New York Journal of Mathematics

New York J. Math. 15 (2009) 485–503.

Entropy of shifts on topological graph C^{∗}-algebras

Valentin Deaconu

Abstract. We give entropy estimates for two canonical noncommutative shifts on C^{∗}-algebras associated to some topological graphs

E = (E^{0}, E^{1}, s, r),

defined using a basis of the corresponding Hilbert bimodule H(E). We compare their entropies with the growth entropies associated directly to the topological graph. We illustrate with some examples of topological graphs considered by Katsura, where the vertex and the edge spaces are unions of unit circles and more detailed computations can be done.

Contents

1. Introduction 485 2. Topological graphs and their C^{∗}-algebras 487 3. Finitely generated bimodules and noncommutative shifts 489 4. Growth entropies of topological graphs 492 5. Entropy of noncommutative shifts 495 6. Examples 498 References 502

1. Introduction

The topological entropy for automorphisms of nuclear C^{∗}-algebras was introduced by Voiculescu in [Vo] and extended by Brown to exact C^{∗}-algebras in [Br]. It was computed for many examples of automorphisms, endomorphisms, and completely positive (cp) maps by several authors, often using the fact that the restriction to a commutative C^{∗}-algebra is a map for which

Received December 31, 2008; revised October 31, 2009. Mathematics Subject Classification. Primary 46L05; Secondary 46L55. Key words and phrases. C^{∗}-algebra, noncommutative shift, entropy, topological graph. Research supported by a UNR JFR Grant.

485^{ISSN 1076-9803/09}

the entropy is known. For a comprehensive treatment of various techniques of computation of this entropy, we refer to [NS].

Recall that the map

Φ : O_{n} → O_{n}, Φ(c) = ∑_{i=1}^{n} S_{i}cS_{i}^{∗},

where O_{n} is the Cuntz algebra with generators S_{1}, …, S_{n}, is an endomorphism and has topological entropy log n, see [Ch]. The map Φ leaves invariant both the AF-core F_{n} ≅ UHF (n^{∞}), which is generated by monomials S_{α}S_{β}^{∗} with |α| = |β|, and the abelian algebra D_{n}, which is generated by monomials S_{α}S_{α}^{∗}. Recall that for a word α = α_{1} ⋯ α_{k} ∈ {1, 2, …, n}^{k}, k ∈ ℕ, we define S_{α} := S_{α_{1}} ⋯ S_{α_{k}} and |α| = k. It is known that D_{n} is isomorphic to C(X), where X = {1, 2, …, n}^{ℕ}, and Φ is called a noncommutative shift, since Φ |_{D_{n}} is conjugate to the map σ̃ : C(X) → C(X), σ̃(f) = f ∘ σ, where σ is the unilateral shift σ : X → X, σ(x_{0}x_{1}x_{2} …) = x_{1}x_{2} . . ..

In the case of the Cuntz–Krieger algebra O_{Λ}, where Λ is an incidence matrix, the corresponding map Φ is no longer multiplicative, but it is a unital completely positive map. Boca and Goldstein (see [BG]) proved that the topological entropy of Φ is log ρ(Λ), where ρ(Λ) is the spectral radius of Λ. This coincides with the classical topological entropy of the underlying Markov shift (X_{Λ}, σ). Similar results are obtained for graph C^{∗}-algebras by Jeong and Park, see [JP1, JP2, JP3], and for higher-rank graph C^{∗}-algebras by Skalski and Zacharias, see [SZ]. The Boca-Goldstein technique was also used by Kerr and Pinzari to analyze the noncommutative pressure and the variational principle in Cuntz–Krieger type C^{∗}-algebras, see [KP].

Let H be a full Hilbert bimodule over a C^{∗}-algebra A with a basis ξ_{1}, …, ξ_{n}, in the sense that for all ξ ∈ H,

ξ = ∑_{i=1}^{n} ξ_{i}⟨ξ_{i}, ξ⟩.

The Cuntz–Pimsner algebra O_{H} is generated by A and S_{i} = S_{ξ_{i}}, 1 ≤ i ≤ n with relations

∑_{i=1}^{n} S_{i}S_{i}^{∗} = 1, S_{i}^{∗}S_{j} = ⟨ξ_{i}, ξ_{j}⟩, a · S_{j} = ∑_{i=1}^{n} S_{i}⟨ξ_{i}, a · ξ_{j}⟩, a ∈ A,

see [KPW]. We consider the unital completely positive map (ucp map)

Φ : O_{H} → O_{H}, Φ(c) = ∑_{i=1}^{n} S_{i}cS_{i}^{∗},

Entropy of shifts on topological graph C^{∗}-algebras 487

and call it a noncommutative shift. Notice that Φ leaves invariant the core F_{H}, generated by monomials S_{α}aS_{β}^{∗} with a ∈ A and |α| = |β|, and also leaves invariant the subalgebra C_{H} generated by monomials S_{α}aS_{α}^{∗}. If H = ℂ^{n}, we recover the canonical endomorphism considered by Choda, since O_{H} = O_{n}, the Cuntz algebra generated by n isometries S_{1}, …, S_{n}. It is our goal to study the map Φ and its entropy for more general finitely generated Hilbert bimodules.

In Section 2 we recall the definition of the C^{∗}-algebras C^{∗}(E) and F_{E} associated to a topological graph E. In Section 3 we give more details about the structure of these C^{∗}-algebras, in the presence of a basis {ξ_{i}}_{1≤i≤n} of the Hilbert bimodule. We also define the ucp map Φ associated with such a basis. In the context of topological graph C^{∗}-algebras there is another candidate for the notion of noncommutative shift, denoted by Ψ. This is defined only on the core algebra F_{E}, using the embeddings K(H^{⊗k}) → K(H^{⊗k+1}) for some particular topological graphs. For these graphs, we can associate an étale groupoid as in [De2], and the map Ψ restricted to the diagonal D_{E} = C(E^{∞}) coincides with the shift map on the unit space of the groupoid identified with the space of infinite paths E^{∞}. In Section 4, we define the loop and block entropies for topological graphs E and compute them for several examples. In Section 5 we study the entropy of the noncommutative shifts Φ and Ψ, and the relationship with these growth entropies. The main result computes the entropy of Φ in terms of the spectral radius of an incidence matrix. The entropy of Ψ is bounded below by the loop entropy. Compared to the situation of discrete graphs or higher rank graphs, new phenomena occur, since Φ |_{F_{E}} and Ψ are different and they may have different entropies. We illustrate this with several examples in Section 6. We also recover earlier computations from [De1].

Our results are similar to results of Pinzari, Watatani and Yonetani in [PWY]. In the final stages of writing this paper, we also learned that Yamashita obtained similar entropy computations for a particular class of toplogical graphs, called circle correspondences [Y].

Acknowledgements. This research was supported by a University of Nevada JFR Grant.

2. Topological graphs and their C^{∗}-algebras

Topological graphs were studied in [De2] under the name continuous graphs, and generalized by several authors. We use the terminology and many facts from papers of Katsura, see [Ka1]. Let E = (E^{0}, E^{1}, s, r) be a topological graph. Recall that E^{0}, E^{1} are locally compact spaces, and s, r : E^{1} → E^{0} are continuous maps such that s is a local homeomorphism. We think of points in E^{0} as vertices, and of elements e ∈ E^{1} as edges from s(e) to r(e). Several examples of topological graphs will be considered in Section 4.

Definition 2.1. The C^{∗}-algebra of a topological graph E, denoted C^{∗}(E) is defined to be the Cuntz–Pimsner algebra O_{H}, where the Hilbert bimodule H = H(E) over the C^{∗}-algebra A = C_{0}(E^{0}) is obtained as the completion of C_{c}(E^{1}) using the inner product

⟨ξ, η⟩(v) = ∑_{s(e)=v} \overline{ξ(e)}η(e), ξ, η ∈ C_{c}(E^{1})

and the multiplications

(ξ · f)(e) = ξ(e)f(s(e)), (f · ξ)(e) = f(r(e))ξ(e).

The core algebra F_{E} is the fixed point algebra under the gauge action, see below.

Recall that a Hilbert bimodule over a C^{∗}-algebra A (sometimes called a C^{∗}-correspondence from A to A) is a Hilbert A-module H with a left action of A given by a homomorphism φ : A → L(H), where L(H) denotes the C^{∗}-algebra of all adjointable operators on H. A Hilbert bimodule is called full if the inner products generate A. For n ≥ 0 we denote by H^{⊗n} the Hilbert bimodule obtained by taking the tensor product of n copies of H, balanced over A (for n = 0, H^{⊗0} = A). For n = 2, the inner product is given by

⟨ξ ⊗ η, ξ′ ⊗ η′⟩ = ⟨η, φ(⟨ξ, ξ′⟩)η′⟩,

and it is inductively defined for general n.

A Toeplitz representation of a Hilbert bimodule H over A in a C^{∗}-algebra C is a pair (τ, π) with τ : H → C a linear map and π : A → C a ∗- homomorphism, such that

τ(ξa) = τ(ξ)π(a), τ(ξ)^{∗}τ(η) = π(⟨ξ, η⟩), τ(φ(a)ξ) = π(a)τ(ξ).

The C^{∗}-algebra generated by the images of π and τ in C is denoted by C^{∗}(τ, π). The corresponding universal C^{∗}-algebra is called the Toeplitz algebra of H, denoted by T_{H}. If H is full, then T_{H} is generated by elements τ^{n}(ξ)τ^{m}(η)^{∗}, m, n ≥ 0, where τ^{0} = π and for n ≥ 1, τ^{n}(ξ_{1} ⊗ ⋯ ⊗ ξ_{n}) = τ(ξ_{1}) ⋯ τ(ξ_{n}) is the extension of τ to H^{⊗n}. Note that A is isomorphic to a subalgebra of T_{H}.

The rank one operators θ_{ξ,η} given by θ_{ξ,η}(ζ) = ξ⟨η, ζ⟩ generate an essential ideal of L(H), denoted K(H). It is known that K(H) ≅ H ⊗ H^{∗}, where H^{∗} is the dual of H, regarded as a left A-module. The rank one operator θ_{ξ,η} ∈ K(H) is identified with ξ⊗η^{∗} ∈ H⊗H^{∗}. The map τ defines a homomorphism ψ : K(H) → C such that ψ(ξ ⊗ η^{∗}) = τ(ξ)τ(η)^{∗}. A representation (τ, π) is Cuntz–Pimsner covariant if π(a) = ψ(φ(a)) for all a in the ideal

I_{H} = φ^{−1}(K(H)) ∩ (ker φ)^{⊥}.

The Cuntz–Pimsner algebra O_{H} is universal with respect to the covariant representations, and it is a quotient of T_{H}. The universal properties allow us to define a gauge action of 𝕋 on T_{H} and O_{H} such that

z · (τ^{n}(ξ)τ^{m}(η)^{∗}) = z^{n−m}τ^{n}(ξ)τ^{m}(η)^{∗}, z ∈ 𝕋.

The core C^{∗}-algebra F_{H} is the fixed point algebra O_{H}^{𝕋}, and it is generated by the union of the algebras K(H^{⊗k}), k ≥ 0. For more details about the algebras F_{H}, T_{H}, and O_{H} we refer to the papers of Pimsner ([Pi]) and Katsura ([Ka1]).

Remark 2.2. Notice that, since Hilbert bimodules over commutative C^{∗}- algebras are associated to continuous fields of Hilbert spaces, it follows that for H = H(E), where E = (E^{0}, E^{1}, s, r) is a topological graph with s surjective, K(H) is a continuous trace C^{∗}-algebra over E^{0}. The elements of K(H) can be thought as compact operator valued continuous functions on E^{0} which vanish at infinity, and the elements of L(H) can be considered as bounded operator valued continuous functions on E^{0}.

3. Finitely generated bimodules and noncommutative shifts

Definition 3.1. The full Hilbert bimodule H over A is finitely generated if it has a basis {ξ_{i}}_{1≤i≤n}, in the sense that for all ξ ∈ H,

ξ = ∑_{i=1}^{n} ξ_{i}⟨ξ_{i}, ξ⟩.

It is easy to check that, in this case, H^{⊗k} has basis ξ_{i_{1}} ⊗ ⋯ ⊗ ξ_{i_{k}}, where i_{1}, …, i_{k} ∈ {1, 2, …, n}. Denote by S_{i} the image of ξ_{i} in the Cuntz–Pimsner algebra O_{H}. Then O_{H} is generated by A and S_{i}, 1 ≤ i ≤ n with relations

∑_{i=1}^{n} S_{i}S_{i}^{∗} = 1, S_{i}^{∗}S_{j} = ⟨ξ_{i}, ξ_{j}⟩, a · S_{j} = ∑_{i=1}^{n} S_{i}⟨ξ_{i}, a · ξ_{j}⟩, a ∈ A,

see [KPW]. For a word α = α_{1} ⋯ α_{k} ∈ {1, 2, …, n}^{k} of length |α| = k, let S_{α} = S_{α_{1}} ⋯ S_{α_{k}}.

Lemma 3.2. Assume that the full Hilbert bimodule H over A has a basis {ξ_{i}}_{1≤i≤n} as above. Then K(H) = L(H), and the Cuntz–Pimsner algebra O_{H} is the closed linear span of monomials S_{α}aS_{β}^{∗}, where α, β are arbitrary words and a ∈ A. The core algebra F_{H} is the closed linear span of monomials S_{α}aS_{β}^{∗} with |α| = |β|.

Proof. Since ∑_{i=1}^{n} S_{i}S_{i}^{∗} = 1, it is clear that K(H) is unital, and therefore K(H) = L(H). It suffices to show that the product of two monomials as above is a sum of monomials of the same form. Consider the product S_{α}aS_{β}^{∗}S_{γ}bS_{δ}^{∗} for some words α, β, γ, δ and a, b ∈ A. Using the relations S_{i}^{∗}S_{j} = ⟨ξ_{i}, ξ_{j}⟩ ∈ A and a · S_{j} = ∑_{i=1}^{n} S_{i}⟨ξ_{i}, a · ξ_{j}⟩ repeatedly, for |β| = |γ| we get that S_{β}^{∗}S_{γ} belongs to A; for |β| > |γ| we get S_{β}^{∗}S_{γ} = S_{β′}^{∗}c with |β′| = |β| − |γ| and c ∈ A; and for |β| < |γ| we get S_{β}^{∗}S_{γ} = dS_{γ′} with |γ′| = |γ| − |β| and d ∈ A. In the first case we are done. In the second and the third case, we use again the relation a·S_{j} = ∑_{i=1}^{n} S_{i}⟨ξ_{i}, a·ξ_{j}⟩ repeatedly to write

S_{β′}^{∗}c = (c^{∗}S_{β′})^{∗} = (∑_{j} S_{μ^{j}}c_{j})^{∗} = ∑_{j} c_{j}^{∗}S_{μ^{j}}^{∗}

and respectively

dS_{γ′} = ∑_{i} S_{ν^{i}}d_{i},

where μ^{j}, ν^{i} are words, and c_{j}, d_{i} ∈ A. We get

S_{α}aS_{β}^{∗}S_{γ}bS_{δ}^{∗} = ∑_{j} S_{α}ac_{j}^{∗}S_{μ^{j}}^{∗}bS_{δ}^{∗}

and

S_{α}aS_{β}^{∗}S_{γ}bS_{δ}^{∗} = ∑_{i} S_{α}aS_{ν^{i}}d_{i}bS_{δ}^{∗}.

We can use again the above relations to write S_{μ^{j}}^{∗}b as ∑_{k} b_{jk}S_{μ^{jk}}^{∗} with b_{jk} ∈ A and aS_{ν^{i}} = ∑_{l} S_{ν^{il}}a_{il} with a_{il} ∈ A to get the desired result. For the core algebra we can use the same method, keeping track of the lengths of the involved words. □

Corollary 3.3. If in addition ⟨ξ_{i}, ξ_{j}⟩ = 0 for all 1 ≤ i ≠ j ≤ n, then S_{i} are partial isometries. Set Q_{i} := S_{i}^{∗}S_{i} and P_{i} := S_{i}S_{i}^{∗} for i = 1, …, n. Suppose that Q_{i} = ∑_{j=1}^{n} Λ(i, j)P_{j} for some incidence matrix Λ. Then the nonzero elements S_{μ}P_{i}S_{ν}^{∗} for |μ| = |ν| = k and i = 1, …, n form a system of matrix units generating a finite-dimensional C^{∗}-algebra F_{k}. If, moreover, A is commutative and unital with A ⊂ K(H), then there is an isomorphism F_{k} ⊗ A ≅ K(H^{⊗k}) ≅ L(H^{⊗k}), and the Cuntz–Pimsner algebra O_{H} contains a copy of the Cuntz–Krieger algebra O_{Λ}. The core F_{H} is isomorphic to the inductive limit \underrightarrow{lim} K(H^{⊗k}) ≅ \underrightarrow{lim} F_{k} ⊗ A, where the embeddings are determined by the left action of A, and it contains a copy of the stationary AF-algebra determined by Λ. There is a commutative subalgebra C_{H} ⊂ F_{H}, generated by monomials S_{α}aS_{α}^{∗} with a ∈ A, which contains o copy of C(X_{Λ}), where X_{Λ} = {(x_{k}) ∈ {1, 2, …, n}^{ℕ} : Λ(x_{k}, x_{k+1}) = 1}.

Proof. Since ⟨ξ_{i}, ξ_{j}⟩ = 0 for i ≠ j, we get ξ_{i} = ξ_{i}⟨ξ_{i}, ξ_{i}⟩, which implies S_{i} = S_{i}S_{i}^{∗}S_{i} for all i. It follows that each S_{i} is a partial isometry, and therefore P_{i}, Q_{i} are projections. Under our assumptions, it is clear that S_{i} generate a copy of the Cuntz–Krieger algebra O_{Λ}. The isomorphism F_{k} ⊗ A ≅ K(H^{⊗k}) is given by the map S_{μ}P_{i}S_{ν}^{∗} ⊗ a ↦ S_{μ}aP_{i}S_{ν}^{∗}. The rest follows from the previous lemma. □

Definition 3.4. The canonical noncommutative shift associated to a basis {ξ_{i}}_{1≤i≤n} of the Hilbert bimodule H over A is the ucp map

Φ : O_{H} → O_{H}, Φ(c) = ∑_{i=1}^{n} S_{i}cS_{i}^{∗}.

Remark 3.5. Notice that Φ leaves invariant the core algebra F_{H}, generated by monomials S_{α}aS_{β}^{∗} with a ∈ A and |α| = |β|, and also leaves invariant the subalgebra C_{H} generated by monomials S_{α}aS_{α}^{∗}. In the hypotheses of the above corollary, the map Φ also leaves invariant the subalgebra C(X_{Λ}), and its restriction is conjugate to the map induced by the Markov shift σ_{Λ} : X_{Λ} → X_{Λ}, σ_{Λ}(x_{0}x_{1}x_{2} ⋯) = x_{1}x_{2} · · ·.

In our examples of topological graphs, E^{0} and E^{1} are compact and s and r are surjective local homeomorphisms. In that case, it follows that the left action of A = C(E^{0}) on H(E) is by compact operators. Moreover, the Hilbert module H = H(E) will have a basis ξ_{1}, ξ_{2}, …, ξ_{n} satisfying

⟨ξ_{i}, ξ_{j}⟩ = δ_{ij}Q_{i},

where Q_{i} ∈ A are projections. It follows from the corollary that the C^{∗}- algebra C^{∗}(E) = O_{H} is generated by A = C(E^{0}) and n partial isometries S_{1}, S_{2}, …, S_{n} with orthogonal ranges satisfying some commutation relations determined by the range map r.

For these topological graphs, there is also a groupoid approach for C^{∗}(E). For k ≥ 2, let’s define E^{k} to be the space of paths of length k in the topological graph and E^{∞} = \underleftarrow{lim} E^{k} to be the space of infinite paths. For k ≥ 0, we define the equivalence relation R_{k} on E^{k} by

R_{k} = {(e, f) ∈ E^{k} × E^{k} : s(e) = s(f)},

with the induced product topology. Note that R_{0} is just the diagonal in E^{0} × E^{0}. The set

Γ = Γ(E) = {(x, p − q, y) ∈ E^{∞} × ℤ × E^{∞} : σ^{p}(x) = σ^{q}(y)},

with natural operations (x, k, y)(y, l, z) = (x, k+l, z), (x, k, y)^{−1} = (y, −k, x) and appropriate topology, becomes an amenable étale groupoid, and its C^{∗}- algebra is isomorphic to C^{∗}(E), see [De2].

Definition 3.6. The map φ : C(E^{0}) → C^{∗}(R_{1}) induces embeddings Ψ_{k} : C^{∗}(R_{k}) → C^{∗}(R_{k+1}) for each k ≥ 0, which define another noncommutative shift Ψ : F_{E} → F_{E}. This is a ∗-homomorphism.

Remark 3.7. Recall that we have isomorphisms C^{∗}(R_{k}) ≅ K(H^{⊗k}). There is a natural diagonal isomorphic to C(E^{∞}) inside \underrightarrow{lim} C^{∗}(R_{k}) ≅ F_{E}, and the restriction Ψ |_{C(E^{∞})} is conjugate to the map defined by the shift

σ : E^{∞} → E^{∞},
σ(e_{1}e_{2}e_{3} ⋯) = e_{2}e_{3} ⋯,

which is a surjective local homeomorphism.

4. Growth entropies of topological graphs

Recall that a path of length k in a topological graph E = (E^{0}, E^{1}, s, r) is a concatenation e_{1}e_{2} ⋯ e_{k} of edges in E^{1} such that s(e_{i}) = r(e_{i+1}) for all i. The maps s and r extend naturally to E^{k}. A vertex v ∈ E^{0} is viewed as a path of length 0. For each vertex v ∈ E^{0} and k ≥ 1 let

E_{s}^{k}(v) := {e ∈ E^{k} | s(e) = v}, E_{r}^{k}(v) := {e ∈ E^{k} | r(e) = v}

and

E_{ℓ}^{k}(v) := {e ∈ E^{k} | s(e) = r(e) = v},

the set of loops of length k based at v. We will be mostly interested in locally finite topological graphs in the sense that each vertex emits and receives a finite number of edges. In that case, the sets E_{s}^{k}(v), E_{r}^{k}(v) and E_{ℓ}^{k}(v) are finite for each k and v. If both maps s and r are onto, then E_{s}^{k}(v), E_{r}^{k}(v) are nonempty for all k and v, but E_{ℓ}^{k}(v) may be empty. Define

E_{ℓ}^{k} := ⋃_{v∈E^{0}} E_{ℓ}^{k}(v).

Definition 4.1. Assume E is a locally finite topological graph. The loop entropy h_{ℓ}(E) and the block entropy h_{b}(E) of E are defined by

h_{ℓ}(E) = lim sup_{k→∞} \frac{1}{k} log |E_{ℓ}^{k}|, h_{b}(E) = sup_{v∈E^{0}} lim sup_{k→∞} \frac{1}{k} log |E_{r}^{k}(v)|,

where |L| denotes the cardinality of L.

Remark 4.2. For a topological graph E where r is also a local homeomorphism, we can interchange s and r to get a new topological graph, denoted E^{t} and called the transposed graph. We have h_{ℓ}(E) = h_{ℓ}(E^{t}) but h_{b}(E) ≠ h_{b}(E^{t}) in general. It is known that the loop entropy and the block entropy make sense for discrete graphs, and that these entropies can be computed in terms of the incidence matrix. For example, if the irreducible finite graph E is given by a matrix Λ_{E}, then it is known that h_{ℓ}(E) = h_{b}(E) = log ρ(Λ_{E}), where ρ(Λ_{E}) denotes the spectral radius. Since ρ(Λ_{E}) = ρ(Λ_{E}^{t}), for finite graphs we have h_{b}(E) = h_{b}(E^{t}). This equality fails for infinite graphs, as was shown by Jeong and Park in Example 3.3 of [JP2]. For topological graphs, see the next examples.

Example 4.3. Let X be a compact metric space, and σ : X → X a continuous surjective map. Define the topological graph E = E(X, σ) = (E^{0}, E^{1}, s, r), where E^{0} = E^{1} = X, s = id and r = σ. For k ≥ 2, the space

E^{k} = {(x_{1}, …, x_{k}) ∈ X × ⋯ × X | x_{j} = σ(x_{j+1}), j = 1, …, k − 1}

is homeomorphic to X. For a fixed v ∈ E^{0} = X we can identify E_{s}^{k}(v) with {v}, E_{r}^{k}(v) with σ^{−k}(v) and E_{ℓ}^{k} with {v ∈ X | σ^{k}(v) = v}. Hence

h_{ℓ}(E) = lim sup_{k→∞} \frac{1}{k} log |\{x ∈ X | σ^{k}(x) = x\}|,
h_{b}(E) = sup_{v∈X} lim sup_{k→∞} \frac{1}{k} log |σ^{−k}(v)|.

Recall that for an expansive map σ : X → X of a metric space X,

h_{top}(σ) ≥ lim sup_{k→∞} \frac{1}{k} log |\{x ∈ X | σ^{k}(x) = x\}|,

see Theorem 8.16 in [Wa], which gives the inequality

h_{top}(σ) ≥ h_{ℓ}(E(X, σ)).

Corollary 4.4. A topological graph E with E^{1} a compact metric space determines a local homeomorphism

σ = σ_{E} : E^{∞} → E^{∞},
σ(e_{1}e_{2}e_{3} …) = e_{2}e_{3} …,

where

E^{∞} = {e_{1}e_{2} ⋯ ∈ (E^{1})^{ℕ} | s(e_{j}) = r(e_{j+1})}

is the space of infinite paths. Then the set of periodic points of σ coincides with the union of loops ⋃_{k≥1} E_{ℓ}^{k}. If σ is expansive, it follows that

h_{top}(σ) ≥ h_{ℓ}(E).

Example 4.5. Consider the topological graphs E_{p,q} from Example A.6 of [Ka3]. Recall that the vertex and edge spaces are copies of the unit circle 𝕋, the source and range maps are s(z) = z^{p} and r(z) = z^{q}, where p, q are integers with p ≥ 1 and q ≠ 0. Let’s assume that gcd(p, q) = 1. Then it is easy to see that

|E_{s}^{k}(v)| = p^{k}, |E_{r}^{k}(v)| = |q|^{k}, |E_{ℓ}^{k}| = |p^{k} − |q|^{k}|.

It follows that for this topological graph, h_{b}(E) = log |q|, h_{b}(E^{t}) = log p, and h_{ℓ}(E) = log max{p, |q|}. The space of infinite paths E_{p,q}^{∞} is homeomorphic to

Σ(p, q) = {(z_{0}, z_{1}, z_{2}, …) ∈ 𝕋^{ℕ} | z_{k}^{p} = z_{k+1}^{q}},

which is a 1-dimensional solenoid. In particular, the shift

σ : Σ(p, q) → Σ(p, q),
σ(z_{0}, z_{1}, z_{2}, …) = (z_{1}, z_{2}, …)

has entropy ≥ log max{p, |q|}. For a more general result about the entropy of automorphisms of solenoids, see [Kw] and [LW].

Example 4.6. Consider the topological graph E = (E^{0}, E^{1}, s, r) where E^{0} = 𝕋, E^{1} = 𝕋 × {1, 2}, s(z, 1) = z^{2}, r(z, 1) = z, s(z, 2) = z, r(z, 2) = z^{3}. The space of infinite paths is the generalized solenoid

E^{∞} = {(z_{k}, a_{k})_{k≥0} ∈ (𝕋 × {1, 2})^{ℕ} |

a_{k} = 1 ⇒ z_{k+1}^{2} = z_{k}, a_{k} = 2 ⇒ z_{k+1} = z_{k}^{3}}.

Counting all the possible words (z_{1}, a_{1})(z_{2}, a_{2}) ⋯ (z_{m}, a_{m}) for a fixed v = r(z_{1}, a_{1}) ∈ 𝕋, we get |E_{r}^{m}(v)| = 4^{m}. It follows that h_{b}(E) = log 4. Counting all the possible words (z_{1}, a_{1})(z_{2}, a_{2}) ⋯ (z_{m}, a_{m}) for a fixed v = s(z_{m}, a_{m}) ∈ 𝕋, we get |E_{s}^{m}(v)| = 3^{m}, hence h_{b}(E^{t}) = log 3. Counting all the loops (z_{1}, a_{1})(z_{2}, a_{2}) ⋯ (z_{m}, a_{m}), we get

|E_{ℓ}^{m}| = ∑_{k=0}^{m} \binom{m}{k}|2^{m−k} − 3^{k}|.

It follows that h_{ℓ}(E) = log 4, since

4^{m} − 3^{m} = |∑_{k=0}^{m} \binom{m}{k}(2^{m−k} − 3^{k})|

≤ ∑_{k=0}^{m} \binom{m}{k}|2^{m−k} − 3^{k}|

≤ ∑_{k=0}^{m} \binom{m}{k}3^{k} = 4^{m}.

Example 4.7. More generally, for a discrete graph E = (E^{0}, E^{1}, s, r) with no sinks, Katsura considers in [Ka3] the topological graph E ×_{p,q} 𝕋 with vertex space E^{0} × 𝕋, edge space E^{1} × 𝕋, source map s(e, z) = (s(e), z^{p(e)}) and range map r(e, z) = (r(e), z^{q(e)}), where p : E^{1} → {1, 2, 3, … } and q : E^{1} → ℤ are two maps. The infinite path space is the generalized solenoid

(E ×_{p,q} 𝕋)^{∞}

= {(e_{1}, e_{2}, …; z_{1}, z_{2}, …) ∈ E^{∞} × 𝕋^{∞} | z_{k}^{p(e_{k})} = z_{k+1}^{q(e_{k+1})} for k = 1, 2, … } .

Define the matrices P, Q, where

P (v, w) = ∑_{e∈s^{−1}(v)∩r^{−1}(w)} p(e), Q(v, w) = ∑_{e∈s^{−1}(v)∩r^{−1}(w)} q(e)

for v, w ∈ E^{0}.

We conjecture that, for E finite, under certain conditions on the maps p and q, we have

h_{b}(E ×_{p,q} 𝕋) = log ρ(Q),
h_{b}((E ×_{p,q} 𝕋)^{t}) = log ρ(P),
h_{ℓ}(E ×_{p,q} 𝕋) = log max{ρ(P), ρ(Q)},

but we were unable to prove it at this time.

5. Entropy of noncommutative shifts

We recall the definition and a few useful facts concerning the topological entropy of completely positive maps. The reader may consult [Br] or [NS] for an extensive treatment. Let A be an exact C^{∗}-algebra, π : A → L(H) a faithful *-representation on a Hilbert space H, and ω ⊂ A a finite subset. For δ > 0 we put

CPA(π, A) = {(ϕ, ψ, B) | ϕ : A → B, ψ : B → L(H) contractive cp maps, dim(B) < ∞},

rcp(π, ω, δ) = inf{rank(B) | (ϕ, ψ, B) ∈ CPA(π, A), ||ψ ∘ ϕ(a) − π(a)|| < δ ∀ a ∈ ω},

where rank(B) denotes the dimension of a maximal abelian subalgebra of B. Since the completely positive rank rcp(π, ω, δ) is independent of the choice of π, we may write rcp(ω, δ) instead of rcp(π, ω, δ).

Definition 5.1. Let A ⊂ L(H) be a C^{∗}-algebra, Φ : A → A a cp map, ω ⊂ A finite and δ > 0. Put

ht(Φ, ω, δ) = lim sup_{k→∞} \frac{1}{k} log (rcp ( ⋃_{i=0}^{k−1} Φ^{i}(ω), δ))
ht(Φ, ω) = sup_{δ} ht(Φ, ω, δ)
ht(Φ) = sup_{ω} ht(Φ, ω).

The number ht(Φ) ∈ [0, ∞] is called the topological entropy of Φ.

Remark 5.2. We collect here some useful facts:

(1) If A_{0} ⊂ A is a Φ-invariant C^{∗}-subalgebra of A, then

ht(Φ |_{A_{0}}) ≤ ht(Φ).

(2) If {ω_{k}} is an increasing sequence of finite subsets of A such that the linear span of ⋃_{k,m≥0} Φ^{m}(ω_{k}) is dense in A, then ht(Φ) = sup_{k} ht(Φ, ω_{k}).

(3) If σ : X → X is a continuous map on a compact metric space X, then ht(σ̃) = h_{top}(σ), where σ̃ : C(X) → C(X), σ̃(f) = f ∘ σ, and h_{top} is the classical topological entropy defined in Chapter 7 of [Wa].

For the remainder of this paper, we consider a topological graph E = (E^{0}, E^{1}, s, r) such that E^{0} and E^{1} are compact metric spaces, and s, r are surjective local homeomorphisms. In this case, it follows that the left action of A = C(E^{0}) on H = H(E) is by compact operators. Moreover, we assume that the Hilbert module H = H(E) has a basis ξ_{1}, ξ_{2}, …, ξ_{n} satisfying

⟨ξ_{i}, ξ_{j}⟩ = δ_{ij}Q_{i},

where Q_{i} ∈ A = C(E^{0}) are projections. Suppose that Q_{i} = ∑_{j=1}^{n} Λ(i, j)P_{j} for some incidence matrix Λ, where P_{i} = S_{i}S_{i}^{∗}. Recall that the C^{∗}-algebra C^{∗}(E) = O_{H} is generated by A and n partial isometries S_{1}, S_{2}, …, S_{n} with orthogonal ranges satisfying some commutation relations determined by the range map r. It contains a copy of the Cuntz–Krieger algebra O_{Λ}. Denote by G = G(Λ) the finite graph defined by the incidence matrix Λ. Notice that S_{μ} for μ ∈ {1, 2, …, n}^{k} is nonzero precisely when μ is a path of length k in this finite graph. We are interested in determining the entropy of the noncommutative shifts Φ and Ψ defined in Section 3 for such a topological graph E.

Lemma 5.3. Consider a topological graph E as above. For each m ≥ 1 there is a ∗-homomorphism χ_{m} : O_{H} → M_{w(m)} ⊗ O_{H} given by

χ_{m}(x) = ∑_{|μ|=|ν|=m} S_{μ}S_{ν}^{∗} ⊗ S_{μ}^{∗}xS_{ν},

where w(m) denotes the number of paths of length m in the associated graph G.

Proof.

χ_{m}(x)χ_{m}(y) = (∑_{μ,ν} S_{μ}S_{ν}^{∗} ⊗ S_{μ}^{∗}xS_{ν})(∑_{μ′,ν′} S_{μ′}S_{ν′}^{∗} ⊗ S_{μ′}^{∗}yS_{ν′})

= ∑_{μ,ν,μ′,ν′} S_{μ}S_{ν}^{∗}S_{μ′}S_{ν′}^{∗} ⊗ S_{μ}^{∗}xS_{ν}S_{μ′}^{∗}yS_{ν′}

= ∑_{μ,ν′} S_{μ}S_{ν′}^{∗} ⊗ S_{μ}^{∗}x(∑_{ν} S_{ν}S_{ν}^{∗})yS_{ν′}

= χ_{m}(xy),

since ∑_{|ν|=m} S_{ν}S_{ν}^{∗} = 1. □

Theorem 5.4. Consider E = (E^{0}, E^{1}, s, r) a topological graph with E^{0} and E^{1} compact metric spaces such that s and r are surjective local homeomorphisms. Assume that A = C(E^{0}) is finitely generated, in the sense that there are a_{1}, a_{2}, …, a_{q} ∈ A such that polynomials in a_{j} and a_{j}^{∗} are dense in A.

Assume also that there are elements ξ_{1}, ξ_{2}, …, ξ_{n} in H = C(E^{1}) such that

ξ = ∑_{i=1}^{n} ξ_{i}⟨ξ_{i}, ξ⟩ for all ξ ∈ H, and
⟨ξ_{i}, ξ_{j}⟩ = δ_{ij}Q_{i} for some projections Q_{i} ∈ A, i = 1, …, n.

Let S_{i} be the image of ξ_{i} in O_{H}, and let P_{i} = S_{i}S_{i}^{∗} for i = 1, …, n. Assume Q_{i} = ∑_{j=1}^{n} Λ(i, j)P_{j}, for Λ = Λ_{E}, the incidence matrix of the associated finite graph. Define the canonical ucp map

Φ : C^{∗}(E) → C^{∗}(E), Φ(c) = ∑_{i=1}^{n} S_{i}cS_{i}^{∗}.

If C^{∗}(E) is simple, then ht(Φ) = log ρ(Λ_{E}), where ρ(Λ_{E}) denotes the spectral radius of the matrix Λ_{E}.

Proof. Since C^{∗}(E) contains a copy of O_{Λ} such that Φ(O_{Λ}) ⊂ O_{Λ}, it follows that ht(Φ) ≥ log ρ(Λ_{E}), (see also Section 6 of [PWY]). For the other inequality, we use the maps χ_{m} from the previous lemma. Since C^{∗}(E) is simple, it follows that χ_{m} : C^{∗}(E) → χ_{m}(C^{∗}(E)) ⊂ M_{w(m)} ⊗ C^{∗}(E) is a ∗-isomorphism. Note that for l ≥ 1 we have

Φ^{l}(c) = ∑_{|γ|=l} S_{γ}cS_{γ}^{∗}, c ∈ C^{∗}(E).

For k ≥ 1, |β| ≤ |α| ≤ k_{0}, m ≥ k + k_{0}, l ≤ k − 1 and a = a_{1}^{p_{1}} ⋯ a_{q}^{p_{q}}, 0 ≤ p_{j} ≤ k_{0}, j = 1, …, q, one has as in [BG] (see also [SZ])

(χ_{m} ∘ Φ^{l})(S_{α}aS_{β}^{∗}) = ∑_{j=1}^{h(m,k)} y_{j} ⊗ c_{j},

where y_{j} ∈ M_{w(m)} are partial isometries, and c_{j} ∈ C^{∗}(E).

For any k ≥ 1 consider

ω(k) = {S_{α}aS_{β}^{∗} : |β| ≤ |α| ≤ k, a = a_{1}^{p_{1}} ⋯ a_{q}^{p_{q}}, 0 ≤ p_{j} ≤ k, j = 1, …, q}.

Notice that ω_{k} = ω(k) ∪ ω(k)^{∗} is an increasing sequence of finite subsets of C^{∗}(E) such that the span of their union is dense in C^{∗}(E). For a fixed k_{0} ≥ 1 and δ > 0 we have

lim sup_{k→∞} \frac{1}{k} log rcp ( ⋃_{i=0}^{k−1} Φ^{i}(ω(k_{0})), δ) ≤ log ρ(Λ_{E}).

Indeed, fix k ≥ 1, and let m = k + k_{0}. Since C^{∗}(E) is nuclear, there exists (ϕ_{0}, ψ_{0}, M_{m_{0}}) ∈ CPA(id, C^{∗}(E)) such that

∥ψ_{0}(ϕ_{0}(c_{j})) − c_{j}∥ < \frac{δ}{h(m, k)}, j = 1, …, h.

Consider B = M_{w(m)} ⊗ M_{m_{0}} and let H be a Hilbert space on which C^{∗}(E) acts faithfully. The ∗-isomorphism χ_{m}^{−1} : χ_{m}(C^{∗}(E)) → C^{∗}(E) extends to a cp map ψ_{m} : M_{w(m)} ⊗ C^{∗}(E) → L(H) with ∥ψ_{m}∥ = 1. Consider the cp maps ϕ = (id ⊗ ϕ_{0}) ∘ χ_{m} : C^{∗}(E) → B and ψ = ψ_{m} ∘ (id ⊗ ψ_{0}) : B → L(H). For c = S_{α}aS_{β}^{∗} ∈ ω(k_{0}) we get

∥ψ(ϕ(Φ^{l}(c))) − Φ^{l}(c)∥ = ∥ψ_{m}(id ⊗ ψ_{0}ϕ_{0})((χ_{m} ∘ Φ^{l})(c)) − Φ^{l}(c)∥

= ∥ψ_{m}(id ⊗ ψ_{0}ϕ_{0})((χ_{m} ∘ Φ^{l})(c)) − ψ_{m}((χ_{m} ∘ Φ^{l})(c))∥

≤ ∥(id ⊗ ψ_{0}ϕ_{0})((χ_{m} ∘ Φ^{l})(c)) − (χ_{m} ∘ Φ^{l})(c)∥

= ∥ ∑_{j=1}^{h(m,k)} y_{j} ⊗ (ψ_{0}ϕ_{0}(c_{j}) − c_{j})∥ < δ.

Hence

rcp ( ⋃_{i=0}^{k−1} Φ^{i}(ω(k_{0})), δ) ≤ m_{0}w(m) = m_{0}w(k + k_{0}),

which gives

lim sup_{k→∞} \frac{1}{k} log ( ⋃_{i=0}^{k−1} Φ^{i}(ω(k_{0})), δ) ≤ lim sup_{k} \frac{1}{k} log w(k) = log ρ(Λ_{E}).

We conclude that ht(Φ) = log ρ(Λ_{E}). □

Proposition 5.5. Consider E = (E^{0}, E^{1}, s, r) a topological graph as in the previous theorem. Assume that the shift σ_{E} : E^{∞} → E^{∞} is expansive. Then the noncommutative shift Ψ : F_{E} → F_{E} defined by the maps Ψ_{k} : C^{∗}(R_{k}) → C^{∗}(R_{k+1}) has entropy ht(Ψ) ≥ h_{ℓ}(E).

Proof. The natural diagonal C(E^{∞}) of the groupoid Γ(E) defined at the end of Section 3 is invariant under the map Ψ and its restriction Ψ |_{C(E^{∞})} is conjugate to the map induced by shift σ_{E} : E^{∞} → E^{∞}, hence ht(Ψ) ≥ h_{top}(σ). Now apply Corollary 4.4. □

Corollary 5.6. The noncommutative shifts Φ, Ψ : F_{E} → F_{E} may have different entropies.

Proof. Since F_{E} is invariant under Φ, from Theorem 5.4 we get

ht(Φ |_{F_{E}}) = log ρ(Λ_{E}).

From the above proposition, ht(Ψ) ≥ h_{ℓ}(E). Examples 6.2 and 6.3 in the next section show that ht(Φ) and ht(Ψ) indeed may be different. □

6. Examples

Example 6.1. Let X be a compact space, let σ : X → X a continuous surjective map, and let E = E(X, σ) = (E^{0}, E^{1}, s, r) be the topological graph we defined earlier, where E^{0} = E^{1} = X, s = id and r = σ. We identify the Hilbert bimodule H = H(E) and its tensor powers H^{⊗n} with C(X). The endomorphism σ̃ : C(X) → C(X), σ̃(f) = f ∘ σ coincides with the left action φ_{r} : C(X) → K(H) defined by r = σ, after the identification of K(H) with C(X). Consider (τ, π) a Toeplitz representation of H. Then the constant function 1 in C(X) determines an element U in C^{∗}(τ, π) satisfying U^{∗}U = 1, Uπ(f) = τ(f) and π(f)U = τ(σ̃(f)) for f ∈ C(X). The map ψ : K(H) → C^{∗}(τ, π) can be expressed as ψ(f) = Uπ(f)U^{∗}, hence ψ(φ_{r}(f)) = Uπ(σ̃(f))U^{∗}. The C^{∗}-algebra O_{E} is the universal C^{∗}-algebra generated by a copy of C(X) and a unitary u satisfying u^{∗}fu = σ̃(f). In particular, for σ a homeomorphism, O_{E} is isomorphic to the crossed product C(X) ⋊_{σ} ℤ. For details, see the results following Example 2 in [Ka1].

The Hilbert bimodule H has basis {ξ_{1}}, where ξ_{1} is determined by the constant function 1. The canonical map Φ : O_{E} → O_{E} is given by Φ(c) = ucu^{∗}. Note that Φ(σ̃(f)) = f, so Φ is a left inverse for σ̃. If σ is a homeomorphism, then it is known that ht(Φ) = h_{top}(σ).

Example 6.2. Consider again the topological graphs E_{p,q} of Katsura [Ka3], where p and q are integers with p ≥ 1 and q ≠ 0, E^{0} = E^{1} = 𝕋, s(z) = z^{p} and r(z) = z^{q}, as in Example 4.5. The Hilbert bimodule H_{p,q} = H(E_{p,q}) = C(E^{1}) has a basis \{ξ_{1}, ξ_{2}, …, ξ_{p}\}, where ξ_{k}(z) = \frac{1}{\sqrt{p}}z^{k−1}, k = 1, …, p. We have

⟨ξ_{j}, ξ_{k}⟩(z) = \frac{1}{p} ∑_{w^{p}=z} \overline{w^{j}}w^{k} = δ_{jk} · 1 and ∑_{j=1}^{p} θ_{ξ_{j},ξ_{j}} = 1.

Indeed, let z = exp(it). Then w^{p} = z has solutions

w_{m} = exp(i(t + 2mπ)/p),

m = 0, …, p − 1. We have

∑_{w^{p}=z} \overline{w^{j}}w^{k} = ∑_{m=0}^{p−1} exp(i(k − j)(t + 2mπ)/p)

= exp(i(k − j)t/p) ∑_{m=0}^{p−1}(exp(i(k − j)2π/p))^{m}

= δ_{kj} · 1.

A similar computation gives

∑_{j=1}^{p} ξ_{j}⟨ξ_{j}, η⟩(z) = ∑_{j=1}^{p} ξ_{j}(z) ∑_{w^{p}=z^{p}} \overline{ξ_{j}(w)}η(w) = η(z).

It follows that the C^{∗}-algebra O_{E_{p,q}} is generated by the unitary u ∈ C(E^{0}) = C(𝕋), where u(z) = z and p isometries S_{1}, S_{2}, …, S_{p} satisfying the relations

S_{j}^{∗}S_{k} = δ_{jk} · 1, ∑_{j=1}^{p} S_{j}S_{j}^{∗} = 1, uS_{k} = S_{k+q},

where S_{k+q} = S_{l}u^{m} using the unique l ∈ {1, 2, …, p} with k + q = l + pm. The isometries S_{1}, S_{2}, …, S_{p} generate a copy of the Cuntz algebra O_{p}, and the finite graph Λ associated to the topological graph E_{p,q} has one vertex and p edges. From Theorem 5.4, we get

ht(Φ) = log p.

We have L(H_{p,q}) = K(H_{p,q}) ≅ 𝕄_{p} ⊗ C(𝕋). Let q = pm + r, where r ∈ {0, 1, …, p − 1}. If 1 ≤ r + k ≤ p, then u · ξ_{k} = ξ_{r+k} · u^{m}, and therefore the inclusion Ψ_{0} = φ : C(E^{0}) → K(H_{p,q}) is given by

u ↦ ∑_{k=1}^{p} S_{r+k}u^{m}S_{k}^{∗};

if p + 1 ≤ r + k ≤ 2p − 1, then u · ξ_{k} = ξ_{r+k−p} · u^{m+1}, and Ψ_{0} it is given by

u ↦ ∑_{k=1}^{p} S_{r+k−p}u^{m+1}S_{k}^{∗}.

The C^{∗}-algebra F_{E_{p,q}} is isomorphic to the inductive limit

C(𝕋) →^{Ψ_{0}} 𝕄_{p} ⊗ C(𝕋) →^{Ψ_{1}} 𝕄_{p^{2}} ⊗ C(𝕋) →^{Ψ_{2}} ⋯,

where the embeddings Ψ_{k} are obtained from Ψ_{0} by tensoring with the identity. Recall from Example 4.5 that the space of infinite paths E_{p,q}^{∞} is homeomorphic to

Σ(p, q) = {(z_{0}, z_{1}, z_{2}, …) ∈ 𝕋^{ℕ} | z_{k}^{p} = z_{k+1}^{q}},

which is a 1-dimensional solenoid if gcd(p, q) = 1. The map

Ψ : F_{E_{p,q}} → F_{E_{p,q}}

determined by the maps Ψ_{k}, when restricted to C(E_{p,q}^{∞}), is conjugate with the shift σ, which has entropy h_{top}(σ) ≥ max{log p, log |q|} for gcd(p, q) = 1. It follows that ht(Ψ) ≥ log max{p, |q|}.

Example 6.3. Consider the topological graph E = (E^{0}, E^{1}, s, r) from Example 4.6, where E^{0} = 𝕋, E^{1} = 𝕋 × {1, 2}, s(z, 1) = z^{2}, r(z, 1) = z, s(z, 2) = z, r(z, 2) = z^{3}. The Hilbert bimodule H = H(E) = C(E^{1}) has a basis \{ξ_{1}, ξ_{2}, ξ_{3}\}, where ξ_{k}(z, 1) = \frac{1}{\sqrt{2}}z^{k−1}, ξ_{k}(z, 2) = 0 for k = 1, 2 and ξ_{3}(z, 1) = 0, ξ_{3}(z, 2) = 1. We have ⟨ξ_{j}, ξ_{k}⟩ = δ_{jk} · 1. The C^{∗}-algebra O_{E} is generated by C(E^{0}) = C(𝕋) and three isometries S_{1}, S_{2}, S_{3} satisfying the relations

∑_{j=1}^{3} S_{j}S_{j}^{∗} = 1, S_{j}^{∗}S_{k} = δ_{jk} · 1, f · S_{j} = ∑_{k=1}^{3} S_{k}⟨ξ_{k}, f · ξ_{j}⟩,

where (f ·ξ_{j})(z, 1) = f(z)\frac{1}{\sqrt{2}}z^{j−1}, j = 1, 2, (f ·ξ_{3})(z, 2) = f(z^{3}) for f ∈ C(𝕋). It follows from Theorem 5.4 that the C^{∗}-algebra O_{E} contains a copy of the Cuntz algebra O_{3} and ht(Φ) = log 3. Let u ∈ C(𝕋) be the generator u(z) = z. We get the relations

uS_{1} = S_{2}, uS_{2} = S_{1}u, uS_{3} = S_{3}u^{3}.

We have L(H) = K(H) ≅ 𝕄_{3} ⊗ C(𝕋). The inclusion Ψ_{0} : C(E^{0}) → L(H) is determined by the map

u ↦ ( \begin{matrix}0 & u & 0 \\ 1 & 0 & 0 \\ 0 & 0 & u^{3} \end{matrix} ) .

Indeed, the element u ∈ C(E^{0}) is sent to the compact operator

ξ_{2} ⊗ ξ_{1}^{∗} + ξ_{1}u ⊗ ξ_{2}^{∗} + ξ_{3}u^{3} ⊗ ξ_{3}^{∗},

because

(u · ξ_{1})(z, 1) = zξ_{1}(z, 1) = \frac{1}{\sqrt{2}}z = ξ_{2}(z, 1),

(u · ξ_{2})(z, 1) = zξ_{2}(z, 1) = \frac{1}{\sqrt{2}}z^{2} = (ξ_{1} · u)(z, 1),

(u · ξ_{3})(z, 2) = z^{3}ξ_{3}(z, 2) = z^{3} = (ξ_{3} · u^{3})(z, 2).

The C^{∗}-algebra F_{E} is isomorphic to the inductive limit

C(𝕋) → 𝕄_{3} ⊗ C(𝕋) → 𝕄_{3^{2}} ⊗ C(𝕋) → ⋯

The first inclusion is given by u ↦ S_{2}S_{1}^{∗} + S_{1}uS_{2}^{∗} + S_{3}u^{3}S_{3}^{∗}, and the other inclusions are obtained by iterating this formula. The restriction of Ψ : F_{E} → F_{E} to the diagonal C(E^{∞}) is conjugate to the map induced by the shift σ on the generalized solenoid

E^{∞} = {(z_{k}, a_{k})_{k≥0} ∈ (𝕋 × {1, 2})^{ℕ} |

a_{k} = 1 ⇒ z_{k+1}^{2} = z_{k}, a_{k} = 2 ⇒ z_{k+1} = z_{k}^{3}},

and has entropy ht(Ψ) ≥ log 4, see Example 4.6.

Example 6.4. Consider the topological graph from Example 4.7. In the case E^{0} and E^{1} are finite, the Hilbert bimodule H(p, q) = H(E ×_{p,q} 𝕋) has a basis \{ξ_{e,k} | e ∈ E^{1}, k = 1, …, p(e)\}, where ξ_{e,k}(e, z) = \frac{1}{\sqrt{p(e)}}z^{k−1} and

ξ_{e,k}(e′, z) = 0 for e′ ≠ e. For each v ∈ E^{0}, let u_{v} ∈ O_{E×_{p,q}𝕋} be the image of the generating unitary of C({v} × 𝕋) ⊂ C(E^{0} × 𝕋). Let S_{e,k} ∈ O_{E×_{p,q}𝕋} be the image of ξ_{e,k}. Then O_{E×_{p,q}𝕋} is generated by the family {u_{v}}_{v∈E^{0}} of partial unitaries with orthogonal ranges and the family {S_{e,k}}_{e∈E^{1},1≤k≤p(e)} of partial isometries with orthogonal ranges satisfying the relations

S_{e,k}^{∗}S_{e,k} = u_{s(e)}^{∗}u_{s(e)} for e ∈ E^{1} and 1 ≤ k ≤ p(e),
u_{r(e)}S_{e,k} = S_{e,k+q(e)} for e ∈ E^{1} and 1 ≤ k ≤ p(e),

u_{v}^{∗}u_{v} = ∑_{r(e)=v} S_{e,k}S_{e,k}^{∗} for v ∈ E^{0},

502 Valentin Deaconu

where S_{e,k+q(e)} = S_{e,k′}u_{s(e)}^{l} for the unique k′ ∈ {1, 2, …, p(e)} and l ∈ ℤ with k + q(e) = k′ + p(e)l, see Appendix A in [Ka3]. Consider the matrices P, Q, where

P (v, w) = ∑_{e∈s^{−1}(v)∩r^{−1}(w)} p(e), Q(v, w) = ∑_{e∈s^{−1}(v)∩r^{−1}(w)} q(e)

for v, w ∈ E^{0}. The partial isometries S_{e,k} generate a copy of the Cuntz– Krieger algebra O_{Λ} inside O_{E×_{p,q}𝕋}, and by Theorem 5.4 we get

ht(Φ) = log ρ(Λ).

If the formulas conjectured in Example 4.7 were true, this would imply that, under certain conditions on the maps p and q,

ht(Ψ) ≥ log max{ρ(P), ρ(Q)}.

References

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Department of Mathematics, University of Nevada, Reno NV 89557-0084, USA vdeaconu@unr.edu

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