IMPROPER FILTRATIONS FOR C^{∗}-ALGEBRAS:

SPECTRA OF UNILATERAL TRIDIAGONAL OPERATORS

William Arveson

Department of Mathematics University of California Berkeley, CA 94720 USA

To Béla Szökefalvi-Nagy on the occasion of his eightieth birthday

Abstract. Let A ⊆ B(H) be a C^{∗}-algebra of operators and let P_{1} ≤ P_{2} ≤ … be an increasing sequence of finite dimensional projections in B(H). In a previous paper [3] we developed methods for computing the spectrum of self adjoint operators T ∈ A in terms of the spectra of the associated sequence of finite dimensional compressions P_{n}T P_{n}. In a suitable context, we showed that this is possible when P_{n} increases to

1. In this paper we drop that hypothesis and obtain an appropriate generalization of the main results of [3].

Let P_{+} = lim_{n} P_{n}, H_{+} = P_{+}H. The set A_{+} ⊆ B(H_{+}) of all compact perturbations of operators P_{+}T ↾_{H_{+}}, T ∈ A, is a C^{∗}-algebra which is somewhat analogous to the Toeplitz C^{∗}-algebra acting on H^{2}. Indeed, in the most important examples A is a simple unital C^{∗}-algebra having a unique tracial state, the operators in A are “bilateral”, those in A_{+} are “unilateral”, and there is a short exact sequence of C^{∗}-algebras

0 → K → A_{+} → A → 0

whose features are central to this problem of approximating spectra of operators in A in terms of the eigenvalues of their finite dimensional compressions along the given filtration.

This work was undertaken in order to develop an efficient method for computing the spectra of discretized Hamiltonians of one dimensional quantum systems in terms of “unilateral” tridiagonal n × n matrices. The solution of that problem is presented in Theorem 3.4.

1991 Mathematics Subject Classification. Primary 46L40; Secondary 81E05. Key words and phrases. spectrum, matrix approximations, quantum mechanics, tridiagonal

operators. This research was supported in part by NSF grants DMS89-12362 and DMS-9212893

Typeset by AMS-TEX

1. Introduction.

There is a natural way to discretize the Hamiltonian of a one dimensional quantum system in such a way that a) the uncertainty principle is preserved, and b) the discretized operator is in a form appropriate for carrying out numerical studies [1,2]. In a suitable representation, this discretized Hamiltonian is a bilateral tridiagonal operator T of the form

1.1  T e_{n} = e_{n−1} + v(cos(nθ))e_{n} + e_{n+1}

where {e_{n} : n ∈ ℤ} is an orthonormal basis for a Hilbert space H, and v is a real function in C[−1, +1]. v is a rescaled version of the potential of the system. The number θ is related to the numerical step size, and may properly be considered an irrational multiple of π [1].

The spectra of these operators cannot be computed explicitly. For example, even in the simplest case where v(x) = 2x (in physical terms, the case of a one dimensional harmonic oscillator) it is not yet known if the spectrum of T is totally disconnected. Choi, Elliott and Lui have shown that this is the case when θ/π is a Liouville number [9], but despite significant progress since the early eighties this ‘Ten Martini’ problem of Mark Kac remains open in general [5,6,7,8,9,14]. To our knowledge, essentially nothing is known in the case of an arbitrary continuous potential v. Thus it is of interest to understand how one might carry out numerical calculations of the spectrum using established techniques for computing eigenvalues of self-adjoint n × n matrices, and to describe the precise sense in which these eigenvalues converge to the spectrum of T as n → ∞. A general approach to this problem was worked out in [3].

Here we will consider operators T of the following somewhat more general tridiagonal form

1.2  T e_{n} = e_{n−1} + d_{n}e_{n} + e_{n+1}

where {d_{n} : n ∈ ℤ} is a bounded bilateral sequence of real numbers which is almost periodic but not periodic. A sequence d_{n} = v(cos(nθ)) as in formula 1.1 will satisfy this hypothesis whenever θ/π is irrational and v ∈ C[−1, +1] is not a constant. In [3], we showed that such a program will succeed for operators of the form 1.1 by establishing a formula which shows how the spectrum of T is determined by the eigenvalues of the sequence of 2n + 1 × 2n + 1 matrices

( \begin{matrix}d_{−n} & 1 & 0 & … & 0 & 0 \\ 1 & d_{−n+1} & 1 & … & 0 & 0 \\ 0 & 1 & d_{−n+2} & … & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ & ⋮ \\ 0 & 0 & 0 & … & d_{n−1} & 1 \\ 0 & 0 & 0 & … & 1 & d_{n} \end{matrix} ) .

However, it is more convenient to compute with the sequence of “unilateral” n × n compressions of T

1.3  T_{n} = ( \begin{matrix}d_{1} & 1 & 0 & … & 0 & 0 \\ 1 & d_{2} & 1 & … & 0 & 0 \\ 0 & 1 & d_{3} & … & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ & ⋮ \\ 0 & 0 & 0 & … & d_{n−1} & 1 \\ 0 & 0 & 0 & … & 1 & d_{n} \end{matrix} ) .

IMPROPER FILTRATIONS FOR C^{∗}-ALGEBRAS 3

In fact, the latter has been implemented for the Ten Martini operators of Mark Kac in a Macintosh program [4], and it is effective. In this paper we will prove that this procedure will always be successful by establishing a variation of the formula [3, formula 5.1] which is applicable to the sequence of unilateral compressions 1.3.

In this setting the formula takes the following form. Let T_{n} be as in 1.3, n = 1, 2, … and let λ_{1}^{n} < λ_{2}^{n} < ⋯ < λ_{n}^{n} be the eigenvalue list of T_{n} (recall that the eigenvalues are distinct because T_{n} is tridiagonal [12, p. 124]). Then there is a probability measure μ_{T}, whose closed support is precisely the spectrum of T, such that for every continuous function f ∈ C_{0}(ℝ) we have

1.4  lim_{n→∞} \frac{1}{n}(f(λ_{1}^{n}) + f(λ_{2}^{n}) + ⋯ + f(λ_{n}^{n})) = ∫_{−∞}^{+∞} f(x) dμ_{T} (x).

It follows that a real number λ is in the spectrum of T iff the eigenvalues of the sequence of matrices T_{1}, T_{2}, … accumulate rapidly enough in any fixed neighborhood U of λ so that their density in U is positive. We also describe the measure μ_{T} in terms of of the unique tracial state on a certain C^{∗}-algebra associated with the operator T. See Theorem 3.4 and the remarks following it.

Formula 1.4 shows quite explicitly how one should go about calculating approximations to the measure μ_{T}, and it is these calculations that have been implemented in the program [4] cited above.

2. Improper filtrations.

In this section we analyze the situation described above in a rather abstract context, and establish an appropriate generalization of the main result of [3]. Let us first recall some terminology from [3]. Let H be a separable infinite dimensional Hilbert space. A filtration of H is an increasing sequence H_{1} ⊆ H_{2} ⊆ … of finite dimensional subspaces of H with the property that ∪_{n}H_{n} is dense in H.

Here, we will simply drop the latter condition on the sequence H_{n}. In order to avoid annoying trivialities, we do require that the closed subspace

H_{+} = \overline{∪_{n}H_{n}}

generated by the filtration should be infinite dimensional. The filtration is called proper or improper according as H_{+} = H or H_{+} ≠ H. Let P_{n} denote the projection onto H_{n}. As in [3] we may introduce the notion of the degree of an operator T ∈ B(H) relative to a filtration,

deg(T) = sup_{n} rank(P_{n}T − T P_{n}).

The degree of an operator is a nonnegative integer or +∞, and properties established in [3] such as deg(ST) ≤ deg(S)+deg(T) persist in this more general context.

For example, if {e_{1}, e_{2}, … } is an infinite orthonormal sequence in H and H_{n} is the linear span of {e_{1}, …, e_{n}}, then {H_{1} ⊆ H_{2} ⊆ … } is a filtration which is typically improper. As we pointed out in [3] in the case of proper filtrations, any operator whose matrix relative to this orthonormal set is band-limited must have finite degree. Thus, it is as appropriate here as it was in [3] to think of finite degree operators as abstractions of band-limited matrices. Any self-adjoint operator T determines a multitude of filtrations relative to which it has degree 1: choose any vector e in H such that {e, T e, T^{2}e, … } is linearly independent and put H_{n} = [e, T e, T^{2}e, …, T^{n−1}e]. This filtration is proper iff e is a cyclic vector for T. Of course, in general there can be very irregular jumps in the dimensions of the subspaces of a filtration.

Finally, if A is a C^{∗}-algebra of operators acting on H and F = {H_{1} ⊆ H_{2} ⊆ … } is a filtration of H, then F is called an A-filtration if the set of all finite degree operators in A is norm-dense in A. If A is generated by a set of finite degree operators, then F is called an A-filtration [3]. We begin with two propositions which concern compact perturbations.

Proposition 2.1. Let A ⊆ B(H) be a C^{∗}-algebra, let {H_{1} ⊆ H_{2} ⊆ … } be an A-filtration and let P_{+} be the projection onto H_{+}. Then P_{+}A − AP_{+} is compact for every A ∈ A.

proof. Since the finite degree operators in A are norm-dense and P_{+}A − AP_{+} is norm-continuous in A, it suffices to show that

rank(P_{+}A − AP_{+}) ≤ deg(A) < ∞

for every A ∈ A having finite degree. Since

rank(P_{n}A − AP_{n}) ≤ deg(A)

for every n = 1, 2, … and since P_{n}A − AP_{n} converges to P_{+}A − AP_{+} in the strong operator topology, the propostion follows from the fact that the rank function is weakly lower semicontinuous on B(H).

Since we lack an appropriate reference for the latter, we sketch a short proof for completeness. The assertion is that for every positive integer k, the set of operators

S_{k} = {T ∈ B(H) : rank(T) ≤ k}

is weakly closed. Let K be the Hilbert space antisymmetric tensor product of k + 1 copies of H. A subspace M of H has dimension at most k iff

M^{∧k+1} = M ∧ M ∧ ⋯ ∧ M = 0.

Thus, an operator T belongs to S_{k} iff

T ξ_{1} ∧ T ξ_{2} ∧ ⋯ ∧ T ξ_{k+1} = 0

for every choice of vectors ξ_{1}, ξ_{1}, … ξ_{k+1} in H. Using the fact that K is spanned by vectors of the form η_{1} ∧ η_{2} ∧ ⋯ ∧ η_{k+1} we see that T belongs to S_{k} iff for every set of 2k + 2 vectors ξ_{1}, …, ξ_{k+1}, η_{1}, … η_{k+1} in H, the polynomial p(T) defined by

p(T) =< T ξ_{1} ∧ T ξ_{2} ∧ ⋯ ∧ T ξ_{k+1}, η_{1} ∧ η_{2} ∧ ⋯ ∧ η_{k+1} >

is zero. Taking note of the inner product in K, we see that p(T) is the determinant of the k + 1 × k + 1 matrix whose ijth term is < T ξ_{i}, η_{j} >, and is therefore weakly continuous. Hence S_{k} is weakly closed □

If one is given a positive linear functional ρ on a C^{∗}-algebra C, then the GNS construction gives rise to a representation π_{ρ} of C on a Hilbert space H_{ρ}. We will say that ρ is infinite dimensional if H_{ρ} is an infinite dimensional Hilbert space.

Proposition 2.2. Let A ⊆ B(H) be a unital C^{∗}-algebra having a unique tracial state τ, and assume τ is infinite dimensional. Then τ(K) = 0 for every compact operator K ∈ A.

proof. Let π be the representation of A obtained via the GNS construction on τ. Then M = π(A)′′ is a von Neumann algebra with the property that the natural extension τ̃ of τ to M is a normal tracial state of M.

Notice that M is a factor. Indeed, if e is any nonzero central projection in M then note first that t = τ̃(e) > 0. Indeed, by the Schwarz inequality, τ̃(e) = 0 implies τ̃(aeb) = τ(eba) = 0 for every a, b ∈ M, and hence e = 0 because τ̃ is a vector state of M which is associated with a cyclic vector. In order to show that e = 1, we define a tracial state ρ of A by

ρ(A) = \frac{1}{t}τ̃(π(A)e).

Because of uniqueness, we have ρ = τ. From this, together with the fact that π(A) is strongly dense in M, it follows that τ̃(ae) = tτ̃(a) for every a ∈ M. By an argument similar to the one given above to show that e ≠ 0, we find that e−t1 = 0, hence e = t1. Thus t = 1 and e = 1.

We may conclude that M is a finite factor. Now let J be the ideal of all compact operators in A. J is a C^{∗}-algebra of compact operators, and hence it must decompose uniquely into a C^{∗}-algebraic direct sum of elementary C^{∗}-algebras

J = ∑_{k} J_{k}.

It suffices to show that τ(J_{k}) = 0 for every k. Contrapositively, assume τ(J_{k}) ≠ 0 for some k. Then π(J_{k}) must be nonzero too, and its weak closure in M must be all of M since it is a nonzero weakly closed ideal in a factor. Because τ is an infinite dimensional state M is infinite dimensional and therefore so is J_{k}. But an infinite-dimensional elementary C^{∗}-algebra has no nonzero bounded traces. Hence the restriction of τ to J_{k} must be zero □

Remark. Perhaps it is worth pointing out that Proposition 2.2 becomes false if one deletes the hypothesis that τ is an infinite dimensional state. For example, choose any C^{∗}-algebra of the form A = B ⊕ C where B is a unital C^{∗}-algebra having no nonzero finite traces (such as a Cuntz algebra) and C is an n × n matrix algebra. Certainly A has a unique tracial state τ, and there is an obvious faithful representation of A which contains finite rank operators not annihilated by τ.

Throughout the remainder of this section, we assume that we are given a unital C^{∗}-algebra A which has a unique tracial state τ. Every self-adjoint operator A ∈ A gives rise to a probability measure μ_{A} on the real line by way of

∫_{−∞}^{+∞} f(x) dμ_{A}(x) = τ(f(A)), f ∈ C_{0}(ℝ).

μ_{A} is called the spectral distribution of A [3, section 4]. If τ is faithful then the closed support of μ_{A} is precisely the spectrum of A. Let F = {H_{1} ⊆ H_{2} … } be a perhaps improper A-filtration and let H_{+} and P_{+} have the meaning assigned above. The following result is a generalization of theorem 4.5 of [3] to the case of improper filtrations.

Theorem 2.3. Let A ⊆ B(H) be a C^{∗}-algebra having a unique tracial state, let {H_{1} ⊆ H_{2} ⊆ … } be an A-filtration, and assume that the subspace H_{+} has the following property

2.4  A↾_{H_{+}}= compact ⟹ A = compact,

for every A ∈ A. Let μ_{A} be the spectral distribution of a self-adjoint operator A ∈ A, and let [a, b] be the smallest closed interval containing σ(A). For each n, let d_{n} = dim H_{n} and let λ_{1}^{n}, λ_{2}^{n}, …, λ_{d_{n}}^{n} be the eigenvalue list of A_{n} = P_{n}A ↾_{H_{n}}. Then for every f ∈ C[a, b],

lim_{n→∞} \frac{1}{d_{n}} (f(λ_{1}^{n}) + f(λ_{2}^{n}) + ⋯ + f(λ_{d_{n}}^{n})) = ∫_{a}^{b} f(x) dμ_{A}(x).

Remarks. We will write K and K_{+} for the respective algebras of compact operators on H and H_{+}. Notice that the hypothesis 2.4 is satisfied for trivial reasons if the filtration is proper, and in that case Theorem 2.3 reduces to [3, Theorem 4.5]. 2.4 is also automatic when A is simple. Indeed, if T is an operator in A whose restriction to H_{+} is compact, then T belongs to the kernel of the ∗-homomorphism ω : A → B(H_{+})/K_{+} given by ω(X) = P_{+}X ↾_{H_{+}} + K_{+}. Since ω is not the zero homomorphism (H_{+} is infinite dimensional), it must be injective by simplicity. Hence T = 0.

proof of Theorem 2.3. Let A_{+} be the set of operators in B(H_{+}) of the form

A_{+} = P_{+}A↾_{H_{+}} + K,

where A ∈ A and K is a compact operator on H_{+}. We will show first that 2.5 A_{+} is a C^{∗}-algebra algebra having a unique tracial state τ_{+}. 2.6 τ_{+} is related to the tracial state τ of A by way of

τ_{+}(P_{+}A↾_{H_{+}} + K) = τ(A), A ∈ A, K ∈ K_{+}.

If A and B are two operators in A with A_{+} and B_{+} denoting their respective compressions to H_{+}, then from Proposition 2.1 we see that

A_{+}B_{+} = P_{H_{+}}AB ↾_{H_{+}} + compact.

It follows that A_{+} is a unital ∗-subalgebra of B(H_{+}) which contains all compact operators. Thus, to prove that A_{+} is a C^{∗}-algebra, it suffices to show that its image in the Calkin algebra B(H_{+})/K_{+} of H_{+} is norm-closed. But the map π : A → B(H_{+})/K_{+} defined by

π(A) = P_{H_{+}}A↾_{H_{+}} + K_{+}

is clearly a unital ∗-homomorphism whose range is the image of A_{+} in B(H_{+})/K_{+}. Since the range of π must be norm-closed, we conclude that A_{+} is norm-closed.

In order to prove 2.5, we first exhibit a tracial state of A_{+}. Let σ : A → B(H_{+})/K_{+} be the map defined by

σ(A) = P_{+}A↾_{H_{+}} + K_{+}.

By Proposition 2.1, σ is a ∗-homomorphism of unital C^{∗}-algebras, and its range is precisely the image of A_{+} in the Calkin algebra. By 2.4, the kernel of σ is A ∩ K; indeed σ(A) = 0 iff σ(A^{∗}A) = 0 iff the restriction of A to H_{+} is compact iff A is compact. Hence σ determines an isomorphism σ̇ of C^{∗}-algebras

σ̇ : A/A ∩ K → A_{+}/K_{+}.

On the other hand, because of Proposition 2.2, τ vanishes on A∩K and thus can be promoted to a tracial state τ̇ on the quotient A/A∩K by way of τ̇(A+A∩K) = τ(A). Thus the composition τ̇ ∘ σ̇^{−1} defines a tracial state of A_{+}/K_{+}. If we now define τ_{+} on A_{+} by τ_{+}(A) = τ̇ ∘ σ̇^{−1}(A + K_{+}), then τ_{+} is certainly a tracial state on A_{+}. After unravelling its definition we find that τ is related to τ_{+} by formula 2.6

2.7  τ_{+}(P_{+}A↾ + K_{+}) = τ(A),  A ∈ A.

It remains to show that τ_{+} is the only tracial state on A_{+}. To see that, let ρ be a tracial state on A_{+}. It is clear that the restriction of ρ to K_{+} must be zero, because there are no nonzero bounded traces on the algebra of compact operators on an infinite dimensional Hilbert space. Hence we may promote ρ to a tracial state ρ̇ on A_{+}/K_{+} by

ρ̇(T + K_{+}) = ρ(T), T ∈ A_{+}.

Making use of the isomorphism σ : A/A ∩ K → A_{+}/K_{+} exhibited above, we define a state ρ_{0} on A by

2.8  ρ_{0}(A) = ρ̇ ∘ σ(A + A ∩ K) = ρ(P_{+}A↾_{H_{+}} + K_{+}).

ρ_{0} is clearly a trace on A, hence ρ_{0} = τ. 2.7 and 2.8 together show that ρ is the trace τ_{+} exhibited above. Hence both assertions 2.5 and 2.6 are established.

It remains to deduce Theorem 2.3 from these two assertions and the results of [3]. For that, we view F_{+} = {H_{1} ⊆ H_{2} ⊆ … } as a proper filtration of H_{+}. Notice that if A is any finite degree operator in A and B is an operator in A_{+} of the form B = P_{+}A ↾_{H_{+}} +F where F is a finite rank operator on H_{+}, then for every n = 1, 2, … we have P_{n}B − BP_{n} = P_{+}(P_{n}A − AP_{n})↾_{H_{+}} +P_{n}F − F P_{n}. Hence

rank(P_{n}B − BP_{n}) ≤ sup_{n} rank(P_{n}A − AP_{n}) + 2rank(F) ≤ deg(A) + 2rank(F) < ∞.

It follows that the degree of B is finite. Such operators B are norm-dense in A_{+} since the finite degree operators of A are norm dense in A. We conclude that F is a proper A–filtration of H_{+}.

Thus we are in position to apply [3, Theorem 4.5] to the operator A_{+} = P_{+}A↾_{H_{+}}, the C^{∗}-algebra A_{+} and the filtration F. From [3, Theorem 4.5] we conclude that for every f ∈ C_{0}(ℝ),

2.9  lim_{n→∞} \frac{1}{d_{n}} (f(λ_{1}^{n}) + f(λ_{2}^{n}) + ⋯ + f(λ_{d_{n}}^{n})) = τ_{+}(f(A_{+})).

From Propostion 2.1 and the fact that f is continuous we have

f(A_{+}) = P_{+}f(A)↾_{H_{+}} + compact.

Using the formula 2.6, we can rewrite the right side of 2.9 as follows

τ_{+}(P_{+}f(A)↾_{H_{+}} + compact) = τ(f(A)) = ∫_{a}^{b} f(x) dμ_{A}(x),

which establishes Theorem 2.3 □ 3. Applications.

In order apply the general results of section 2 to establish the formula 1.4, we recall some of the lore of almost periodic sequences of complex numbers. Consider the commutative C^{∗}-algebra l^{∞} = l^{∞}(ℤ). The additive group ℤ acts naturally on l^{∞} by translations, giving a C^{∗}-dynamical system. An element a ∈ l^{∞} is almost periodic if the set of all of ℤ-translates of a is a relatively norm-compact subset of l^{∞}. The set AP (ℤ) of all almost periodic sequences is a unital C^{∗}-algebra-subalgebra of l^{∞} which is invariant under the action of ℤ.

The characters of ℤ are the sequences

e_{λ}(n) = λ^{n}, n ∈ ℤ

where λ is a complex number of absolute value 1. Every e_{λ} is almost periodic, and in fact AP (ℤ) is precisely the closed linear span of all characters. More generally, if S is any translation-invariant linear subspace of AP (ℤ), then the spectrum of S is defined as the following subset of the unit circle

σ(S) = {λ ∈ 𝕋 : e_{λ} ∈ S}.

σ(S) is nonempty if S ≠ 0, and more generally every norm-closed translationinvariant subspace S of almost periodic sequences obeys uniform spectral synthesis in that

3.1  S = \overline{span} \{e_{λ} : λ ∈ σ(S)\}.

If, in addition, S is a unital C^{∗}-algebra subalgebra of AP (ℤ) then σ(S) is a subgroup of the circle group 𝕋. Typically, σ(S) fails to be closed in the usual topology of 𝕋; indeed, every subset of 𝕋 arises as the spectrum of some S (so long as one is working with an underlying set theory that is appropriate for functional analysis). It follows easily from this summary that an almost periodic sequence a is periodic iff σ(a) generates a finite subgroup of 𝕋.

Finally, the norm-closed convex hull of the set of translates of any almost periodic sequence a contains exactly one constant sequence M(a)1, and the scalar M(a) is the von Neumann mean of a:

M(a) = lim_{n→∞} \frac{1}{2n + 1}(a_{−n} + a_{−n+1} + ⋯ + a_{n}).

Moreover, the convergence to M(a) is uniform in that

sup_{k∈ℤ} |M(a) − \frac{1}{2n + 1}(a_{k−n} + a_{k−n+1} + ⋯ + a_{k+n})| → 0

as n → ∞. M defines a translation-invariant state of the C^{∗}-algebra AP (ℤ). We may conclude from these facts that if C is any unital translation-invariant C^{∗}– subalgebra of AP (ℤ), then the restriction of M to C is the only translation-invariant state on C. A convenient reference for most of the above is [11].

The following result is a consequence of standard techniques in operator algebras, together with the preceding observations. Since it is the essential link between the applications and the material of section 2, we include a proof.

Proposition 3.2. Let d = {d_{n} : n ∈ ℤ} be a bounded real sequence which is almost periodic but not periodic and let D be the corresponding diagonal operator acting on l^{2}(ℤ). Then the C^{∗}-algebra generated by D and the bilateral shift is a simple C^{∗}-algebra possessing a unique tracial state.

proof. Let U be the bilateral shift acting on l^{2}(ℤ), Uf(n) = f(n − 1), n ∈ ℤ. For f ∈ l^{2}(ℤ) we will write M_{f} for the obvious diagonal operator. Let A be the C^{∗}- algebra generated by M_{d} = D and U. We first give a description of A which is more convenient for our purposes.

Let Δ be the translation-invariant C^{∗}-subalgebra of AP (ℤ) generated by d and the constant sequence 1, and let D be the set of all diagonal operators M_{f}, f ∈ Δ. Since U^{n}M_{f}U^{−n} = M_{f_{n}} where f_{n} is the translate of f by n, it follows that A is the norm-closure of the set of all operators which are finite sums of the form

T = D_{−n}U^{−n} + D_{−n+1}U^{−n+1} + ⋯ + D_{n}U^{n}

where each D_{k} belongs to D and n = 1, 2, . . .. A is therefore the image of the C^{∗}- algebraic crossed product ℤ × Δ under the obvious representation it has on l^{2}(ℤ). Thus, to show that A is simple it suffices to show that ℤ × Δ is simple. By the characterization of simplicity of discrete crossed products given in [13, Theorem 8.11.12], it is enough to show that

(1) the only closed translation-invariant nonzero ideal in Δ is Δ itself.

(2) The Connes spectrum of the action of ℤ on Δ is all of 𝕋.

(1) follows immediately from the preceding remarks. Indeed, the preceding remarks imply that any translation-invariant ideal must contain enough unitary elements to span itself. Hence if it is nonzero it must be all of Δ. To prove (2), note that since (1) is valid and Δ is abelian, the Connes spectrum is identical with the Arveson spectrum, and the latter is simply the closure in 𝕋 of the subgroup

σ(Δ) = {λ ∈ 𝕋 : e_{λ} ∈ Δ}.

But since Δ contains the sequence d which is not periodic, σ(Δ) cannot be a finite subgroup of 𝕋; hence its closure is 𝕋.

It remains to show that the crossed product ℤ × Δ has a unique tracial state. Indeed, the translation-invariant state of Δ gives rise to a tracial state τ of ℤ × Δ, and therefore of A. On the other hand, suppose that ρ is an arbitrary tracial state on A. For each λ ∈ σ(Δ), consider the unitary operator V_{λ} = M_{e_{λ}}. V is a unitary representation of the discrete abelian group σ(Δ), D is spanned by {V_{λ} : λ ∈ σ(Δ)}, and moreover V satisfies the following commutation relation with U

V_{λ}U = λUV_{λ}, λ ∈ σ(Δ).

This implies that for every n ∈ ℤ and D ∈ D we have

ρ(DU^{n}) = ρ(V_{λ}DU^{n}V_{−λ}) = λ^{n}ρ(DU^{n}),

for all λ ∈ σ(Δ). If n ≠ 0 then it follows that ρ(DU^{n}) = 0 = τ(DU^{n}). If n = 0, then the restriction of ρ to D gives rise to a translation-invariant state of Δ and hence it agrees with the restriction of τ to D. Hence ρ = τ □

We can now deduce applications to the problem of computing the spectra of discretized Hamiltonians. Let d = {d_{n} : n ∈ ℤ} be an almost periodic sequence of reals which is not periodic. Consider the Hilbert space H = l^{2}(ℤ), let {e_{n} : n ∈ ℤ} be the obvious basis for H, and let T be the tridiagonal operator defined by 1.2. Let A ⊆ B(H) be the C^{∗}-algebra generated by the diagonal operator M_{d} and the bilateral shift. According to 3.2, A is a simple C^{∗}-algebra with a unique tracial state τ. T certainly belongs to A, and we may consider its spectral distribution μ_{T}, defined by

3.3  ∫_{−∞}^{+∞} f(x) dμ_{T} (x) = τ(f(T)),  f ∈ C_{0}(ℝ).

Theorem 3.4. The spectrum of T is the closed support of μ_{T}. Moreover, for every positive integer n let λ_{1}^{n} < λ_{2}^{n} < ⋯ < λ_{n}^{n} be the eigenvalue list of the symmetric n × n matrix

3.5  ( \begin{matrix}d_{1} & 1 & 0 & … & 0 & 0 \\ 1 & d_{2} & 1 & … & 0 & 0 \\ 0 & 1 & d_{3} & … & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ & ⋮ \\ 0 & 0 & 0 & … & d_{n−1} & 1 \\ 0 & 0 & 0 & … & 1 & d_{n} \end{matrix} ) .

Then for every f ∈ C_{0}(ℝ) we have

lim_{n→∞} \frac{1}{n}(f(λ_{1}^{n}) + f(λ_{2}^{n}) + …, +f(λ_{n}^{n})) = ∫_{−∞}^{+∞} f(x) dμ_{T} (x).

proof. Since A is a simple C^{∗}-algebra, τ must be a faithful trace. Thus it is apparent from definition 3.3 that μ_{T} must be supported precisely on the spectrum of T.

Let F = {H_{1} ⊆ H_{2} ⊆ … } be the filtration of H defined by H_{n} = [e_{1}, e_{2}, …, e_{n}]. Then F is an improper filtration with respect to which the bilateral shift has degree 1 and every diagonal operator has degree 0. Hence F is an A–filtration. The hypothesis 2.4 of Theorem 2.3 is automatic in this case because of the simplicity of A. Thus Theorem 3.4 follows after an application of Theorem 2.3 and its accompanying remarks □

Remarks. We reiterate some remarks from [3] which reveal the significance of Theorem 3.4 for computations. For every n = 1, 2, … and any Borel set S ⊆ ℝ, let N_{n}(S) be the number of eigenvalues of the matrix 2.5 which belong to S. N_{n} is a positive integer-valued measure having total mass n. Theorem 3.5 asserts that the sequence of probability measures n^{−1}N_{n} converges to μ_{T} in the weak^{∗}-topology of C_{0}(ℝ). Notice that this is the same notion of convergence that one has in the central limit theorem, and one may apply similar interpretations in the context of this paper.

For example, suppose that λ is a point in the spectrum of T. Then for every open neighborhood I of λ whose endpoints have μ_{T} –measure zero, and for any positive numbers α, β satisfying α < μ_{T} (I) < β we have

nα ≤ N_{n}(I) ≤ nβ

for sufficiently large n. Notice that since α and β may be chosen arbitrarily close to μ_{T} (I), we have a precise estimate of the rate of growth of N_{n}(I) with n. If, on the other hand, λ is not in the spectrum of T, then for a sufficiently small interval I containing λ we will have

lim_{n→∞} \frac{1}{n}N_{n}(I) = μ_{T} (I) = 0.

Actually, in this case one can say more, namely that the numbers N_{1}(I), N_{2}(I), … are uniformly bounded. Indeed, since A is simple it cannot contain any nonzero compact operators, and hence the spectrum of T is the same as its essential spectrum. The assertion follows from [3, Theorem 3.8]. In light of the fact that there are extremely fast algorithms for calculating the value of N_{n}(S) for large but fixed values of n whenever S has the form S = (−∞, x] for x ∈ ℝ, these remarks provide the basis for a very efficient method of calculating the spectrum of T [4].

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