Algebras of approximation sequences: Fredholm theory in fractal algebras

Steffen Roch

Abstract The present paper is a continuation of [5, 7] where a Fredholm theory for approximation sequences is proposed and some of its properties and consequences are studied. Here this theory is specified to the class of fractal approximation methods. The main result is a formula for the socalled α-number of an approximation sequence (A_{n}) which is the analogue of the kernel dimension of a Fredholm operator.

1 Introduction

Let H be a complex Hilbert space, L(H) the C^{∗}-algebra of the linear and bounded operators on H, and K(H) the ideal of the linear compact operators on H. Further let (P_{n}) a sequence of orthogonal projections on H which converges strongly to the identity operator I on H:

s-lim P_{n} = I ⇔ P_{n}x → x for all x ∈ H,

and assume that dim Im P_{n} = n. Thus, Im P_{n} and L(Im P_{n}) can be identified with the linear space ℂ^{n} and with the algebra ℂ^{n×n} = L(ℂ^{n}), respectively. The n × n identity matrix will be denoted by I_{n}.

Let A ∈ L(H). An approximation method for A is a sequence (A_{n}) of matrices A_{n} ∈ ℂ^{n×n} such that A_{n}P_{n} → A and A_{n}^{∗}P_{n} → A^{∗} strongly as n → ∞. This method converges, if the equations

A_{n}x^{(n)} = P_{n}y

possess unique solutions x^{(n)} ∈ Im P_{n} for all sufficiently large n and all right hand sides y ∈ H, and if these solutions converge in the norm of H to a solution of the equation

Ax = y.

By the Banach-Steinhaus theorem, the method (A_{n}) for A is applicable if and only if the sequence (A_{n}) is stable in the sense that the matrices A_{n} are invertible for all sufficiently large n and that

sup ∥A_{n}^{−1}∥ < ∞.

For another characterization of stability, introduce the set F of all bounded sequences (A_{n}) of matrices A_{n} ∈ ℂ^{n×n}. Provided with elementwise operations and the supremum norm, this set becomes a C^{∗}-algebra with identity element (I_{n}), and the set G of all sequences (A_{n}) ∈ F with lim ∥A_{n}∥ = 0 forms a closed twosided ideal in F. A Neumann series argument shows that the sequence (A_{n}) ∈ F is stable if and only if its coset (A_{n})+G is invertible in the quotient algebra F/G.

The role of the algebra F/G in numerical analysis can be compared with the role of the algebra L(H) in operator theory. In this sense, the stable sequences in F correspond to the invertible operators on H. For operators, there is a useful generalization of the notion invertibility: one considers operators A ∈ L(H) which are almost invertible in the sense that both their kernel Ker A and their cokernel Coker A := H/Im A have finite dimension. Operators with this property are called Fredholm operators, and the quantity ind A := dim Ker A − dim Coker A is referred to as the index of A.

In [7, 5] there is introduced and studied an analogous notion of a Fredholm approximation sequence. Thereby, the kernel dimension condition for a Fredholm operator is replaced by the following condition for a Fredholm sequence (A_{n}) ∈ F: There is an ε > 0 and a fixed number k such that at most k singular values of A_{n} are less than ε for all sufficiently large n. The smallest possible number k with this property is the analogue of the kernel dimension for the Fredholm sequence (A_{n}), and we call it the α-number of that sequence. One can also define the index of a Fredholm sequence in a natural way, but it turns out that this index is always zero. This is a consequence of the fact that the approximation operators A_{n} which we consider act on finite dimensional spaces. Exact definitions will be given in Section 2.

So the most important quantity related with a Fredholm sequence seems to be its α-number. In [7], we derived a simple and practicable formula to compute this number for large classes of concrete approximation sequences. (In fact, for all sequences which lie in a standard algebra in the terminology of [7]. This class includes, e.g., projection methods with spline ansatz functions for singular integral operators or quadrature formula methods for Mellin operators, to mention only a few.) It has been one goal of [5] to generalize this formula to the class of all fractal approximation methods. This class has been introduced in [6, 4]. Roughly speaking, a sequence (A_{n}) is fractal (or self-similar) if (A_{n}) can be completely reconstructed from each of its infinite subsequences up to a sequence tending to zero in the norm. The precise definition is in Section 3. The class of the fractal sequences includes all sequences in standard algebras.

For the formula for the α-numbers derived in [7] one needs an additional hypothesis which is satisfied for all sequences in standard algebras but not for all fractal sequences. In the present paper, we will set forth a specification of the general Fredholm theory (including a formula for the α-numbers) to fractal approximation methods without any additional (and hard to check) hypotheses (Section 4).

For applications of this Fredholm theory to the analysis of concrete approximation methods we refer to the already mentioned papers [7, 8, 5] and to the textbook [2]. Nice applications to a special class of approximation sequences (the finite section method for Toeplitz operators) are in [1].

2 Compact and Fredholm sequences

In this section we recall the definition of a compact and a Fredholm approximation sequence from [5], and we derive some characterizations of compact sequences which are basic in what follows.

2.1 The ideal of the compact sequences

Sequences of rank one matrices. A sequence (K_{n}) in F is a sequence of rank one matrices if every matrix K_{n} has range dimension less than or equal to one. The smallest closed ideal of F which contains all sequences of rank one matrices will be denoted by K. The product of a sequence of rank one matrices with another sequence in F is again a sequence of rank one matrices. Hence, the set K_{0} of all finite sums of sequences of rank one matrices forms an (in general, non-closed) ideal of F the closure of which is just the ideal K. Consequently, a sequence (A_{n}) ∈ F belongs to K if and only if, for every ε > 0, there is a sequence (K_{n}) ∈ F such that

sup_{n} ∥A_{n} − K_{n}∥ < ε and sup_{n} dim Im K_{n} < ∞.  (1)

We refer to the elements of K as compact sequences . The role of the ideal K of the compact sequences in numerical analysis can be compared with the role of the ideal of the compact operators in operator theory.

Characterization of compact sequences via singular values. Given a sequence (K_{n}) ∈ F, denote by

0 ≤ σ_{1}^{(n)} ≤ σ_{2}^{(n)} ≤ … ≤ σ_{n}^{(n)}  (2)

the singular values of K_{n}, i.e. the non-negative square roots of the eigenvalues of K_{n}^{∗}K_{n}. Further set Σ_{k}^{(n)} := σ_{n−k+1}^{(n)} for k = 1, …, n. Clearly, ∥K_{n}∥ = σ_{n}^{(n)} = Σ_{1}^{(n)}.

Theorem 2.1 A sequence (K_{n}) ∈ F is compact if and only if

lim_{k→∞} sup_{n≥k} Σ_{k}^{(n)} = 0.  (3)

Proof. Let (K_{n}) ∈ F be a sequence such that (3) holds, and let (with unitary matrices U_{n}, V_{n})

K_{n} = U_{n} diag (Σ_{1}^{(n)}, …, Σ_{n}^{(n)})V_{n}^{∗}

be the singular value decomposition of K_{n}. For every k, n ≥ 1, set

K_{n}^{(k)} := \{ \begin{matrix}U_{n} diag (Σ_{1}^{(n)}, …, Σ_{k−1}^{(n)}, 0, …, 0)V_{n}^{∗} & if & 1 < k ≤ n \\ 0 & if & 1 = k ≤ n \\ K_{n} & if & n < k. \end{matrix}

Then, for all k, n ≥ 1,

∥K_{n} −K_{n}^{(k)}∥ = \{ \begin{matrix}∥U_{n} diag (0, …, 0, Σ_{k}^{(n)}, …, Σ_{n}^{(n)})V_{n}^{∗}∥ = Σ_{k}^{(n)} & if & 1 < k ≤ n \\ ∥K_{n}∥ = Σ_{1}^{(n)} & if & 1 = k ≤ n \\ 0 & if & n < k. \end{matrix}

Hence,

∥(K_{n})_{n≥1} − (K_{n}^{(k)})_{n≥1}∥_{F} = sup_{n≥k} Σ_{k}^{(n)},

which together with hypothesis (3) implies

lim_{k→∞} ∥(K_{n})_{n≥1} − (K_{n}^{(k)})_{n≥1}∥_{F} = lim_{k→∞} sup_{n≥k} Σ_{k}^{(n)} = 0.

Thus, the sequence (K_{n}) is the limit as k → ∞ of the sequences (K_{n}^{(k)})_{n≥1}. Since dim Im K_{n}^{(k)} ≤ k−1, each of the sequences (K_{n}^{(k)})_{n≥1} lies in K_{0}, and the sequence (K_{n}) is compact.

Let now (K_{n}) be a compact sequence. Then the sequence (sup_{n≥k} Σ_{k}^{(n)})_{k≥1} is monotonically decreasing and bounded below (by zero) and, hence, convergent. Assume that the limit of this sequence is positive. Then there is a C > 0 such that

sup_{n≥k} Σ_{k}^{(n)} ≥ C for all k ≥ 1.

Hence, for each k ≥ 1, there is an n_{k} ≥ k such that

Σ_{k}^{(n_{k})} ≥ C for all k ≥ 1  (4)

Further, since (K_{n}) is compact, there is a secuence (R_{n}) ∈ F with

sup_{n} dim Im R_{n} < ∞ and sup_{n} ∥K_{n}^{∗}K_{n} − R_{n}^{∗}R_{n}∥ < C

(compare (1)). Set r := sup_{n} dim Im R_{n} and choose k > r. Thus, in particular,

∥K_{n_{k}}^{∗}K_{n_{k}} − R_{n_{k}}^{∗}R_{n_{k}}∥ < C.  (5)

Let now

K_{n_{k}}^{∗}K_{n_{k}} = U_{n_{k}} diag (Σ_{1}^{(n_{k})}, …, Σ_{n_{k}}^{(n_{k})})U_{n_{k}}^{∗}

with a unitary matrix U_{n_{k}}, and set

P_{k} := diag (1, …, 1, 0, …, 0)

with k times a 1 and n_{k} − k times a 0. Then, obviously, the matrix

P_{k}U_{n_{k}}^{∗}K_{n_{k}}^{∗}K_{n_{k}}U_{n_{k}}P_{k} = diag (Σ_{1}^{(n_{k})}, …, Σ_{k}^{(n_{k})}, 0, …, 0)

is invertible on Im P_{k}, and

∥(P_{k}U_{n_{k}}^{∗}K_{n_{k}}^{∗}K_{n_{k}}U_{n_{k}}P_{k}|_{Im P_{k}})^{−1}∥ = max \{\frac{1}{Σ_{1}^{(n_{k})}} , …, \frac{1}{Σ_{k}^{(n_{k})}} \} = \frac{1}{Σ_{k}^{(n_{k})}}

whence

∥(P_{k}U_{n_{k}}^{∗}K_{n_{k}}^{∗}K_{n_{k}}U_{n_{k}}P_{k}|_{Im P_{k}})^{−1}∥^{−1} = Σ_{k}^{(n_{k})} ≥ C

due to (4). Since, by (5),

∥P_{k}U_{n_{k}}^{∗}K_{n_{k}}^{∗}K_{n_{k}}U_{n_{k}}P_{k} − P_{k}U_{n_{k}}^{∗}R_{n_{k}}^{∗}R_{n_{k}}U_{n_{k}}P_{k}∥ < C,

a Neumann series argument yields the invertibility of P_{k}U_{n_{k}}^{∗}R_{n_{k}}^{∗}R_{n_{k}}U_{n_{k}}P_{k}, considered as an operator on Im P_{k}. The invertibility of this operator implies that

dim Im R_{n_{k}}^{∗}R_{n_{k}} ≥ dim Im P_{k} = k

which contradicts our assumption. Hence, the sequence (sup_{n≥k} Σ_{k}^{(n)})_{k≥1} cannot have a positive limit, i.e. the sequence (K_{n}) satisfies (3).

Corollary 2.2 Let (K_{n}) ∈ K. Then the limit lim_{n→∞} σ_{k}^{(n)} exists and is equal to 0 for every k.

Proof. Let ε > 0. By Theorem 2.1, there is a k_{0} such that sup_{n≥k_{0}} Σ_{k_{0}}^{(n)} < ε. Then, for all n ≥ n_{0} := k_{0} + k − 1,

σ_{k}^{(n)} = Σ_{n−k+1}^{(n)} ≤ Σ_{k_{0}}^{(n)} ≤ sup_{n≥k_{0}} Σ_{k_{0}}^{(n)} < ε.

Thus, lim_{n→∞} σ_{k}^{(n)} = 0.

Further characterizations of the ideal K. We proceed with two further characterizations of the ideal K which are partially based on the results of the preceding paragraph. One aim of these characterizations is to show that the above definition of compact sequences is in some sense the only possible one. Indeed, suppose for a moment that we have no idea of what a compact sequence might be and that we would like to introduce an ideal of F which corresponds to the ideal of the compact operators on a Hilbert space. The concrete approximation methods considered and mentioned in [6] suggest that the constant sequence (P_{1}) where P_{1} is the diagonal matrix diag (1, 0, …, 0) should be considered as compact in any case. Thus, a minimal candidate for the desired ideal of compact sequences is the smallest closed ideal of F which contains the sequence (P_{1}). There is also a natural largest candidate: the ideal of all sequences (K_{n}) such that W (K_{n}) is a compact operator for every irreducible representation of F. The characterizations below show that actually both candidates coincide with the ideal K defined before.

Theorem 2.3 K is the smallest closed ideal of F which contains the constant sequence (P_{1}).

Proof. Let, for a moment, K′ stand for the smallest closed ideal of F which contains the constant sequence (P_{1}). It is evident that K′ ⊆ K. For the reverse inclusion, we have to verify that every sequence (K_{n}) ∈ F of rank one matrices belongs to K′.

Let (K_{n}) be an arbitrary bounded sequence of matrices with dim Im K_{n} ≤ 1. Define a new sequence (K′_{n}) by

K′_{n} := \{ \begin{matrix}∥K_{n}^{∗}K_{n}∥^{−1}K_{n}^{∗}K_{n} & if & K_{n} ≠ 0 \\ P_{1} & if & K_{n} = 0. \end{matrix}

Every matrix K′_{n} is an orthogonal projection with range dimension 1. Hence, there are unitary matrices U_{n} ∈ ℂ^{n×n} such that

K′_{n} = U_{n}^{∗} diag (1, 0, …, 0) U_{n} = U_{n}^{∗}P_{1}U_{n}.

The sequence (U_{n}) is bounded and belongs therefore to F. Hence, (K′_{n}) = (U_{n}^{∗})(P_{1})(U_{n}) belongs to K′. Because of

(K_{n}) = (λ_{n})(K_{n})(K′_{n}) with λ_{n} = \{ \begin{matrix}1 & if & K_{n} ≠ 0 \\ 0 & if & K_{n} = 0, \end{matrix}

this implies that (K_{n}) ∈ K′, too.

Theorem 2.4 (a) If (K_{n}) ∈ F is a sequence of rank one matrices and (H, π) is an irreducible representation of F, then π(K_{n}) is an operator with rank not greater than 1.

(b) A sequence (K_{n}) ∈ F belongs to the ideal K if and only if π(K_{n}) is compact for every irreducible representation (H, π) of F.

Proof. Assertion (a) is proved in [5] (Proposition 2 and Theorem 3). For a proof of (b), let K′ stand for the set of all sequences (K_{n}) in F such that π(K_{n}) is compact for every irreducible representation π of F. The inclusion K ⊆ K′ is an immediate consequence of assertion (a). For the proof of the reverse inclusion, we will show that, for every sequence (K_{n}) ∈ F \ K, there is an irreducible representation π of F such that π(K_{n}) is not compact.

Let (K_{n}) ∈ F be a sequence which is not in K, and let Id (K_{n}) refer to the smallest closed ideal of F which contains the sequence (K_{n}). Further, let Λ_{1}^{(n)} ≥ Λ_{2}^{(n)} ≥ … ≥ Λ_{n}^{(n)} ≥ 0 denote the eigenvalues of K_{n}^{∗}K_{n}, and let U_{n} be a unitary matrix such that

U_{n}^{∗}K_{n}^{∗}K_{n}U_{n} = diag (Λ_{1}^{(n)}, …, Λ_{n}^{(n)}).

Since (K_{n}) ∉ K, Theorem 2.1 implies that lim_{k→∞} sup_{n≥k} Λ_{k}^{(n)} ≠ 0. Thus, as we checked in the proof of Theorem 2.1, there is a C > 0 as well as a sequence (n_{k})_{k≥1} with n_{k} ≥ k such that

Λ_{k}^{(n_{k})} ≥ C for all k.  (6)

The sequence (n_{k}) can be chosen strictly monotonically increasing. Indeed, suppose that we have already determined n_{1} < n_{2} < … < n_{k}. Choose k′ > n_{k} and a corresponding n′ ≥ k′ such that Λ_{k′}^{(n′)} ≥ C which is possible due to (6). Then n_{k+1} := n′ satisfies n_{k+1} = n′ ≥ n_{k} + 1 ≥ k + 1, yielding the desired strict monotonicity.

So let (n_{k}) be a strictly monotonically increasing sequence satisfying (6). Set

L_{n} := diag (1/Λ_{1}^{(n_{k})}, …, 1/Λ_{k}^{(n_{k})}, 0, …, 0)

if n = n_{k} for some k and L_{n} := 0 if n ≠ n_{k}. The sequence (L_{n}) is bounded because of

(Λ_{j}^{(n_{k})})^{−1} ≤ 1/C for j = 1, …, k.

Hence, the sequence (D_{n}) with

D_{n} := L_{n}U_{n}^{∗}K_{n}^{∗}K_{n}U_{n} = \{ \begin{matrix}0 & ifn & ≠ n_{k} \\ diag (1, …, 1, 0, …, 0) & if & n = n_{k} \end{matrix}

with k times a 1 and n_{k} − k times a 0 belongs to the ideal Id (K_{n}). Then all sequences

(D_{n}P_{n}AP_{n}D_{n}) with A ∈ L(l^{2})  (7)

lie in Id(K_{n}), and we consider the smallest closed C^{∗}-subalgebra B of Id(K_{n}) which contains all sequences (7). Clearly, the strong limit s-lim B_{n_{k}} as k → ∞ exists for every sequence (B_{n}) ∈ B and, in particular,

s-lim_{k→∞} D_{n_{k}}P_{n_{k}}AP_{n_{k}}D_{n_{k}} = A

for all sequences of the form (7). The mapping

W : B → L(l^{2}), (B_{n}) ↦ s-lim_{k→∞} B_{n_{k}}

is an irreducible representation of B (all compact operators lie in the range of W) which maps (D_{n}) to the identity operator on l^{2}. From Proposition 4.1.8 in [3] we know that there exists an irreducible representation π : F → L(H), a closed subspace H_{1} of H, and a bijective isometry U : H_{1} → l^{2} such that

W (B_{n}) = Uπ(B_{n})|_{H_{1}}U^{∗} for all (B_{n}) ∈ B.  (8)

The operator π(K_{n}) cannot be compact. Indeed, suppose for a moment that π(K_{n}) is a compact operator. Then the operators π(R_{n}) are compact for all sequences (R_{n}) ∈ Id(K_{n}) whence the compactness of π(D_{n}). Then, by (8), W (D_{n}) must be compact. This is impossible since W (D_{n}) is the identity operator on l^{2} as we already observed. Thus, (K_{n}) ∉ K′, yielding the inclusion K′ ⊆ K.

2.2 Fredholm sequences

Corresponding to the ideal K we introduce an appropriate class of Fredholm sequences by calling a sequence (A_{n}) ∈ F Fredholm if it is invertible modulo K (see [5]). The following properties of Fredholm sequences are obvious.

– Every stable sequence is Fredholm.

– The adjoint of a Fredholm sequence is Fredholm.

– The product of Fredholm sequences is Fredholm.

– If (A_{n}) is Fredholm and (K_{n}) ∈ K, then (A_{n} + K_{n}) is Fredholm.

– The set of the Fredholm sequences is open in F. For another characterization of Fredholm sequences, let 0 ≤ σ_{1}^{(n)} ≤ … ≤ σ_{n}^{(n)} denote the singular values of a sequence A_{n} ∈ F.

Theorem 2.5 Each of the following conditions is equivalent to the Fredholmness of a sequence (A_{n}) ∈ F:

(a) There is a sequence (B_{n}) ∈ F and a sequence (J_{n}) ∈ K with sup_{n} dim Im J_{n} < ∞ such that

B_{n}A_{n}^{∗}A_{n} = I_{n} + J_{n}.  (9)

(b) There is a k such that

lim inf_{n→∞} σ_{k+1}^{(n)} > 0.  (10)

The preceding theorem suggests to introduce the α-number α(A_{n}) of a Fredholm sequence (A_{n}) (corresponding to the kernel dimension of a Fredholm operator) as the smallest number k for which (10) is true. Equivalently, α(A_{n}) is the smallest number for which there exists a sequence (B_{n}) ∈ F as well as a sequence (J_{n}) ∈ K such that B_{n}A_{n}^{∗}A_{n} = I_{n} + J_{n} and lim sup_{n→∞} dim Im J_{n} = α(A_{n}). The index of a Fredholm sequence is the quantity

ind (A_{n}) := α(A_{n}) − α(A_{n}^{∗}).

Observe that, in the case at hand, this index is always zero. This is a consequence of the fact that the entries of the sequences under consideration are finite-dimensional operators and, hence, the matrices A_{n}^{∗}A_{n} and A_{n}A_{n}^{∗} have the same eigenvalues even with respect to their multiplicity. So the most interesting quantity associated with a Fredholm sequence of matrices seems to be its α-number. On the other hand, the vanishing of the index of (A_{n}) has remarkable consequences as has been pointed out in [5].

Let us still emphasize that, in general, the singular values of a compact or a Fredholm sequence do not show the behaviour one would expect due to the knowledge on the singular values of a compact or a Fredholm operator. Indeed, if K is a compact operator on a Hilbert space, then there are at most countably many singular values, and all non-zero singular values are isolated points. Similarly, if A is a Fredholm operator on a Hilbert space, then its range is closed and A is Moore-Penrose invertible. The latter property is satisfied if and only if either A^{∗}A is invertible or if 0 is an isolated point in the spectrum of A^{∗}A. Thus, one might expect that the non-zero singular values of the coset (K_{n}) + G of a compact sequence (K_{n}) or the singular value 0 of the coset (A_{n}) + G of a nonstable Fredholm sequence (A_{n}) are isolated. But this is not true as the following simple example shows.

Example. Let (a_{n}) be an enumeration of the rational numbers in [0, 1], and set

K_{n} := a_{n}P_{n}P_{1}P_{n} = diag (a_{n}, 0, …, 0)

and

A_{n} := K_{n} + P_{n}(I − P_{1})P_{n} = diag (a_{n}, 1, …, 1).

The sequence (K_{n}) consists of rank one matrices; hence (K_{n}) is a compact and (A_{n}) is a Fredholm sequence. But both the largest singular values of the matrices K_{n} and the smallest singular values of the matrices K_{n} lie dense in [0, 1]. Thus, the spectrum of both cosets (K_{n}^{∗}K_{n}) + G and (A_{n}^{∗}A_{n}) + G is [0, 1].

We will see in the following sections that compact and Fredholm sequences in fractal subalgebras show a behaviour of the singular values which is in perfect accordance with the behaviour of the singular values of compact and Fredholm operators.

3 Compact sequences in fractal algebras

In this section, we consider compact sequences (K_{n}) in fractal algebras. In particular, the announced behaviour of the singular values of the matrices K_{n} will be established. We start with recalling the definition of a fractal algebra and of some properties of fractal algebras from [7, 5].

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}. Further, 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 3.1 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), (A_{n}) ↦ (A_{n}) + (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 in A ∩ G. This assumption is very natural for sequences arising from discretization procedures, and it is indeed satisfied for a lot of concrete approximation sequences; see e.g. [6]. On the other hand, the algebra F of all bounded sequences fails to be fractal.

Here are a few properties of fractal sequences the proofs of which can be found in [5].

Theorem 3.2 A C^{∗}-subalgebra A of F is fractal if and only if the following implication holds for every element (A_{n}) ∈ A and every strongly monotonically increasing sequence η:

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

Theorem 3.3 A sequence (A_{n}) ∈ F of self-adjoint matrices is fractal if and only if lim inf σ(A_{n}) = lim sup σ(A_{n}) where σ(A_{n}) stands for the spectrum of A_{n}.

If the limes superior lim sup M_{n} and the limes inferior lim inf M_{n} of a sequence (M_{n}) of compact subsets of the complex plane coincide, then the sequence (M_{n}) converges with respect to the Hausdorff distance, and we write lim M_{n} for the limit in the Hausdorff sense.

Proposition 3.4 If (A_{n}) ∈ F is a fractal sequence, then the limit lim ∥A_{n}∥ exists and it is equal to the norm of the coset (A_{n}) + G in the quotient algebra F/G.

3.1 Singular values of fractal compact sequences.

Let (K_{n}) ∈ F be a fractal sequence (not necessarily compact in the moment), and let Σ_{1}^{(n)} ≥ … Σ_{n}^{(n)} ≥ 0 denote the singular values of K_{n}. Further we write σ_{2}(A) for the set of the singular values of an operator A.

Lemma 3.5 The sequence (Σ_{1}^{(n)}) of the largest singular values of K_{n} converges.

Proof. The assertion is an immediate consequence of Proposition 3.4 and of the identity Σ_{1}^{(n)} = ∥K_{n}∥. Another short proof based on Theorem 3.3 runs as follows. Assume that the sequence (Σ_{1}^{(n)}) possesses two limiting points, say α and β with α > β. Then β is the limit of a subsequence (Σ_{1}^{(n_{r})})_{r≥1} of (Σ_{1}^{(n)}). The point α cannot belong to the partial limiting set lim sup_{r→∞} σ_{2}(K_{n_{r}}) (the Σ_{1}^{(n_{r})} are the largest singular values of K_{n_{r}}, and they converge to β which is less than α). Hence,

α ∈ lim sup σ_{2}(K_{n}) \ lim inf σ_{2}(K_{n}),

in contradiction to Theorem 3.3.

Observe that the sequence (Σ_{2}^{(n)}) needs not to converge even if (K_{n}) is a fractal sequence as the example

K_{n} := \{ \begin{matrix}diag (1, 0, 0, …, 0) & if n is even \\ diag (1, 1, 0, …, 0) & if n is odd \end{matrix}

shows. But, as in the second proof of Lemma 3.5, one can easily verify that the sequence (Σ_{2}^{(n)}) can possess at most two limiting points, at most one of which can be different from lim Σ_{1}^{(n)}. More general, the following is true, where we write Π_{k} for the set of all partial limits of the sequence (Σ_{k}^{(n)})_{n≥k}.

Proposition 3.6 If (K_{n}) ∈ F be fractal, then Π_{k+1} \ Π_{k} contains at most one element.

Proof. We start with showing the identity

Π_{1} ∪ … ∪ Π_{k} = lim sup_{n→∞}{Σ_{1}^{(n)}, …, Σ_{k}^{(n)}} for every k.  (11)

The inclusion ⊆ is evident. Conversely, if λ ∈ lim sup_{n→∞}{Σ_{1}^{(n)}, …, Σ_{k}^{(n)}}, then there are a subsequence (n_{r}) of ℕ and numbers k_{r} in {1, …, k} such that

λ = lim_{r→∞} Σ_{k_{r}}^{(n_{r})}.

Since the k_{r} can take only finitely many values, there is a k_{0} between 1 and k such that

λ = lim_{r→∞} Σ_{k_{0}}^{(n_{r})}.

Hence, λ ∈ Π_{k_{0}}, which shows (11).

Now we proceed as in the second proof of Lemma 3.5. Assume that the sequence (Σ_{k+1}^{(n)}) possesses two limiting points α and β with α > β, both not belonging to Π_{k}. Choose a subsequence (Σ_{k+1}^{(n_{r})})_{r≥1} of (Σ_{k+1}^{(n)}) which converges to β as r → ∞. Then α cannot belong to the partial limiting set lim sup_{r→∞} σ_{2}(K_{n_{r}}). Indeed, since α ∉ Π_{k} by assumption, α cannot belong to

Π_{1} ∪ … ∪ Π_{k} = lim sup_{n→∞}{Σ_{1}^{(n)}, …, Σ_{k}^{(n)}}

due to monotony reasons. But the k + 1 th singular values of K_{n_{r}} converge to β < α. Hence,

α ∈ lim sup σ_{2}(K_{n}) \ lim inf σ_{2}(K_{n}),

in contradiction to Theorem 3.3.

Corollary 3.7 If (K_{n}) ∈ F is fractal, then lim sup_{n→∞}{Σ_{1}^{(n)}, …, Σ_{k}^{(n)}} contains at most k elements.

Now we can show that, with respect to the singular values, fractal compact sequences behave as compact operators on Hilbert space.

Corollary 3.8 Let (K_{n}) ∈ K be a fractal sequence and ε > 0. Then there are only finitely many points in lim σ_{2}(K_{n}) with absolute value larger than ε.

Proof. By Theorem 2.1, lim_{k→∞} sup_{n≥k} Σ_{k}^{(n)} = 0. Thus, there is a k_{0} such that sup_{n≥k_{0}} Σ_{k_{0}}^{(n)} ≤ ε, whence

lim sup_{n→∞}{Σ_{k_{0}}^{(n)}, …, Σ_{n}^{(n)}} ⊆ [0, ε].

Since

lim_{n→∞} σ_{2}(K_{n}) = lim sup_{n→∞}{Σ_{1}^{(n)}, …, Σ_{k_{0}−1}^{(n)}} ∪ lim sup_{n→∞}{Σ_{k_{0}}^{(n)}, …, Σ_{n}^{(n)}},

it is obvious that the points in lim σ_{2}(K_{n}) with absolute value larger than ε lie in

lim sup_{n→∞}{Σ_{1}^{(n)}, …, Σ_{k_{0}−1}^{(n)}}.

This set is finite by Corollary 3.7.

Corollary 3.9 Let (K_{n}) ∈ K be fractal. Then 0 ∈ lim σ_{2}(K_{n}), the set lim σ_{2}(K_{n}) is at most countable, and 0 is the only limiting point of that set.

3.2 The ideal (A ∩ K)/G

Throughout what follows let A be a unital and fractal C^{∗}-subalgebra of F which contains the ideal G. We will now apply the results of the preceding section to analyse the ideal A ∩ K of the compact sequences in A.

Generators of (A ∩ K)/G. Our first goal is to show that the ideal A ∩ K is generated by its projections. Later on we will see that it is even generated by its minimal projections.

Theorem 3.10 The ideal A∩K is generated (as C^{∗}-algebra or as an ideal of A) by its projections, i.e. by the sequences (Q_{n}) ∈ A ∩ K where each matrix Q_{n} is a self-adjoint idempotent.

Proof. Every sequence in A ∩ K is a linear combination of four non-negative sequences in A ∩ K, and every non-negative sequence in A ∩ K is of the form (K_{n}^{∗}K_{n}) with a sequence (K_{n}) ∈ A ∩ K. The assertion will follow once we have shows that, for every sequence (K_{n}) ∈ A ∩ K, the sequence (K_{n}^{∗}K_{n}) can be approximated in the norm of F by linear combinations of projections in A ∩ K as closely as desired.

By Corollary 3.9, there are (finitely or countably many) numbers λ_{1} > λ_{2} > … > 0 such that

σ_{F}((K_{n}^{∗}K_{n})) = {λ_{1}, λ_{2}, …} ∪ {0} =: Λ.

The smallest C^{∗}-subalgebra K′ of F which contains the sequences (K_{n}^{∗}K_{n}) and (I_{n}) is ^{∗}-isometric to C(Λ), and the Gelfand transform maps (K_{n}^{∗}K_{n}) to the identity mapping on Λ. For every λ_{k}, let P_{k} be the sequence in K′ = C(Λ) the Gelfand transform of which is 1 at λ_{k} and 0 on Λ\{λ_{k}} (this function is obviously continuous on Λ). Each of the sequences P_{k} is a projection in K′ ⊆ A ∩ K, and

∥(K_{n}^{∗}K_{n}) − ∑_{i=1}^{r} λ_{i}P_{i}∥_{F} = \{ \begin{matrix}λ_{k+1} & if Λ has at least k + 2 elements \\ 0 & if Λ has at most k + 1 elements. \end{matrix}

Since λ_{k} → 0 as k → ∞, this implies the assertion.

Due to Theorem 3.3, one of the following possibilities is satisfied for every fractal sequence (Q_{n}) of projections: either σ(Q_{n}) = {1}, or σ(Q_{n}) = {0}, σ(Q_{n}) = {0, 1} for all sufficiently large n. In the first case we have Q_{n} = I_{n} for all sufficiently large n, and the sequence (Q_{n}) cannot belong to K (Corollary 2.2). In the second case, Q_{n} = 0 for all large n, and the sequence (Q_{n}) lies in G.

We call a sequence projections (Q_{n}) ∈ F essential if 1 ∈ σ(Q_{n}) for all n.

Proposition 3.11 The ideal A ∩ K is generated (as ideal in A) by its essential projections.

Proof. For a moment, let K′ stand for the smallest closed ideal of A which contains all essential projections from A ∩ K. Evidently, K′ ⊆ A ∩ K. For the reverse inclusion, we have to show that K′ contains all projections in A ∩ K (Theorem 3.10). For this goal, we first check that

G ⊆ K′.  (12)

Let r ∈ ℕ and (Q_{n}) ∈ A ∩ K be an essential sequence of projections. The sequence (0, …, 0, I_{r}, 0, …) with I_{r} standing at the r th place lies in A; hence, the sequence (0, …, 0, Q_{r}, 0, …) belongs to K′. Since Q_{r} ≠ 0, the smallest closed ideal of ℂ^{r×r} which contains Q_{r} is equal to ℂ^{r×r}, which shows that K′ contains all sequences (0, …, 0, G_{r}, 0, …) with G_{r} ∈ ℂ^{r×r}. Consequently, all sequences (G_{1}, G_{2}, …, G_{k}, 0, 0, …) belong to K′. Since G is the closure of the set of all sequences of this kind, (12) holds.

Let now (R_{n}) ∈ A ∩ K be a projection. By the above classification, either σ(R_{n}) = {0} or σ(R_{n}) = {0, 1} for all sufficiently large n. In the first case, (R_{n}) ∈ G, whence (R_{n}) ∈ K′ due to (12). In the second case, at most finitely many of the matrices R_{n} are equal to zero. We replace these matrices by I_{n} and get a new sequence (R′_{n}). This sequence is an essential projection and, hence, in K′ by definition, and the difference (R′_{n}) − (R_{n}) is in G by construction and in K′ due to (12).

Later, we will need the following simple consequence of the preceding proposition.

Corollary 3.12 If q ∈ (A ∩ K)/G is a non-zero projection, then there is an essential projection (Q_{n}) ∈ A ∩ K such that (Q_{n}) + G = q.

We also say that q can be lifted to an essential projection. For the proof note that every projection q ∈ F/G can be lifted to a projection (R_{n}) ∈ F (see, e.g., [7]). Since q ≠ 0 and q ∈ K/G, we have (R_{n}) ∉ G and (R_{n}) ∉ (P_{n}) + G. Thus, σ(R_{n}) = {0, 1} for all sufficiently large n, and (R_{n}) can be made to an essential projection by adding a sequence in G as in the proof of Proposition 3.11.

An order relation for projections. In what follows we will have to compare projections in (A ∩ K)/G. The needed definitions will be given in a more general context.

Let 𝔅 be a C^{∗}-algebra, and let q, r be projections (i.e. self-adjoint idempotents) in 𝔅. We say that r ⪯ q if rqr = r. If r ⪯ q and r ≠ q, then we write r ≺ q. Obviously, 0 ⪯ q and, if 𝔅 has an identity element e, q ⪯ e for every projection q.

If rqr = r for projections q, r, then (i(qr − rq))^{3} = 0 as one easily verifies. Since i(qr − rq) is self-adjoint, this observation implies that qr = rq whenever rqr = r. With this commutativity, it is trivial to check that the relation ⪯ is reflexive, symmetric and transitive, i.e. an order relation.

Lemma 3.13 Let q, r be projections in a C^{∗}-algebra 𝔅 such that r ⪯ q. Then q − r is also a projection, and q − r ⪯ q.

Proof. We have already remarked that r ⪯ q implies rqr = rq = qr = r. Hence,

(q − r)^{2} = q^{2} − qr − rq + r^{2} = q − r

and

(q − r)q(q − r) = q^{3} − q^{2}r − rq^{2} + rqr = q − qr − rq + r = q − r.

The first identity shows that q − r is a projection, the second that q − r ⪯ q.

We call a projection p ∈ 𝔅 minimal if, for every projection q ∈ 𝔅 with q ⪯ p, q = 0 or q = p. Clearly, 0 is a minimal projection, the trivial minimal projection. The quotient l^{∞}/c_{0} is an example for a C^{∗}-algebra without non-trivial minimal projections. Further, an element r ∈ 𝔅 is said to be of rank one if, for every b ∈ 𝔅, there is a β ∈ ℂ such that rbr = βr.

Proposition 3.14 Let 𝔅 be a C^{∗}-algebra. Then every projection of rank one is minimal.

Proof. Let p ∈ 𝔅 be a projection of rank one. If p = 0, then nothing is to prove. So let p ≠ 0, and let q ∈ 𝔅 be a projection with q ⪯ p, i.e. qpq = q. Since p is of rank one, there is a μ ∈ ℂ such that pqp = μp. So one has

q = q^{2} = (qpq)^{2} = q(pqp)q = μqpq = μq,

whence q = 0 or μ = 1. If μ = 1, then pqp = p, i.e. p ⪯ q. Due to the symmetry of ⪯, this implies p = q. Thus, in any case, q = 0 or q = p, i.e. p is minimal.

The converse of Proposition 3.14 is not true. If, for instance, 𝔅 = C([0, 1]∪[2, 3]), then the function which is 1 on [0, 1] and 0 on [2, 3] is a minimal projection in 𝔅 which is not of rank one. We will see now that for the algebras we shall be concerned with the classes of the minimal projections and of the rank one projections coincide.

Theorem 3.15 Let A be a unital and fractal C^{∗}-subalgebra of F which contains the ideal G. Then every minimal projection in (A∩K)/G is a rank one projection in A/G.

Proof. Let q be a minimal projection in (A ∩ K)/G. If q = 0, then q is of rank one. So let q ≠ 0 and a ∈ A/G. We have to show that there is an α ∈ ℂ such that qaq = αq. Since a can be written as a linear combination of four non-negative elements, it is sufficient to verify that, for every b ∈ A/G, there is a β ∈ ℂ such that

qb^{∗}bq = βq.  (13)

Let C stand for the smallest C^{∗}-subalgebra of A/G which contains the elements q and qb^{∗}bq. This algebra is commutative, has q as its identity element, and its maximal ideal space is homeomorphic to σ_{C}(qb^{∗}bq).

Choose sequences (Q_{n}) ∈ A ∩ K and (B_{n}) ∈ A with (Q_{n}) + G = q and (B_{n}) + G = b. From Corollary 3.9 we infer that

σ_{A}((Q_{n}B_{n}^{∗}B_{n}Q_{n})_{n≥1}) = {0} ∪ {λ_{1}, λ_{2}, …} =: Λ

with numbers λ_{1} > λ_{2} > … > 0 having 0 as their only possible accumulation point. Hence, σ_{A/G}(qb^{∗}bq) ⊆ Λ. We claim that

σ_{C}(qb^{∗}bq) ⊆ σ_{A/G}(qb^{∗}bq)  (14)

what clearly implies that

σ_{C}(qb^{∗}bq) ⊆ Λ.  (15)

To prove (14), let λ ∈ ℂ \ σ_{A/G}(qb^{∗}bq). Then there is a c in A/G such that (qb^{∗}bq − λe)c = e. Multiplying this identity by q from both sides we get (qb^{∗}bq − λq)qcq = q, i.e. qb^{∗}bq−λq is invertible in the C^{∗}-algebra q(A/G)q with identity q. But then, due to the inverse closedness of C^{∗}-algebras, qb^{∗}bq−λq is also invertible in the C^{∗}-subalgebra C of q(A/G)q. Hence, λ ∈ ℂ \ σ_{C}(qb^{∗}bq) which yields (14).

Now assume that σ_{C}(qb^{∗}bq) contains at least two points. Then at least one of these points, λ_{0} say, is not 0. We consider the element p ∈ C the Gelfand transform of which satisfies

p(λ_{0}) = 1 and p(λ) = 0 for all λ ∈ σ_{C}(qb^{∗}bq) \ {λ_{0}}

(this function is continuous on σ_{C}(qb^{∗}bq) due to (15) and the choice of λ_{0}). The element p is a projection in C ⊆ (A ∩ K)/G, and p ⪯ q since q(λ) = 1 for all λ ∈ σ_{C}(qb^{∗}bq). But, evidently, p is neither 0 (since p(λ_{0}) = 1) nor q (since σ_{C}(qb^{∗}bq) contains at least two points by assumption). This contradicts the minimality of q. Hence, the maximal ideal space σ_{C}(qb^{∗}bq) of C consists of exactly one point. But then every element of C is a multiple of the identity element whence the existence of a β ∈ ℂ such that (13).

It is also clear that β ≥ 0: Since σ_{A/G}(q) = {0, 1}, one has σ_{A/G}(βq) = {0, β} and, consequently,

σ_{A/G}(qb^{∗}bq) = {0, β}.

Because of qb^{∗}bq ≥ 0, this implies β ≥ 0.

Minimal generators. The next goal is to show that the ideal (A ∩ K)/G not only contains minimal projections, but is even generated by its minimal projections. We start with considering the consequences of the relation r ⪯ q between two projections q, r ∈ (A ∩ K)/G for the representatives of these cosets.

Theorem 3.16 Let A be as in the preceding theorem, and let q, r ∈ (A ∩ K)/G be projections with r ⪯ q. Let further q = (Q_{n}) + G and r = (R_{n}) + G with projections (Q_{n}), (R_{n}) ∈ A ∩ K (these projections exist and can be chosen as essential projections due to Corollary 3.12). If r ≠ q, then

dim Im R_{n} < dim Im Q_{n}

for all n ∈ ℕ with at most finitely many exceptions.

In the proof, we will make use of the following simple observation: Let P be an orthogonal projection on a Hilbert space H, and let A ∈ L(H) be an operator such that ∥P − A∥ < 1. Then ∥P − P AP ∥ = ∥P (P − A)P ∥ ≤ ∥P − A∥ < 1. Hence, by Neumann series, the operator

P AP = P − (P − P AP) : Im P → Im P

is invertible, whence

dim Im P AP = dim Im P.  (16)

Proof. Since rqr = r, one has r = rq = qr = rqr. Consequently,

∥R_{n} − Q_{n}R_{n}Q_{n}∥ → 0,  (17)
∥R_{n} − R_{n}Q_{n}∥ → 0 and ∥R_{n} − Q_{n}R_{n}∥ → 0 as n → ∞.  (18)

Due to (17),

∥R_{n} − Q_{n}R_{n}Q_{n}∥ < 1 for all sufficiently large n.

Thus, by (16),

dim Im R_{n} = dim Im R_{n}Q_{n}R_{n} ≤ dim Im Q_{n}

for all sufficiently large n. Suppose that dim Im R_{n} = dim Im Q_{n} for all n in an infinite subset η of ℕ. Then, for every n ∈ η, there is a unitary matrix U_{n} : ℂ^{n} → ℂ^{n} such that R_{n} = U_{n}Q_{n}U_{n}^{∗}. (Indeed, choose orthonormal bases {e_{1}, …, e_{k}} and {f_{1}, …, f_{k}} of Im Q_{n} and Im R_{n}, complete these bases to orthonormal bases {e_{1}, …, e_{n}} and {f_{1}, …, f_{n}} of ℂ^{n}, and set U_{n}e_{r} := f_{r} for 1 ≤ r ≤ n.) Thus, for every n ∈ η,

∥R_{n} − R_{n}Q_{n}∥ = ∥U_{n}Q_{n}U_{n}^{∗} − U_{n}Q_{n}U_{n}^{∗}Q_{n}∥
= ∥Q_{n} − Q_{n}U_{n}^{∗}Q_{n}U_{n}∥ = ∥Q_{n} − Q_{n}R_{n}∥.

Together with the first assertion of (18), this shows that (Q_{n} − Q_{n}R_{n})_{n∈η} ∈ G_{η} and, together with the second assertion of (18), we even get (R_{n} − Q_{n})_{n∈η} ∈ G_{η}. Via Theorem 3.2, the fractality of A implies that (R_{n} − Q_{n})_{n∈ℕ} ∈ G, i.e. q = r in contradiction to the hypotheses.

Corollary 3.17 Let A be as in the preceding theorem. Then every chain q_{0} ≻ q_{1} ≻ q_{2} ≻ … of projections q_{i} ∈ (A ∩ K)/G is finite.

Proof. Let (Q_{n}) ∈ A ∩ K be a projection such that (Q_{n}) + G = q_{0}. By the definition of K, there is a sequence (K_{n}) ∈ F with l := sup_{n} dim Im R_{n} < ∞ such that ∥Q_{n} − R_{n}∥ < 1 for all n. Then, by (16),

dim Im Q_{n} = dim Im Q_{n}R_{n}Q_{n} ≤ dim Im R_{n} ≤ l

for all n. In particular, l_{0} := sup_{n} dim Im Q_{n} < ∞. For j ≥ 1, let (Q_{n}^{(j)}) ∈ A ∩ K be a projection such that (Q_{n}^{(j)}) + G = q_{j}. Repeated application of Theorem 3.16 shows that

dim Im Q_{n}^{(1)} ≤ l_{0} − 1 for almost all n,
dim Im Q_{n}^{(2)} ≤ l_{0} − 2 for almost all n, …

where for almost all n means for all n with finitely many exceptions. Thus, after at most l_{0} steps, dim Im Q_{n}^{(1)} = 0 for almost all n, i.e. q_{l_{0}} = 0.

Corollary 3.18 For every non-trivial projection q ∈ (A ∩ K)/G, there is a nontrivial minimal projection r ∈ (A ∩ K)/G such that r ⪯ q.

Proof. If q is not minimal, then there is a projection q_{1} ∈ (A ∩ K)/G such that q_{1} ≺ q. If also q_{1} is not minimal, then there is a projection q_{2} ∈ (A ∩ K)/G such that q_{2} ≺ q_{1}, etc. By Corollary 3.17, the chain q ≻ q_{1} ≻ q_{2} ≻ … has a smallest non-trivial entry which is a minimal projection.

Corollary 3.19 Every projection q ∈ (A ∩ K)/G is a finite sum of minimal projections in (A ∩ K)/G.

Proof. If q = 0 then q is minimal. So let q ∈ (A ∩ K)/G be a non-trivial projection. If q is minimal, then again nothing is to prove. If q is not minimal, then there exists a non-trivial minimal projection p_{1} ≺ q by Corollary 3.18. The element q − p_{1} is a projection by Lemma 3.13. If q − p_{1} is minimal, then q = p_{1} + (q − p_{1}) is a sum of two minimal projections, and we are done. If q − p_{1} is not minimal, then (again by Corollary 3.18) there a non-trivial minimal projection p_{2} ≺ q − p_{1}. If q − p_{1} − p_{2} is minimal, then q = p_{1} + p_{2} + (q − p_{1} − p_{2}) is a sum of three minimal projections. Otherwise, we proceed in this manner to find a non-trivial projection p_{3} such that p_{3} ≺ q − p_{1} − p_{2}. Since the chain

q ≻ q − p_{1} ≻ q − p_{1} − p_{2} ≻ …

is finite by Corollary 3.17, this process stops at some point, i.e. there is a nontrivial minimal projection q−p_{1}−…−p_{r}. Then q = p_{1}+…+p_{r}+(q−p_{1}−…−p_{r}) is the sum of r + 1 non-trivial minimal projections.

Combining this result with Theorem 3.10 we get:

Theorem 3.20 Let A be a unital and fractal C^{∗}-subalgebra of F which contains the ideal G. Then the ideal (A ∩ K)/G is (as ideal of A/G) generated by its non-trivial minimal projections.

Equivalently, (A ∩ K)/G is (as ideal of A/G) generated by its non-trivial projections of rank one (Proposition 3.14 and Theorem 3.15).

Structure of (A ∩ K)/G. To get a complete description of the structure of the ideal (A ∩ K)/G in case A is a fractal algebra, we need the following result a proof of which is in [5], Theorem 4.

Theorem 3.21 Let 𝔅 be a unital C^{∗}-algebra and k a non-zero rank one element of 𝔅. Further let Id (k) denote the smallest closed ideal of 𝔅 which contains k. Then there exists an irreducible representation (H, π) of 𝔅 such that

π(Id (k)) = K(H) and Ker(π|_{Id (k)}) = {0}.

Consequently, if k_{1} and k_{2} are non-trivial rank one elements in a C^{∗}-algebra 𝔅, then

either Id (k_{1}) = Id (k_{2}) or Id (k_{1}) ∩ Id (k_{2}) = {0}.  (19)

This alternative suggests to introduce an equivalence relation ∼ in the set of all non-trivial rank one elements in (A ∩ K)/G as follows:

k_{1} ∼ k_{2} if Id (k_{1}) = Id (k_{2}).

Let T refer to the set of all equivalence classes with respect to this relation and, for each t ∈ T, let I_{t} stand for the smallest closed ideal of A/G which is generated by a non-trivial rank one element k ∈ t.

Theorem 3.22 Let A be a unital and fractal C^{∗}-subalgebra of F which contains the ideal G.

(a) Each ideal I_{t} is ^{∗}-isometric to the ideal K(H_{t}) of all compact operators on a Hilbert space H_{t}.

(b) If t_{1}, …, t_{m} ∈ T and t_{i} ≠ t_{j} for i ≠ j, then (I_{t_{1}} + … + I_{t_{m−1}}) ∩ I_{t_{m}} = {0}.

(c) (A ∩ K)/G is the smallest closed ideal of A/G which contains every ideal I_{t}.

(d) There is a bijection between T and the spectrum of (A ∩ K)/G.

Proof. Assertion (a) follows from Theorem 3.21, assertion (b) is proved in [5], and (c) is a consequence of (a) and of Theorem 3.20.

Now let (H, W) be an irreducible representation of (A ∩ K)/G. Then there exists a non-trivial minimal projection p ∈ (A ∩ K)/G such that W does not vanish on Id (p) (otherwise W would be the zero representation due to (c)). Hence, W |_{Id (p)} is an irreducible representation of Id (p). Suppose that W (q) ≠ 0 for another non-trivial minimal projection q ≁ p. Then Id (p) ∩ Id (q) = {0}, i.e. qk = kq = 0 for all k ∈ Id (p). Consequently,

W (k)W (q) = W (kq) = 0 = W (qk) = W (q)W (k),

i.e. Im W (q) is an invariant subspace for W (Id (p)). Since W |_{Id (p)} is an irreducible representation, this implies that Im W (q) = {0} (which is in contradiction with our assumption W (q) ≠ 0) or that Im W (q) = H. The latter is impossible if dim H ≥ 2, since W (q) is an operator of rank one (cp. [5], Theorem 3).

It remains to consider the case when Im W (q) = H and dim H = 1. Then, necessarily, W (p) = 1 and W (q) = 1, hence W (pq) = 1 in contradiction to pq = 0.

Thus, for every irreducible representation (H, W) of (A ∩ K)/G, there is exactly one t ∈ T such that W |_{I_{t}} ≠ 0. Conversely, since I_{t} ≅ K(H) by (a), it is evident that each ideal I_{t} determines (up to unitary equivalence) exactly one irreducible representation (H, W) of (A ∩ K)/G with W |_{I_{t}} ≠ 0.

We still mention the following consequence of the preceding theorem. Its proof is in [5], Theorems 5 and 6. For t ∈ T, we let W_{t} denote the (essentially unique) irreducible representation from I_{t} to the ideal K(H_{t}) the existence of which is guaranteed by assertion (a) of the preceding theorem. The extensions of W_{t} onto the algebras A/G and A will again be denoted by W_{t} for simplicity.

Theorem 3.23 (Lifting theorem.) Let A and T be as before.

(a) An element a ∈ A/G is invertible in A/G if and only if the operators W_{t}(a) are invertible in L(H_{t}) for every t ∈ T and if the coset a+(A∩K)/G is invertible in the quotient algebra (A/G)/((A ∩ K)/G).

(b) Let a ∈ A/G be invertible modulo (A ∩ K)/G. Then all operators W_{t}(a) are Fredholm, and there are only finitely many t ∈ T for which W_{t}(a) is not invertible.

4 Fredholm theory in fractal algebras

Again let A be a unital and fractal C^{∗}-subalgebra of F which contains the ideal G. The aim of this section is to specify the general Fredholm theory to the case of fractal algebras and to derive a formula for the α-number of a fractal Fredholm sequence. Our starting point is the following result on the lifting of minimal projections.

Let p ∈ (A ∩ K)/G be a non-trivial minimal projection. Then there is an essential projection (Π_{n}) ∈ A ∩ K which lifts p (Corollary 3.12). We will see now that the behaviour of the sequence (dim Im Π_{n}) for large n depends only on the coset of p with respect to the equivalence relation ∼.

Theorem 4.1 Up to a sequence of integers which tends to zero, the sequence (dim Im Π_{n})_{n≥1} is uniquely defined by the coset of p with respect to the equivalence relation ∼.

We prepare the proof by two simple lemmata.

Lemma 4.2 Let 𝔅 be a C^{∗}-algebra, and let p, q be rank one projections in 𝔅 with q ∈ Id (p). Then there is a c ∈ Id (p) such that q = c^{∗}pc.

Proof. By Theorem 3.21, there is a ^{∗}-isomorphism W from Id (p) onto K(H) for some Hilbert space H, and W (p) is a non-zero projection of rank one on H (see [5], Theorem 3). Thus, there is a y ∈ H with ∥y∥ = 1 such that

W (p)x = ⟨x, y⟩y for all x ∈ H.

Analogously (recall that Id (p) = Id (q) by (19)), there is a z ∈ H with ∥z∥ = 1 such that

W (q)x = ⟨x, z⟩z for all x ∈ H.

Define C ∈ K(H) by

Cx := ⟨x, z⟩y for x ∈ H.

Then C^{∗}x = ⟨x, y⟩z and C^{∗}W (p)Cx = W (q)x for all x ∈ H as one easily checks. Since W is a bijection, there is exactly one c ∈ Id (p) with W (c) = C, and this element c satisfies q = c^{∗}pc.

Lemma 4.3 Let P and Q be orthogonal projections on a Hilbert space H and let A, B ∈ L(H) be operators such that ∥Q − AP B∥ < 1. Then

dim Im Q ≤ dim Im P.

Proof. By (16), dim Im Q = dim Im QAP BQ, whence the assertion.

Proof of Theorem 4.1. What we have to show is: If p ∼ q ∈ (A ∩ K)/G are non-trivial minimal projections with liftings (Π_{n}^{p}) and (Π_{n}^{q}), respectively, then

dim Im Π_{n}^{p} = dim Im Π_{n}^{q} for all sufficiently large n.

First let (Π_{n}^{1}) and (Π_{n}^{2}) be liftings of one and the same minimal projection p. Then Π_{n}^{1} = Π_{n}^{2} + G_{n} with a sequence (G_{n}) ∈ G. From (16) we conclude that dim Im Π_{n}^{1} = dim Im Π_{n}^{2} for all n with ∥G_{n}∥ < 1.

Let now (Π_{n}^{p}) and (Π_{n}^{q}) be liftings of p ≠ q. By Lemma 4.2, there is a c ∈ Id (p) = Id (q) such that q = c^{∗}pc. Let (C_{n}) ∈ A ∩ K be a lifting of c. Then, clearly,

Π_{n}^{q} = C_{n}^{∗}Π_{n}^{p}C_{n} + G_{n}

with a sequence (G_{n}) ∈ G. For n sufficiently large, ∥G_{n}∥ < 1, and for these n we conclude from the preceding lemma that dim Im Π_{n}^{q} ≤ dim Im Π_{n}^{p}. Now change the roles of p and q to get the reverse inequality.

Thus, with every t ∈ T = Spec (A ∩ K)/G, we can associate a sequence (α_{n}^{t})_{n≥1} by choosing a non-trivial minimal projection p ∈ I_{t} with lifting (Π_{n}^{p}) and by setting α_{n}^{t} := dim Im Π_{n}^{p}. Recall that this sequence is uniquely determined up to a sequence tending to zero.

Let now A := (A_{n}) ∈ A be a Fredholm sequence, that is, (A_{n}) is invertible modulo K. Then the sequence A is also invertible modulo A ∩ K, and from Theorem 3.23 (b) we conclude that the operators W_{t}(A) are Fredholm for every t ∈ T and that only a finite number of them is not invertible. Thus, the following sums are actually finite:

α_{n}(A) := ∑_{t∈T} α_{n}^{t} dim Ker W_{t}(A).  (20)

Evidently, (20) determines the sequence (α_{n}(A)) only up to a sequence tending to zero. The main result of the present paper is the following.

Theorem 4.4 Let A be a unital and fractal C^{∗}-subalgebra of F which contains the ideal G. If A = (A_{n}) is a Fredholm sequence in A, then

lim_{n→∞} σ_{α_{n}(A)}^{(n)} = 0,  lim inf_{n→∞} σ_{α_{n}(A)+1}^{(n)} > 0  (21)

where 0 ≤ σ_{1}^{(n)} ≤ … ≤ σ_{n}^{(n)} are the singular values of A_{n}.

Let us add a few comments on this result before its proof starts.

• For every Fredholm sequence A = (A_{n}) ∈ A, the number

α(A) := lim sup_{n→∞} α_{n}(A)  (22)

is finite. Since (α_{n}(A)) is a sequence of non-negative integers, there is an infinite subsequence of (α_{n}(A)) the entries of which are constant and equal to α(A). This shows that

lim inf_{n→∞} σ_{α(A)}^{(n)} = 0,  lim inf_{n→∞} σ_{α(A)+1}^{(n)} > 0,  (23)

i.e. the number α(A) defined by (22) is exactly the α-number of the Fredholm sequence A as defined in Section 2. In that sense, one can consider (21) as an essential refinement of (10).

• The appearence of the limit in (21) shows that the singular values of a fractal Fredholm sequence (A_{n}) possess the splitting property, i.e. the singular values of A_{n} are contained in [0, ε_{n}] ∪ [δ, ∞) where ε_{n} → 0 as n → ∞ and where δ > 0. Thus, the coset (A_{n}) + G is Moore-Penrose invertible in F/G which is in perfect accordance with the Moore-Penrose invertibility (or, equivalently, with the closedness of the range) of a Fredholm operator on a Hilbert space.

• Observe that the number of the singular values of A_{n} which lie in [0, ε_{n}] can depend on n (it is just given by the number α_{n}(A) in (21)). It is not hard to construct fractal algebras with arbitrarily prescribed sequences (α_{n}^{t}). Consider, for example, for every c ∈ ℂ and every K ∈ K(l^{2}), the sequence (A_{n}) with

A_{n} := \{ \begin{matrix}cP_{n} + P_{n}KP_{n} & if n is even \\ cP_{n} + P_{n}KP_{n} + R_{n}KR_{n} & if n is odd \end{matrix}  (24)

where P_{n}, R_{n} : l^{2} → l^{2} are the operators mapping (x_{1}, x_{2}, …) to

(x_{1}, …, x_{n}, 0, 0, …) and (x_{n}, …, x_{1}, 0, 0, …),

respectively. The smallest closed subalgebra A of F which contains all of these sequences is unital and symmetric, and it contains the ideal G. Further, the mapping

W : A → L(l^{2}), (A_{n}) ↦ s-lim A_{n}P_{n}

is a representation of A which maps the sequence (24) to the operator cI + K. In particular, W is an irreducible representation of A. It is also easy to see that the invertibility of cI + K implies the stability of the sequence (24). Since every sequence in A is a sum of a sequence of the form (24) on of a sequence in G, we conclude that a sequence (A_{n}) ∈ A is stable if and only if W (A_{n}) is invertible. This shows that A is a fractal algebra and that the quotient algebra A/G is ^{∗}-isometric to the algebra ℂI +K(l^{2}).

Let (Π_{n}) be the sequence of the form (24) where c = 0 and K = P_{1}. Then p := (Π_{n}) + G is a non-trivial minimal projection in (A ∩ K)/G, and one has

dim Im Π_{n} := \{ \begin{matrix}1 & if n is even \\ 2 & if n is odd. \end{matrix}

• In [7, 8], we exclusively considered algebras A of approximation sequences having the property that every non-trivial minimal projection in (A∩K)/G lifts to a sequence of projections of rank one. Similarly, in [5], we studied the Fredholm theory for sequences in fractal algebras under the additional hypothesis that the ideal A ∩ K is generated by sequences of projections of rank one. Thus, in both cases, the numbers α_{n}^{t} are independent on n and can be chosen to be 1 for all n. Under these assumptions, the formula (20) reduces to

α(A) := ∑_{t∈T} dim Ker W_{t}(A)

as it occured in [7, 5].

The remainder of this section is devoted to the proof of Theorem 4.4. Once we have derived the following proposition, the main steps in the proof of Theorem 4.4 will be as in [5].

Proposition 4.5 Let q, r ∈ F/G be projections with qr = rq = 0, and let (R_{n}) ∈ F be a lifting of r (i.e. all matrices R_{n} are orthogonal projections, and (R_{n}) + G = r). Then there exists a projection (Q_{n}) which lifts q such that

R_{n}Q_{n} = Q_{n}R_{n} = 0 for all sufficiently large n.  (25)

Proof. Let (Q′_{n}) be a lifting of q (which exists by Corollary 3.12) and consider the matrices S_{n} := (I_{n} − R_{n})Q′_{n}(I_{n} − R_{n}). The coset modulo G of the sequence (S_{n}) is

(e − r)q(e − r) = q − rq − qr + rqr = q.  (26)

Hence ((e − r)q(e − r))^{2} = (e − r)q(e − r), i.e. ∥S_{n} − S_{n}^{2}∥ → 0. Choose n_{0} such that ∥S_{n} − S_{n}^{2}∥ < 1/4 for all n ≥ n_{0}. Then, by Proposition 3 in [7], for all n ≥ n_{0} there is a G_{n} which belongs to the C^{∗}-algebra generated by I_{n} − R_{n} and S_{n} such that S_{n} + G_{n} is a projection and

∥G_{n}∥ ≤ 2 ∥S_{n} − S_{n}^{2}∥.

Define Q_{n} := Q′_{n} if n < n_{0} and Q_{n} := S_{n} + G_{n} if n ≥ n_{0}. Then, for n ≥ n_{0},

∥Q_{n} − Q′_{n}∥ = ∥S_{n} − Q′_{n} + G_{n}∥ ≤ ∥(I_{n} − R_{n})Q′_{n}(I_{n} − R_{n}) − Q′_{n}∥ + ∥G_{n}∥ → 0

because of (26). Hence, (Q_{n}) is also a lifting of q, and this lifting satisfies (25):

R_{n}Q_{n} = R_{n}(I_{n} − R_{n})Q′_{n}(I_{n} − R_{n}) + R_{n}G_{n} = 0

(recall that R_{n}G_{n} = 0 since G_{n} ∈ alg (I_{n} − R_{n}, S_{n})).

Let now (A_{n}) ∈ A be a Fredholm sequence, i.e. (A_{n}) is invertible modulo the ideal A ∩ K. Then, by Theorem 3.23, the operators W_{t}(A_{n}) are Fredholm for every t ∈ T and there are only finitely many of these operators which are not invertible. Let P_{Ker W_{t}(A_{n})} denote the orthogonal projection from H_{t} onto the kernel of W_{t}(A_{n}) (only a finite number of these projections is not zero). Decompose each of these projections into a sum of dim Ker W_{t}(A_{n}) orthogonal projections of range dimension one:

P_{Ker W_{t}(A_{n})} = ∑_{i=1}^{dim Ker W_{t}(A_{n})} P_{i,t}

such that P_{i,t}P_{j,t} = P_{j,t}P_{i,t} = 0 whenever i ≠ j (for example, by choosing an orthonormal basis in Ker W_{t}(A_{n}) and by defining P_{i,t} as the orthogonal projection onto the ith element of this basis). Since W_{t} is an isomorphism between I_{t} and K(H_{t}), every projection P_{i,t} corresponds uniquely to a coset p_{i,t} ∈ I_{t}. Clearly, p_{i,t}p_{j,t} = p_{j,t}p_{i,t} = 0 if i ≠ j. Since I_{s} ∩ I_{t} = {0} for all s, t ∈ T with s ≠ t, one has moreover

p_{i,t}p_{j,s} = p_{j,s}p_{i,t} = 0 whenever (i, t) ≠ (j, s).  (27)

Using Proposition 4.5 it is not hard to see that the cosets p_{i,t} can be lifted to sequences (P_{n}^{i,t}) in such a way that

P_{n}^{i,t} P_{n}^{j,s} = P_{n}^{j,s} P_{n}^{i,t} = 0 whenever (i, t) ≠ (j, s)  (28)

for all sufficiently large n. Indeed, suppose we have already lifted a finite number of the projections p_{i,t} in such a way that (28) holds. Let (R_{n}) be the sum of these liftings. Then every R_{n} with n large enough is an orthogonal projection due to (28). If q is an element of the family of the projections p_{i,t} which has not yet been lifted, then we construct a lifting (Q_{n}) of q such that R_{n}Q_{n} = Q_{n}R_{n} = 0 for all sufficiently large n which is possible due to Proposition 4.5. It is obvious that the new set of all liftings (enlarged by (Q_{n})) again satisfies the condition (28). This procedure stops since there is only a finite number of projections to lift.

The orthogonality (28) ensures that the operators

R_{n} := ∑_{t∈T} ∑_{i=1}^{dim Ker W_{t}(A_{n})} P_{n}^{i,t}

are orthogonal projections for all sufficiently large n and that

dim Im R_{n} = ∑_{t∈T} ∑_{i=1}^{dim KerW_{t}(A_{n})} α_{n}^{t} = ∑_{t∈T} α_{n}^{t} dim Ker W_{t}(A) = α_{n}(A)  (29)

with A = (A_{n}) for all sufficiently large n. As in [5] one can now easily check that

(A_{n})^{∗}(A_{n}) + (R_{n}) is a stable sequence and (A_{n})^{∗}(A_{n})(R_{n}) ∈ G.  (30)

Thus, the coset (A_{n}) + G is Moore-Penrose invertible in A/G, and (R_{n}) + G is the associated Moore-Penrose projection. It is shown in Theorem 3 in [7] that then there is a sequence (Π_{n}) of orthogonal projections such that ∥Π_{n} − R_{n}∥ → 0 and that every projection Π_{n} belongs to the C^{∗}-algebra generated by A_{n}^{∗}A_{n} and by the identity matrix I_{n}. From ∥Π_{n} − R_{n}∥ → 0 we conclude that

dim Im Π_{n} = dim Im R_{n} = α_{n}(A) for all sufficiently large n,

and the property Π_{n} ∈ alg (A_{n}^{∗}A_{n}, I_{n}) ensures that the matrices A_{n}^{∗}A_{n} and Π_{n} can be diagonalized simultaneously:

U_{n}^{∗}A_{n}^{∗}A_{n}U_{n} = diag (λ_{1}^{(n)}, λ_{2}^{(n)}, …, λ_{n}^{(n)}),
U_{n}^{∗}Π_{n}U_{n} = diag (π_{1}^{(n)}, π_{2}^{(n)}, …, π_{n}^{(n)})

with unitary matrices U_{n} which we choose in such a way that

0 ≤ λ_{1}^{(n)} ≤ λ_{2}^{(n)} ≤ … ≤ λ_{n}^{(n)}.

The conditions (30) (with Π_{n} in place of R_{n}) imply the existence of a C > 0 such that

λ_{i}^{(n)} + π_{i}^{(n)} ≥ C > 0 for all n ≥ n_{0} and i = 1, …, n,  (31)
λ_{i}^{(n)} · π_{i}^{(n)} < C for all n ≥ n_{0} and i = 1, …, n,  (32)

Let i = i_{n} be such that

λ_{1}^{(n)} ≤ … ≤ λ_{i}^{(n)} < c ≤ λ_{i+1}^{(n)} ≤ … ≤ λ_{n}^{(n)}.

Then, for (31) to hold it is necessary that

π_{1}^{(n)} = … = π_{i}^{(n)} = 1

whereas (32) requires that

π_{i+1}^{(n)} = … = λ_{n}^{(n)} = 0.

Hence, i_{n} = α_{n}(A) and

U_{n}^{∗}Π_{n}U_{n} = diag (1, …, 1, 0, …, 0)

with α_{n}(A) times a 1 and n − α_{n}(A) times a 0. Now the identity (21) follows from the diagonalized form of (30) with Π_{n} in place of R_{n}. This finishes the proof of Theorem 4.4.

References

[1] A. Böttcher, B. Silbermann, Introduction to Large Truncated Toeplitz Matrices. – Springer-Verlag, New York, Berlin, Heidelberg 1999.

[2] R. Hagen, S. Roch, B,. Silbermann, C^{∗}-Algebras and Numerical Analysis. – Marcel Dekker, Inc., New York 2000.

[3] G. K. Pedersen, C^{∗}-Algebras and Their Automorphism Groups. – Academic Press, London, New York, San Francisco 1979.

[4] S. Roch, Algebras of approximation sequences: Fractality. – Preprint 2052(1999) TU Darmstadt, submitted to: Proceedings of the 11. TMP Conference, Chemnitz 1999.

[5] S. Roch, Algebras of approximation sequences: Fredholmness. – Preprint 2048(1999) TU Darmstadt, submitted to: J. Oper. Theory.

[6] S. Roch, B. Silbermann, C^{∗}-algebra techniques in numerical analysis. – J. Oper. Theory 35(1996), 2, 241 – 280.

[7] S. Roch, B. Silbermann, Index calculus for approximation methods and singular value decomposition. – J. Math. Anal. Appl. 225(1998), 401 – 426.

[8] S. Roch, B. Silbermann, A note on singular values of Cauchy-Toeplitz matrices. – Linear Algebra Appl. 275-276(1998), 531 – 536.

AMS Subject classification: primary 46N40, secondary 65R20.

Author’s address:

Technische Universität Darmstadt

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