Flow equivalence of topological Markov shifts and Ruelle algebras

New York Journal of Mathematics

New York J. Math.27(2021) 1375–1414.

Flow equivalence of topological Markov shifts and Ruelle algebras

Kengo Matsumoto

Abstract. In this paper we study discrete flow equivalence of two-sided topological Markov shifts by using extended Ruelle algebra. We characterize flow equivalence of two-sided topological Markov shifts in terms of conjugacy of certain actions weighted by ceiling functions of two-dimensional torus on the stabilized extended Ruelle algebras for the Markov shifts.

Contents

1. Introduction 1375

2. Preliminaries 1379

3. Bilateral dimension groups 1382

4. Dimension quadruplets and AF-algebras 1386

5. Gauge actions with potentials 1394

6. Flow equivalence 1400

7. Flow equivalence and topological conjugacy 1408

References 1412

1. Introduction

Flow equivalence relation in two-sided topological Markov shifts is one of the most interesting and important equivalence relations in symbolic dynam- ics as seen in many papers [2], [3], [9], [22], etc. Let ( ̄𝑋_𝐴, ̄𝜎_𝐴)be the two- sided topological Markov shift defined by an 𝑁 × 𝑁 irreducible matrix𝐴 = [𝐴(𝑖, 𝑗)]^𝑁_𝑖,𝑗=1with entries in{0, 1}.The shift space𝑋̄_𝐴consists of bi-infinite se- quences (𝑥_𝑛)_𝑛∈ℤ ∈ {1, … , 𝑁}^ℤ of {1, … , 𝑁} such that𝐴(𝑥_𝑛, 𝑥_𝑛+1) = 1for all 𝑛 ∈ ℤ. Take and fix a real number𝜆_◦ such as0 < 𝜆_◦ < 1. The space𝑋̄_𝐴 is a

Received October 18, 2018.

2010Mathematics Subject Classification. Primary 37A55, 46L35; Secondary, 37B10, 46L55.

Key words and phrases. Topological Markov shift, flow equivalence, topological conjugacy, Ruelle algerba, Cuntz–Krieger algebra.

The author would like to thank the referee for his careful reading, comments and sugges- tions on the first draft of the paper. This work was supported by JSPS KAKENHI Grant Number 15K04896, 19K03537.

ISSN 1076-9803/2021

1375

compact metric space by the metric defined by for𝑥 = (𝑥_𝑛)_𝑛∈ℤ, 𝑦 = (𝑦_𝑛)_𝑛∈ℤ with𝑥 ≠ 𝑦

𝑑(𝑥, 𝑦) = {1 if𝑥₀≠ 𝑦₀,

(𝜆_◦)^𝑚 if𝑚 = Max{𝑛 ∣ 𝑥_𝑘 = 𝑦_𝑘for all𝑘with|𝑘| < 𝑛}.

The homeomorphism of the shift transformation𝜎̄_𝐴on𝑋̄_𝐴is defined by

̄

𝜎_𝐴((𝑥_𝑛)_𝑛∈ℤ) = (𝑥_𝑛+1)_𝑛∈ℤ.

Two topological Markov shifts( ̄𝑋_𝐴, ̄𝜎_𝐴)and( ̄𝑋_𝐵, ̄𝜎_𝐵)are said to be flow equiv- alent if they are realized as cross sections with their first return maps of a com- mon flow space. Parry–Sullivan in [22] proved that( ̄𝑋_𝐴, ̄𝜎_𝐴)and( ̄𝑋_𝐵, ̄𝜎_𝐵)are flow equivalent if and only if they are realized as discrete cross sections with their first return maps of a common topological Markov shift. Cuntz–Krieger have first found that there is an interesting relation between flow equivalence of topological Markov shifts and certain purely infinite simple𝐶^∗-algebras called Cuntz–Krieger algebras that they introduced in [7]. For an irreducible ma- trix𝐴 with entries in{0, 1}, let 𝒪_𝐴 be the Cuntz–Krieger algebra and𝒟_𝐴 its canonical maximal abelian𝐶^∗-subalgebra of𝒪_𝐴. We denote by𝒦 and𝒞 the 𝐶^∗-algebra of compact operators on the separable infinite dimensional Hilbert space𝓁²(ℕ)and its commutative𝐶^∗-subalgebra of diagonal operators on𝓁²(ℕ), respectively. Cuntz–Krieger proved that for irreducible non-permutation ma- trices 𝐴 and 𝐵, if( ̄𝑋_𝐴, ̄𝜎_𝐴) and ( ̄𝑋_𝐵, ̄𝜎_𝐵) are flow equivalent, then there ex- ists an isomorphism Φ ∶ 𝒪_𝐴 ⊗ 𝒦 ⟶ 𝒪_𝐵 ⊗ 𝒦 of 𝐶^∗-algebras such that Φ(𝒟_𝐴⊗ 𝒞) = 𝒟_𝐵 ⊗ 𝒞.Its converse implication holds by [21] (for more gen- eral matrices a similar assertion is shown in [5]). They also proved in [7] that the extension groupExt(𝒪_𝐴),which is isomorphic to the𝐾-group𝐾₀(𝒪_𝐴)as groups, appears as the Bowen–Franks groupBF(𝐴)defined by Bowen–Franks in [2], that is an invariant of flow equivalence of( ̄𝑋_𝐴, ̄𝜎_𝐴)([2]).

There is another kind of construction of𝐶^∗-algebras from two-sided topolog- ical Markov shifts by using groupoids and regarding the Markov shifts as Smale spaces ([1], [30], [33], etc. ). The construction was initiated by D. Ruelle [30], [31] and I. Putnam [23], [24]. I. Putnam in [23] constructed several kinds of groupoids from each Smale space. Each of the groupoids yields a𝐶^∗-algebra.

In this paper, we focus on asymptotic groupoids𝐺_𝐴^𝑎 among several groupoids studied in [12], [23], [24], [25], etc. and their semi-direct products defined be- low. The asymptotic étale groupoid𝐺_𝐴^𝑎 for( ̄𝑋_𝐴, ̄𝜎_𝐴)is defined by

𝐺_𝐴^𝑎 ∶= {(𝑥, 𝑦) ∈ ̄𝑋_𝐴× ̄𝑋_𝐴∣

𝑛→∞lim 𝑑(𝜎_𝐴^𝑛(𝑥), 𝜎_𝐴^𝑛(𝑦)) = lim

𝑛→−∞𝑑(𝜎^𝑛_𝐴(𝑥), 𝜎^𝑛_𝐴(𝑦)) = 0}

with natural groupoid operations and topology (see [23]). It has been shown in [24] that the groupoid𝐺_𝐴^𝑎 is amenable and its𝐶^∗-algebra𝐶^∗(𝐺_𝐴^𝑎)is stably isomorphic to the tensor productℱ_𝐴𝑡 ⊗ ℱ_𝐴 of the canonical AF-subalgebras ℱ_𝐴^𝑡 andℱ_𝐴inside the Cuntz–Krieger algebras𝒪_𝐴^𝑡 and𝒪_𝐴, respectively. The

semi-direct product𝐺_𝐴^𝑎⋊ ℤis defined by

𝐺^𝑎_𝐴⋊ ℤ ∶= {(𝑥, 𝑘 − 𝑙, 𝑦) ∈ ̄𝑋_𝐴× ℤ × ̄𝑋_𝐴∣ ( ̄𝜎_𝐴^𝑘(𝑥), ̄𝜎_𝐴^𝑙 (𝑥)) ∈ 𝐺^𝑎_𝐴}

with natural groupoid operations and topology (see [23]). It is étale and amen- able. The groupoid 𝐶^∗-algebra 𝐶^∗(𝐺_𝐴^𝑎 ⋊ ℤ) is called the Ruelle algebra for the Markov shift( ̄𝑋_𝐴, ̄𝜎_𝐴)and writtenℛ_𝐴. Since the unit space(𝐺_𝐴^𝑎 ⋊ ℤ)^◦ is {(𝑥, 0, 𝑥) ∈ 𝐺^𝑎_𝐴⋊ ℤ ∣ 𝑥 ∈ ̄𝑋_𝐴}that is identified with the shift space𝑋̄_𝐴, the alge- braℛ_𝐴has the commutative𝐶^∗-algebra𝐶( ̄𝑋_𝐴)of continuous functions on𝑋̄_𝐴 as a maximal commutative𝐶^∗-subalgebra. It is the crossed product𝐶^∗(𝐺_𝐴^𝑎)⋊ℤ of 𝐶^∗(𝐺_𝐴^𝑎)induced by the automorphism of the shift 𝜎̄_𝐴, and hence has the dual action written𝜌^𝐴_𝑡 , 𝑡 ∈ 𝕋. See [26] for the construction of𝐶^∗-algebras from groupoids.

Following [23], let us consider the groupoids 𝐺_𝐴^𝑠 and 𝐺^𝑢_𝐴 defined by sta- ble equivalence relation and unstable equivalence relation on( ̄𝑋_𝐴, ̄𝜎_𝐴), respec- tively, which are defined by

𝐺_𝐴^𝑠 ={(𝑥, 𝑦) ∈ ̄𝑋_𝐴× ̄𝑋_𝐴∣ lim

𝑛→∞𝑑( ̄𝜎_𝐴^𝑛(𝑥), ̄𝜎_𝐴^𝑛(𝑦) = 0}, 𝐺_𝐴^𝑢 ={(𝑥, 𝑦) ∈ ̄𝑋_𝐴× ̄𝑋_𝐴∣ lim

𝑛→−∞𝑑( ̄𝜎^𝑛_𝐴(𝑥), ̄𝜎^𝑛_𝐴(𝑦) = 0}.

In [19] and [20], the author introduced the groupoid𝐺_𝐴^𝑠,𝑢⋊ ℤ²defined by 𝐺_𝐴^𝑠,𝑢⋊ ℤ²∶= {(𝑥, 𝑝, 𝑞, 𝑦) ∈ ̄𝑋_𝐴× ℤ × ℤ × ̄𝑋_𝐴∣

( ̄𝜎_𝐴^𝑝(𝑥), 𝑦) ∈ 𝐺_𝐴^𝑠, ( ̄𝜎^𝑞_𝐴(𝑥)), 𝑦) ∈ 𝐺^𝑢_𝐴}

which has a natural groupoid operations and topology making it étale and amen- able. The groupoid𝐶^∗-algebra𝐶^∗(𝐺_𝐴^𝑠,𝑢⋊ ℤ²)is called the extended Ruelle al- gebra writtenℛ˜_𝐴.Since the unit space(𝐺_𝐴^𝑠,𝑢⋊ ℤ²)^◦is{(𝑥, 0, 0, 𝑥) ∈ 𝐺_𝐴^𝑠,𝑢⋊ ℤ² ∣ 𝑥 ∈ ̄𝑋_𝐴}that is identified with the shift space𝑋̄_𝐴, the algebraℛ˜_𝐴has𝐶( ̄𝑋_𝐴)as a maximal abelian𝐶^∗-subalgebra. As in [19] and [20], there exists a projection 𝐸_𝐴in the tensor product𝒪_𝐴^𝑡⊗ 𝒪_𝐴such that𝐸_𝐴(𝒪_𝐴^𝑡⊗ 𝒪_𝐴)𝐸_𝐴is naturally iso- morphic to the algebraℛ˜_𝐴, so that the𝐶^∗-algebraℛ˜_𝐴is regarded as a version of the bilateral Cuntz–Krieger algebra. Let𝛼_𝐴denote the gauge action on the Cuntz–Krieger algebra𝒪_𝐴. Under the identification between𝐸_𝐴(𝒪_𝐴^𝑡⊗ 𝒪_𝐴)𝐸_𝐴 and ℛ˜_𝐴, the tensor product𝛼^𝐴_𝑟^𝑡 ⊗ 𝛼_𝑠^𝐴 for (𝑟, 𝑠) ∈ 𝕋² yields an action of 𝕋² written 𝛾^𝐴_(𝑟,𝑠), (𝑟, 𝑠) ∈ 𝕋².In [20, Theorem 1.1], it was shown that the triplet ( ˜ℛ_𝐴, 𝐶( ̄𝑋_𝐴), 𝛾^𝐴)is a complete invariant for the topological conjugacy class of ( ̄𝑋_𝐴, ̄𝜎_𝐴). For a continuous function𝑓 ∶ ̄𝑋_𝐴 ⟶ ℕ,we may define an action 𝛾^𝐴,𝑓weighted by𝑓onℛ˜_𝐴.In this paper, we will characterize the flow equiva- lence class of( ̄𝑋_𝐴, ̄𝜎_𝐴)in terms of the stabilized version ofℛ˜_𝐴with the weighted action𝛾^𝐴,𝑓.The continuous function𝑓 ∶ ̄𝑋_𝐴 ⟶ ℕexactly corresponds to a ceiling function of a discrete suspension. The main result of this paper is the following theorem.

Theorem 1.1(Theorem6.8). Let𝐴, 𝐵be irreducible, non-permutation matri- ces with entries in{0, 1}.The two-sided topological Markov shifts( ̄𝑋_𝐴, ̄𝜎_𝐴)and

( ̄𝑋_𝐵, ̄𝜎_𝐵)are flow equivalent if and only if there is an irreducible non-permutation matrix𝐶with entries in{0, 1}and continuous functions𝑓_𝐴, 𝑓_𝐵 ∶ ̄𝑋_𝐶 ⟶ ℕwith values in the positive integers such thatℛ˜_𝐴⊗ 𝒦andℛ˜_𝐵⊗ 𝒦 are isomorphic to ℛ˜_𝐶⊗ 𝒦via isomorphismsΦ_𝐴andΦ_𝐵satisfying

Φ_𝐴◦(𝛾_(𝑟,𝑠)^𝐴 ⊗ id) = (𝛾_(𝑟,𝑠)^{𝐶,𝑓𝐴}⊗ id)◦Φ_𝐴, Φ_𝐵◦(𝛾^𝐵_(𝑟,𝑠)⊗ id) = (𝛾^𝐶,𝑓_(𝑟,𝑠)^𝐵⊗ id)◦Φ_𝐵 for(𝑟, 𝑠) ∈ 𝕋².

The above statement exactly corresponds to the situation that the topological Markov shift( ̄𝑋_𝐴, ̄𝜎_𝐴)is realized as a discrete suspension of( ̄𝑋_𝐶, ̄𝜎_𝐶)by ceiling function𝑓_𝐴, and( ̄𝑋_𝐵, ̄𝜎_𝐵)is realized as a discrete suspension of ( ̄𝑋_𝐶, ̄𝜎_𝐶) by ceiling function𝑓_𝐵. As a corollary we have the following.

Corollary 1.2(Corollary6.9). Let𝐴, 𝐵be irreducible, non-permutation matrices with entries in{0, 1}.Two-sided topological Markov shifts( ̄𝑋_𝐴, ̄𝜎_𝐴)and( ̄𝑋_𝐵, ̄𝜎_𝐵) are flow equivalent if and only if there exist continuous functions𝑓_𝐴∶ ̄𝑋_𝐴⟶ ℕ and𝑓_𝐵 ∶ ̄𝑋_𝐵 ⟶ ℕwith values in the positive integers, and an isomorphism Φ ∶ ˜ℛ_𝐴⊗ 𝒦 ⟶ ˜ℛ_𝐵⊗ 𝒦of𝐶^∗-algebras such that

Φ◦(𝛾^𝐴,𝑓_(𝑟,𝑠)^𝐴⊗ id) = (𝛾^𝐵,𝑓_(𝑟,𝑠)^𝐵⊗ id)◦Φ, (𝑟, 𝑠) ∈ 𝕋². The organization of the paper is the following.

In Section 2, we will briefly recall basic notation and terminology on groupoid 𝐶^∗-algebras, Cuntz-Krieger algebras, Ruelle algebras and flow equivalence of topological Markov shifts.

In Section 3, a bilateral version of the Krieger’s dimension group for topo- logical Markov shifts will be studied and called the dimension quadruplet that will be shown to be invariant for shift equivalence of the underlying matrices.

In Section 4, the dimension quadruplet is described by the K-group of the AF-algebra𝐶^∗(𝐺_𝐴^𝑎)of the groupoid𝐺_𝐴^𝑎.As a result, a sufficient condition under which the two-sided Markov shifts( ̄𝑋_𝐴, ̄𝜎_𝐴)and( ̄𝑋_𝐵, ̄𝜎_𝐵)are flow equivalent is given in terms of the stabilized action𝛾^𝐴⊗ idof𝕋²onℛ˜_𝐴⊗ 𝒦(Proposition 4.6).

In Section 5, the action𝛾^𝐴,𝑓 with potential function𝑓on the algebraℛ˜_𝐴is introduced.

In Section 6, we characterize the flow equivalence of two-sided topological Markov shifts in terms of the actions with potential functions of two dimen- sional torus on the extended Ruelle algebrasℛ˜_𝐴.

In Section 7, we reformulate Theorem1.1 and Corollary1.2 by describing their statements including not only flow equivalence but also topological con- jugacy of two-sided topological Markov shifts (Theorem7.2and Theorem7.3).

Throughout the paper, we denote byℤ₊andℕthe set of nonnegative integers and the set of positive integers, respectively.

2. Preliminaries

In this section, we briefly recall basic notation and terminology on the𝐶^∗- algebras of étale groupoids, Cuntz-Krieger algebras, Ruelle algebras and flow equivalence of topological Markov shifts. In what follows, a square matrix𝐴 = [𝐴(𝑖, 𝑗)]^𝑁_𝑖,𝑗=1is assumed to be an𝑁 × 𝑁 irreducible, non-permutation matrix with entries in{0, 1}.

2.1. 𝑪^∗-algebras of étale groupoids. Let us construct𝐶^∗-algebras from étale groupoids. The general theory of the construction of groupoid𝐶^∗-algebras was initiated and studied by Renault [26] (see also [27], [28]). The construction will be used in the following sections. Let𝐺be an étale groupoid with its unit space 𝐺^◦and range map, source map𝑟, 𝑠 ∶ 𝐺 ⟶ 𝐺^◦and𝐶_𝑐(𝐺)denote the∗-algebra of continuous functions on𝐺with compact support having its product and∗- involution defined by

(𝑓 ∗ 𝑔)(𝛾) = ∑

𝜂;𝑟(𝛾)=𝑟(𝜂)

𝑓(𝜂)𝑔(𝜂⁻¹𝛾), 𝑓^∗(𝛾) = 𝑓(𝛾⁻¹)

for𝑓, 𝑔 ∈ 𝐶_𝑐(𝐺), 𝛾 ∈ 𝐺.We denote by𝐶₀(𝐺^◦)the commutative𝐶^∗-algebra of continuous functions on𝐺^◦ vanishing at infinity. The algebra𝐶_𝑐(𝐺) has a structure of right𝐶₀(𝐺^◦)-module with𝐶₀(𝐺^◦)-valued right inner product given by

(𝜉𝑔)(𝛾) = 𝜉(𝛾)𝑔(𝑠(𝛾)), < 𝜉, 𝜁 > (𝑡) = ∑

𝜂;𝑡=𝑠(𝜂)

𝜉(𝜂)𝜁(𝜂)

for𝜉, 𝜁 ∈ 𝐶_𝑐(𝐺), 𝑔 ∈ 𝐶_𝑐(𝐺^◦), 𝛾 ∈ 𝐺, 𝑡 ∈ 𝐺^◦. The completion of𝐶_𝑐(𝐺) by the norm defined by the above inner product is denoted by𝓁²(𝐺),which is a Hilbert 𝐶^∗-right module over 𝐶₀(𝐺^◦). The algebra𝐶_𝑐(𝐺) is represented on 𝓁²(𝐺) as bounded adjointable𝐶₀(𝐺^◦)-right module maps by𝜋(𝑓)𝜉 = 𝑓 ∗ 𝜉 for𝑓 ∈ 𝐶_𝑐(𝐺), 𝜉 ∈ 𝓁²(𝐺).The closure of𝜋(𝐶_𝑐(𝐺))by the operator norm on 𝓁²(𝐺) is denoted by 𝐶_𝑟^∗(𝐺) and called the (reduced) groupoid 𝐶^∗-algebra for the étale groupoid𝐺.The completeion of𝐶_𝑐(𝐺) by the universal𝐶^∗-norm is called the (full) groupoid 𝐶^∗-algebra for𝐺. Now we treat the three kinds of groupoids 𝐺^𝑎_𝐴, 𝐺_𝐴^𝑎 ⋊ ℤ, 𝐺_𝐴^𝑠,𝑢 ⋊ ℤ². They are all étale and amenable, so that the two groupoid𝐶^∗-algebras𝐶_𝑟^∗(𝐺)and𝐶^∗(𝐺)are canonically isomorphic for such groupoids. We do not distinguish them, and write them as𝐶^∗(𝐺)for𝐺 = 𝐺_𝐴^𝑎, 𝐺^𝑎_𝐴⋊ ℤ, 𝐺_𝐴^𝑠,𝑢⋊ ℤ².

2.2. Cuntz-Krieger algebras, Ruelle algebras and extended Ruelle alge- bras. The Cuntz–Krieger algebra𝒪_𝐴introduced by Cuntz–Krieger [7] is a uni- versal unique𝐶^∗-algebra generated by partial isometries𝑆₁, … , 𝑆_𝑁subject to the relations:

∑𝑁 𝑗=1

𝑆_𝑗𝑆_𝑗^∗ = 1, 𝑆^∗_𝑖𝑆_𝑖 =

∑𝑁 𝑗=1

𝐴(𝑖, 𝑗)𝑆_𝑗𝑆_𝑗^∗, 𝑖 = 1, … , 𝑁. (2.1)

By the universality for the relations (2.1) of operators, the correspondence𝑆_𝑖 ⟶ exp(2𝜋

√

−1𝑡)𝑆_𝑖, 𝑖 = 1, … , 𝑁for each𝑡 ∈ ℝ∕ℤ = 𝕋yields an automorphism written𝛼^𝐴_𝑡 on the𝐶^∗-algebra𝒪_𝐴.The automorphisms𝛼^𝐴_𝑡 , 𝑡 ∈ 𝕋define an ac- tion of𝕋on𝒪_𝐴called the gauge action. It is well-known that the fixed point algebra(𝒪_𝐴)^𝛼𝐴of𝒪_𝐴under the gauge action is an AF-algebra writtenℱ_𝐴. Let us denote by𝐵_𝑚( ̄𝑋_𝐴) the set of admissible words in𝑋̄_𝐴 of length 𝑚 and by 𝐵_∗( ̄𝑋_𝐴)the set of all admissible words of𝑋̄_𝐴. For𝜇 = (𝜇₁, … , 𝜇_𝑚) ∈ 𝐵_𝑚( ̄𝑋_𝐴), we write𝑆_𝜇 = 𝑆_𝜇1⋯ 𝑆_𝜇𝑚. We denote by𝒟_𝐴the𝐶^∗-subalgebra ofℱ_𝐴generated by projections𝑆_𝜇𝑆_𝜇^∗, 𝜇 ∈ 𝐵_∗( ̄𝑋_𝐴).

As in [7] and [6] (cf. [18], [29]), the crossed product𝒪_𝐴⋊_𝛼𝐴𝕋is stably isomor- phic to the AF-algebraℱ_𝐴. Hence the dual action ̂𝛼^𝐴on𝒪_𝐴⋊_𝛼^𝐴𝕋induces an automorphism on𝐾₀(ℱ_𝐴), that is written𝛿_𝐴. The triplet(𝐾₀(ℱ_𝐴), 𝐾⁺₀(ℱ_𝐴), 𝛿_𝐴) appears as the (future) dimension triplet written (∆_𝐴, ∆⁺_𝐴, 𝛿_𝐴) for 𝐴 defined by W. Krieger [16]. For the transposed matrix𝐴^𝑡 of𝐴, we similarly consider the Cuntz–Krieger algebra𝒪_𝐴^𝑡 and its AF-subalgebraℱ_𝐴^𝑡. Let us denote by 𝑇₁, … , 𝑇_𝑁the generating partial isometries of𝒪_𝐴𝑡 which satisfy the relations:

∑𝑁 𝑖=1

𝑇_𝑖𝑇^∗_𝑖 = 1, 𝑇^∗_𝑗𝑇_𝑗 =

∑𝑁 𝑖=1

𝐴(𝑖, 𝑗)𝑇_𝑖𝑇_𝑖^∗, 𝑗 = 1, … , 𝑁. (2.2) For𝜉 = (𝜉₁, … , 𝜉_𝑘) ∈ 𝐵_𝑘( ̄𝑋_𝐴),we denote by𝜉̄the transposed word(𝜉_𝑘, … , 𝜉₁) which belongs to𝐵_𝑘( ̄𝑋_𝐴^𝑡),and write𝑇𝜉̄ = 𝑇_𝜉𝑘⋯ 𝑇_𝜉1.

Define the projection𝐸_𝐴∈ ℱ_𝐴^𝑡 ⊗ ℱ_𝐴by setting 𝐸_𝐴=

∑𝑁 𝑗=1

𝑇_𝑗^∗𝑇_𝑗⊗ 𝑆_𝑗𝑆_𝑗^∗ which coincides with∑𝑁

𝑖=1𝑇_𝑖𝑇_𝑖^∗⊗ 𝑆^∗_𝑖𝑆_𝑖because of the equalities (2.1) and (2.2).

Let𝐺_𝐴^𝑎, 𝐺_𝐴^𝑎⋊ℤ, 𝐺_𝐴^𝑠,𝑢⋊ℤ²denote the étale amenable groupoids stated in Section 1. For reference, we state [20, Proposition 2.1] as

Lemma 2.1.

(i) The groupoid 𝐶^∗-algebra 𝐶^∗(𝐺_𝐴^𝑎) is canonically isomorphic to the𝐶^∗- subalgebra ofℱ_𝐴𝑡 ⊗ ℱ_𝐴generated by elements𝑇𝜉̄𝑇^∗_𝜂̄⊗ 𝑆_𝜇𝑆^∗_𝜈 where𝜇 = (𝜇₁, … , 𝜇_𝑚), 𝜈 = (𝜈₁, … , 𝜈_𝑛) ∈ 𝐵_∗( ̄𝑋_𝐴), ̄𝜉 = (𝜉_𝑘, … , 𝜉₁), ̄𝜂 = (𝜂_𝑙, … , 𝜂₁) ∈ 𝐵_∗( ̄𝑋_𝐴𝑡)satisfying𝐴(𝜉_𝑘, 𝜇₁) = 𝐴(𝜂_𝑙, 𝜈₁) = 1and𝑘 = 𝑙, 𝑚 = 𝑛.Hence 𝐶^∗(𝐺_𝐴^𝑎)is canonically isomorphic to the𝐶^∗-algebra𝐸_𝐴(ℱ_𝐴^𝑡⊗ ℱ_𝐴)𝐸_𝐴. (ii) The Ruelle algebraℛ_𝐴 = 𝐶^∗(𝐺_𝐴^𝑎 ⋊ ℤ)is canonically isomorphic to the

𝐶^∗-subalgebra of𝒪_𝐴𝑡⊗𝒪_𝐴generated by elements𝑇𝜉̄𝑇_𝜂^∗_̄⊗𝑆_𝜇𝑆_𝜈^∗where𝜇 = (𝜇₁, … , 𝜇_𝑚), 𝜈 = (𝜈₁, … , 𝜈_𝑛) ∈ 𝐵_∗( ̄𝑋_𝐴), ̄𝜉 = (𝜉_𝑘, … , 𝜉₁), ̄𝜂 = (𝜂_𝑙, … , 𝜂₁) ∈ 𝐵_∗( ̄𝑋_𝐴𝑡)satisfying𝐴(𝜉_𝑘, 𝜇₁) = 𝐴(𝜂_𝑙, 𝜈₁) = 1and𝑚 + 𝑘 = 𝑛 + 𝑙.

(iii) The extended Ruelle algebraℛ˜_𝐴= 𝐶^∗(𝐺^𝑠,𝑢_𝐴 ⋊ ℤ²)is canonically isomor- phic to the𝐶^∗-subalgebra of𝒪_𝐴𝑡⊗𝒪_𝐴generated by elements𝑇𝜉̄𝑇^∗_𝜂̄⊗𝑆_𝜇𝑆_𝜈^∗ where𝜇 = (𝜇₁, … , 𝜇_𝑚), 𝜈 = (𝜈₁, … , 𝜈_𝑛) ∈ 𝐵_∗( ̄𝑋_𝐴), ̄𝜉 = (𝜉_𝑘, … , 𝜉₁), ̄𝜂 =

(𝜂_𝑙, … , 𝜂₁) ∈ 𝐵_∗( ̄𝑋_𝐴𝑡)satisfying𝐴(𝜉_𝑘, 𝜇₁) = 𝐴(𝜂_𝑙, 𝜈₁) = 1.Henceℛ˜_𝐴is canonically isomorphic to the𝐶^∗-algebra𝐸_𝐴(𝒪_𝐴^𝑡⊗ 𝒪_𝐴)𝐸_𝐴.

Under the identification between𝐸_𝐴(𝒪_𝐴𝑡⊗ 𝒪_𝐴)𝐸_𝐴andℛ˜_𝐴, the tensor prod- uct𝛼^𝐴_𝑟^𝑡 ⊗ 𝛼_𝑠^𝐴 of gauge actions on𝒪_𝐴𝑡 and 𝒪_𝐴 yields an action of 𝕋² onℛ˜_𝐴 written𝛾_(𝑟,𝑠)^𝐴 , (𝑟, 𝑠) ∈ 𝕋²,because𝛾_(𝑟,𝑠)^𝐴 (𝐸_𝐴) = 𝐸_𝐴.We write

𝛿^𝐴_𝑡 ∶= 𝛼^𝐴_𝑡^𝑡⊗ 𝛼^𝐴_𝑡 , 𝜌^𝐴_𝑡 ∶= 𝛾^𝐴

(−^𝑡

2,^𝑡

2), 𝑡 ∈ 𝕋.

Lemma 2.2.

(i) The restriction of the action𝜌^𝐴_𝑡 , 𝑡 ∈ 𝕋to the subalegeraℛ_𝐴is regarded as the dual action onℛ_𝐴under a natural identification betweenℛ_𝐴and the crossed product𝐶^∗(𝐺_𝐴^𝑎) ⋊ ℤ. Hence the fixed point algebra(ℛ_𝐴)^𝜌𝐴is isomorphic to𝐶^∗(𝐺_𝐴^𝑎).

(ii) The fixed point algebra( ˜ℛ_𝐴)^𝛿𝐴ofℛ˜_𝐴under𝛿^𝐴is isomorphic toℛ_𝐴, so that the fixed point algebra( ˜ℛ_𝐴)^𝛾𝐴ofℛ˜_𝐴under𝛾^𝐴is isomorphic to𝐶^∗(𝐺_𝐴^𝑎).

2.3. Suspension and flow equivalence. We will briefly review discrete sus- pensions of topological Markov shifts. Let𝑓 ∶ ̄𝑋_𝐴 ⟶ ℕ be a continuous function on the shift space𝑋̄_𝐴with values in the positive integers. Let𝑓( ̄𝑋_𝐴) = {1, 2, … , 𝐿}.Put𝑋_𝑗 = {𝑥 ∈ ̄𝑋_𝐴 ∣ 𝑓(𝑥) = 𝑗}, 𝑗 = 1, … , 𝐿.Define the suspension space𝑋̄_𝐴,𝑓 = ∪^𝐿_𝑗=1𝑋_𝑗× {0, 1, … , 𝑗 − 1}with transformation𝜎̄_𝐴,𝑓on𝑋̄_𝐴,𝑓by

̄

𝜎_𝐴,𝑓([𝑥, 𝑘]) = {[𝑥, 𝑘 + 1] if0 ≤ 𝑘 ≤ 𝑗 − 2, [ ̄𝜎_𝐴(𝑥), 0] if𝑘 = 𝑗 − 1

for[𝑥, 𝑘] ∈ 𝑋_𝑗 × {0, 1, … , 𝑗 − 1}. The resulting topological dynamical system ( ̄𝑋_𝐴,𝑓, ̄𝜎_𝐴,𝑓)is called the discrete suspension of( ̄𝑋_𝐴, ̄𝜎_𝐴)by the ceiling function 𝑓, which is homeomorphic to a topological Markov shift. If, in particular, the function𝑓 ∶ ̄𝑋_𝐴 ⟶ ℕ depends only on the0th coordinate of𝑋̄_𝐴, then 𝑓 is written𝑓 = ∑𝑁

𝑗=1𝑓_𝑗𝜒_𝑈

𝑗(0) for some integers𝑓_𝑗 ∈ ℕ,where 𝜒_𝑈

𝑗(0) is the characteristic function of the cylinder set

𝑈_𝑗(0) = {(𝑥_𝑛)_𝑛∈ℤ ∈ ̄𝑋_𝐴∣ 𝑥₀ = 𝑗}, 𝑗 = 1, … , 𝑁.

Put𝑚_𝑗 = 𝑓_𝑗− 1for𝑗 = 1, … , 𝑁.Let𝒢 = (𝒱, ℰ)be the directed graph defined by the matrix𝐴with the vertex set𝒱 = {1, 2, … , 𝑁}.An edge of𝒢is defined by a pair(𝑖, 𝑗)of vertices𝑖, 𝑗 = 1, … , 𝑁such that𝐴(𝑖, 𝑗) = 1,whose source is𝑖and the terminal is𝑗.The set of such pairs(𝑖, 𝑗)is the edge setℰ.Construct a new graph𝒢_𝑓 = (𝒱_𝑓, ℰ_𝑓)with its transition matrix𝐴_𝑓 from the graph𝒢 = (𝒱, ℰ) and the function𝑓such that𝒱_𝑓 = ∪^𝑁_𝑗=1{𝑗₀, 𝑗₁, 𝑗₂, … , 𝑗_𝑚𝑗}and if𝐴(𝑗, 𝑘) = 1, then

𝐴_𝑓(𝑗₀, 𝑗₁) = 𝐴_𝑓(𝑗₁, 𝑗₂) = ⋯ = 𝐴_𝑓(𝑗_𝑚

𝑗−1, 𝑗_𝑚

𝑗) = 𝐴_𝑓(𝑗_𝑚

𝑗, 𝑘₀) = 1. (2.3) For other pairs(𝑗_𝑖, 𝑗_𝑖^′′) ∈ 𝒱_𝑓× 𝒱_𝑓,we define𝐴_𝑓(𝑗_𝑖, 𝑗_𝑖^′′) = 0.Hence the size of the matrix𝐴_𝑓is(𝑓₁+ 𝑓₂+ ⋯ + 𝑓_𝑁) × (𝑓₁+ 𝑓₂+ ⋯ + 𝑓_𝑁). Then the discrete

suspension( ̄𝑋_𝐴,𝑓, ̄𝜎_𝐴,𝑓)is nothing but the topological Markov shift( ̄𝑋_𝐴𝑓, ̄𝜎_𝐴𝑓) defined by the matrix𝐴_𝑓.

Two topological Markov shifts are said to be flow equivalent if they are real- ized as cross sections with their first return maps of a common one-dimensional flow space. Parry–Sullivan in [22] proved that( ̄𝑋_𝐴, ̄𝜎_𝐴)and( ̄𝑋_𝐵, ̄𝜎_𝐵)are flow equivalent if and only if there exist another topological Markov shift( ̄𝑋_𝐶, ̄𝜎_𝐶) for some matrix𝐶and continuous maps𝑓_𝐴, 𝑓_𝐵 ∶ ̄𝑋_𝐶⟶ ℕsuch that( ̄𝑋_𝐴, ̄𝜎_𝐴) is topologically conjugate to the discrete suspension( ̄𝑋_{𝐶,𝑓𝐴}, ̄𝜎_{𝐶,𝑓𝐴})and( ̄𝑋_𝐵, ̄𝜎_𝐵) is topologically conjugate to the discrete suspension( ̄𝑋_{𝐶,𝑓𝐵}, ̄𝜎_{𝐶,𝑓𝐵}).

Cuntz and Krieger were the first to find interesting relations between flow equivalence of topological Markov shifts and Cuntz–Krieger algebras in [7]. Re- call that𝒦and𝒞are the𝐶^∗-algebra of compact operators on the separable in- finite dimensional Hilbert space𝓁²(ℕ)and its commutative𝐶^∗-subalgebra of diagonal operators on𝓁²(ℕ). Cuntz and Krieger proved that for irreducible non-permutation matrices𝐴and𝐵, if( ̄𝑋_𝐴, ̄𝜎_𝐴)and( ̄𝑋_𝐵, ̄𝜎_𝐵)are flow equiva- lent, then there exists an isomorphismΦ ∶ 𝒪_𝐴⊗ 𝒦 ⟶ 𝒪_𝐵⊗ 𝒦of𝐶^∗-algebras such thatΦ(𝒟_𝐴⊗𝒞) = 𝒟_𝐵⊗𝒞.Its converse implication holds by [21] (for more general matrices a similar assertion is shown in [5]).

In this paper, we will study flow equivalence of topological Markov shifts in terms of the extended Ruelle algebras with its action𝛾^𝐴of𝕋².

3. Bilateral dimension groups

We keep an irreducible, non-permutation matrix𝐴 = [𝐴(𝑖, 𝑗)]^𝑁_𝑖,𝑗=1with en- tries in{0, 1}.Following W. Krieger [16] (cf. [14], [15], [8], etc.), the dimension group(∆_𝐴, ∆⁺_𝐴)for the matrix𝐴are defined as an ordered group by the induc- tive limits

∆_𝐴= ℤ^{𝑁 𝐴}

𝑡

⟶ ℤ^{𝑁 𝐴}

𝑡

⟶ ⋯ , ∆⁺_𝐴= ℤ^𝑁₊ ^𝐴

𝑡

⟶ ℤ^𝑁₊ ^𝐴

𝑡

⟶ ⋯ .

The group∆_𝐴 is identified with the equivalence classes of ∪^∞_𝑛=0{(𝑣, 𝑛) ∣ 𝑣 ∈ ℤ^𝑁, 𝑛 ∈ ℤ₊} by the equivalence relation generated by(𝑣, 𝑛) ∼ (𝐴^𝑡𝑣, 𝑛 + 1). The equivalence class of(𝑣, 𝑛)is denoted by[𝑣, 𝑛].The dimension drop auto- morphism𝛿_𝐴on(∆_𝐴, ∆⁺_𝐴)is defined by𝛿_𝐴([𝑣, 𝑛]) = [(𝑣, 𝑛 + 1)]for[𝑣, 𝑛] ∈ ∆_𝐴. The triplet (∆_𝐴, ∆⁺_𝐴, 𝛿_𝐴) is called the (future) dimension triplet for the topo- logical Markov shift( ̄𝑋_𝐴, ̄𝜎_𝐴). We similarly have the (future) dimension triplet (∆_𝐴𝑡, ∆⁺_𝐴𝑡, 𝛿_𝐴𝑡)for the topological Markov shift( ̄𝑋_𝐴𝑡, ̄𝜎_𝐴𝑡)for the matrix𝐴^𝑡, which is called the (past) dimension triplet for( ̄𝑋_𝐴, ̄𝜎_𝐴).Hence we have two dimen- sion triplets(∆_𝐴, ∆⁺_𝐴, 𝛿_𝐴)and(∆_𝐴𝑡, ∆⁺_𝐴𝑡, 𝛿_𝐴𝑡)for the matrix𝐴.

Let𝑒_𝑖 ∈ ℤ^𝑁be the vector ofℤ^𝑁whose𝑖th component is1,other components are zeros. We will define a specific element𝑢̃_𝐴in∆_𝐴𝑡⊗ ∆_𝐴by setting

̃ 𝑢_𝐴∶=

∑𝑁 𝑖,𝑗=1

[𝑒_𝑗, 1] ⊗ 𝐴(𝑗, 𝑖)[𝑒_𝑖, 1] ∈ ∆_𝐴^𝑡 ⊗ ∆_𝐴.

We then see that

̃ 𝑢_𝐴=

∑𝑁 𝑗=1

⎛

⎜

⎝

[𝑒_𝑗, 1] ⊗ [

∑𝑁 𝑖=1

𝐴(𝑗, 𝑖)𝑒_𝑖, 1]⎞

⎟

⎠

=

∑𝑁 𝑗=1

[𝑒_𝑗, 1] ⊗ [𝐴^𝑡𝑒_𝑗, 1]

=(id ⊗ 𝛿_𝐴⁻¹)

∑𝑁 𝑗=1

([𝑒_𝑗, 1] ⊗ [𝑒_𝑗, 1])

and

̃ 𝑢_𝐴=

∑𝑁 𝑖=1

⎛

⎜

⎝ [

∑𝑁 𝑗=1

𝐴(𝑗, 𝑖)𝑒_𝑗, 1] ⊗ [𝑒_𝑖, 1]⎞

⎟

⎠

=

∑𝑁 𝑖=1

[𝐴𝑒_𝑖, 1] ⊗ [𝑒_𝑖, 1]

=(𝛿_𝐴⁻¹_𝑡 ⊗ id)

∑𝑁 𝑖=1

([𝑒_𝑖, 1] ⊗ [𝑒_𝑖, 1]).

Define an automorphism𝛿̃_𝐴 ∶ ∆_𝐴𝑡⊗ ∆_𝐴 ⟶ ∆_𝐴𝑡 ⊗ ∆_𝐴by𝛿̃_𝐴 = 𝛿⁻¹_𝐴𝑡 ⊗ 𝛿_𝐴. It satisfies

𝛿̃_𝐴([𝑢, 𝑛] ⊗ [𝑣, 𝑚]) = [𝐴𝑢, 𝑛] ⊗ [𝑣, 𝑚 + 1], [𝑢, 𝑛] ⊗ [𝑣, 𝑚] ∈ ∆_𝐴𝑡⊗ ∆_𝐴. We set the abelian group∆̃_𝐴= ∆_𝐴𝑡⊗ ∆_𝐴with its positive cone∆̃⁺_𝐴 = ∆⁺_𝐴𝑡⊗ ∆⁺_𝐴. Definition 3.1. The quadruplet( ̃∆_𝐴, ̃∆⁺_𝐴, ̃𝛿_𝐴, ̃𝑢_𝐴)is called thedimension quadru- pletfor the two-sided topological Markov shift( ̄𝑋_𝐴, ̄𝜎_𝐴).

We note that a bilateral version of the dimension groups first appeared in Krieger’s paper [14] (cf. [15], [16]).

Lemma 3.2. 𝛿̃_𝐴( ̃𝑢_𝐴) = ̃𝑢_𝐴. Proof. Since

̃

𝑢_𝐴= (id ⊗ 𝛿⁻¹_𝐴 )

∑𝑁 𝑗=1

([𝑒_𝑗, 1] ⊗ [𝑒_𝑗, 1]) = (𝛿_𝐴⁻¹𝑡 ⊗ id)

∑𝑁 𝑖=1

([𝑒_𝑖, 1] ⊗ [𝑒_𝑖, 1]), and𝛿̃_𝐴 = 𝛿_𝐴⁻¹_𝑡 ⊗ 𝛿_𝐴,the assertion is immediate.

We will next show that the dimension quadruplet( ̃∆_𝐴, ̃∆⁺_𝐴, ̃𝛿_𝐴, ̃𝑢_𝐴)is invari- ant under shift equivalence of the underlying matrices𝐴. The notion of shift equivalence in square matrices with entries in nonnegative integers has been introduced by W. F. Williams [34]. Two matrices𝐴and𝐵are said to be shift equivalent if there exist rectangular matrices𝐻, 𝐾with entries in nonnegative integers and a positive integer𝓁such that

𝐴^𝓁= 𝐻𝐾, 𝐵^𝓁= 𝐾𝐻, 𝐴𝐻 = 𝐻𝐵, 𝐾𝐴 = 𝐵𝐾. (3.1) W. Krieger has proved in [16] that two matrices𝐴and𝐵are shift equivalent if and only if their dimension triplet(∆_𝐴, ∆⁺_𝐴, 𝛿_𝐴)and(∆_𝐵, ∆⁺_𝐵, 𝛿_𝐵)are isomorphic.

The following result has been already proved by C. G. Holton [10, Proposition

6.7] for primitive matrices by using Rohlin property of automorphisms on the AF-algebras𝐶^∗(𝐺_𝐴^𝑎). The proof given below does not use𝐶^∗-algebra theory, nor does it assume that the matrices are primitive.

Proposition 3.3 (C. G. Holton [10, Proposition 6.7]). Suppose that 𝐴and 𝐵 are shift equivalent. Then there exists an isomorphismΦ ∶ ̃∆_𝐴 ⟶ ̃∆_𝐵 which yields an isomorphism between the dimension quadruplets( ̃∆_𝐴, ̃∆⁺_𝐴, ̃𝛿_𝐴, ̃𝑢_𝐴)and ( ̃∆_𝐵, ̃∆⁺_𝐵, ̃𝛿_𝐵, ̃𝑢_𝐵).

Proof. Let𝐴and𝐵be𝑁 × 𝑁matrix and𝑀 × 𝑀matrix, respectively. Assume that there exist rectangular matrices𝐻, 𝐾with entries in nonnegative integers and a positive integer𝓁satisfying (3.1). Define

Φ₊∶ ∆_𝐴⟶ ∆_𝐵 by Φ₊([𝑣, 𝑘]) = [𝐻^𝑡𝑣, 𝑘], Φ₋∶ ∆_𝐴^𝑡 ⟶ ∆_𝐵^𝑡 by Φ₋([𝑣, 𝑘]) = [𝐾𝑣, 𝑘 + 𝓁],

so that

Φ⁻¹₊ ∶ ∆_𝐵 ⟶ ∆_𝐴 satisfies Φ⁻¹₊ ([𝑢, 𝑗]) = [𝐾^𝑡𝑢, 𝑗 + 𝓁], Φ⁻¹₋ ∶ ∆_𝐵𝑡 ⟶ ∆_𝐴𝑡 satisfies Φ⁻¹₋ ([𝑢, 𝑗]) = [𝐻𝑢, 𝑗].

As in [16],Φ₊∶ ∆_𝐴⟶ ∆_𝐵andΦ₋∶ ∆_𝐴𝑡 ⟶ ∆_𝐵𝑡yield isomorphisms for each such that

Φ₊(∆⁺_𝐴) = ∆⁺_𝐵, Φ₊◦𝛿_𝐴= 𝛿_𝐵◦Φ₊, Φ₋(∆⁺_𝐴𝑡) = ∆⁺_𝐵𝑡, Φ₋◦𝛿_𝐴^𝑡 = 𝛿_𝐵^𝑡◦Φ₋.

Hence they induce isomorphisms

Φ₊∶ (∆_𝐴, ∆⁺_𝐴, 𝛿_𝐴) ⟶ (∆_𝐵, ∆⁺_𝐵, 𝛿_𝐵), Φ₋∶ (∆_𝐴𝑡, ∆⁺_𝐴𝑡, 𝛿_𝐴𝑡) ⟶ (∆_𝐵𝑡, ∆⁺_𝐵𝑡, 𝛿_𝐵𝑡).

We defineΦ = Φ₋⊗ Φ₊ ∶ ̃∆_𝐴 ⟶ ̃∆_𝐵.Let𝑓_𝑙 ∈ ℤ^𝑀 be the vector whose𝑙th component is1, and other components are zeros. It then follows that

Φ( ̃𝑢_𝐴) =

∑𝑁 𝑖,𝑗=1

Φ₋([𝑒_𝑗, 1]) ⊗ Φ₊(𝐴(𝑗, 𝑖)[𝑒_𝑖, 1])

=

∑𝑁 𝑖,𝑗=1

[𝐾𝑒_𝑗, 1 + 𝓁] ⊗ 𝐴(𝑗, 𝑖)[𝐻^𝑡𝑒_𝑖, 1]

=

∑𝑁 𝑗=1

[𝐾𝑒_𝑗, 1 + 𝓁] ⊗ [(𝐴𝐻)^𝑡𝑒_𝑗, 1]

=

∑𝑁 𝑗=1

[𝐾𝑒_𝑗, 1 + 𝓁] ⊗ [

∑𝑀 𝑙=1

(𝐴𝐻)(𝑗, 𝑙)𝑓_𝑙, 1]

=

∑𝑀 𝑙=1

∑𝑁 𝑗=1

[

⎡

⎢⎢

⎢

⎣

𝐾(1, 𝑗)(𝐴𝐻)(𝑗, 𝑙) 𝐾(2, 𝑗)(𝐴𝐻)(𝑗, 𝑙)

⋮

𝐾(𝑀, 𝑗)(𝐴𝐻)(𝑗, 𝑙)

⎤

⎥⎥

⎥

⎦

, 1 + 𝓁] ⊗ [𝑓_𝑙, 1]

=

∑𝑀 𝑙=1

[(𝐾𝐴𝐻)𝑓_𝑙, 1 + 𝓁] ⊗ [𝑓_𝑙, 1]

=

∑𝑀 𝑙=1

[(𝐵𝐾𝐻)𝑓_𝑙, 1 + 𝓁] ⊗ [𝑓_𝑙, 1]

=

∑𝑀 𝑙=1

[𝐵^𝓁+1𝑓_𝑙, 1 + 𝓁] ⊗ [𝑓_𝑙, 1]

=

∑𝑀 𝑙=1

[𝐵𝑓_𝑙, 1] ⊗ [𝑓_𝑙, 1] = ̃𝑢_𝐵.

R. F. Williams characterized topological conjugacy of two-sided topological Markov shifts( ̄𝑋_𝐴, ̄𝜎_𝐴)and( ̄𝑋_𝐵, ̄𝜎_𝐵)in terms of an equivalence relation of its underlying matrices, called strong shift equivalence ([34]). Two square ma- trices𝐴and𝐵with entries in nonnegative integers are said to be elementary equivalent if there exist rectangular matrices𝐶, 𝐷with entries in nonnegative integers such that𝐴 = 𝐶𝐷, 𝐵 = 𝐷𝐶. If two matrices are connected by a finite chain of elementary equivalences, they are said to be strong shift equivalent.

Williams proved that two-sided topological Markov shift( ̄𝑋_𝐴, ̄𝜎_𝐴)and( ̄𝑋_𝐵, ̄𝜎_𝐵) are topologically conjugate if and only if the matrices𝐴and𝐵are strong shift equivalent ([34]). Since shift equivalence is weaker than strong shift equiva- lence, by virtue of the Williams’ result, we have

Proposition 3.4. The dimension quadruplet( ̃∆_𝐴, ̃∆⁺_𝐴, ̃𝛿_𝐴, ̃𝑢_𝐴)is invariant under topological conjugacy of the two-sided topological Markov shift( ̄𝑋_𝐴, ̄𝜎_𝐴).

4. Dimension quadruplets and AF-algebras

In this section, we will study the dimension quadruplet( ̃∆_𝐴, ̃∆⁺_𝐴, ̃𝛿_𝐴, ̃𝑢_𝐴)by using K-theory for𝐶^∗-algebras. D. B. Killough and I. F. Putnam in [13] have deeply studied ring and module structure of the AF-algebras𝐶^∗(𝐺^𝑠_𝐴)as well as 𝐶^∗(𝐺_𝐴^𝑎)from a different view point from ours below. Recall that𝒦denotes the 𝐶^∗-algebra of compact operators on the separable infinite dimensional Hilbert space𝐻 = 𝓁²(ℕ).

Lemma 4.1. Let 𝐴be an irreducible, non-permutation matrix with entries in {0, 1}.

(i) There exists a projection𝑝₀in the crossed productℛ˜_𝐴⋊_𝛾𝐴𝕋²ofℛ˜_𝐴by𝛾^𝐴 such that𝑝₀( ˜ℛ_𝐴⋊_𝛾^𝐴𝕋²)𝑝₀is isomorphic to𝐶^∗(𝐺_𝐴^𝑎).Henceℛ˜_𝐴⋊_𝛾^𝐴𝕋²is stably isomorphic to the AF-algebra𝐶^∗(𝐺_𝐴^𝑎).

(ii) The inclusion𝜄_𝐴 ∶ 𝑝₀( ˜ℛ_𝐴⋊_𝛾𝐴𝕋²)𝑝₀ ↪ ˜ℛ_𝐴⋊_𝛾𝐴𝕋²induces an isomor- phism

𝜄_𝐴∗ ∶ 𝐾₀(𝐶^∗(𝐺_𝐴^𝑎)) ⟶ 𝐾₀( ˜ℛ_𝐴⋊_𝛾^𝐴𝕋²)

on K-theory where𝐶^∗(𝐺_𝐴^𝑎)is identified with𝑝₀( ˜ℛ_𝐴⋊_𝛾^𝐴𝕋²)𝑝₀.

Proof. (i) The fixed point algebra( ˜ℛ_𝐴)^𝛾𝐴 ofℛ˜_𝐴under 𝛾^𝐴 coincides with the fixed point algebra(𝐸_𝐴(𝒪_𝐴^𝑡 ⊗ 𝒪_𝐴)𝐸_𝐴)^{𝛼𝐴𝑡⊗𝛼𝐴} which is nothing but𝐸_𝐴(ℱ_𝐴^𝑡 ⊗ ℱ_𝐴)𝐸_𝐴. Hence ( ˜ℛ_𝐴)^𝛾𝐴 is identified with 𝐶^∗(𝐺_𝐴^𝑎). Let 𝑝₀ be the projection in

∈ 𝐿¹(𝕋², ˜ℛ_𝐴)defined by𝑝₀(𝑟, 𝑠) = 1for all(𝑟, 𝑠) ∈ 𝕋².We know that𝑝₀is a full projection inℛ˜_𝐴⋊_𝛾𝐴𝕋²and

𝑝₀( ˜ℛ_𝐴⋊_𝛾^𝐴𝕋²)𝑝₀= ( ˜ℛ_𝐴)^𝛾𝐴 = 𝐶^∗(𝐺_𝐴^𝑎)

by [29] or a manner similar to [18]. This shows that the algebraℛ˜_𝐴⋊_𝛾𝐴𝕋²is stably isomorphic to the AF-algebra𝐶^∗(𝐺^𝑎_𝐴)by [4].

(ii) By [4], there exists a partial isometry𝑣_𝐴in the multiplier algebra𝑀( ˜ℛ_𝐴⋊_𝛾𝐴

𝕋²⊗ 𝒦)ofℛ˜_𝐴⋊_𝛾𝐴𝕋²⊗ 𝒦such that𝑣_𝐴^∗𝑣_𝐴= 𝑝₀, 𝑣_𝐴𝑣_𝐴^∗ = 1.Put𝜓_𝐴 = Ad(𝑣_𝐴) ∶ 𝑝₀( ˜ℛ_𝐴⋊_𝛾^𝐴𝕋²)𝑝₀⊗ 𝒦 ⟶ ˜ℛ_𝐴⋊_𝛾^𝐴𝕋²⊗ 𝒦,which is an isomorphism of𝐶^∗- algebras. We then have for a projection𝑝₀𝑓𝑝₀⊗ 𝑞 ∈ ˜ℛ_𝐴⋊_𝛾𝐴𝕋²⊗ 𝒦,

(𝜄_𝐴⊗ id)_∗([𝑝₀𝑓𝑝₀⊗ 𝑞]) =[𝑝₀𝑓𝑝₀⊗ 𝑞]

=[𝑣_𝐴^∗𝑣_𝐴(𝑝₀𝑓𝑝₀⊗ 𝑞)𝑣^∗_𝐴𝑣_𝐴]

=[𝑣_𝐴(𝑝₀𝑓𝑝₀⊗ 𝑞)𝑣_𝐴^∗]

=𝜓_𝐴∗([𝑝₀𝑓𝑝₀⊗ 𝑞]).

Hence𝜄_𝐴∗= 𝜓_𝐴∗∶ 𝐾₀(𝐶^∗(𝐺_𝐴^𝑎)) ⟶ 𝐾₀( ˜ℛ_𝐴⋊_𝛾^𝐴𝕋²)is an isomorphism.