Algebras of approximation sequences: Structure of fractal algebras

Steffen Roch

Abstract

A basic idea to study the stability of an approximation sequence is to translate the stability problem into an invertibility problem in a suitably defined C^{∗}-algebra which offers the applicability of C^{∗}-tools to problems in numerical analysis. This approach associates with every class L of operators and with every discretization procedure D for the operators in L a concrete C^{∗}-algebra 𝔄(L, D). The surprising result of an analysis of these algebras is that, for many apparently quite different choices of L and D, they show a common structure which might be formally summarized in the so-called standard model.

The goal of this paper is to analyse how this formal structure arises from intrinsic properties of the algebras 𝔄. The basic of these properties are the fractality of the algebra and a certain maximality condition for an ideal in 𝔄 which is intimately related with the Fredholm theory in 𝔄. Here fractality of 𝔄 means that every sequence in 𝔄 can be completely reconstructed from each of its subsequences modulo a sequence tending to zero in the norm, whereas the Fredholm property of a sequence (A_{n}) is closely related to the asymptotic behaviour of the smallest singular values of the matrices A_{n}.

1 Introduction

Nowadays, it is a standard approach to study the stability of (a class of) approximation sequences for operators on a Hilbert space by putting all sequences under consideration into a C^{∗}-algebra, A, and by studying its irreducible representations. Thereby, the primary interest is not to determine all irreducible representations of A, but to single out a family of irreducible representations which is sufficient in the sense that an element a ∈ A is invertible if and only if W (a) is invertible for all representations W in that family, and which is as small as possible.

More presicely: Let H be a Hilbert space and, for every n, let H_{n} be an ndimensional subspace of H. We suppose that the orthogonal projections P_{n} from H onto H_{n} converge strongly to the identity operator on H. By an approximation sequence for an operator A ∈ L(H) we mean a sequence of operators A_{n} : H_{n} → H_{n} which converges strongly to A as n → ∞. The sequence (A_{n}) is called stable if the operators A_{n} are invertible for n large enough and if the norms of their inverses are uniformly bounded. The stability of an sequence can be translated into an algebraic language as follows: we let F stand for the set of all bounded sequences (A_{n}) of operators A_{n} : H_{n} → H_{n} and provide this set with elementwisely defined operations and with the supremum norm. This makes F to a C^{∗}- algebra and the set G of all sequences (G_{n}) ∈ F with lim ∥G_{n}P_{n}∥ = 0 to a closed ideal of F. The Banach-Steinhaus theorem implies the uniform boundedness of every approximation sequence (A_{n}), whence (A_{n}) ∈ F, and a simple Neumann series argument shows that a sequence (A_{n}) ∈ F is stable if and only if its coset (A_{n}) + G is invertible in the quotient algebra F/G. Thus, to examine a class of concrete approximation sequences, one considers the smallest closed subalgebra A of F which contains all sequences under consideration, and what one is looking for is a family (W_{t}) of irreducible representations of A such that G ∩ A is in the kernel of W_{t} for every t and such that a sequence (A_{n}) ∈ A is stable (i.e. invertible modulo G ∩ A) if and only if all operators W_{t}(A_{n}) are invertible.

This programm has been carried out for several concrete classes of operators and discretization methods. Here are two examples which have played a pioneering role in the development of that approach.

Finite section method of Toeplitz operators. Given an L^{∞}(𝕋) function a with kth Fourier coefficient a_{k}, let T (a) denote the Toeplitz operator generated by a, i.e. the operator which is defined by its matrix representation (a_{i−j})_{i,j=1}^{∞} with respect to the standard basis of l^{2}. Toeplitz operators with bounded generating function are bounded. Further write P_{n} for the projection operator on l^{2} which maps the sequence (x_{i})_{i=1}^{∞} to (x_{1}, x_{2}, …, x_{n}, 0, 0, …), and consider the sequence (P_{n}T (a)P_{n})_{n≥1} of the finite sections of T (a). The operators P_{n}T (a)P_{n} can be identified with finite Toeplitz matrices acting on Im P_{n} ≅ ℂ^{n}.

Suppose we are interested in the case when a is a piecewise continuous function. The associated algebra F consists of sequences of operators A_{n} : Im P_{n} → Im P_{n}, and we let A stand for the smallest closed subalgebra of F which contains all sequences (P_{n}T (a)P_{n})_{n≥1} with a a piecewise continuous function on 𝕋. Further, write R_{n} for the reflection operators (x_{i}) ↦ (x_{n}, x_{n−1}, …, x_{1}, 0, 0, …) on l^{2}. It has been shown in [2] that A contains all sequences of the form (P_{n}KP_{n} + R_{n}LR_{n} + G_{n}) with K, L compact operators on l^{2} and with (G_{n}) ∈ G, that the set J of all these sequences forms a closed ideal of A and that, for every sequence (A_{n}) ∈ A, the strong limits

W (A_{n}) := s-lim A_{n} and W̃ := s-lim R_{n}A_{n}R_{n}.  (1)

exist. The main result from [2] reads as follows.

Theorem 1 (a) A coset (A_{n})+J ∈ A/J is invertible if and only if the operators W (A_{n}) and W̃ (A_{n}) are Fredholm.

(b) A sequence (A_{n}) ∈ A is stable if and only if the operators W (A_{n}) and W̃ (A_{n}) are invertible.

Observe that, indeed, the mappings W and W̃ are irreducible representations of the algebra A. This is a consequence of the identities

W (P_{n}KP_{n}) = K and W̃ (R_{n}LR_{n}) = L

which hold for arbitrary compact operators K and L and which show that the ideal K(l^{2}) of all compact operators on l^{2} lies in the range of both W and W̃.

Finite section method of operators in the Toeplitz algebra. In the next example we consider the same approximation method for a larger class of operators. Let T stand for the smallest closed subalgebra of L(l^{2}) which contains all Toeplitz operators T (a) with piecewise continuous generating function a. Again we discretize operators A in T by the finite section method. Accordingly, let B refer to the smallest closed subalgebra of F which contains all sequences (P_{n}AP_{n}) with A ∈ T . It is not hard to prove that the strong limits W (A_{n}) and W̃ (A_{n}) (defined as in (1)) exist for every sequence (A_{n}) ∈ B. Thus, the invertibility of the operators W (A_{n}) and W̃ (A_{n}) is also a necessary condition for the stability of the sequence (A_{n}). But it turns out that these conditions are no longer sufficient (see [11] for a quite simple argument). A detailed analysis (performed by Werbitzky, Rathsfeld, Böttcher, Silbermann and others) yields the following stability result for sequences in B where, besides the invertibility of W (A_{n}) and W̃ (A_{n}), certain local stability conditions occur. Let S_{n} refer to the subspace of the Hilbert space L^{2}([0, 1]) which is spanned by the functions

φ_{k,n}(x) := \{ \begin{matrix}\sqrt{n} & if & x ∈ (k/n, (k + 1)/n) \\ 0 & if & x ∈ (0, k/n) ∪ ((k + 1)/n, 1) \end{matrix}

with k = 0, …, n − 1, write E_{n} for the operators

E_{n} : Im P_{n} → S_{n}, (x_{1}, …, x_{n}, 0, …) ↦ ∑_{k=1}^{n} x_{k}φ_{k,n},

and define E_{−n} : S_{n} → Im P_{n} by E_{−n} := (E_{n})^{−1}. Finally, for t ∈ 𝕋, let Y_{t} stand for the operator

Y_{t} : l^{2} → l^{2}, (x_{k})_{k=1}^{∞} ↦ (t^{−k}x_{k})_{k=1}^{∞}.

One can show that, for every sequence (A_{n}) ∈ B and for every t ∈ 𝕋, the strong limit

W_{t}(A_{n}) := s-lim_{n→∞}E_{n}Y_{t^{−1}}A_{n}Y_{t}E_{−n}

exists and that it defines a bounded linear operator W_{t}(A_{n}) on L^{2}([0, 1]). Moreover, for every t ∈ 𝕋 and every compact operator K ∈ K(L^{2}([0, 1])), the sequence (P_{n}Y_{t}E_{−n}L_{n}KE_{n}Y_{t^{−1}}P_{n}) belongs to B, and the sets

J_{t} := {(P_{n}Y_{t}E_{−n}L_{n}KE_{n}Y_{t^{−1}}P_{n} + G_{n}) : K ∈ K(L^{2}([0, 1])), (G_{n}) ∈ G}

form closed ideals of B.

Theorem 2 A sequence (A_{n}) ∈ B is stable if and only if the operators W (A_{n}), W̃ (A_{n}) and W_{t}(A_{n}) are invertible for every t ∈ 𝕋.

For a proof see, e.g., [5], Theorem 4.1. Again, the mappings W, W̃ and W_{t} are irreducible representations which follows for W and W̃ as above and for W_{t} from

W_{s}(P_{n}Y_{t}E_{−n}L_{n}KE_{n}Y_{t^{−1}}P_{n}) = \{ \begin{matrix}K & if & s = t \\ 0 & if & s ≠ t \end{matrix}

whence K(L^{2}([0, 1])) ⊂ Im W_{t}.

The standard model. A similar analysis as in these examples has been performed for numerous approximation methods including, e.g., Galerkin, collocation and qualocation methods with spline ansatz functions and quadrature methods for singular integral operators and Mellin operators. For a more detailed overview see the monographs [5, 7]. The surprising and to some extend miraculous result of this analysis is that (almost) all of the investigated algebras (which arise from very different approximation procedures for very different classes of operators) exhibit one and the same structure. The attempt to formalize this structure leads to a standard model for algebras arising from approximation sequences. A C^{∗}-subalgebra A of F is called a standard algebra if it satisfies the following axioms:

1. There is a (possibly infinite) set T, and for every t ∈ T, there is a Hilbert space H^{t} and a sequence (E_{n}^{t}) of partial isometries E_{n}^{t} : H^{t} → H such that - the initial projections (E_{n}^{t})^{∗}E_{n}^{t} converge strongly to the identity on H^{t},

- the range projections E_{n}^{t}(E_{n}^{t})^{∗} coincide with P_{n}, and

- the separation condition holds:

(E_{n}^{s})^{∗}E_{n}^{t} → 0 weakly for all s ≠ t.  (2)

2. For every t ∈ T and every (A_{n}) ∈ A, there exists the strong limit

s-lim (E_{n}^{t})^{∗}A_{n}E_{n}^{t} =: W_{t}(A_{n}).  (3)

3. A is unital and contains all sequences (E_{n}^{t}K(E_{n}^{t})^{∗}) with K compact on H^{t} as well as all sequences (G_{n}) ∈ G. The closed linear span of all of these sequences forms a closed ideal J of A.

4. A sequence (A_{n}) ∈ A is stable if and only if all associated operators W_{t}(A_{n}) are invertible on H^{t}.

The algebras which result from applying the finite section method to Toeplitz operators or to operators in the Toeplitz algebra are obvious examples for standard algebras.

It is evident that these axioms provide only a formal description of the structure of the occurring algebras which says nothing about the reasons which force this structure. It is the goal of the present paper to show how this structure arises from more intrinsic properties of the algebra. The main of these properties are the fractality of the algebra and the fact that a certain ideal which is intimately related with the Fredholm theory in the algebra is large in some sense. The notions of fractality and Fredholmness for approximation sequances were introduced in [10, 11, 12] and will be briefly recalled in Section 2. It will turn out that these properties are naturally satisfied for many algebras generated by commonly used approximation processes, although not for all. For a first discussion of a class of non-fractal approximation methods see [8].

2 Fractality and Fredholmness of approximation sequences

Fractal algebras. This class has been introduced and studied in [10, 12]. Here are the definition as well as a few properties of fractal algebras which we will need in the sequel.

Let F be as above. Given a strongly monotonically increasing sequence η : ℕ → ℕ, let F_{η} refer to the C^{∗}-algebra of all bounded sequences (A_{n}) with A_{n} ∈ ℂ^{η(n)×η(n)}, and write G_{η} for the ideal of all sequences (A_{n}) ∈ F_{η} which tend to zero in the norm. Further, let R_{η} stand for the restriction mapping R_{η} : F → F_{η}, (A_{n}) ↦ (A_{η(n)}). This mapping is a ^{∗}-homomorphism from F onto F_{η} which moreover maps G onto G_{η}. Given a C^{∗}-subalgebra A of F, let A_{η} denote the image of A under R_{η} which is a C^{∗}-algebra again.

Definition 1 Let A be a C^{∗}-subalgebra of the algebra F.

(a) A ^{∗}-homomorphism W : A → B of A into a C^{∗}-algebra B is fractal if, for every strongly monotonically increasing sequence η, there is a ^{∗}-homomorphism W_{η} : A_{η} → B such that W = W_{η}R_{η}.

(b) The algebra A is fractal if the canonical homomorphism π : A → A/(A ∩ G) is fractal.

(c) A sequence (A_{n}) ∈ F is fractal if the smallest C^{∗}-subalgebra of F which contains (A_{n}) is fractal.

Thus, given a subsequence (a_{η(n)}) of a sequence (a_{n}) which belongs to a fractal algebra A, it is possible to reconstruct the original sequence (a_{n}) from this subsequence modulo sequences tending to zero in the norm. This assumption is very natural for sequences arising from discretization procedures. For example, employing the following theorem, one easily gets the fractality of the algebra of the finite section method for operators in the Toeplitz algebra. On the other hand, the algebra F of all bounded sequences fails to be fractal.

Theorem 3 Let A be a unital C^{∗}-subalgebra of F which contains the ideal G. The algebra A is fractal if and only if there exists a family {W_{t}}_{t∈T} of unital and fractal ^{∗}-homomorphisms from A into unital C^{∗}-algebras B_{t} such that the following equivalence holds for every sequence (A_{n}) ∈ A: The coset (A_{n}) + G is invertible in A/G if and only if W_{t}(a_{n}) is invertible in B_{t} for every t ∈ T.

The representations W, W̃ and W_{t} occurring in Theorems 1 and 2 are given by strong limits which immediately implies their fractality. Hence, by Theorem 3, the algebras of the finite section methods for Toeplitz operators and operators in the Toeplitz algebra are fractal. Another useful criterion for fractality is the following.

Theorem 4 A C^{∗}-subalgebra A of F is fractal if and only if the following implication holds for every sequence (A_{n}) ∈ A and every strongly monotonically increasing sequence η:

R_{η}(A_{n}) ∈ G_{η} ⇒ (A_{n}) ∈ A ∩ G.

Corollary 1 (a) Every C^{∗}-subalgebra of a fractal algebra A ⊆ F is fractal.

(b) A C^{∗}-subalgebra A of F is fractal if and only if each of its elements is fractal.

The main feature of fractality is that it makes certain limiting processes uniform. For example, the following result for the norms of the approximation operators ([12, 10]) holds.

Proposition 1 (a) For every sequence (A_{n}) ∈ F,

lim sup_{n→∞} ∥A_{n}∥ = ∥(A_{n}) + G∥.

(b) If (A_{n}) ∈ F is a fractal sequence, then lim_{n→∞} ∥A_{n}∥ exists.

Similar effects can be observed for the limiting sets of the eigenvalues, pseudoeigenvalues, singular values or Raileigh quotients of the A_{n}.

Fredholm sequences. Let J (F) stand for the smallest closed subset of F which contains all sequences (K_{n}) for which sup dim Im K_{n} is finite. The set J (F) is a closed two-sided ideal of F which contains the ideal G of all sequences tending to zero in the norm. A sequence (A_{n}) ∈ F is called a Fredholm sequence if it is invertible modulo the ideal J (F). If (A_{n}) is a Fredholm sequence then there is a number k such that lim inf_{n→∞} σ_{k+1}^{(n)} > 0 (Theorem 2 in [11]). The smallest number k with this property is called the α-number of the sequence (A_{n}) and will be denoted by α(A_{n}). This number plays the same role in the Fredholm theory of approximation sequences as the number dim Ker A plays in the common Fredholm theory for operators A on a Hilbert space.

Sequences of rank one. Let A be a unital and fractal C^{∗}-subalgebra of F which contains the ideal G. A sequence (K_{n}) in A is said to be of rank one if every matrix K_{n} has range dimension less than or equal to one. Clearly, if (K_{n}) ∈ A is a rank one sequence then, for every sequence (A_{n}) ∈ A, there is a bounded sequence (α_{n}) of complex numbers such that

K_{n}A_{n}K_{n} = α_{n}K_{n} for all n.  (4)

For the boundedness of (α_{n}) observe that we can choose α_{n} = 0 if K_{n} = 0 and that |α_{n}| ≤ ∥K_{n}∥ ∥A_{n}∥ in case K_{n} ≠ 0. The smallest closed ideal of A which contains all sequences of rank one will be denoted by K(A).

Further, we call a sequence (K_{n}) ∈ A of central rank one if, for every sequence (A_{n}) ∈ A, there is a convergent sequence (α_{n}) such that (4) holds. The smallest closed two-sided ideal of A which contains all sequences of central rank one will be denoted by J_{cen}(A). Finally, a central rank one sequence (K_{n}) of A is said to be of essential rank one if it does not belong to the ideal G and if K_{n} ≠ 0 for all n. We denote the smallest closed ideal of A which contains all sequences of essential rank one by J_{ess}(A). In case A does not possess essential rank one sequences, we simply define J_{ess}(A) := G.

It is easy to see that these definitions imply the inclusions

G ⊆ J_{ess}(A) ⊆ J_{cen}(A) ⊆ K(A).  (5)

Proposition 2 If A is a unital and fractal C^{∗}-subalgebra of F which contains the ideal G, then

J_{ess}(A) = J_{cen}(A) = K(A).

Proof. The inclusions ⊆ hold by definition. Let (K_{n}) be a rank one sequence in A. If (K_{n}) ∈ G, then (K_{n}) ∈ J_{ess}(A) due to (5). Thus, let (K_{n}) ∉ G. Then, by Proposition 1, the limit lim ∥K_{n}∥ =: α exists and is not zero. We will first show that then (K_{n}) is a central rank one sequence.

Assume {K_{n}} is not of central rank one. Then there are a sequence {A_{n}} ∈ A and a non-convergent bounded sequence (α_{n}) such that

K_{n}A_{n}K_{n} = α_{n}K_{n} for all n.

Choose two partial limits β ≠ γ of the sequence (α_{n}) as well as two subsequences μ and η of the the positive integers such that

α_{μ(n)} → β and α_{η(n)} → γ as n → ∞.

Then both sequences {α_{μ(n)}K_{μ(n)} − βK_{μ(n)}} and {α_{η(n)}K_{η(n)} − γK_{η(n)}} tend to zero. Hence, by Theorem 4, both sequences {α_{n}K_{n} − βK_{n}} and {α_{n}K_{n} − γK_{n}} are zero sequences. But then, also their difference (β − γ){K_{n}} goes to zero. Since ∥K_{n}∥ → α ≠ 0, this implies β = γ in contradiction to the choice of β and γ. Hence, (K_{n}) is a central rank one sequence.

Further, α ≠ 0 implies that K_{n} ≠ 0 for all sufficiently large n. Thus, there is a sequence (G_{n}) ∈ G (with a finite number of non-vanishing entries) such that (K_{n} + G_{n}) is an essential rank one sequence. Since (G_{n}) ∈ J_{ess}(A), this shows that (K_{n}) ∈ J_{ess}(A).

The algebra A is called Fredholm inverse closed in F if J_{ess}(A) = A ∩ J (F).

Fredholm theory in fractal and Fredholm inverse closed algebras. For every essential rank one sequence (K_{n}), let Id (K_{n}) refer to the smallest closed ideal of A which contains the sequence (K_{n}). It is easy to see that G ⊂ Id (K_{n}). In [11] it is shown that, if (K_{n}) and (L_{n}) are sequences of essential rank one in A, then either Id (K_{n}) = Id (L_{n}) or Id (K_{n}) ∩ Id (L_{n}) = G. We call (K_{n}) and (L_{n}) equivalent in the first case and denote by S the set of all equivalence classes with respect to this relation. With every s ∈ S, there is associated a unique irreducible representation W_{s} of A into the algebra L(H_{s}) for some Hilbert space H_{s} such that the ideal Id (K_{n}) is mapped onto the ideal K(H_{s}) of the compact operators on H_{s} and that the kernel of the mapping W_{s} : Id (K_{n}) → K(H_{s}) is G. The main result of [11] reads as follows:

Theorem 5 Let A be a unital, fractal and Fredholm inverse closed C^{∗}-subalgebra of F which contains the ideal G.

(a) If (A_{n}) ∈ A is a Fredholm sequence, then the operators W_{s}(A_{n}) are Fredholm operators for every s ∈ S, and α(A_{n}) = ∑_{s∈S} dim Ker W_{s}(A_{n}).

(b) If (A_{n}) ∈ A is Fredholm and k = α(A_{n}) > 0, then lim_{n→∞} σ_{k}^{(n)} = 0.

(c) Assume that family (W_{s})_{s∈S} is sufficient for the stability of sequences in A. If, for a sequence (A_{n}) ∈ A, all operators W_{s}(A_{n}) are Fredholm and if only a finite number of them is not invertible, then (A_{n}) is a Fredholm sequence.

3 Structure of certain fractal homomorphisms

The assumption made in the introduction that the orthogonal projections P_{n} : H → H_{n} converge strongly to the identity operator ensures that H is a separable Hilbert space. Further, to avoid trivialities, we will also assume that H has infinite dimension. Thus, without essential loss of generality, we can identify H with l^{2} = l^{2}(ℤ^{+}) in what follows and H_{n} with the subspace of H consisting of all sequences of the form (x_{1}, x_{2}, …, x_{n}, 0, 0, …). In particular, also the algebra F is defined with respect to this special choice. The orthogonal projection from H onto H_{n} will be denoted by P_{n}, and the sequence (0, …, 0, 1, 0, …) with the 1 standing at the ith place will be abbreviated to e_{i}. Thus, the vectors (e_{i})_{i≥1} form an orthonormal basis of l^{2}.

Let us emphasize once more that the Fredholm theory proposed in [10] and briefly sketched in the previous section is based on a decomposition of the ideal of the centrally compact sequences into ideals Id (K_{n}) which are generated by an essential rank one sequence and that the quotients Id (K_{n})/G are ^{∗}-isomorphic to the ideal K(H) of all compact operators on some Hilbert space H. It is the goal of this section to give a precise description of all sequences in Id (K_{n}) in case H is a separable Hilbert space. The main result of this section is the following.

Theorem 6 Let A be a unital and fractal C^{∗}-subalgebra of F which contains the ideal G, let (K_{n}) be an essential rank one sequence in A, and suppose that there is a ^{∗}-homomorphism W from Id (K_{n}) onto the ideal K(H) of all compact operators on a separable Hilbert space H with Ker W = G. Then there exist partial isometries E_{n} : H → l^{2} with E_{n}E_{n}^{∗} = P_{n} such that

Id (K_{n}) = {(E_{n}KE_{n}^{∗} + G_{n}) : K ∈ K(H), (G_{n}) ∈ G}.  (6)

Before going into the details of the proof, let us consider an example which shows that the assertion of the theorem becomes false if the ideal Id (K_{n}) of A is replaced by an ideal which is not generated by a rank one sequence.

Example. Let the operators P_{n} and R_{n} and the further notations be as in the example of the finite section method of Toeplitz matrices, and consider the sets of sequences

A := {(αP_{n} + P_{n}KP_{n} + R_{n}KR_{n} + G_{n}) : K ∈ K(l^{2}), (G_{n}) ∈ G},
J := {(P_{n}KP_{n} + R_{n}KR_{n} + G_{n}) : K ∈ K(l^{2}), (G_{n}) ∈ G}.

It is not hard to check that A is a ^{∗}-subalgebra of F and J a closed ideal of A. For example, if K and L are compact, and if we set Q_{n} := I − P_{n} for brevity, then

(P_{n}KP_{n} + R_{n}KR_{n})(P_{n}LP_{n} + R_{n}LR_{n}) =
= P_{n}KLP_{n} + R_{n}KLR_{n} − P_{n}KQ_{n}LP_{n} − R_{n}KQ_{n}LR_{n} + P_{n}KR_{n}LR_{n}
+ R_{n}KR_{n}LP_{n}
= P_{n}KLP_{n} + R_{n}KLR_{n} + G_{n}

with (G_{n}) ∈ G. Since A is a subalgebra of the algebra related with the finite section method for Toeplitz operators, and since

W (αP_{n} + P_{n}KP_{n} + R_{n}KR_{n} + G_{n}) = αI + K = W̃ (αP_{n} + P_{n}KP_{n} + R_{n}KR_{n} + G_{n})

we conclude from Theorem 1 that a sequence (A_{n}) ∈ A is stable if and only if the operator W (A_{n}) is invertible. Via Theorem 3, this implies the fractality of the algebra A, and it is also clear that the homomorphism W maps the ideal J onto K(l^{2}) and that the kernel of that homomorphism is G.

But J does not contain sequences of essential rank one. Indeed, every such sequence would be of the form (P_{n}KP_{n} + R_{n}KR_{n} + G_{n}) with K ≠ 0. Choose x ∈ l^{2} such that ∥Kx∥ = 1 and set

y_{n} := (P_{n}KP_{n} + R_{n}KR_{n} + G_{n})P_{n}x and z_{n} := (P_{n}KP_{n} + R_{n}KR_{n} + G_{n})R_{n}x.

Then y_{n} → Kx and R_{n}z_{n} → Kx, whence

∥y_{n}∥ → ∥Kx∥ = 1 and ∥z_{n}∥ = ∥R_{n}z_{n}∥ → 1.  (7)

Moreover, z_{n} ⇁ 0 weakly, which implies

⟨y_{n}, z_{n}⟩ = ⟨y_{n} − Kx, z_{n}⟩ + ⟨Kx, z_{n}⟩ → 0.  (8)

Now suppose that there are numbers α_{n} such that α_{n}y_{n} = z_{n}. Then |α_{n}| ∥y_{n}∥ = ∥z_{n}∥ whence, due to (7), |α_{n}| → 1. On the other hand, we get from (8)

\overline{α_{n}} ∥y_{n}∥^{2} = \overline{α_{n}} ⟨y_{n}, y_{n}⟩ = ⟨y_{n}, z_{n}⟩ → 0

which yields α_{n} → 0. Contradiction.

We prepare the proof of Theorem 6 by the following propositions.

Proposition 3 Let the algebra A be as in Theorem 6. Then, for every essential rank one sequence (K_{n}) ∈ A, there exists an essential rank one sequence (Π_{n}) ∈ A of projections such that Id (K_{n}) = Id (Π_{n}).

Proof. By definition, there is a sequence (μ_{n}) ∈ c such that K_{n}K_{n}^{∗}K_{n} = μ_{n}K_{n}, whence (K_{n}^{∗}K_{n})^{2} = μ_{n}K_{n}^{∗}K_{n}. Since ∥K_{n}^{∗}K_{n}∥ = ∥K_{n}∥^{2} ≠ 0 belongs to the spectrum of the non-negative matrix K_{n}^{∗}K_{n}, the functional calculus for self-adjoint matrices yields ∥K_{n}^{∗}K_{n}∥^{2} = μ_{n}∥K_{n}^{∗}K_{n}∥ and, thus, μ_{n} = ∥K_{n}^{∗}K_{n}∥. Since (K_{n}) is not in G, the limit of the sequence (μ_{n}) cannot be zero. Hence, the sequence (μ_{n}^{−1}P_{n}) belongs to A, too. The matrices

Π_{n} := μ_{n}^{−1}P_{n}K_{n}^{∗}K_{n} = ∥K_{n}^{∗}K_{n}∥^{−1}K_{n}^{∗}K_{n}

are projections of rank one as one easily checks. Further, (Π_{n}) is a central rank one sequence in A. Indeed, given a sequence (A_{n}) ∈ A, there is a convergent sequence (α_{n}) such that K_{n}(A_{n}K_{n}^{∗})K_{n} = α_{n}K_{n}, whence Π_{n}A_{n}Π_{n} = α_{n}Π_{n}. Thus, (Π_{n}) is an essential rank one sequence in A. Finally, the equalities

(Π_{n}) = (μ_{n}^{−1}P_{n})(K_{n})^{∗}(K_{n}) and (K_{n}) = (μ_{n}^{−1}K_{n}K_{n}^{∗}K_{n}) = (μ_{n}^{−2}P_{n})(K_{n})(Π_{n})

show that Id (Π_{n}) ⊆ Id (K_{n}) and Id (K_{n}) ⊆ Id (Π_{n}), respectively.

Proposition 4 Let the coset p ∈ F/G be a projection. Then there exists a sequence of projections (P_{n}) ∈ F such that (P_{n}) + G = p.

The proof rests on an application of the Gelfand-Naimark theorem for commutative C^{∗}-algebras and can be found, e.g., in [13]. In the present context, this result can be essentially completed. For, recall that an element k of a C^{∗}-algebra B is of rank one if, for every b ∈ B, there is a μ ∈ ℂ such that kbk = μk. Further we agree upon calling two projections p_{1}, p_{2} ∈ B orthogonal if p_{1}p_{2} = p_{2}p_{1} = 0.

Proposition 5 Let A and (K_{n}) be as in Theorem 6, let (P_{n}^{(1)}) be a sequence of projections in Id (K_{n}), and set p^{(1)} := (P_{n}^{(1)}) + G. Further suppose that p^{(2)} ∈ Id (K_{n})/G is a projection of rank one such that the cosets p^{(1)} and p^{(2)} are orthogonal. Then there is a sequence (P_{n}^{(2)}) in Id (K_{n}) such that (P_{n}^{(2)}) + G = p^{(2)} and that P_{n}^{(2)} is a projection which is orthogonal to P_{n}^{(1)} for all sufficiently large n.

For a proof see Propositions 10 and 11 in [11].

Proof of Theorem 6. We will prove the theorem in case H is an infinitedimensional Hilbert space in which case we can assume that H = l^{2}. The proof is subdivided into several steps. Step 1. Let (K_{n}) be an essential rank one sequence in A, and let W be a ^{∗}- homomorphism W from Id (K_{n}) onto K(l^{2}) with Ker W = G. Due to Proposition 3, we can choose an essential rank one sequence (Π_{n}) ∈ A of projections such that Id (K_{n}) = Id (Π_{n}). Observe further that W is an irreducible representation of Id (Π_{n}) since the range of W contains the ideal of all compact operators. The ideal Id (Π_{n}) will be abbreviated to J^{(0)} in what follows.

Our first claim is that W (Π_{n}) is a non-zero rank one projection. Clearly, W (Π_{n}) is a projection which cannot be zero since, otherwise, W (J^{(0)}) = {0} in contrast to the assumptions. We will check that the range dimension of W (Π_{n}) is at most one. Given a compact operator T on l^{2}, let (T_{n}) be a sequence in J^{(0)} such that W (T_{n}) = T, and let (μ_{n}) be a sequence in c with limit μ such that Π_{n}T_{n}Π_{n} = μ_{n}Π_{n}. Since G in in the kernel of W, we find

W (Π_{n})T W (Π_{n}) = W (Π_{n}T_{n}Π_{n}) = W (μΠ_{n}) + W ((μ_{n} − μ)Π_{n}) = μW (Π_{n}).

Hence, the operator W (Π_{n}) has rank one.

Next we will show that, by a unitary transformation, we can always reach that W (Π_{n}) = P_{1}. Such transformations will be used several times in what follows, so we will go a little bit more into the details here.

Let f_{1} ∈ Im W (Π_{n}) be a vector of length one. If Im W (Π_{n}) = span e_{1}, then we choose f_{1} := e_{1}. Otherwise, we have f_{1} ∉ span e_{1}, and the vectors f_{1} and e_{1} are linearly independent. Decompose l^{2} into the orthogonal sum

l^{2} = span {e_{1}, f_{1}} ⊕ H_{1}

and choose vectors ê_{1} and f̂_{1} in span {e_{1}, f_{1}} such that both {e_{1}, ê_{1}} and {f_{1}, f̂_{1}} become orthogonal bases of span {e_{1}, f_{1}}. Thus, every element h ∈ l^{2} can be written uniquely as

h = αf_{1} + βf̂_{1} + h_{1} with α, β ∈ ℂ, h_{1} ∈ H_{1}.

Define a mapping J_{1} : l^{2} → l^{2} by

J_{1}h = J_{1}(αf_{1} + βf̂_{1} + h_{1}) := αe_{1} + βê_{1} + h_{1}.

Clearly, W is a unitary operator on l^{2} which maps f_{1} onto e_{1}, and

W_{1} : J^{(0)} → K(l^{2}), (A_{n}) ↦ J_{1}W (A_{n})J_{1}^{−1}

is an irreducible representation of J^{(0)} with kernel G and range K(l^{2}) which is unitarily equivalent to W. Thus, we can suppose without loss that already W (Π_{n}) = P_{1}.

An analogous transformation can be performed for every projection Π_{n} (recall that the Π_{n} are non-zero rank one projections by their definition). Thus, there exist unitary n × n matrices E_{n}^{(1)} : L(Im P_{n}) → L(Im P_{n}) such that

(E_{n}^{(1)})^{∗}Π_{n}E_{n}^{(1)} = P_{1} for all n.

So, the outcome of the first step is that the ideal J^{(0)} of A is ^{∗}-isomorphic to

J^{(1)} := {((E_{n}^{(1)})^{∗}A_{n}E_{n}^{(1)}) : (A_{n}) ∈ J^{(0)}},

that the isomorphism can be chosen such that it maps the sequence (Π_{n}) onto the constant sequence (P_{1}), and that the mapping

W^{(1)} : J^{(1)} → K(l^{2}), (A_{n}) → W (E_{n}^{(1)}A_{n}(E_{n}^{(1)})^{∗})

is an irreducible representation of J^{(1)} which maps the sequence (P_{1}) onto the operator P_{1} and which has kernel G. The quotient homomorphism

J^{(1)}/G → K(l^{2}), (A_{n}) + G → W^{(1)}(A_{n})

will be denoted by W_{π}^{(1)}.

Step 2. Let us fix some notations. The orthogonal projection from l^{2} onto ℂe_{k} will be denoted by S_{k} (i.e. S_{k} = P_{k} − P_{k−1} for k ≥ 2), and we let F_{ij} ∈ L(l^{2}) stand for the operator which satisfies

S_{i}F_{ij} = F_{ij}S_{j} = F_{ij}

(i.e. the representation of F_{ij} with respect to the basis (e_{k})_{k≥1} of l^{2} has a 1 at the ijth entry whereas all other entries are zero).

What we will show in this step is that the ideal J^{(0)} is ^{∗}-isomorphic to an ideal J^{(2)} which contains the sequences (P_{n}KP_{n}) for K = S_{1}, S_{2}, F_{12}, F_{21} and which, thus, contains a copy of ℂ^{2×2}.

Since S_{2} is a rank one projection in K(l^{2}), there is a uniquely determined rank one coset S_{2}^{π} in J^{(1)}/G such that W_{π}^{(1)}(S_{2}^{π}) = S_{2}. From Proposition 5 (which we here apply in the special case P_{n}^{(1)} = P_{1}) we infer that there is a sequence (S_{n}^{(2)}) ∈ J^{(1)} which owns the following properties:

• (S_{n}^{(2)}) + G = S_{2}^{π}, whence W^{(1)}(S_{n}^{(2)}) = S_{2},

• S_{1}^{(2)} = 0, and S_{n}^{(2)} is a projection with one-dimensional range for every n ≥ 2,

• P_{1} and S_{n}^{(2)} are orthogonal for all n.

The orthogonality of P_{1} and S_{n}^{(2)} ensures that the ranges of these projections are mutually orthogonal subspaces of l^{2}. Thus, as in step 1, we can find unitary matrices Ê_{n}^{(2)} : Im P_{n} → Im P_{n} such that

(Ê_{n}^{(2)})^{∗}P_{1}Ê_{n}^{(2)} = P_{1} and (Ê_{n}^{(2)})^{∗}S_{n}^{(2)}Ê_{n}^{(2)} = S_{2}.  (9)

The Ê_{n}^{(2)} are chosen such that they act on the orthogonal complement of the linear span of Im S_{n}^{(2)} and e_{2} as the identity.

On defining

Ẽ_{n}^{(2)} := \{ \begin{matrix}E_{n}^{(1)} & if n = 1 \\ E_{n}^{(1)}Ê_{n}^{(2)} & if n ≥ 2 \end{matrix}

we see that the ideal J^{(0)} is ^{∗}-isomorphic to

J̃^{(2)} := {((Ẽ_{n}^{(2)})^{∗}A_{n}Ẽ_{n}^{(2)}) : (A_{n}) ∈ J^{(0)}}

with the isomorphism mapping the sequence (Π_{n}) onto the constant sequence (P_{1}), that (P_{n}S_{2}P_{n}) ∈ J̃^{(2)}, and that

W̃^{(2)} : J̃^{(2)} → K(l^{2}), (A_{n}) ↦ W (Ẽ_{n}^{(2)}A_{n}(Ẽ_{n}^{(2)})^{∗})

is an irreducible representation of J̃^{(2)} which maps (S_{1}) onto S_{1} and (P_{n}S_{2}P_{n}) onto S_{2}.

Further, since W̃^{(2)} maps J̃^{(2)} onto the compact operators, there is a sequence (F̃_{n}^{12}) in J̃^{(2)} such that W̃^{(2)}(F̃_{n}^{12}) = F_{12}. Then, clearly, W̃^{(2)}(P_{n}S_{1}F̃_{n}^{12}S_{2}P_{n}) = F_{12}, there are complex numbers α_{n} such that

P_{n}S_{1}F̃_{n}^{12}S_{2}P_{n} = α_{n}P_{n}F_{12}P_{n} for all n,

and the sequence (α_{n}P_{n}F_{12}P_{n}) belongs to J̃^{(2)}. The fractality of A (and, hence, of J̃^{(2)}) implies the existence of the limit

lim_{n→∞} ∥α_{n}P_{n}F_{12}P_{n}∥ = lim_{n→∞} |α_{n}|,

and from

lim_{n→∞} ∥α_{n}P_{n}F_{12}P_{n}∥ = ∥(α_{n}P_{n}F_{12}P_{n}) + G∥ = ∥W̃^{(2)}(α_{n}P_{n}F_{12}P_{n})∥ = ∥F_{12}∥ = 1

we conclude that lim |α_{n}| = 1. Write α_{n} as

α_{n} = r_{n}e^{iφ_{n}} with r_{n} ≥ 0 and φ_{n} ∈ [0, 2π).

Because of

α_{n} − e^{iφ_{n}} = (r_{n} − 1)e^{iφ_{n}} → 0,

the sequence ((α_{n} − e^{iφ_{n}})P_{n}F_{12}P_{n}) belongs to G, whence (e^{iφ_{n}}P_{n}F_{12}P_{n}) ∈ J̃^{(2)}. For n ≥ 2, one has

e^{iφ_{n}}P_{n}F_{12}P_{n} = e^{iφ_{n}}F_{12} = ( \begin{matrix}0 & e^{iφ_{n}} \\ 0 & 0 \end{matrix} ) .

Thus, defining unitary diagonal matrices

D_{n}^{(2)} := diag (1, e^{−iφ_{n}}, 1, …, 1) ∈ ℂ^{n×n}

for n > 1, we find

(D_{n}^{(2)})^{∗}P_{n}S_{1}P_{n}D_{n}^{(2)} = P_{n}S_{1}P_{n}, (D_{n}^{(2)})^{∗}P_{n}S_{2}P_{n}D_{n}^{(2)} = P_{n}S_{2}P_{n}

and

(D_{n}^{(2)})^{∗}P_{n}e^{iφ_{n}}F_{12}P_{n}D_{n}^{(2)} = P_{n}F_{12}P_{n}

for all n > 1. If we finally set

E_{n}^{(2)} := \{ \begin{matrix}Ẽ_{n}^{(2)} & if n = 1 \\ Ẽ_{n}^{(2)}D_{n}^{(2)} & if n > 1, \end{matrix}

then we can summarize the result of step 2 as follows: The ideal J^{(0)} of A is ∗-isomorphic to

J^{(2)} := {((E_{n}^{(2)})^{∗}A_{n}E_{n}^{(2)}) : (A_{n}) ∈ J^{(0)}},

the isomorphism can be chosen such that it maps the sequence (Π_{n}) onto the constant sequence (P_{1}), and the mapping

W^{(2)} : J^{(2)} → K(l^{2}), (A_{n}) → W (E_{n}^{(2)}A_{n}(E_{n}^{(2)})^{∗})

is an irreducible representation of J^{(2)} which has kernel G. Besides the sequence (P_{1}), the algebra J^{(2)} also contains the sequences (P_{n}S_{2}P_{n}), (P_{n}F_{12}P_{n}) and (P_{n}F_{21}P_{n}) = (P_{n}F_{12}P_{n})^{∗}, and the representation W^{(2)} maps these sequences onto the operators P_{1}, S_{2}, F_{12} and F_{21}, respectively.

Step 3. Now we proceed per induction. Suppose that, for k ≥ 1, we are given unitary operators E_{n}^{(k−1)} with the related algebra

J^{(k−1)} := {((E_{n}^{(k−1)})^{∗}A_{n}E_{n}^{(k−1)}) : (A_{n}) ∈ Id (Π_{n})}

and with the corresponding irreducible representation

W^{(k−1)} : J^{(k−1)} → K(l^{2}), (A_{n}) → W (E_{n}^{(k−1)}A_{n}(E_{n}^{(k−1)})^{∗})

having G as its kernel, and suppose furthermore that the algebra J^{(k−1)} contains all sequences (P_{n}KP_{n}) with K ∈ ℂ^{(k−1)×(k−1)} = L(Im P_{k−1}) and that W^{(k−1)}(P_{n}KP_{n}) = K for all these sequences.

As in step 2, there is a sequence (S_{n}^{(k)}) ∈ J^{(k−1)} such that

• W^{(k−1)}(S_{n}^{(k)}) = S_{k},

• S_{n}^{(k)} = 0 for n < k and S_{n}^{(k)} is a projection with one-dimensional range for every n ≥ k,

• P_{k−1} and S_{n}^{(2)} are orthogonal for all n.

Further, we can choose unitary matrices Ê_{n}^{(k)} : Im P_{n} → Im P_{n} such that

• (Ê_{n}^{(k)})^{∗}P_{n}KP_{n}Ê_{n}^{(k)} = P_{n}KP_{n} for every K ∈ ℂ^{(k−1)×(k−1)} and that

• (Ê_{n}^{(k)})^{∗}S_{n}^{(k)}Ê_{n}^{(k)} = P_{n}S_{k}P_{n}.

Consider the matrices

Ẽ_{n}^{(k)} := \{ \begin{matrix}E_{n}^{(k−1)} & if n < k \\ E_{n}^{(k−1)}Ê_{n}^{(k)} & if n ≥ k, \end{matrix}

and introduce the related algebra

J̃^{(k)} := {((Ẽ_{n}^{(k)})^{∗}A_{n}Ẽ_{n}^{(k)}) : (A_{n}) ∈ J^{(0)}}

with corresponding irreducible representation

W̃^{(k)} : J̃^{(k)} → K(l^{2}), (A_{n}) ↦ W (Ẽ_{n}^{(k)}A_{n}(Ẽ_{n}^{(k)})^{∗}).

In J̃^{(k)}, we find a sequence (F̃_{n}^{1k}) such that W̃^{(k)}(F̃_{n}^{1k}) = F_{1k}. The same reasoning as in step 2 reveals the existence of unimodular numbers such that the sequence (e^{iφ_{n}}P_{n}F_{1k}P_{n}) lies in J̃^{(k)}. For n ≥ k, set

D_{n}^{(k)} := diag (1, …, e^{−iφ_{n}}, 1, …, 1) ∈ ℂ^{n×n}

with e^{−iφ_{n}} standing at the kth place, and define

E_{n}^{(k)} := \{ \begin{matrix}Ẽ_{n}^{(k)} & if n < k \\ Ẽ_{n}^{(k)}D_{n}^{(k)} & if n ≥ k. \end{matrix}

With these sequences, we relate the algebra

J^{(k)} := {((E_{n}^{(k)})^{∗}A_{n}E_{n}^{(k)}) : (A_{n}) ∈ J^{(0)}}

and its irreducible representation

W^{(k)} : J^{(k)} → K(l^{2}), (A_{n}) → W (E_{n}^{(k)}A_{n}(E_{n}^{(k)})^{∗})

which has kernel G. We claim that J^{(k)} contains all sequences (P_{n}KP_{n}) with K ∈ ℂ^{k×k} and that W^{(k)}(P_{n}KP_{n}) = K for these sequences.

Indeed, all sequences (P_{n}F_{ij}P_{n}) with i, j ≤ k − 1 lie already in J^{(k−1)}. Since the unitary operators E_{n}^{(k)} act on Im P_{k−1} as the identity, these sequences also belong to J^{(k)}. Further, by construction, we have included the sequences (P_{n}S_{k}P_{n}) and (P_{n}F_{1k}P_{n}) and, hence, also the sequence (P_{n}F_{k1}P_{n}) = (P_{n}F_{1k}P_{n})^{∗} into J^{(k)}. Consequently, for l = 2, …, k − 1, the sequences

(P_{n}F_{lk}P_{n}) = (P_{n}F_{l1}P_{n})(P_{n}F_{1k}P_{n}) and (P_{n}F_{kl}P_{n}) = (P_{n}F_{lk}P_{n})^{∗}

lie in J^{(k)}, too, which proves our claim.

Final step. For every n, let E_{n} be the unitary matrix E_{n}^{(n)}, and consider the algebra

J^{∞} := {((E_{n})^{∗}A_{n}E_{n}) : (A_{n}) ∈ J^{(0)}}

which has

W^{∞} : J^{∞} → K(l^{2}), (A_{n}) → W (E_{n}A_{n}(E_{n})^{∗})

as its irreducible representation.

Let k be a fixed positive integer and K ∈ L(Im P_{k}). We know from the previous step that there is a sequence (K_{n}) in the ideal Id (Π_{n}) of A such that

((E_{n}^{(k)})^{∗}K_{n}E_{n}^{(k)}) = P_{n}KP_{n} or, equivalently, (K_{n}) = (E_{n}^{(k)}P_{n}KP_{n}(E_{n}^{(k)})^{∗}).

Due to the definitions of E_{n}^{(k)} and E_{n},

E_{n}^{(k)}P_{k} = E_{n}^{(n)}P_{k} = E_{n}P_{k} for all n ≥ k

and, consequently, K_{n} = E_{n}P_{n}KP_{n}(E_{n})^{∗} for all n ≥ k. This shows that, for every K ∈ ℂ^{k×k}, the sequence (E_{n}P_{n}KP_{n}(E_{n})^{∗}) belongs to J^{(0)} and that W (E_{n}P_{n}KP_{n}(E_{n})^{∗}) = K. A simple approximation argument shows that then, for every compact operator K on l^{2},

(E_{n}P_{n}KP_{n}(E_{n})^{∗}) ∈ J^{(0)} and W (E_{n}P_{n}KP_{n}(E_{n})^{∗}) = K.

It is elementary to check that

J := {((E_{n}P_{n}KP_{n}(E_{n})^{∗} + G_{n}) : K ∈ K(l^{2}), (G_{n}) ∈ G}

is a closed subalgebra of J^{(0)} and that W maps J onto K^{(}l^{2}) in such a way that the only sequences in J which are mapped onto 0 are the sequences in G. Consequently, J = J^{(0)}.

To get the assertion in the form stated in the theorem, we extend E_{n} and (E_{n})^{∗} to partial isometries on l^{2} by letting them act on Im (I − P_{n}) as the zero operator.

Remark 1. In case the Hilbert space H has finite dimension r, the induction process in the preceding proof terminates after r steps, and one can choose E_{n} := E_{n}^{(r)} for all n.

Remark 2. Observe that Theorem 6 remains valid if we only assume the fractality of the ideal Id (K_{n}) instead of the fractality of A.

4 The first half of the standard model

Now we will see how the results of the preceding section imply the first three axioms of the standard model.

Let E_{n} : H → l^{2} be partial isometries with range projections P_{n} and such that the initial projections P′_{n} := E_{n}^{∗}E_{n} converge strongly to the identity operator on H. Further set

J (E_{n}) := {(E_{n}KE_{n}^{∗}) + (G_{n}) with K ∈ K(H), (G_{n}) ∈ G}.  (10)

Theorem 7 Let A be a C^{∗}-subalgebra of F which contains J (E_{n}). Then the following assertions are equivalent.

(a) J (E_{n}) is an ideal of A.

(b) For any sequence (A_{n}) ∈ A, there exists the strong limit s-lim E_{n}^{∗}A_{n}E_{n}.

Proof. (b) ⇒ (a): Given a sequence (A_{n}) in A, let W (A_{n}) denote the strong limit s-lim E_{n}^{∗}A_{n}E_{n}. Then, for arbitrary operators K ∈ K(H) and sequences (G_{n}) ∈ G,

A_{n}(E_{n}KE_{n}^{∗} + G_{n}) = E_{n}E_{n}^{∗}A_{n}E_{n}KE_{n}^{∗} + A_{n}G_{n}
= E_{n}W (A_{n})KE_{n}^{∗} + E_{n}(E_{n}^{∗}A_{n}E_{n} − W (A_{n}))KE_{n}^{∗} + A_{n}G_{n}.  (11)

The operator W (A_{n})K is compact, and the strong convergence of (E_{n}^{∗}A_{n}E_{n}) to W (A_{n}) together with the compactness of K imply the norm convergence of (E_{n}^{∗}A_{n}E_{n} − W (A_{n})) to 0. Hence, the right hand side of (11) belongs to J (E_{n}), and J (E_{n}) is a left ideal of A. The other side of the ideal property can be checked analogously.

(a) ⇒ (b): For the reverse implication, let (A_{n}) ∈ A and K ∈ K(H). Since J (E_{n}) is an ideal of A, there are an operator R ∈ K(H) as well as a sequence (G_{n}) ∈ G such that

A_{n}E_{n}KE_{n}^{∗} = E_{n}RE_{n}^{∗} + G_{n}.

Multiplication by E_{n}^{∗} from the left and by E_{n} from the right yields

E_{n}^{∗}A_{n}E_{n}K = P′_{n}RP′_{n} + E_{n}^{∗}G_{n}E_{n} = R + G′_{n}

with a sequence (G′_{n}) tending to zero in the norm (recall that the projections P′_{n} converge strongly to the identity operator on H by hypothesis). Thus, the sequence (E_{n}^{∗}A_{n}E_{n}K) is norm convergent for every compact operator K ∈ K(H).

Let x ∈ H, and write P_{x} for the (compact) orthogonal projection from H onto its one-dimensional subspace ℂx. Then the sequence (E_{n}^{∗}A_{n}E_{n}P_{x}) is norm convergent as we have just seen, and from

E_{n}^{∗}A_{n}E_{n}x = E_{n}^{∗}A_{n}E_{n}P_{x}x

we conclude that the sequence (E_{n}^{∗}A_{n}E_{n}x) converges in the norm of H for every x, i.e. (U_{n}^{∗}A_{n}U_{n}) is a strongly convergent sequence.

The restriction of the mapping A → L(H), (A_{n}) ↦ s-lim E_{n}^{∗}A_{n}E_{n} onto J (E_{n}) is evidently a homomorphism from J (E_{n}) onto K(H) which has G as its kernel.

Now we consider the case where two (or more) ideals of the form (10) occur and ask for specifications of these ideals and of the related strong limits which are involved by the separation property.

Theorem 8 Let H_{1} and H_{2} be Hilbert spaces and let E_{n} and F_{n} be partial isometries from H_{1} and H_{2} into l^{2} with range projections P_{n} and such that their initial projections converge strongly to the identity operators on H_{1} and H_{2}, respectively. Further, let A be a C^{∗}-subalgebra of F which contains the sets

J (E_{n}) := {(E_{n}KE_{n}^{∗} + G_{n}) with K ∈ K(H_{1}) and (G_{n}) ∈ G},
J (F_{n}) := {(F_{n}KF_{n}^{∗} + G_{n}) with K ∈ K(H_{2}) and (G_{n}) ∈ G}

as its ideals, and let W_{1} denote the homomorphism (A_{n}) ↦ s-lim E_{n}^{∗}A_{n}E_{n}. Then the following assertions are equivalent.

(a) The separation property W_{1}(J (F_{n})) = {0} holds.

(b) The sequence (F_{n}^{∗}E_{n}) converges weakly to zero.

Proof. The separation property (a) is equivalent to

s-lim E_{n}^{∗}F_{n}KF_{n}^{∗}E_{n} = 0 for all K ∈ K(H_{2}).

Thus, for every K ∈ K(H_{2}) and x ∈ H_{1}, one has

lim ⟨E_{n}^{∗}F_{n}K^{∗}KF_{n}^{∗}E_{n}x, x⟩ = lim ⟨KF_{n}^{∗}E_{n}x, KF_{n}^{∗}E_{n}x⟩ = 0

and, consequently, ∥KV_{n}^{∗}U_{n}x∥ → 0. Given y ∈ H_{2}, let P_{y} refer to the orthogonal projection from H_{2} onto its subspace ℂy. The operator P_{y} is compact, hence, ∥P_{y}F_{n}^{∗}E_{n}x∥ → 0 and

⟨F_{n}^{∗}E_{n}x, y⟩ = ⟨F_{n}^{∗}E_{n}x, P_{y}y⟩ = ⟨P_{y}F_{n}^{∗}E_{n}x, y⟩ → 0

for every x ∈ H_{1} and y ∈ H_{2}. Thus, the sequence (F_{n}^{∗}E_{n}) converges weakly to zero.

For the reverse implication note that the weak convergence of (F_{n}^{∗}E_{n}) to zero involves the strong convergence of (KF_{n}^{∗}E_{n}) to zero for every compact operator K and, thus, the strong convergence to zero of the sequence (F_{n}^{∗}E_{n}KF_{n}^{∗}E_{n}). But this is the separation property.

In algebras arising from concrete approximation methods, one usually has the existence of the strong limit s-lim A_{n} for every sequence in the algebra, i.e. one can choose E_{n} := P_{n} in the preceding theorem. In this case, the separation condition implies that the F_{n} converge weakly to 0.

5 The second half of the standard model

Up to now, we have associated with the ideal J_{ess}(A) of the fractal algebra A a family {W_{t}} of homomorphisms which are given in terms of strong limits. Now we will discuss the forth axiom of the standard model, i.e. we ask whether the invertibility of all operators W_{t}(A_{n}) is sufficient for the stability of the sequence (A_{n}) ∈ A. Since W_{t}(A_{n}) depends on the coset (A_{n}) + G only, one can consider the quotient mappings

A/G → L(H_{t}), (A_{n}) + G ↦ W_{t}(A_{n}).  (12)

These mappings are irreducible representations of A/G; the irreducibility is a consequence of the fact that the ideal K(H_{t}) of the compact operators lies in the range of W_{t}. It is also evident from the construction of the W_{t} that the restrictions of these representations to J_{ess}(A)/G are irreducible representations of this ideal. Moreover, these restrictions already exhaust the spectrum of the C^{∗}-algebra J_{ess}(A)/G.

These observations suggest to consider a more general situation. Let 𝔄 be a C^{∗}-algebra with identity element e and with spectrum (= set of all equivalence classes of irreducible representations) Spec 𝔄, and let 𝔍 be a closed two-sided ideal of 𝔄. Then every irreducible representation W of 𝔍 determines uniquely an irreducible representation of 𝔄 which we denote by W again. Now our problem reads as follows: If a ∈ 𝔄 and W (a) is invertible for all W ∈ Spec 𝔍, is then a invertible? Let us call a subset M of Spec 𝔄 a sufficient family of irreducible representations if the equivalence

a invertible ⟺ W (a) invertible for all W ∈ M

holds for every a ∈ 𝔄, and call M a weakly sufficient family if

a invertible ⟺ W (a) invertible for all W ∈ M and sup_{W ∈M} ∥W (a)^{−1}∥ < ∞

for all a ∈ 𝔄. These definitions make sense since both the invertibility and the norm of W (a) depend on the equivalence class of the representation W only.

Since equivalent representations have the same kernel, there is a natural surjective mapping

Π : Spec 𝔄 → Prim 𝔄, W ↦ ker W

where Prim 𝔄 stands for the space of the primitive ideals of 𝔄. This space carries a natural topology, the so-called hull-kernel or Jacobson topology. For definitions see, e.g., [4, 6].

Weakly sufficient families of representations. We start with criteria for weak sufficiency of a family M of irreducible representations.

Theorem 9 Let 𝔄 be a unital C^{∗}-algebra. The following conditions are equivalent for a subset M of Spec 𝔄.

(a) The family M is weakly sufficient.

(b) For every a ∈ 𝔄, W (a) = 0 for every W ∈ M implies that a = 0.

(c) For every a ∈ 𝔄, ∥a∥ = sup_{W ∈M} ∥W (a)∥.

(d) Π(M) is dense in Prim 𝔄.

Proof. (a) ⇒ (b): Let a ∈ 𝔄 be an element such that W (a) = 0 for all W ∈ M, and let b be an arbitrary invertible element of 𝔄. Then, evidently, W (b) is invertible for all W ∈ M, and the norms ∥W (b)^{−1}∥ are uniformly bounded by ∥b^{−1}∥. Then also the operators W (a + b) are invertible for all W ∈ M, and the norms of their inverses are uniformly bounded, too. By (a), the element a + b is invertible. Hence, a lies in the radical of 𝔄 which consists of the zero element only.

(b) ⇒ (c): Let 𝔉 stand for the set of all bounded functions on M which take at W ∈ M a value in W (𝔄). Provided with pointwisely defined operations and the supremum norm, the set 𝔉 becomes a C^{∗}-algebra, and the mapping

S : 𝔄 → 𝔉, a ↦ (W ↦ W (a))

is a ^{∗}-homomorphism. By (b), this homomorphism has kernel {0} and, hence, is an isometry. Thus, (c) follows.

(c) ⇒ (a): Let a ∈ 𝔄 be an element for which the operators W (a) are invertible for all W ∈ M and the norms ∥W (a)^{−1}∥ are uniformly bounded. Then S(a) is invertible in 𝔉 and, due to the inverse closedness of C^{∗}-algebras, also in S(𝔄). But S is an isometry due to (c) and, consequently, an isomorphism between 𝔄 and S(𝔄). Hence, a is invertible in 𝔄.

(b) ⇔ (d): Let M′ := {ker W : W ∈ M} ⊆ Prim 𝔄. The condition (b) implies that ∩_{J∈M′}J = {0}. Hence, the kernel of M′ is {0} and, consequently, the hull of M′ is all of Prim 𝔄. Thus, M′ is dense in the primitive ideal space of 𝔄.

Let, conversely, M′ be dense in Prim 𝔄. Then the kernel of M′ is contained in every primitive ideal of 𝔄 and, hence, in the kernel of Prim 𝔄. Since the intersection of all primitive ideals of a C^{∗}-algebra is the zero ideal, we have ∩_{J∈M′}J = {0}, whence (b).

Essential ideals. Now we are going to single out a class of ideals 𝔍 of 𝔄 having the property that Spec 𝔍 is a weakly sufficient family of representations for 𝔄. Recall that an ideal 𝔍 of 𝔄 is called essential if the intersection of 𝔍 with every non-zero ideal of 𝔄 is non-zero. Further we need the notion of the annulator of an ideal which is defined by

Ann 𝔍 := {a ∈ 𝔄 : a𝔍 = 𝔍a = {0}}.

Lemma 1 Let 𝔄 be a C^{∗}-algebra with identity e and 𝔍 a closed two-sided ideal of 𝔄. Then

Ann 𝔍 = {a ∈ 𝔄 : W (a) = 0 for all W ∈ Spec 𝔄}.

Here we use again the same letter W for an irreducible representation of 𝔍 and its extension to 𝔄.

Proof. Let W (a) = 0 for all W ∈ Spec 𝔄. Suppose there is a j ∈ 𝔍 such that aj ≠ 0. Then there is an irreducible representation W of 𝔍 such that W (aj) ≠ 0. For this representation, one cannot have W (a) = 0.

Conversely, let aj = 0 for all j ∈ 𝔍, and let W : 𝔍 → L(H) be in Spec 𝔍. By [4], Proposition 2.10.4, the set W (𝔍) is dense in W (𝔄) with respect to the strong operator topology on H. Since W maps the identity element e onto the identity operator I, there is a net (j_{t}) ⊂ 𝔍 such that

W (j_{t}) → I strongly as t → ∞.

Then

0 = W (aj_{t}) = W (a)W (j_{t}) → W (a) strongly as t → ∞,

whence W (a) = 0.

Theorem 10 Let 𝔄 be a unital C^{∗}-algebra. The following conditions are equivalent for a closed two-sided ideal 𝔍 of 𝔄.

(a) 𝔍 is an essential ideal of 𝔄.

(b) Ann 𝔍 = {0}.

(c) Spec 𝔍 is weakly sufficient for 𝔄.

(d) For every a ∈ 𝔄, W (a) = 0 for every W ∈ Spec 𝔍 implies that a = 0.

(e) For every a ∈ 𝔄, ∥a∥ = sup_{W ∈Spec 𝔍} ∥W (a)∥.

(f) Prim 𝔍 is dense in Prim 𝔄.

The assertion (f) has to be interpreted as follows. Set

Spec^{𝔍}𝔄 := {W ∈ Spec 𝔄 : W |_{𝔍} ≠ 0}, Prim^{𝔍}𝔄 := {I ∈ Prim 𝔄 : 𝔍 ⊈ I}.

It is well known that the mappings W ↦ W |_{𝔍} and I ↦ I ∩ 𝔍 are homemorphisms from Spec^{𝔍}𝔄 onto Spec 𝔍 and from Prim^{𝔍}𝔄 onto Prim 𝔍, respectively (see [4], Sections 2.11 and 3.2), and we have identified the corresponding sets in the formulation of (f). In this sense, the statement of (f) is equivalent to:

Π(Spec^{𝔍}𝔄) (which evidently coincides with Prim^{𝔍}𝔄) is dense in Prim 𝔄.

Proof of Theorem 10. The simple proof of (a) ⇔ (b) can be found, e.g., in [9], Lemma 2.36. The equivalence between (b) and (d) follows from the preceding lemma, and the equivalence of (c) − (f) is a consequence of Theorem 9.

Recall that a C^{∗}-algebra 𝔄 is called postliminal if K(H) ∈ W (𝔄) for every irreducible representation (H, W) of 𝔄. One feature of postliminal algebras is that the mapping Π is a bijection from Spec 𝔄 onto Prim 𝔄.

It turns out that every postliminal C^{∗}-algebra contains some canonical essential ideals. Following Pedersen, we call an element a of a C^{∗}-algebra 𝔄 abelian if a ≥ 0 and if the algebra a𝔄a is commutative. By J_{abel}(𝔄) we denote the smallest closed two-sided ideal of 𝔄 which contains all abelian elements of 𝔄.

Theorem 11 If 𝔄 is postliminal, then J_{abel}(𝔄) is an essential ideal of 𝔄.

Proof. In [6], Theorem 6.2.11, it is shown that every postliminal algebra possesses an essential ideal with continuous trace. Further, by Theorem 6.1.11 in [6], every algebra with continuous trace is generated by its abelian elements and, by Proposition 6.1.7 of [6], J_{abel}(𝔄) is the largest ideal of 𝔄 which is generated by its abelian elements. Hence, this ideal contains an essential ideal, and so it is essential, too.

Further, let J_{com}(𝔄) stand for the set of all elements a ∈ 𝔄 such that W (a) is a compact operator for every irreducible representation W of 𝔄. Similarly to the preceding theorem, one has:

Theorem 12 If 𝔄 is postliminal, then J_{com}(𝔄) is an essential ideal of 𝔄.

Proof. From Theorem 11 we know that J_{abel}(𝔄) is an essential ideal of 𝔄. Since J_{abel}(𝔄) is a liminal ideal (Corollary 6.1.6 in [6]) and J_{com}(𝔄) is the largest liminal ideal of 𝔄 (Lemma 6.2.2 in [6]), we conclude that J_{abel}(𝔄) ⊆ J_{com}(𝔄) and. hence, J_{com}(𝔄) is essential, too.

Essential dual ideals. In general, the irreducible representations of an essential ideal yield an weakly sufficient family only. To force the sufficiency of that family, one needs stronger properties of the ideal. One version which fits to our purposes is as follows. A C^{∗}-algebra is called elementary if it is ^{∗}-isomorphic to the ideal K(H) of the compact operators on some Hilbert space H. A C^{∗}-algebra is dual if it is the restricted direct sum of elementary algebras. This is not the original definition of a dual algebra, but it is an equivalent one (for this and other equivalent descriptions of dual algebras see [1]).

Theorem 13 Let 𝔄 be a unital C^{∗}-algebra and 𝔍 be an essential dual ideal of 𝔄. Then Spec 𝔍 is a sufficient family of representations for 𝔄.

For the proof, we need the following result.

Theorem 14 Let 𝔄 be a unital C^{∗}-algebra, and let M ⊆ Spec 𝔄 be a family of representations. This family is sufficient if and only if, for every a ∈ 𝔄, there is a W_{0} ∈ M such that ∥a∥ = ∥W_{0}(a)∥.

Proof. Let M be a sufficient family, but assume contrary to what we want that there is an a ∈ 𝔄 such that

∥W_{0}(a)∥ < sup_{W ∈M} ∥W (a)∥ for all W_{0} ∈ M.  (13)

Since ∥W (a)∥ = ∥W ((a^{∗}a)^{1/2})∥ by the C^{∗}-axiom, we can without loss of generality suppose that the element a in (13) is self-adjoint and non-negative. The norm of the self-adjoint element a coincides with its spectral radius ρ(a). Thus, (13) can be rewritten as

ρ(W_{0}(a)) < sup_{W ∈M} ρ(W (a)) for all W_{0} ∈ M.  (14)

Denote the supremum on the right hand side of (14) by C and set c = a−Ce. The elements W (c) = W (a)−CW (e) are invertible for all W ∈ M since ρ(W (a)) < C, and the hypothesis yields the invertibility of c = a − Ce. Then, clearly, ba − ce is invertible for all c belonging to some neighborhood U of C. On the other hand, since sup_{W ∈M} ρ(W (a)) = C, for every neighborhood U of C there is an W_{U} ∈ M such that m_{U} := ρ(W_{U}(a)) ∈ U. The element W_{U}(a)−m_{U}W_{U}(e) is not invertible, because the spectral radius of a non-negative element belongs to the spectrum of this element. Hence, a − m_{U}e is not invertible. This contradiction proves the only-if-part of the assertion.

To verify its if-part, let a ∈ 𝔄 be a non-invertible element. We have to show the existence of a W_{0} ∈ M such that W_{0}(a) is not invertible.

If a is not invertible, then at least one of the self-adjoint elements aa^{∗} or a^{∗}a is not invertible, for definiteness say a^{∗}a. Since a^{∗}a is non-negative, a simple application of the Gelfand-Naimark theorem for commutative C^{∗}-algebras shows that

∥ ∥a^{∗}a∥e − a^{∗}a∥ = ∥a^{∗}a∥.  (15)

Set b := ∥a^{∗}a∥e−a^{∗}a. By hypothesis, there is a W_{0} ∈ M such that ∥W_{0}(b)∥ = ∥b∥, which together with (15) implies

∥ ∥a^{∗}a∥W_{0}(e) − W_{0}(a^{∗}a)∥ = ∥W_{0}(∥a^{∗}a∥e − a^{∗}a)∥ = ∥ ∥a^{∗}a∥e − a^{∗}a∥ = ∥a^{∗}a∥.

Since ∥W_{0}(a^{∗}a)∥ ≤ ∥a^{∗}a∥, we can apply the Gelfand-Naimark theorem once more to obtain the non-invertibility of W_{0}(a^{∗}a) and, thus, of W_{0}(a).

Proof of Theorem 13. By the preceding theorem, what we have to prove is that, for every a ∈ 𝔄, there is a W_{0} ∈ Spec 𝔄 such that ∥a∥ = ∥W_{0}(a)∥. For a = 0, this is trivially satisfied. Let ∥a∥ > 0. Since 𝔍 is an essential ideal, the set

M_{0} := {W ∈ Spec 𝔍 : ∥W (a)∥ ≥ ∥a∥/2}

is not empty due to Theorem 10(e). Further, this set is finite since 𝔍 is a dual ideal. Thus, there is a W_{0} ∈ M_{0} such that

∥W_{0}(a)∥ ≥ ∥W (a)∥ for all W ∈ M_{0}.

Since

sup{∥W (a)∥ : W ∈ M_{0}} = sup{∥W (a)∥ : W ∈ Spec 𝔍} = ∥a∥

by Theorem 10(e) again, we have ∥W_{0}(a)∥ = ∥a∥.

Applications to algebras of approximation sequences. Now we are in a position to establish an equivalent characterization of standard algebras.

Theorem 15 Let A be a unital and fractal C^{∗}-subalgebra of F which contains the ideal G of the zero sequences. If J_{ess}(A) is separable and J_{ess}(A)/G is an essential ideal in A/G (for example, if 𝔄 is postliminal and J_{ess}(A)/G = J_{abel}(A/G)), then A is a standard algebra.

Proof. The ideal J_{ess}(A)/G is generated by the ideals Id (K_{n})/G where (K_{n}) runs through the esential rank one sequences of A. Since each of the ideals Id (K_{n})/G is ^{∗}-isomorphic to K(H) for some Hilbert space H (see [11]), J_{ess}(A)/G is a dual ideal in A/G. By hypothesis, this ideal is also essential. Hence, by Theorem 13, the set T := Spec J_{ess}(A)/G provides us with sufficient family of representations for the invertibility in A/G and, thus, for the stability of sequences in A.

Further, it is easy to see that the spectrum T of J_{ess}(A)/G is in one-to one correspondence with the set S of equivalence classes of ideals Id (K_{n})/G introduced before Theorem 5. Hence, with each of these ideals, there is associated a representation W_{t} ∈ T with

W_{t}(Id (K_{n})/G) = K(H_{t}).

The separability assumption ensures that every ideal Id (K_{n})/G is separable. Consequently, the associated Hilbert space H_{t} is separable, too, and from Theorem 6 we conclude that

Id (K_{n}) = {(E_{n}^{t}K(E_{n}^{t})^{∗}) + G_{n}) : K ∈ K(H_{t}), (G_{n}) ∈ G}

with certain partial isometries E_{n}^{t} : H_{t} → l^{2} satisfying E_{n}^{t}(E_{n}^{t})^{∗} = P_{n}. Now the first three conditions in the definition of a standard algebra follow from Theorems 7 and 8.

References

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Author’s address:

Technische Universität Darmstadt Fachbereich Mathematik Schlossgartenstrasse 7 D 64289 Darmstadt FRG E-mail: roch@mathematik.tu-darmstadt.de