New York Journal of Mathematics

New York J. Math. 8 (2002) 145–159.

Inductive Limit Algebras from Periodic Weighted Shifts on Fock Space

David W. Kribs

Abstract. Noncommutative multivariable versions of weighted shift operators arise naturally as ‘weighted’ left creation operators acting on the Fock space Hilbert space. We identify a natural notion of periodicity for these Ntuples, and then find a family of inductive limit algebras determined by the periodic weighted shifts which can be regarded as noncommutative multivariable generalizations of the Bunce-Deddens C^{∗}-algebras. We establish this by proving that the C^{∗}-algebras generated by shifts of a given period are isomorphic to full matrix algebras over Cuntz-Toeplitz algebras. This leads to an isomorphism theorem which parallels the Bunce-Deddens and UHF classification scheme.

Contents

1. Introduction 146 2. Noncommutative weighted shifts 147 3. Fock space trees and periodicity 149 4. Main Theorem 151 5. Classification 156 References 158

The primary goal of this paper is to initiate the study of noncommutative multivariable weighted shifts. Almost three decades ago, Bunce and Deddens [3, 4] introduced a family of inductive limit C^{∗}-algebras generated by periodic unilateral weighted shift operators. On the other hand, we now know that noncommutative multivariable versions of unilateral shifts arise in theoretical physics and free probability theory as the so-called left creation operators acting on the full Fock space Hilbert space. There is now an extensive body of research for these operators and the algebras they generate (see [1, 9, 11, 10, 21, 23, 24] for example).

Received July 2, 2002. Mathematics Subject Classification. 46L05, 46L35, 47B37, 47L40. Key words and phrases. Hilbert space, operator, weighted shift, noncommutative multivariable

operator theory, Fock space, creation operators, Cuntz-Toeplitz C*-algebras, K-theory. Partially supported by a Canadian NSERC Post-doctoral Fellowship.

145^{ISSN 1076-9803/02}

In this paper, we introduce a family of C^{∗}-algebras which can be regarded as noncommutative multivariable generalizations of the Bunce-Deddens algebras. In accomplishing this, based on the creation operators, we introduce the concept of a noncommutative multivariable weighted shift and discover a satisfying notion of periodicity based on the structure of Fock space. We characterize these algebras in terms of inductive limits of full matrix algebras over the Cuntz-Toeplitz and Cuntz algebras. This leads to a classification theorem which parallels the classification of UHF algebras by Glimm [16], and the Bunce-Deddens algebras classification [3, 4], by supernatural numbers.

In the opening section we recall the formulation of Fock space and the creation operators. We also quickly review the basics of the Cuntz and Cuntz-Toeplitz algebras. In the second section we introduce noncommutative weighted shifts and investigate their basic structure. The third section describes a pictorial method for thinking of these shifts, by using the Fock space ‘tree’ structure. This leads to a natural notion of periodicity, and then we define the C^{∗}-algebras we study in the rest of the paper. The final two sections consist of an in-depth analysis of these algebras. Most importantly, we prove they are isomorphic to inductive limits of full matrix algebras of distinguished sizes over Cuntz and Cuntz-Toeplitz algebras. Using this characterization we establish a classification theorem based on K-theory for the Cuntz algebras.

1. Introduction

We begin by recalling the formulation of the full Fock space Hilbert space and its associated creation operators. For N ≥ 2, let 𝔽_{N}^{+} be the unital free semigroup on N noncommuting letters {1, 2, …, N}. We denote the unit in 𝔽_{N}^{+} by e. One way to realize N-variable Fock space is as H_{N} = ℓ^{2}(𝔽_{N}^{+}). From this point of view, the vectors {ξ_{w} : w ∈ 𝔽_{N}^{+}} form an orthonormal basis for H_{N} which can be thought of as a generalized Fourier basis. The left creation operators (also known as the Cuntz-Toeplitz isometries we will see below) L = (L_{1}, …, L_{N}) are defined on H_{N} by their actions on basis vectors,

L_{i}ξ_{w} = ξ_{iw}, for 1 ≤ i ≤ N and w ∈ 𝔽_{N}^{+}.

The L_{i} are isometries with pairwise orthogonal ranges for which the sum of the range projections satisfies ∑_{i=1}^{N} L_{i}L_{i}^{∗} = I − P_{e}, where P_{e} = ξ_{e}ξ_{e}^{∗} is the rank one projection onto the span of the vacuum vector ξ_{e}. We will discuss a helpful pictorial method for thinking of the actions of these operators in Section 3.

Most importantly for our purposes, this N-tuple forms the noncommutative multivariable version of a unilateral shift. This claim is well supported by a number of facts. For instance, the unilateral shift is obtained for N = 1, and otherwise each of the L_{i} is unitarily equivalent to a shift of infinite multiplicity. Further, the study of the L_{i} in operator theory and operator algebras was at least partly initiated by the dilation theorem of Frazho [15], Bunce [2] and Popescu [22], which provided the noncommutative multivariable version of Sz.-Nagy’s classical minimal isometric dilation of a contraction [14]. Namely, every row contraction of operators on Hilbert space has a minimal joint dilation to isometries, acting on a larger space, with pairwise orthogonal ranges. The classical Wold decomposition shows that every isometry breaks up into an orthogonal direct sum of a unitary together with copies of the shift. Analogously, Popescu’s version [22] shows that every Ntuple of isometries with pairwise orthogonal ranges decomposes into an orthogonal direct sum of isometries which form a representation of the Cuntz C^{∗}-algebra O_{N} (see below), together with copies of L = (L_{1}, …, L_{N}).

In addition, the wot-closed nonselfadjoint algebras generated by the L_{i} have been shown by Davidson, Pitts, Arias, Popescu and others to be the appropriate noncommutative analytic Toeplitz algebras (see [1, 11, 10, 21, 24]). We also mention that the wot-closed nonselfadjoint algebras generated by the weighted shifts discussed here have been investigated in [19], where a number of results from the single variable setting have been generalized, at the same time exposing new noncommutative phenomena. Finally, we note that compressing the creation operators to symmetric Fock space yields the commutative multivariable shift. The C^{∗}-algebras generated by weighted versions of which were studied in [6] for instance.

The C^{∗}-algebras determined by the isometries L = (L_{1}, …, L_{N}) have also been studied extensively. The C^{∗}-algebra generated by L_{1}, …, L_{N} is called the Cuntz-Toeplitz algebra and is denoted E_{N}. The ideal generated by the rank one projection I − ∑_{i=1}^{N} L_{i}L_{i}^{∗} in E_{N} yields a copy of the compact operators. When this ideal is factored out, the C^{∗}-algebra obtained is the Cuntz algebra O_{N}. It is the universal C^{∗}-algebra generated by the relations

s_{i}^{∗}s_{j} = δ_{ij}1 for 1 ≤ i, j ≤ N and ∑_{i=1}^{N} s_{i}s_{i}^{∗} = 1.

Up to isomorphism, O_{N} is the C^{∗}-algebra generated by any N isometries S = (S_{1}, …, S_{N}) which satisfy these relations, since it is simple.

The K-theory for a C^{∗}-algebra consists of a series of invariants which hold information on equivalence classes of projections in the matrix algebras over the algebra. The K-theory for O_{N} was worked out by Cuntz [5]. In particular, its K_{0} group is the finite abelian group K_{0}(O_{N}) = ℤ/(N − 1)ℤ. In connection with classification results for inductive limits of Cuntz algebras we mention work of Rørdam [25]. We also note that our isomorphism theorem has overlap with work of Evans [13].

2. Noncommutative weighted shifts

From the discussion in the previous section, we are led to the following definition for noncommutative multivariable weighted shifts. We shall drop the multivariable reference for succinctness. We mention that the idea for considering these weighted shifts came during the author’s preparation of [20], where a related class of N-tuples was used in the analysis there.

Definition 2.1. We say that an N-tuple of operators S = (S_{1}, …, S_{N}) acting on a Hilbert space H forms a noncommutative weighted shift if there is a unitary U : H_{N} → H, operators T = (T_{1}, …, T_{N}) on H_{N}, and scalars {λ_{i,w} : 1 ≤ i ≤ N and w ∈ 𝔽_{N}^{+}} such that S_{i} = UT_{i}U^{∗} for 1 ≤ i ≤ N and

T_{i}ξ_{w} = λ_{i,w}ξ_{iw} for 1 ≤ i ≤ N and w ∈ 𝔽_{N}^{+}.

Note 2.2. For brevity, we assume that the weighted shifts T = (T_{1}, …, T_{N}) we consider actually act on Fock space H_{N} = ℓ^{2}(𝔽_{N}^{+}). Further, the proposition below will allow us to make the following simplifying assumption on weights throughout the paper:

Assumption: λ_{i,w} ≥ 0 for 1 ≤ i ≤ N and w ∈ 𝔽_{N}^{+}.

Indeed, every shift is jointly unitarily equivalent to a shift with nonnegative weights:

Proposition 2.3. Suppose T = (T_{1}, …, T_{N}) is a weighted shift with weights {λ_{i,w}}. Then there is a unitary U ∈ B(H_{N}), which is diagonal with respect to the standard basis for H_{N}, such that the weighted shift

(UT_{1}U^{∗}, …, UT_{N}U^{∗})

has weights {|λ_{i,w}|}.

Proof. We build the unitary by inductively choosing scalars μ_{w} and defining Uξ_{w} = μ_{w}ξ_{w}. Put μ_{e} = 1. Let k ≥ 1 and assume the scalars {μ_{w} : |w| = k − 1} corresponding to words of length k − 1 in 𝔽_{N}^{+} (as the empty word, the unit e is taken to have length zero) have been chosen. The scalars {μ_{w} : |w| = k} are obtained in the following manner. For iw ∈ 𝔽_{N}^{+} with |w| = k − 1 and 1 ≤ i ≤ N, choose μ_{iw} ∈ ℂ of modulus one such that

(μ_{w} λ_{i,w}) μ_{iw} ≥ 0.

Now if 1 ≤ i ≤ N and w ∈ 𝔽_{N}^{+} are arbitrary, we have

(UT_{i}U^{∗}) ξ_{w} = μ_{w} UT_{i}ξ_{w} = μ_{w}λ_{i,w}Uξ_{iw}
= (μ_{w}λ_{i,w}μ_{iw}) ξ_{iw}.

This yields the desired conclusion. □

We next present a direct generalization of the factorization of weighted shift operators into products of the unilateral shift and diagonal weight operators.

Proposition 2.4. Let T = (T_{1}, …, T_{N}) be a weighted shift. Then each T_{i} factors as T_{i} = L_{i}W_{i}, where W_{i} is a positive operator which is diagonal with respect to the standard basis for H_{N}. It follows that the norms of the T_{i} and the row matrix T are given by:

(i) ∥T_{i}∥ = sup{λ_{i,w} : w ∈ 𝔽_{N}^{+}} for 1 ≤ i ≤ N. (ii) ∥T ∥ = sup_{1≤i≤N} ∥T_{i}∥ = sup{λ_{i,w} : w ∈ 𝔽_{N}^{+} and 1 ≤ i ≤ N}.

Proof. For 1 ≤ i ≤ N, the operators W_{i} are given by the equation

W_{i}ξ_{w} = (T_{i}ξ_{w}, ξ_{iw})ξ_{w} = λ_{i,w}ξ_{w}.

Since W_{i} ≥ 0, we have T_{i}^{∗}T_{i} = W_{i}^{∗}L_{i}^{∗}L_{i}W_{i} = W_{i}^{2}, which is diagonal. Hence, W_{i} = (T_{i}^{∗}T_{i})^{1/2} and T_{i} = L_{i}W_{i}. Further, this shows that

∥T_{i}∥^{2} = ∥T_{i}^{∗}T_{i}∥ = ∥W_{i}^{2}∥ = sup{∥W_{i}^{2}ξ_{w}∥ : w ∈ 𝔽_{N}^{+}}
= sup{λ_{i,w}^{2} : w ∈ 𝔽_{N}^{+}}.

On the other hand, the entries of the N × N matrix T^{∗}T consist of T_{i}^{∗}T_{i}’s down the diagonal and zero off the diagonal, since the ranges of the T_{i} are pairwise orthogonal. Hence, from the above computation

∥T ∥ = ∥T^{∗}T ∥^{1/2} = sup_{1≤i≤N} ∥T_{i}∥ = sup_{1≤i≤N}{λ_{i,w} : w ∈ 𝔽_{N}^{+}},

which completes the proof. □

We finish this section by observing that the C^{∗}-algebra generated by a noncommutative weighted shift, which is bounded below in an appropriate sense, contains the Cuntz-Toeplitz algebra.

Definition 2.5. Let T = (T_{1}, …, T_{N}) be a weighted shift. If each W_{i} is bounded away from zero, in other words if

inf{λ_{i,w} : 1 ≤ i ≤ N and w ∈ 𝔽_{N}^{+}} > 0,

we say that T is bounded below.

Corollary 2.6. The C^{∗}-algebra C^{∗}(T_{1}, …, T_{N}) generated by the operators {T_{1}, …, T_{N}} from a weighted shift T = (T_{1}, …, T_{N}) contains E_{N} = C^{∗}(L_{1}, …, L_{N}) when T is bounded below.

Proof. From the proof of the previous proposition, we see that W_{i} is invertible precisely when inf{λ_{i,w} : w ∈ 𝔽_{N}^{+}} > 0. Thus, T being bounded below implies that each W_{i} is invertible. However, W_{i} = (T_{i}^{∗}T_{i})^{1/2} belongs to C^{∗}(T_{i}), and hence to C^{∗}(T_{1}, …, T_{N}), thus so does L_{i} = T_{i}W_{i}^{−1} for 1 ≤ i ≤ N. □

Note 2.7. We mention that the C^{∗}-algebras C^{∗}(T_{1}, …, T_{N}) generated by the T_{i} from a single weighted shift T = (T_{1}, …, T_{N}) are the focus of analysis in [7].

3. Fock space trees and periodicity

In this section we aim to convey to the reader a helpful pictorial method for thinking of noncommutative weighted shifts. In doing so, we introduce what seems to be a natural notion of periodicity for these operators. We also define the operator algebras which will be studied in the rest of the paper.

Recall that N-variable Fock space H_{N} = ℓ^{2}(𝔽_{N}^{+}) has the orthonormal basis {ξ_{w} : w ∈ 𝔽_{N}^{+}}. This basis yields a natural tree structure for Fock space which is traced out by the creation operators, and more generally by weighted shifts.

Definition 3.1. Let T = (T_{1}, …, T_{N}) be a weighted shift. Let F_{T} be the set of vertices {w : w ∈ 𝔽_{N}^{+}}, together with the ‘weighted’ directed edges which correspond to the directions

{λ_{i,w} := w ↦ iw | for 1 ≤ i ≤ N and w ∈ 𝔽_{N}^{+}}.

We regard an edge λ_{i,w} as lying to the left of another edge λ_{j,w} precisely when i < j. We call F_{T} the weighted Fock space tree generated by T.

Pictorially, with N = 2 as an example, a typical weighted Fock space tree is given by the following diagram:

[figure: e; 1; 2; λ_{1,e}; λ_{2,e}; 11; 21; 12; 22; .; .; .; λ_{1,1}; λ_{1,2}; λ_{2,1}; λ_{2,2}]

Notice that this structure is really determined by the operators T = (T_{1}, …, T_{N}). Indeed, given a basis vector ξ_{w}, the directed edge λ_{i,w} corresponds to the action of T_{i} on ξ_{w}, namely mapping it to λ_{i,w}ξ_{iw}. Thus, more generally, we have the following picture for weighted edges leaving a typical vertex in the tree:

[figure: w; 1w; 2w; Nw; λ_{1,w}; λ_{2,w}; λ_{N,w}; .; .; .]

There are a number of conceptual benefits obtained by identifying these trees with weighted shifts. For instance, this point of view leads to the following notion of periodicity.

Definition 3.2. Let k ≥ 1 be a positive integer. We say that a weighted shift T = (T_{1}, …, T_{N}) is of period k if

T_{i}ξ_{w} = λ_{i,u}ξ_{iw} for w ∈ 𝔽_{N}^{+},

where w = uv is the unique decomposition of w with 0 ≤ |u| < k and |v| ≡ 0 (mod k).

Note 3.3. Observe that this says the scalars {λ_{i,u} : 0 ≤ |u| < k} completely determine the shift. They can be thought of as a ‘remainder tree top’. For N = 1 the standard notion of periodicity is recovered, since the tree collapses to a single infinite stalk. For N ≥ 2, it is most satisfying to think of this notion of periodicity in terms of the tree structure: If T = (T_{1}, …, T_{N}) is period k, then the remainder tree top, that is the finite top of the tree determined by vertices {w : |w| < k} and edges {λ_{i,w} : |w| < k}, is ‘repeated’ throughout the entire weighted tree.

In fact, this finite tree top is repeated with a certain exponential growth. For instance, at the level of the tree corresponding to words w ∈ 𝔽_{N}^{+} of length nk for some positive integer n ≥ 1, the top of this finite tree is repeated N^{nk} times, once for every word of length nk.

We mention that related tree top constructions play a key role in the paper [18].

Finally, we introduce the operator algebras which we are interested in studying.

Definition 3.4. For positive integers N ≥ 2 and k ≥ 1, let C_{N}^{∗}(per k) be the C^{∗}-algebra (contained in B(H_{N})) generated by the T_{i} from all weighted shifts T = (T_{1}, …, T_{N}) of period k.

It is clear from the picture given by the Fock space trees that if n_{1}|n_{2}, then C_{N}^{∗}(per n_{1}) is contained in C_{N}^{∗}(per n_{2}). Thus, given an increasing sequence of positive integers {n_{k}}_{k≥1} with n_{k}|n_{k+1} for k ≥ 1, we may consider the inductive limit algebra

𝔄(n_{k}) = \overline{⋃_{k≥1} C_{N}^{∗}(per n_{k})}

determined by this sequence. Let q be the quotient map of B(H_{N}) onto the Calkin algebra. We are also interested in describing the inductive limit algebras q(𝔄(n_{k})).

Note 3.5. The reader may find it helpful to know that C_{N}^{∗}(per k) is generated by the T_{i} from a single weighted shift. This is proved in the next section, using the matrix decompositions obtained there.

4. Main Theorem

The C^{∗}-algebra C_{N}^{∗}(per k) generated by the k-periodic weighted shifts can be described in terms of a full matrix algebra with entries in a Cuntz-Toeplitz algebra. From the discussion in Section 1, recall the Cuntz-Toeplitz algebra E_{N^{k}} is the C^{∗}-algebra generated by the creation operators L = (L_{1}, …, L_{N^{k}}) acting on N^{k}- variable Fock space H_{N^{k}}.

Theorem 4.1. For positive integers N ≥ 2 and k ≥ 1, let d_{N,k} be the total number of words in 𝔽_{N}^{+} of length strictly less than k; that is, d_{N,k} = 1 + N + ⋯ + N^{k−1}. Then the algebra C_{N}^{∗}(per k) of k-periodic weighted shifts is unitarily equivalent to the algebra M_{d_{N,k}}(E_{N^{k}}) of d_{N,k} × d_{N,k} matrices with entries in E_{N^{k}}. Further, this algebra is generated by the T_{i} from a single shift T = (T_{1}, …, T_{N}).

Remark 4.2. At first glance the N^{k} appearing in the theorem may seem somewhat peculiar to the reader. We shall see that it arises from the exponential nature of periodicity here. We mention that the special case N = 2 and k = 2 of the theorem is expanded on in Example 4.7.

We shall prove the theorem in several stages. Throughout, N ≥ 2 and k ≥ 1 will be fixed positive integers. The first step is to decompose Fock space in a manner which will lead to simple matrix representations of the periodic weighted shifts.

Lemma 4.3. For w ∈ 𝔽_{N}^{+} with |w| < k, the subspaces K_{w} of N-variable Fock space H_{N} given by

K_{w} = span{ξ_{wv} : |v| = km, m ≥ 0},

are pairwise orthogonal and

H_{N} = ∑_{|w|<k} ⊕ K_{w}.

Further, for |w| < k, the operators U_{w} : K_{e} → K_{w} defined by U_{w}ξ_{v} = ξ_{wv}, for |v| = km with m ≥ 0, are unitary. Hence

U := ∑_{|w|<k}⊕ U_{w} : K_{e}^{(d_{N,k})} ⟶ H_{N}

is a unitary operator.

Proof. The subspaces K_{w} for |w| < k clearly span H_{N} = ℓ^{2}(𝔽_{N}^{+}) since any word u ∈ 𝔽_{N}^{+} can be written, in fact uniquely, as u = u_{1}u_{2}, where |u_{1}| < k and k divides |u_{2}|. To see orthogonality, let w_{1}, w_{2} be words with |w_{i}| < k, and consider typical basis vectors ξ_{w_{i}v_{i}} for K_{w_{i}}, where |v_{i}| = km_{i} and m_{i} ≥ 0. The only way the inner product (ξ_{w_{1}v_{1}}, ξ_{w_{2}v_{2}}) can be nonzero, is if w_{1}v_{1} = w_{2}v_{2}, and hence by uniqueness of factorization we have w_{1} = w_{2} and v_{1} = v_{2}.

The operators U_{w} as defined are unitary since they send one orthonormal basis to another. Spatially, these unitaries can be thought of as the restrictions of the isometries L_{w}, where L_{w} := L_{i_{1}} … L_{i_{s}} when w is the word w = i_{1} … i_{s}, to a distinguished subspace K_{e} of Fock space. Alternatively, the action of the adjoint U_{w}^{∗} on K_{w} is described by restricting L_{w}^{∗} to K_{w}. The last statement of the lemma is immediate from the spatial decomposition of H_{N}. □

We will distinguish between coordinate spaces of K_{e}^{(d_{N,k})} in the following manner: For w ∈ 𝔽_{N}^{+} with |w| < k, let

{ ξ_{u}^{w} : u ∈ 𝔽_{N}^{+} with |u| = km for m ≥ 0}

be the standard basis for the wth coordinate space of K_{e}^{(d_{N,k})}, which is given by U^{∗}K_{w} = U_{w}^{∗}K_{w}. Notice that for |v|, |w| < k and |u| = km, the vectors ξ_{u}^{w} and ξ_{u}^{v} really correspond to the same vector ξ_{u} in K_{e}. Further, the action of U is described by

Uξ_{u}^{w} = ξ_{wu}, (1)

for w, u ∈ 𝔽_{N}^{+} with |w| < k and |u| = km.

The next step is to point out a relationship between particular Fock space trees.

Definition 4.4. We define a natural bijective correspondence between; the N^{k} words of length k in 𝔽_{N}^{+} on the one hand, and the N^{k} letters which generate 𝔽_{N^{k}}^{+} on the other, through the function

φ : {w ∈ 𝔽_{N}^{+} : |w| = k} ⟶ {w ∈ 𝔽_{N^{k}}^{+} : |w| = 1}

given by

φ(i_{1}i_{2} … i_{k}) = (i_{1} − 1)N^{k−1} + ⋯ + (i_{k−1} − 1)N + i_{k},

for 1 ≤ i_{j} ≤ N and 1 ≤ j ≤ k. This correspondence is also characterized by associating the words {iw ∈ 𝔽_{N}^{+} : |w| = k − 1} with the ‘ith block’ of N^{k−1} letters in the listing {1, 2, …, N^{k}}. Notice with this ordering that the operators {L_{φ(w)} : w ∈ 𝔽_{N}^{+}, |w| = k} are the N^{k} creation operators associated with N^{k}- variable Fock space H_{N^{k}}.

The map φ extends in a natural way to a bijective identification φ_{m} of the set {w ∈ 𝔽_{N}^{+} : |w| = km} with {w ∈ 𝔽_{N^{k}}^{+} : |w| = m} for m ≥ 0. Given w_{1}, …, w_{m} ∈ 𝔽_{N}^{+} with |w_{i}| = k, the extensions are given by

φ_{m}(w_{1} … w_{m}) = φ(w_{1}) … φ(w_{m}),

The units in 𝔽_{N}^{+} and 𝔽_{N^{k}}^{+} are identified with each other. We will use the notation φ for the extended map as well. This ordering leads to the following spatial equivalence.

Lemma 4.5. The map from K_{e} = span{ξ_{w} : w ∈ 𝔽_{N}^{+}, |w| = km, m ≥ 0} to N^{k}- variable Fock space H_{N^{k}} = ℓ^{2}(𝔽_{N^{k}}^{+}) which sends a basis vector ξ_{w_{1}...w_{m}} ∈ K_{e}, where each |w_{i}| = k, to the basis vector ξ_{φ(w_{1})...φ(w_{m})} ∈ H_{N^{k}}, is unitary.

Proof. This follows directly from the definitions of the space K_{e} and the map φ. □

This lemma gives us a tight spatial equivalence between the orthogonal direct sums K_{e}^{(d_{N,k})} and H_{N^{k}}^{(d_{N,k})}, which carries through for the weighted shifts. We wish to preserve the correspondence discussed after Lemma 4.3. In particular, for w ∈ 𝔽_{N}^{+} with |w| < k, we let

{ ξ_{φ(u)}^{w} : u ∈ 𝔽_{N}^{+} with |u| = km for m ≥ 0}

be the standard basis for the wth coordinate space of H_{N^{k}}^{(d_{N,k})}. Once again, with this identification, for |v|, |w| < k and |u| = km, the vectors ξ_{φ(u)}^{w} and ξ_{φ(u)}^{v} correspond

to the same vector ξ_{φ(u)} in H_{N^{k}}. Finally, we let V : H_{N^{k}}^{(d_{N,k})} → K_{e}^{(d_{N,k})} be the unitary operator which encodes this correspondence and the action of the map from the previous lemma. For w, u ∈ 𝔽_{N}^{+} with |w| < k and |u| = km, the action of V is given by

V ξ_{φ(u)}^{w} = ξ_{u}^{w}. (2)

With these Fock space decompositions in hand, we are now ready to focus on the particular actions of weighted shifts.

Lemma 4.6. For w ∈ 𝔽_{N}^{+} with |w| < k, let P_{w} be the orthogonal projection of H_{N^{k}}^{(d_{N,k})} onto the wth coordinate space of H_{N^{k}}^{(d_{N,k})}, so that I = ∑_{|w|<k} ⊕ P_{w}. Let T = (T_{1}, …, T_{N}) be a k-periodic weighted shift acting on H_{N}. Then the operators Ad_{UV} (T_{i}) = V^{∗}U^{∗}T_{i}UV act on H_{N^{k}}^{(d_{N,k})} and have the following block matrix decompositions:

(i) For |w| < k − 1 and |v| < k,

P_{v}(Ad_{UV} (T_{i}))P_{w} = \{ \begin{matrix}λ_{i,w}I_{H_{N^{k}}} & if v = iw \\ 0 & if v ≠ iw. \end{matrix}

(ii) For |w| = k − 1 and |v| < k,

P_{v}(Ad_{UV} (T_{i}))P_{w} = \{ \begin{matrix}λ_{i,w}L_{φ(iw)} & if v = e \\ 0 & if v ≠ e. \end{matrix}

Proof. We first prove case (i). From the preceding discussion, the vectors ξ_{φ(u)}^{w}, where u ∈ 𝔽_{N}^{+} with |u| = km, form an orthonormal basis for the range of P_{w}. Further, from equations (1) and (2) we have

UV ξ_{φ(u)}^{w} = Uξ_{u}^{w} = ξ_{wu}.

Lastly, as T = (T_{1}, …, T_{N}) is k-periodic, we have T_{i}ξ_{wu} = λ_{i,w}ξ_{iwu}. Since |w| < k − 1, these facts lead us to the following computation:

P_{v}(Ad_{UV} (T_{i}))ξ_{φ(u)}^{w} = P_{v}V^{∗}U^{∗}T_{i}ξ_{wu}
= λ_{i,w}P_{v}V^{∗}U^{∗}ξ_{(iw)u}
= λ_{i,w}P_{v}ξ_{φ(u)}^{iw} = λ_{i,w}δ_{v,iw} ξ_{φ(u)}^{iw},

where δ_{v,iw} is equal to 1 if v = iw, and is 0 otherwise. But recall that the vectors ξ_{φ(u)}^{w} and ξ_{φ(u)}^{iw} both correspond to the same vector ξ_{φ(u)} in H_{N^{k}}. Thus, case (i) is established.

Now suppose that |w| = k − 1. Again, k-periodicity gives us T_{i}ξ_{wu} = λ_{i,w}ξ_{iwu}. In this case |iw| = k, hence the definition of U and V from (1) and (2) yields:

P_{v}(Ad_{UV} (T_{i}))ξ_{φ(u)}^{w} = P_{v}V^{∗}U^{∗}T_{i}ξ_{wu}
= λ_{i,w}P_{v}V^{∗}U^{∗}ξ_{(iw)u}
= λ_{i,w}P_{v}V^{∗}ξ_{iwu}^{e}
= λ_{i,w}P_{v}ξ_{φ(iwu)}^{e} = λ_{i,w}δ_{v,e} ξ_{φ(iwu)}^{e}.

However, from the definition of φ, we have φ(iwu) = φ(iw)φ(u), since |iw| = k. Therefore,

P_{v}(Ad_{UV} (T_{i}))ξ_{φ(u)}^{w} = λ_{i,w}δ_{v,e}L_{φ(iw)}ξ_{φ(u)}^{e}.

Once again, the vectors ξ_{φ(u)}^{w} and ξ_{φ(u)}^{e} both correspond to ξ_{φ(u)} in H_{N^{k}}. This establishes case (ii), and completes the proof. □

Proof of Theorem 4.1. We define an injective homomorphism of C^{∗}-algebras π : C_{N}^{∗}(per k) → M_{d_{N,k}}(E_{N^{k}}) by π(T_{i}) = Ad_{UV} (T_{i}), for every k-periodic weighted shift T = (T_{1}, …, T_{N}). The map π is clearly an injective homomorphism since it is a unitary equivalence. Further, it follows from case (i) in Lemma 4.6, that all the matrix units in M_{d_{N,k}}(E_{N^{k}}) can be obtained in the image of π, by judicious choice of scalars λ_{i,w}’s and appropriate matrix multiplication. From case (ii) in that lemma, we see that the N^{k} creation operators which generate E_{N^{k}} can be obtained in certain matrix entries. Since all the matrix units are present in the image, these creation operators can be moved around to every entry. Therefore, it follows that π is also surjective, and hence defines a ∗-isomorphism.

Lastly, it is not hard to see from the matrix decompositions of Lemma 4.6 that the algebra C_{N}^{∗}(per k) is generated by the T_{i} from a single shift T = (T_{1}, …, T_{N}). For instance, from work in [7] it follows that any shift will do for which the N^{k} numeric k-tuples corresponding to the weights on each path of length k in the associated tree are different. □

Before continuing, we discuss a special case of the theorem which may help to clarify some of the technical issues.

Example 4.7. Consider the case when N = 2 and k = 2. Then C_{2}^{∗}(per 2) is the C^{∗}-algebra generated by the T_{i} from all 2-periodic shifts T = (T_{1}, T_{2}). The theorem shows that this algebra is unitarily equivalent to the matrix algebra M_{3}(E_{4}). Let us expand on this point.

Such 2-tuples act on the Fock space H_{2}, which has orthonormal basis {ξ_{w} : w ∈ 𝔽_{2}^{+}}. As in the previous section, the remainder tree top which determines the weighted Fock space tree for a given 2-periodic shift T = (T_{1}, T_{2}) is generated by six scalars {a, b, c, d, e, f} as follows:

T_{1}ξ_{e} = a ξ_{1}, T_{1}ξ_{1} = c ξ_{1^{2}}, T_{1}ξ_{2} = e ξ_{12}.

and

T_{2}ξ_{e} = b ξ_{2}, T_{2}ξ_{1} = d ξ_{21}, T_{2}ξ_{2} = f ξ_{2^{2}}.

Thus, for example, the action of T_{1} on basis vectors is given by

T_{1}ξ_{w} = \{ \begin{matrix}a ξ_{1w} & if |w| is even \\ c ξ_{1w} & if w = 1v and |v| is even \\ e ξ_{1w} & if w = 2v and |v| is even. \end{matrix}

In the proof of the theorem for this case, 2-variable Fock space H_{2} = ℓ^{2}(𝔽_{2}^{+}) decomposes into a direct sum H_{2} = K_{e}⊕K_{1}⊕K_{2} of d_{2,2} = 1+2 = 3 subspaces, each of which may be naturally identified with (2^{2} =)4-variable Fock space H_{4} = ℓ^{2}(𝔽_{4}^{+}). Take K_{e} for example. It is given by

K_{e} = span {ξ_{e}, {ξ_{1^{2}}, ξ_{12}, ξ_{21}, ξ_{2^{2}}}, {ξ_{w} : w ∈ 𝔽_{2}^{+}, |w| = 4}, … }.

The unitary equivalence produced by this spatial identification yields the following block matrix form for our given 2-periodic shift T = (T_{1}, T_{2}), with respect to the decomposition H_{2} = K_{e} ⊕ K_{1} ⊕ K_{2} ≃ H_{4} ⊕ H_{4} ⊕ H_{4},

T_{1} ≃ [\begin{matrix}0 & cL_{1} & eL_{3} \\ aI & 0 & 0 \\ 0 & 0 & 0 \end{matrix}] and T_{2} ≃ [\begin{matrix}0 & dL_{2} & fL_{4} \\ 0 & 0 & 0 \\ bI & 0 & 0 \end{matrix}] ,

where {L_{1}, L_{2}, L_{3}, L_{4}} are the standard creation operators on H_{4}.

Since we have complete freedom in C_{2}^{∗}(per 2) on the choices of scalars a, b, c, d, e, f, it is now easy to see why it is unitarily equivalent to the matrix algebra M_{3}(E_{4}). Further, it follows from these matrix decompositions that C_{2}^{∗}(per 2) is generated, for instance, by {T_{1}, T_{2}} with a = b = 1, c = 1/2, d = 1/4, e = 1/8, f = 1/16.

From Theorem 4.1 it follows that when we factor out the ideal of compact operators from C_{N}^{∗}(per k), simple C^{∗}-algebras are obtained. The key point being that the Cuntz-Toeplitz algebra is the extension of the compacts by the Cuntz algebra.

Corollary 4.8. Let q be the quotient map of B(H_{N}) onto the Calkin algebra. Then for N ≥ 2 and k ≥ 1, the algebra q(C_{N}^{∗}(per k)) is ∗-isomorphic to the matrix algebra M_{d_{N,k}}(O_{N^{k}}). In particular, it is a simple C^{∗}-algebra.

It follows that the inductive limit algebras q(𝔄(n_{k})) defined in the previous section are simple and have real rank zero.

Corollary 4.9. Let N ≥ 2 and let {n_{k}}_{k≥1} be an increasing sequence of positive integers such that n_{k} divides n_{k+1} for k ≥ 1. Then the inductive limit algebra q(𝔄(n_{k})) is simple and has real rank zero.

Proof. Every ideal in q(𝔄(n_{k})) is the closed union of ideals of the subalgebras q(C_{N}^{∗}(per n_{k})). Thus, simplicity follows immediately from the previous corollary. These algebras have real rank zero because, as observed in [25], the class of C^{∗}- algebras of real rank zero is closed under tensoring with finite dimensional C^{∗}- algebras, forming direct sums, and forming inductive limits. The Cuntz algebras O_{N^{n_{k}}} have real rank zero since they are purely infinite. □

We finish this section by pointing out the connection between our results and the single variable setting results of Bunce-Deddens.

Remark 4.10. The focus of this paper is on the noncommutative multivariable setting; however, we remark that the proof of Theorem 4.1 goes through as presented for N = 1. Namely, the C^{∗}-algebra generated by all unilateral weighted shift operators of period k is isomorphic to M_{k}(E_{1}), where E_{1} is the C^{∗}-algebra generated by the unilateral shift, also realized as the algebra of Toeplitz operators with continuous symbol [3, 4]. The proof presented here recaptures this result for N = 1, although from a different perspective. In particular, the Bunce-Deddens proof heavily relies on the associated function theory which is omnipresent in the single variable case. Conceptually speaking the proof here is more spatially oriented.

While more effort is required to prove Theorem 4.1 for N ≥ 2, the simplicity in Corollary 4.6 is more easily obtained as compared to the single variable case. The basic point is that O_{N} is simple for N ≥ 2, while for N = 1 it is the C^{∗}-algebra generated by the bilateral shift operator, the algebra of continuous functions on the unit circle, which is not simple. Nonetheless, the inductive limit algebras q(𝔄(n_{k})) turn out to be simple for N = 1, and thus our results on these algebras for N ≥ 2 can be regarded as a noncommutative multivariable generalization of their result.

5. Classification

In this section, we establish an isomorphism theorem for the limit algebras discussed in the previous two sections. Let {n_{k}}_{k≥1} be an increasing sequence of positive integers with n_{k} dividing n_{k+1} for k ≥ 1. Then for each prime p, there is a unique α_{p} in ℕ ∪ {∞} which is the supremum of the exponents of p which divide n_{k} as k → ∞. The supernatural number determined by the sequence {n_{k}}_{k≥1} is the formal product δ(n_{k}) = ∏_{p prime} p^{α_{p}}. Given two such sequences {n_{k}}_{k≥1} and {m_{j}}_{j≥1}, it follows that δ(n_{k}) = δ(m_{j}) precisely when: for all k ≥ 1, there is a j ≥ 1 with n_{k}|m_{j}; and for all j ≥ 1, there is a k ≥ 1 with m_{j}|n_{k}.

Supernatural numbers have been used to classify UHF algebras [16], and Bunce-Deddens algebras [3, 4]. They also distinguish between the inductive limit algebras of the current paper, as do the associated K_{0} groups.

Theorem 5.1. Let N ≥ 2 be a positive integer. Let {n_{k}}_{k≥1} and {m_{j}}_{j≥1} be increasing sequences of positive integers for which n_{k}|n_{k+1} and m_{j}|m_{j+1} for j, k ≥ 1. Then the following are equivalent:

(i) The supernatural numbers δ(n_{k}) and δ(m_{j}) are the same. (ii) The algebras 𝔄(n_{k}) and 𝔄(m_{j}) are equal. (iii) The algebras q(𝔄(n_{k})) and q(𝔄(m_{j})) are equal. (iv) If 𝔅(n_{k}) is an inductive limit of Cuntz algebras determined by a sequence

B_{n_{1}} → B_{n_{2}} → … such that B_{n_{k}} ≅ M_{d_{N,n_{k}}} (O_{N^{n_{k}}}), and 𝔅(m_{j}) is similarly defined, then 𝔅(n_{k}) and 𝔅(m_{j}) are ∗-isomorphic.

(v) The groups K_{0}(𝔅(n_{k})) and K_{0}(𝔅(m_{j})) are isomorphic.

Proof. To see (i) ⇒ (ii), observe the division property associated with (i) shows that each n_{k} divides some m_{j}. Thus,

C_{N}^{∗}(per n_{k}) ⊆ C_{N}^{∗}(per m_{j}) ⊆ 𝔄(m_{j})

for k ≥ 1. Whence, 𝔄(n_{k}) ⊆ 𝔄(m_{j}). The converse inclusion follows by symmetry. The implications (ii) ⇒ (iii) and (iv) ⇒ (v) are obvious. Since 𝔅(n_{k}) ≅ q(𝔄(n_{k})) and 𝔅(m_{j}) ≅ q(𝔄(m_{j})) by Corollary 4.8, we have (iii) ⇒ (iv).

It remains to establish the implication (v) ⇒ (i). Recall the K_{0} group of O_{N} is the finite abelian group K_{0}(O_{N}) = ℤ/(N − 1)ℤ of order N − 1. Hence,

K_{0}(M_{d_{N,n_{k}}} (O_{N^{n_{k}}})) = K_{0}(O_{N^{n_{k}}}) = ℤ/(N^{n_{k}} − 1)ℤ.

Since 𝔅(n_{k}) is the inductive limit of the algebras B_{n_{k}}, we have

K_{0}(𝔅(n_{k})) = lim_{⟶} K_{0}(B_{n_{k}}).

There are similar facts for K_{0}(𝔅(m_{j})) = lim_{⟶} K_{0}(B_{m_{j}}).

For k ≥ 1, let g_{k} ∈ K_{0}(𝔅(n_{k})) be an element of order N^{n_{k}} − 1. Let Γ : K_{0}(𝔅(n_{k})) → K_{0}(𝔅(m_{j})) be a group isomorphism. Then Γ(g_{k}) ∈ K_{0}(𝔅(m_{j})), and it follows that the order of Γ(g_{k}) must divide N^{m_{j}} − 1 for some j ≥ 1. (Every element of K_{0}(𝔅(m_{j})) has this property.)

However, for positive integers N ≥ 2 and k ≥ 1, recall that

d_{N,k} = 1 + N + ⋯ + N^{k−1} = \frac{N^{k} − 1}{N − 1}.

Thus we have just observed that d_{N,n_{k}} divides d_{N,m_{j}}. But this implies that n_{k} divides m_{j}. Indeed, suppose d_{N,m_{j}} = c d_{N,n_{k}} for some positive integer c. Consider the base N^{n_{k}} expansion of c given by c = c_{0} + c_{1}N^{n_{k}} + ⋯ + c_{l}(N^{n_{k}})^{l}, where 0 ≤ c_{i} < N^{n_{k}} for 0 ≤ i ≤ l. By comparing coefficients in d_{N,m_{j}} = c d_{N,n_{k}}, we get each c_{i} = 1 and m_{j} − 1 = ln_{k} + n_{k} − 1. Whence, m_{j} = (l + 1)n_{k}, and n_{k} divides m_{j}.

By symmetry, every m_{j} divides some n_{k}. Hence by the remarks preceding the theorem, this shows that the supernatural numbers δ(n_{k}) and δ(m_{j}) are identical.

□

Remark 5.2. After preparing this article, the author became aware of related work of Evans [13] on Cuntz-Krieger algebras. The most notable overlap between our papers is that the equivalence of conditions (i) and (v) in the previous theorem follows from the Cuntz algebra case of Theorem 4.3 from [13]. We also point out that a related notion of periodicity is used to define certain inductive limits of Cuntz-Krieger algebras in [13]. In the Cuntz algebra case, we see it is a more restrictive version; requiring scalars {c_{k} : k ≥ 0} such that λ_{i,w} = c_{k} for 1 ≤ i ≤ N and all |w| = k. Thus the periodicity introduced here is new, as is our main result Theorem 4.1.

We finish by pointing out that, not surprisingly, as for the Bunce-Deddens algebras, the algebras here are not almost finite dimensional. We need the following easy generalization of a theorem of Halmos.

Lemma 5.3. For 1 ≤ i ≤ N, the operator L_{i} is not quasitriangular.

Proof. In [17], Halmos proves that the unilateral shift is not quasitriangular. However, as he points out before proving this result, the proof really only depends on the operator of concern being an isometry, with adjoint having nontrivial kernel. The L_{i} clearly have this property since L_{i}^{∗} annihilates the vacuum vector. □

We can follow the lines of the Bunce-Deddens proof to show these algebras are not AF.

Theorem 5.4. The algebras 𝔄(n_{k}) and q(𝔄(n_{k})) are not approximately finite dimensional.

158 D.W. Kribs

Proof. Suppose there are finite dimensional C^{∗}-algebras {𝔅_{l}}_{l≥1} for which 𝔅_{l} ⊆ 𝔅_{l+1} and q(𝔄(n_{k})) = \overline{⋃_{l≥1} 𝔅_{l}}. Then given ε > 0, there is a B_{i} ∈ ⋃_{l≥1} 𝔅_{l} with ∥q(L_{i})−B_{i}∥ < ε. Choose A_{i} ∈ 𝔄(n_{k}) such that q(A_{i}) = B_{i}. Then ∥q(L_{i}−A_{i})∥ < ε, so there is a compact operator C_{i} with ∥L_{i} − A_{i} − C_{i}∥ < ε. But B_{i} belongs to a finite dimensional C^{∗}-algebra, hence there is a nontrivial polynomial p with p(B_{i}) = p(q(A_{i})) = 0. Thus A_{i} is polynomially compact, and as such, it is quasitriangular. (This was proved initially in [12].) In particular, A_{i} + C_{i} is quasitriangular, so that L_{i} belongs to the norm closure of the quasitriangular operators, and is itself quasitriangular [17]. This contradicts the previous lemma. Thus q(𝔄(n_{k})) is not approximately finite dimensional. The proof that 𝔄(n_{k}) is not AF is easier. □

Acknowledgements. The author is grateful to Raul Curto, Ken Davidson, Palle Jorgensen and Paul Muhly for enlightening conversations. Thanks also to the Department of Mathematics at the University of Iowa for kind hospitality during the preparation of this article.

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Department of Mathematics, Purdue University, West Lafayette, IN 47907-2068 kribs@math.purdue.edu

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