} ((e − p)(e − q)) = 0.

Hence, the assumption ∑ |α_{i}|^{2} ≠ 0 enforces that at least one of the cosets ϕ_{x}(e_{i}), i = 1, …, 4, is zero. For definiteness, let ϕ_{x}(e_{1}) = ϕ_{x}(pq) = 0 (the other elements can be treated similarly). Then, for all x ∈ σ_{A}(b),

xϕ_{x}(e) = ϕ_{x}(b) = ϕ_{x} (pqp + (e − p)(e − q)(e − p))
= ϕ_{x}(pq)ϕ_{x}(p) + ϕ_{x} ((e − q)(e − p)) + ϕ_{x}(pq)ϕ_{x}(e − p)
= ϕ_{x} ((e − p)(e − q)) .

First, let x ≠ 0. Then the latter identity shows that ϕ_{x}(e − p) and ϕ_{x}(e − q) are invertible from one side, but a one–sided invertible idempotent is necessarily the identity element. Thus, ϕ_{x}(p) = ϕ_{x}(q) = 0. As in the proof of Proposition 11 this involves ϕ_{x}(e) = ϕ_{x}(0), i.e. A/I_{x} = {0} or I_{x} = A. But this is impossible for x ∈ σ_{A}(b) as we have already remarked. So let x = 0. We conclude from the linear dependence of ϕ_{x}(pq) and ϕ_{x}(pjq) that ϕ_{x}(pjq) = 0 and, consequently,

ϕ_{x} ((e − p)q) = ϕ_{x}(j)ϕ_{x}(pjp) = 0, ϕ_{x} (p(e − q)) = ϕ_{x}(pjq)ϕ_{x}(j) = 0.

Thus,

ϕ_{x}(e) = ϕ_{x} (pq + p(e − q) + (e − p)q + (e − p)(e − q)) = ϕ_{x}(0),

which is impossible again. Hence, ϕ_{x}(e_{1}), …, ϕ_{x}(e_{4}) are linearly independent, and the dimension of A/I_{x} is four.

Now we are in position to derive the wanted local characterization of the components Σ_{1} and Σ_{2} of σ_{A}(b) (recall that σ_{A}(b) = σ_{C}(b)).

Theorem 5 (a) The following assertions are equivalent for x ∈ σ_{A}(b) \ {0, 1}: (a.1) dim A/I_{x} = 8. (a.2) ϕ_{x}(pq) and ϕ_{x}(pjq) are linearly independent. (a.3) ϕ_{x}(b) and ϕ_{x}(c) are linearly independent. (a.4) x ∈ Σ_{2}.

(b) The following assertions are equivalent for x ∈ σ_{A}(b): (b.1) dim A/I_{x} = 4. (b.2) ϕ_{x}(pq) and ϕ_{x}(pjq) are linearly dependent. (b.3) ϕ_{x}(b) and ϕ_{x}(c) are linearly dependent. (b.4) x ∈ Σ_{1}.

Proof: The equivalences (a1)⇔(a2) and (b1)⇔(b2) are shown in Proposition 11 and 12. \underline{(a1), (a2)⇒(a4):} If A/I_{x} is an eight–dimensional vector space, then every coset ϕ_{x}(a) (which can be thought of as a multiplication operator on A/I_{x}) corresponds to a certain complex 8 × 8 matrix. A direct computation yields

ϕ_{x}(e) → diag(1, 1, 1, 1, 1, 1, 1, 1),
ϕ_{x}(p) → diag(1, 0, 1, 0, 1, 0, 1, 0),

ϕ_{x}(j) → ( \begin{matrix}0 & J \\ J & 0 \end{matrix} ) , with J = ( \begin{matrix}0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{matrix} ) ,

and

ϕ_{x}(q) → diag(Q_{1}, Q_{2}, Q_{1}, Q_{2})

with

Q_{1} = ( \begin{matrix}x & 1 − x \\ x & 1 − x \end{matrix} ) , Q_{2} = ( \begin{matrix}x & −x \\ x − 1 & 1 − x \end{matrix} ) .

Thus,

ϕ_{x}(c) = ϕ_{x} ((pq − qp)j) → ( \begin{matrix}0 & 0 & 0 & C_{1} \\ 0 & 0 & C_{2} & 0 \\ 0 & C_{1} & 0 & 0 \\ C_{2} & 0 & 0 & 0 \end{matrix} )

with C_{1} = ( \begin{matrix}1 − x & 0 \\ 0 & −x \end{matrix} ) , C_{2} = ( \begin{matrix}−x & 0 \\ 0 & 1 − x \end{matrix} ) . The eigenvalues of the latter

8 × 8 matrix are y_{1/2} = ±\sqrt{x(x − 1)} as one easily checks. Since x ∉ \{0, 1\} we have y_{1} ≠ y_{2}, i.e. this matrix has two different eigenvalues and the spectrum of ϕ_{x}(c) is a doubleton, that is x ∈ Σ_{2}.

\underline{(b1), (b2)⇒(b3)⇒(b4):} If the ϕ_{x}(e_{i}), i = 1, …, 8, are linearly dependent then (as we have checked in Proposition 11 and its proof) there is an α ∈ C \ {0} such that

αϕ_{x}(pq) = ϕ_{x}(pjq)  (21)
αϕ_{x} ((e − p)(e − q)) = ϕ_{x} ((e − p)j(e − q)) .  (22)

Multiplying (21) resp. (22) by ϕ_{x}(p) resp. ϕ_{x}(e − p) from the right hand side and adding the equalities we find

αϕ_{x}(b) = ϕ_{x} (pqq + (e − p)(e − q)(e − p))
= ϕ_{x} (pjqp + (e − p)j(e − q)(e − p))
= ϕ_{x} ((p(e − q)(e − p) + (e − p)qp) j) = ϕ_{x}(−c)

whence (b3) follows. Now (b4) is almost evident: The spectrum of ϕ_{x}(b) is a singleton (namely {x}), hence σ (ϕ_{x}(c)) is a singleton, too, i.e.x ∈ Σ_{1}.

\underline{(a4)⇒(a1):} Assume (a1) to be violated. Then, necessarily by Proposition 12, (b1) is valid, which implies (b4) as we have just checked. But (b4) contradicts (a4). The implications (b4)⇒(b1) and (a3)⇒(a1) follow analogously.

We have seen in Example 1 that it may happen that c belongs to the algebra generated by e and b. This is not only a C^{∗}–effect but can be observed in many cases of practical importance (Hankel operators with special piecewise constant generating functions belong to algebras generated by Toeplitz operators on l^{p} or L^{p}(R^{r}), for example). On the other hand, if 0 or 1 are inner points of σ_{C}(b) in C, then c cannot be an element of alg(e, b). Indeed, in this case Σ_{1} cannot coincide with σ_{C}(b) since there is no continuous branch of the square root function in the neighbourhood of 0, thus Σ_{2} is not empty, but then ϕ_{x}(c) and ϕ_{x}(b) are linearly independent for every x ∈ Σ_{2} by the preceding theorem.

\underline{Conjecture:} If ϕ_{x}(b) and ϕ_{x}(c) are linearly dependent for every x ∈ σ_{A}(b) then c ∈alg(e, b).

5 Invertibility in the larger algebra

The Theorem 3 provides an effective criterion for the invertibility of an element of A in A. On the other hand, examples from operator theory lead in a natural way to the question, whether an element from A is invertible in a larger algebra (for instance in the algebra of all linear bounded operators ating on some Banach space or in the Calkin algebra). In case A is a C^{∗}–algebra or possesses a dense subalgebra, the elements of which have thin spectra (i.e. spectra without inner points), it is inverse closed. Recall that a closed subalgebra A of B is inverse closed in B if the invertibility of an arbitrary element a ∈ A in B implies its invertibility in A. In general it is not easy to decide whether a Banach algebra is inverse closed.

One way to overcome this difficulty for an algebra, which can be investigated with the help of Allan’s local principle (or another local principle) is the imbedding into a larger algebra and the consideration of the inverse closedness of the corresponding local objects (cf. [7], [9]). We will include into this investigation the case of a non–scalar coefficient algebra. This is basicly motivated by the study of operator equations with matrix–valued coefficients (see [7], [9]).

Let B be a Banach algebra with identity element e ≠ 0. Let p, q, j be elements of B satisfying the axioms (1) and (2). Let G be a unital subalgebra of B the elements of which commute with p, q and j, resp. Let us denote the smallest closed subalgebra of B containing G, p, q and j by A_{G}. We are interested in criteria for the invertibility of elements from A_{G} in B. We imbed it into a larger one. Let E stand for the set

{f ∈ B : fb = bf and fc = cf} ,  (23)

where b and c are defined by (7). E forms a closed subalgebra of B. Moreover, E is inverse closed in B and contains A_{G}. Thus, we have reduced our problem to the study of the invertibility of elements from A_{G} in E. Now we can extend the mapping w from (5) to the algebra E,

w : E → E^{2×2}, a ↦ ( \begin{matrix}w_{11}(a) & w_{12}(a) \\ w_{21}(a) & w_{22}(a) \end{matrix} ) .  (24)

The images of the generating elements of A_{G} are given by (6), Prop. 3(a), and

w(g) = ( \begin{matrix}g & 0 \\ 0 & g \end{matrix} ) , for g ∈ G.  (25)

Let C =alg(e, b, c) be defined as before and let C_{G} be thesmallest closed subalgebra of A_{G} which contains C and G. The following proposition is the analogon to Proposition 3.

Proposition 13 (a) The set K := w_{11}(E) is a closed subalgebra of E which contains e.

(b) The mapping w from (24) is a continuous isomorphism from E onto K^{2×2}. The image of w with respect to A_{G} is C_{G}^{2×2}.

(c) C belongs to the center of E.

Thus, an element a ∈ A_{G} is invertible in E if and only if the matrix w(a) is invertible in K^{2×2}. As a consequence of part (c) of the preceding proposition we get that Allan’s local principle applies with E (resp. K^{2×2}.) and C (resp.

\{( \begin{matrix}c & 0 \\ 0 & c \end{matrix} ) : c ∈ C\} ) in place of A and B, respectively. The maximal ideal

space of B is M. We know the structure of this set from section 2 and 3. Let M_{0} stand for the collection of those (x, y) ∈ M for which I_{(x,y)} ≠ A. Given (x, y) ∈ M_{0}, we denote the related local algebra E/I_{(x,y)} (resp. K^{2×2}/I_{(x,y)}^{2×2}) by E_{(x,y)} (resp. K_{(x,y)}^{2×2}) and the canonical homomorphism from E onto E_{(x,y)} by ϕ_{(x,y)}. Set G_{(x,y)} = ϕ_{(x,y)}(G). Then Allan’s local principle states the following.

Proposition 14 An element a ∈ A_{G} is invertible in E if and only if the function

M_{0} → G_{(x,y)}^{2×2}, (x, y) ↦ ( \begin{matrix}ϕ_{(x,y)} (w_{11}(a)) & ϕ_{(x,y)} (w_{12}(a)) \\ ϕ_{(x,y)} (w_{21}(a)) & ϕ_{(x,y)} (w_{22}(a)) \end{matrix} )

is invertible in K_{(x,y)}^{2×2} at every point (x, y) ∈ M_{0}.

The functions, corresponding to the elements g ∈ G, p, j and q, are given by

(x, y) ↦ ( \begin{matrix}ϕ_{(x,y)} (g) & 0 \\ 0 & ϕ_{(x,y)} (g) \end{matrix} ) ,
(x, y) ↦ ( \begin{matrix}ϕ_{(x,y)} (e) & 0 \\ 0 & 0 \end{matrix} ) ,
(x, y) ↦ ( \begin{matrix}0 & ϕ_{(x,y)} (e) \\ ϕ_{(x,y)} (e) & 0 \end{matrix} ) ,
(x, y) ↦ ( \begin{matrix}xϕ_{(x,y)} (e) & yϕ_{(x,y)} (e) \\ −yϕ_{(x,y)} (e) & (1 − x)ϕ_{(x,y)} (e) \end{matrix} ) .

Before employing this result to a simple coefficient algebra we shall give some characterization of the set M_{0}. The local principle states that

σ_{K}(b + c) = ∪_{(x,y)∈M_{0}}σ_{K_{(x,y)}} (ϕ_{(x,y)}(b + c)) = ∪_{(x,y)∈M_{0}}{x + y},

and, hence, the mapping (11) from Proposition 9 sends M_{0} onto σ_{K}(b + c). But this means nothing but that the restriction of (11) onto M_{0},

M_{0} → C, (x, y) ↦ x + y  (26)

is a bijection between M_{0} and σ_{K}(b + c).

Proposition 15 σ_{K}(b + c) = σ_{E}(b + c) = σ_{B}(b + c).

Proof: The second equality follows from the inverse closedness of E in B. The first one is a consequence of Proposition 13(b) if we take into consideration that

w(b) = ( \begin{matrix}b & 0 \\ 0 & b \end{matrix} ) and w(c) = ( \begin{matrix}c & 0 \\ 0 & c \end{matrix} ) .

We can specify the Propositions 14 and 15 by choosing A_{G} itself as the larger algebra (in place of B). Comparing the invertibility criteria in both cases leads to

Corollary 1 The algebra A_{G} is inverse closed in B if and only if the following two conditions are fulfilled:

σ_{B}(b + c) = σ_{C_{G}}(b + c)  (27)
G_{(x,y)}^{2×2} is inverse closed in K_{(x,y)}^{2×2}.  (28)

In what follows we suppose G to be simple, i.e. G possesses the trivial ideals only. Then the condition (28) proves to be fulfilled. Combining the Propositions 14, 15 and Corollary 1 yields

Theorem 6 Let B, G, A_{G}, E, C and p, q, j be defined as above. Let M be the maximal ideal space of C. Further, let M_{0} consist of all pairs (x, y) ∈ C × C where x^{2} − x = y^{2} and x + y ∈ σ_{B}(b + c).

(a) The set M_{0} coincides with the collection of those (x, y) ∈ M for which I_{(x,y)} ≠ E.

(b) The mapping smb which assigns with g ∈ G, p, j and q a matrix–valued function on M_{0} by

(smb g)(x, y) = ( \begin{matrix}g & 0 \\ 0 & g \end{matrix} ) ,  (smb p)(x, y) = ( \begin{matrix}e & 0 \\ 0 & 0 \end{matrix} )
(smb j)(x, y) = ( \begin{matrix}0 & e \\ e & 0 \end{matrix} ) , (smb q)(x, y) = ( \begin{matrix}xe & ye \\ −ye & (1 − x)e \end{matrix} )

extends to a continuous homomorphism from A_{G} into C(M_{0}, G^{2×2}).

(c) An element a ∈ A_{G} is invertible in B if and only if (smb a)(x, y) is invertible for every (x, y) ∈ M_{0}.

(d) If, additional, the condition (27) holds, then M_{0} = M and A_{G} is inverse closed in B.

Let us point out the case that G is isomorphic to C^{k×k} (and we will identify these two algebras). Then the preceding theorem associate with every element a ∈ A_{G} a matrix symbol of order 2k which answers the invertibility of a in B. There is a somewhat other way to derive this result. Since C^{k×k} is a further full matrix algebra in A_{G} we can define a homomorphism ŵ from E into E^{2k×2k} which maps A_{G} onto C^{2k×2k}. Now Allan’s local principle applies and leads to an analogous result.

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Tilo Finck, Bernd Silbermann Technische Universitt Chemnitz-Zwickau Fakultät für Mathematik D-09107 Chemnitz tfinck@mathematik.tu-chemnitz.de silbermn@mathematik.tu-chemnitz.de

Steffen Roch Universitt Leipzig Mathematisches Institut Augustusplatz 10–11 D-04109 Leipzig roch@miaix550.mathematik.uni-leipzig.de