Topological Conjugacy of Topological Markov Shifts

and Cuntz–Krieger Algebras

Kengo Matsumoto^{1}

Received: June 23, 2016 Revised: February 8, 217

Communicated by Joachim Cuntz

Abstract.

For an irreducible non-permutation matrix A, the triplet (O_{A}, D_{A}, ρ^{A}) for the Cuntz-Krieger algebra O_{A}, its canonical maximal abelian C^{∗}- subalgebra D_{A}, and its gauge action ρ^{A} is called the Cuntz–Krieger triplet. We introduce a notion of strong Morita equivalence in the Cuntz–Krieger triplets, and prove that two Cuntz–Krieger triplets (O_{A}, D_{A}, ρ^{A}) and (O_{B}, D_{B}, ρ^{B}) are strong Morita equivalent if and only if A and B are strong shift equivalent. We also show that the generalized gauge actions on the stabilized Cuntz–Krieger algebras are cocycle conjugate if the underlying matrices are strong shift equivalent. By clarifying K-theoretic behavior of the cocycle conjugacy, we investigate a relationship between cocycle conjugacy of the gauge actions on the stabilized Cuntz–Krieger algebras and topological conjugacy of the underlying topological Markov shifts.

2010 Mathematics Subject Classification: Primary 46L55; Secondary 46L35, 37B10. Keywords and Phrases: Topological Markov shifts, topological conjugacy, strong shift equivalence, Cuntz–Krieger algebras, K-theory, gauge action.

1Supported by JSPS KAKENHI Grant Number 15K04896

Documenta Mathematica 22 (2017) 873–915

1 Introduction and Preliminaries

Let A = [A(i, j)]_{i,j=1}^{N} be an irreducible matrix with entries in {0, 1} with 1 < N ∈ ℕ. We assume that A is not any permutation matrix. In [7], J. Cuntz and W. Krieger have introduced a C^{∗}-algebra O_{A} associated to the topological Markov shift (X_{A}, σ_{A}). The C^{∗}-algebra is called the Cuntz–Krieger algebra, which is a universal unique purely infinite simple C^{∗}-algebra generated by partial isometries S_{1}, …, S_{N} subject to the relations:

∑_{j=1}^{N} S_{j}S_{j}^{∗} = 1,  S_{i}^{∗}S_{i} = ∑_{j=1}^{N} A(i, j)S_{j}S_{j}^{∗}, i = 1, …, N.  (1.1)

For t ∈ ℝ/ℤ = 𝕋, the correspondence S_{i} → e^{2π\sqrt{−1}t}S_{i}, i = 1, …, N gives rise to an automorphism of O_{A} denoted by ρ_{t}^{A}. The automorphisms ρ_{t}^{A}, t ∈ 𝕋 yield an action of 𝕋 on O_{A} called the gauge action. Cuntz and Krieger in [7] have shown that the algebra O_{A} has close relationships with the underlying dynamical system called topological Markov shift. Let us denote by X_{A} the shift space

X_{A} = {(x_{n})_{n∈ℕ} ∈ {1, …, N}^{ℕ} | A(x_{n}, x_{n+1}) = 1 for all n ∈ ℕ}.  (1.2)

Define the shift transformation σ_{A} on X_{A} by σ_{A}((x_{n})_{n∈ℕ}) = (x_{n+1})_{n∈ℕ}, which is a continuous surjection on X_{A}. The topological dynamical system (X_{A}, σ_{A}) is called the one-sided topological Markov shift for matrix A. The two-sided topological Markov shift (X̄_{A}, σ̄_{A}) is defined similarly with the shift space

X̄_{A} = {(x_{n})_{n∈ℤ} ∈ {1, …, N}^{ℤ} | A(x_{n}, x_{n+1}) = 1 for all n ∈ ℤ}  (1.3)

and the shift homeomorphism σ̄_{A}((x_{n})_{n∈ℤ}) = (x_{n+1})_{n∈ℤ} on X̄_{A}. Let us denote by D_{A} the C^{∗}-subalgebra of O_{A} generated by the projections of the form: S_{i_{1}} ⋯ S_{i_{n}}S_{i_{n}}^{∗} ⋯ S_{i_{1}}^{∗}, i_{1}, …, i_{n} = 1, …, N. The subalgebra D_{A} is canonically isomorphic to the commutative C^{∗}-algebra C(X_{A}) of the complex valued continuous functions on X_{A} by identifying the projection S_{i_{1}} ⋯ S_{i_{n}}S_{i_{n}}^{∗} ⋯ S_{i_{1}}^{∗} with the characteristic function χ_{U_{i_{1}···in}} ∈ C(X_{A}) of the cylinder set U_{i_{1}···i_{n}} for the word i_{1} ⋯ i_{n}. Let us denote by K the C^{∗}-algebra K(ℓ^{2}(ℕ)) of compact operators on a separable infinite dimensional Hilbert space ℓ^{2}(ℕ) and by C its maximal abelian C^{∗}-subalgebra of diagonal operators. In [25], R. F. Williams proved that the topological Markov shifts (X̄_{A}, σ̄_{A}) and (X̄_{B}, σ̄_{B}) are topologically conjugate if and only if the matrices A, B are strong shift equivalent. Two nonnegative matrices A, B are said to be elementary equivalent if there exist nonnegative rectangular matrices C, D such that A = CD, B = DC. We write it as A ≈_{C,D} B. If there exists a finite sequence

of nonnegative matrices A_{0}, A_{1}, …, A_{n} such that A = A_{0}, B = A_{n} and A_{i} is elementary equivalent to A_{i+1} for i = 0, 1, 2, …, n−1, then A and B are said to be strong shift equivalent. Hence elementary equivalence generates topological conjugacy of two-sided topological Markov shifts.

Let A be an irreducible non-permutation matrix. The triplet (O_{A}, D_{A}, ρ^{A}) for the Cuntz-Krieger algebra O_{A}, its canonical maximal abelian C^{∗}-subalgebra D_{A}, and its gauge action ρ^{A} is called the Cuntz–Krieger triplet for the matrix A. As pointed out in [11], two elementary equivalence matrices A = CD, B = DC yield an O_{A} − O_{B}-imprimitivity bimodule via the Cuntz–Krieger algebra O_{Z}

for the matrix Z defined by Z = [\begin{matrix}0 & C \\ D & 0 \end{matrix}] .

In the first part of the paper, We will introduce a notion of strong Morita equivalence in the Cuntz–Krieger triplets, and prove the following theorem.

Theorem 1.1 (Corolary 2.19). The Cuntz–Krieger triplets (O_{A}, D_{A}, ρ^{A}) and (O_{B}, D_{B}, ρ^{B}) are strong Morita equivalent if and only if the matrices A and B are strong shift equivalent.

It is well-known that two unital C^{∗}-algebras A and B are strong Morita equivalent if and only if their stabilizations A ⊗ K and B ⊗ K are isomorphic by Brown–Green–Rieffel Theorem [3, Theorem 1.2] (cf. [2], [3], [4]). We will next study relationships between stabilized Cuntz–Krieger algebras with their gauge actions and strong shift equivalence for matrices. We must emphasize that Cuntz and Krieger in [7, Theorem 3.8] and Cuntz in [6, Theorem 2.3] have shown that the stabilized Cuntz–Krieger triplet (O_{A} ⊗ K, D_{A} ⊗ C, ρ^{A} ⊗ id) is invariant under topological conjugacy of the two-sided topological Markov shifts (X̄_{A}, σ̄_{A}). We will investigate stabilizations of generalized gauge actions from a view point of flow equivalence. Let us denote by C(X_{A}, ℤ) the set of ℤ-valued continuous functions on X_{A}. For f ∈ C(X_{A}, ℤ), define a one-parameter unitary group U_{t}(f), t ∈ 𝕋 = ℝ/ℤ in D_{A} by

U_{t}(f) = exp(2π\sqrt{−1}tf),  (1.4)

and an automorphism ρ_{t}^{A,f} on O_{A} for each t ∈ 𝕋 by

ρ_{t}^{A,f}(S_{i}) = U_{t}(f)S_{i},  i = 1, …, N.  (1.5)

For f ≡ 1, the action ρ_{t}^{A,1} is the gauge action denoted by ρ_{t}^{A}. Suppose that A = CD and B = DC for some nonnegative rectangular matrices C, D. Then there exist homomorphisms φ : C(X_{A}, ℤ) → C(X_{B}, ℤ) and ψ : C(X_{B}, ℤ) → C(X_{A}, ℤ) such that

(ψ ∘ φ)(f) = f ∘ σ_{A},  (φ ∘ ψ)(g) = g ∘ σ_{B}  (1.6)

for f ∈ C(X_{A}, ℤ) and g ∈ C(X_{B}, ℤ). Let us denote by (H^{A}, H_{+}^{A}) the ordered cohomology groups for the one-sided topological Markov shift (X_{A}, σ_{A}) which has appeared in [17] by setting

H^{A} = C(X_{A}, ℤ)/{η − η ∘ σ_{A} | η ∈ C(X_{A}, ℤ)}

and its positive cone

H_{+}^{A} = {[η] ∈ H^{A} | η(x) ≥ 0 for all x ∈ X_{A}}.

The ordered cohomology group (H̄^{A}, H̄_{+}^{A}) for (X̄_{A}, σ̄_{A}) has been considered by Y. T. Poon in [19]. The latter ordered group (H̄^{A}, H̄_{+}^{A}) has been proved to be a complete invariant of flow equivalence of the two-sided topological Markov shift (X̄_{A}, σ̄_{A}) by M. Boyle and D. Handelman in [1]. The two ordered groups (H̄^{A}, H̄_{+}^{A}) and (H^{A}, H_{+}^{A}) are actually isomorphic ([17, Lemma 3.1]). In [15], the following result has been proved.

Theorem 1.2 ([15, Corollary 4.4]). Suppose that A and B are strong shift equivalent. Then there exist an isomorphism Φ : O_{A} ⊗ K → O_{B} ⊗ K satisfying Φ(D_{A} ⊗ C) = D_{B} ⊗ C and a homomorphism φ : C(X_{A}, ℤ) → C(X_{B}, ℤ) of ordered groups which induces an isomorphism between (H^{A}, H_{+}^{A}) and (H^{B}, H_{+}^{B}) of ordered groups such that for each function f ∈ C(X_{A}, ℤ) there exists a unitary one-cocycle v_{t}^{f} ∈ U(M(O_{A} ⊗ K)) relative to ρ^{A,f} ⊗ id satisfying

Φ ∘ Ad(v_{t}^{f}) ∘ (ρ_{t}^{A,f} ⊗ id) = (ρ_{t}^{B,φ(f)} ⊗ id) ∘ Φ for t ∈ 𝕋.

In the second part of the present paper, we will study K-theoretic behavior of the above isomorphism Φ : O_{A} ⊗ K → O_{B} ⊗ K. Let us denote by ϵ_{A} : K_{0}(O_{A}) → ℤ^{N}/(id − A^{t})ℤ^{N} the isomorphism defined in [6, Proposition 3.1] satisfying ϵ_{A}([1_{A}]) = [(1, 1, …, 1)], where 1_{A} is the unit of O_{A}. We will prove the following theorem.

Theorem 1.3 (Proposition 3.10 and Theorem 4.6). Let A and B be elementary equivalent matrices, and choose matrices C and D satisfying A = CD and B = DC. Then there exist an isomorphism Φ : O_{A} ⊗ K → O_{B} ⊗ K satisfying Φ(D_{A} ⊗ C) = D_{B} ⊗ C and a unitary representation t ∈ 𝕋 → u_{t}^{f} ∈ M(D_{A} ⊗ C) for each f ∈ C(X_{A}, ℤ) such that

Φ ∘ Ad(u_{t}^{f}) ∘ (ρ_{t}^{A,f} ⊗ id) = (ρ_{t}^{B,φ(f)} ⊗ id) ∘ Φ for f ∈ C(X_{A}, ℤ), t ∈ 𝕋

and the diagram

K_{0}(O_{A})  ⟶^{Φ_{∗}}  K_{0}(O_{B})
ϵ_{A}↓  ↓^{ϵ_{B}}
ℤ^{N}/(id − A^{t})ℤ^{N} ⟶^{Φ_{C^{t}}} ℤ^{M}/(id − B^{t})ℤ^{M}

is commutative, where Φ_{C^{t}} is the isomorphism induced by multiplying by the matrix C^{t}.

In the third part of the paper, we will study the converse of the above theorem for the gauge actions. We will introduce an invariant K_{0}^{SSE}(O_{A}) which is a non-empty subset of K_{0}(O_{A}). The invariant K_{0}^{SSE}(O_{A}) is realized as the subset of ℤ^{N}/(id − A^{t})ℤ^{N} consisting of the classes [v] of vectors v ∈ ℤ^{N} such that v = D_{1}^{t} ⋯ D_{n−1}^{t}D_{n}^{t}[1, 1, …, 1]^{t} for some strong shift equivalences A ≈_{C_{1},D_{1}}

⋯ ≈_{C_{n},D_{n}} D_{n}C_{n} (Proposition 5.7). We will then prove the following theorem.

Theorem 1.4 (Theorem 5.8). Let A, B be irreducible and non-permutation matrices. The following two assertions are equivalent.

(i) Two-sided topological Markov shifts (X̄_{A}, σ̄_{A}) and (X̄_{B}, σ̄_{B}) are topologically conjugate.

(ii) There exist an isomorphism Φ : O_{A} ⊗ K → O_{B} ⊗ K and a unitary representation t ∈ 𝕋 → v_{t}^{A} ∈ M(D_{A} ⊗ C) such that

Φ(D_{A} ⊗ C) = D_{B} ⊗ C,  Φ ∘ Ad(v_{t}^{A}) ∘ (ρ_{t}^{A} ⊗ id) = (ρ_{t}^{B} ⊗ id) ∘ Φ for t ∈ 𝕋,
Φ_{∗}(K_{0}^{SSE}(O_{A})) = K_{0}^{SSE}(O_{B}).

The set K_{0}^{SSE}(O_{A}) is always a non-empty subset of K_{0}(O_{A}). If in particular the condition K_{0}^{SSE}(O_{A}) = K_{0}(O_{A}) holds, the matrix A is said to have full units. In this case, we have the following corollary.

Corollary 1.5 (Corollary 5.12). Suppose that the matrices A and B have full units. Then the two-sided topological Markov shifts (X̄_{A}, σ̄_{A}) and (X̄_{B}, σ̄_{B}) are topologically conjugate if and only if there exist an isomorphism Φ : O_{A} ⊗ K → O_{B} ⊗ K of C^{∗}-algebras and a unitary representation t ∈ 𝕋 → v_{t}^{A} ∈ M(D_{A} ⊗ C) such that

Φ(D_{A} ⊗ C) = D_{B} ⊗ C, Φ ∘ Ad(v_{t}^{A}) ∘ (ρ_{t}^{A} ⊗ id) = (ρ_{t}^{B} ⊗ id) ∘ Φ.

Throughout the paper, we denote by ℕ the set of positive integers and by ℤ_{+} the set of nonnegative integers, respectively. For the one-sided topological Markov shift (X_{A}, σ_{A}), a word μ = (μ_{1}, …, μ_{k}) for μ_{i} ∈ {1, …, N} is said to be admissible for X_{A} if (μ_{1}, …, μ_{k}) = (x_{1}, …, x_{k}) for some element (x_{n})_{n∈ℕ} ∈ X_{A}. The length of μ is denoted by |μ| = k. We denote by B_{k}(X_{A}) the set of all admissible words of length k. We similarly denote by B_{k}(X̄_{A}) the set of admissible words of length k, so that B_{k}(X̄_{A}) = B_{k}(X_{A}). The cylinder set {(x_{n})_{n∈ℕ} ∈ X_{A} | x_{1} = μ_{1}, …, x_{k} = μ_{k}} for μ = (μ_{1}, …, μ_{k}) ∈ B_{k}(X_{A}) is denoted by U_{μ}.

This paper is a revised version of arXiv:1604.02763v1, in which the given proofs of the main results were incorrect.

2 Strong Morita equivalence for Cuntz–Krieger triplets

There is a standard method to associate a Cuntz–Krieger algebra from a square matrix with entries in nonnegative integers as described in [7, Remark 2.16] (see also [23, Section 4]). Now we suppose that A = [A(i, j)]_{i,j=1}^{N} is an N ×N matrix with entries in nonnegative integers. Then the associated graph G_{A} = (V_{A}, E_{A}) consists of the vertex set V_{A} = {v_{1}^{A}, …, v_{N}^{A}} of N vertices and the edge set E_{A} = {a_{1}, …, a_{N_{A}}}, where there are A(i, j) edges from v_{i}^{A} to v_{j}^{A}. Hence the total number of edges is ∑_{i,j=1}^{N} A(i, j) denoted by N_{A}. For a_{i} ∈ E_{A}, denote by t(a_{i}), s(a_{i}) the terminal vertex of a_{i}, the source vertex of a_{i}, respectively. The graph G_{A} has the N_{A} × N_{A} transition matrix A^{G} = [A^{G}(i, j)]_{i,j=1}^{N_{A}} of edges defined by

A^{G}(i, j) = \begin{cases}1 & if t(a_{i}) = s(a_{j}), \\ 0 & otherwise. \end{cases}  (2.1)

The Cuntz–Krieger algebra O_{A} for the matrix A with entries in nonnegative integers is defined as the Cuntz–Krieger algebra O_{A^{G}} for the matrix A^{G} which is the universal C^{∗}-algebra generated by partial isometries S_{a_{i}} indexed by edges a_{i}, i = 1, …, N_{A} subject to the relations:

∑_{j=1}^{N_{A}} S_{a_{j}}S_{a_{j}}^{∗} = 1,  S_{a_{i}}^{∗}S_{a_{i}} = ∑_{j=1}^{N_{A}} A^{G}(i, j)S_{a_{j}}S_{a_{j}}^{∗} for i = 1, …, N_{A}. (2.2)

For a word μ = (μ_{1}, …, μ_{k}), μ_{i} ∈ E_{A}, we denote by S_{μ} the partial isometry S_{μ_{1}} ⋯ S_{μ_{k}}. As in the standard text books [9], [10] of symbolic dynamics, the two-sided topological Markov shift defined by a square matrix with entries in {0, 1} is naturally topologically conjugate to a topological Markov shift of the edge shift defined by the underlying directed graph. In what follows, we consider edge shifts and hence square matrices with entries in nonnegative integers (cf. [9], [10], [25], etc.). Such a matrix is simply called a nonnegative square matrix. For a nonnegative square matrix A, the two-sided shift space X̄_{A} is defined by the two-sided shift space X̄_{A^{G}} for the matrix A^{G} which consists of the two-sided bi-infinite sequences of concatenated edges of the directed graph G_{A}. Let A and B be elementary equivalent matrices, and choose matrices C and D satisfying A = CD and B = DC. The sizes of the matrices A and B are denoted by N and M respectively, so that C is an N × M matrix and D is an

M × N matrix, respectively. We set Z = [\begin{matrix}0 & C \\ D & 0 \end{matrix}] as a block matrix, and we see

Z^{2} = [\begin{matrix}CD & 0 \\ 0 & DC \end{matrix}] = [\begin{matrix}A & 0 \\ 0 & B \end{matrix}] .

For the rectangular matrices C and D, the vertex sets V_{C} and V_{D} are defined by the disjoint union V_{A} ⊔ V_{B}, and C(i, j) directed edges are defined from the vertex v_{i}^{A} to v_{j}^{B}, and D(j, i) directed edges are defined from the vertex v_{j}^{B} to v_{i}^{A}, respectively. The former forms a directed graph written G_{C} = (V_{C}, E_{C}), and the latter forms a directed graph written G_{D} = (V_{D}, E_{D}), Hene we have five directed graphs G_{A} = (V_{A}, E_{A}), G_{B} = (V_{B}, E_{B}), G_{C} = (V_{C}, E_{C}), G_{D} = (V_{D}, E_{D}) and G_{Z} = (V_{Z}, E_{Z}) associated to the nonnegative matrices A, B, C, D and Z, respectively. In the identity

A(i, j) = ∑_{k=1}^{N_{B}} C(i, k)D(k, j) for i, j = 1, …, N_{A},

the left hand side expresses the number of edges in E_{A} starting with v_{i}^{A} and ending with v_{j}^{A}, whereas the right hand side expresses the number of pairs of edges E_{C} and E_{D} starting with v_{i}^{A} through some vertex v_{k}^{B} and ending with v_{j}^{A}. Hence we may take a bijection, which is denoted by φ_{A,CD}, from E_{A} to a subset of E_{C} × E_{D}. The other identity B = DC similarly admits us to take a bijection, which is denoted by φ_{B,DC}, from E_{B} to a subset of E_{D} × E_{C}. Let S_{c}, S_{d}, c ∈ E_{C}, d ∈ E_{D} be the generating partial isometries of the Cuntz– Krieger algebra O_{Z} for the matrix Z, so that ∑_{c∈E_{C}} S_{c}S_{c}^{∗} + ∑_{d∈E_{D}} S_{d}S_{d}^{∗} = 1 and

S_{c}^{∗}S_{c} = ∑_{d∈E_{D}} Z(c, d)S_{d}S_{d}^{∗}, S_{d}^{∗}S_{d} = ∑_{c∈E_{C}} Z(d, c)S_{c}S_{c}^{∗}

for c ∈ E_{C}, d ∈ E_{D}. Since S_{c}S_{d} ≠ 0 (resp. S_{d}S_{c} ≠ 0) if and only if φ_{A,CD}(a) = cd (resp. φ_{B,DC}(b) = dc) for a unique edge a ∈ E_{A} (resp. b ∈ E_{B}), we may identify cd (resp. dc) with a (resp. b) through the map φ_{A,CD} (resp. φ_{B,DC}). We may then write S_{cd} = S_{a} (resp. S_{dc} = S_{b}) where S_{cd} denotes S_{c}S_{d} (resp. S_{dc} denotes S_{d}S_{c}). We define two particular projections P_{C} and P_{D} in D_{Z} by P_{C} = ∑_{c∈E_{C}} S_{c}S_{c}^{∗} and P_{D} = ∑_{d∈E_{D}} S_{d}S_{d}^{∗} so that P_{C} + P_{D} = 1. It has been shown in [11] (cf. [15]) that

P_{C}O_{Z}P_{C} = O_{A}, P_{D}O_{Z}P_{D} = O_{B}, D_{Z}P_{C} = D_{A}, D_{Z}P_{D} = D_{B}. (2.3)

As in [11, Lemma 3.10], both P_{C} and P_{D} are full projections so that P_{C}O_{Z}P_{D} has a natural structure of O_{A}–O_{B} imprimitivity bimodule that makes O_{A} and O_{B} strong Morita equivalent (cf. [16], [21], [22]). Let ρ^{Z}, ρ^{A}, ρ^{B} be the gauge actions of T on O_{Z}, O_{A}, O_{B}, respectively. Since S_{c}S_{d} (resp. S_{d}S_{c}) in O_{Z} is identified with S_{a} in O_{A} (resp. S_{b} in O_{B}) if φ_{A,CD}(a) = cd (resp. φ_{B,DC}(b) = dc, we have

ρ_{t}^{Z}|_{P_{C}O_{Z}P_{C}} = ρ_{2t}^{A} on O_{A},  ρ_{t}^{Z}|_{P_{D}O_{Z}P_{D}} = ρ_{2t}^{B} on O_{B}.  (2.4)

Let A be an irreducible non-permutation matrix. The triplet (O_{A}, D_{A}, ρ^{A}) for the Cuntz-Krieger algebra O_{A}, its canonical maximal abelian C^{∗}-subalgebra D_{A}, and its gauge action ρ^{A} is called the Cuntz–Krieger triplet for the matrix A. In this section we will define the notion of strong Morita equivalence in Cuntz–Krieger triplets. We will then prove that the Cuntz–Krieger triplets (O_{A}, D_{A}, ρ^{A}) and (O_{B}, D_{B}, ρ^{B}) are strong Morita equivalent if and only if the matrices A and B are strong shift equivalent. Let A, B be irreducible nonpermutation matrices.

Definition 2.1. The Cuntz–Krieger triplets (O_{A}, D_{A}, ρ^{A}) and (O_{B}, D_{B}, ρ^{B}) are said to be strong Morita equivalent in 1-step if there exist a Cuntz–Krieger triplet (O_{Z}, D_{Z}, ρ^{Z}) for some nonnegative matrix Z and projections P_{A}, P_{B} ∈ D_{Z} having the following properties:

(1) P_{A} + P_{B} = 1, (2) O_{Z} contains both O_{A} and O_{B} as subalgebras, and

P_{A}O_{Z}P_{A} = O_{A} and P_{B}O_{Z}P_{B} = O_{B},

(3) D_{Z}P_{A} = D_{A} and D_{Z}P_{B} = D_{B},

(4) ρ_{t}^{Z}|_{P_{A}O_{Z}P_{A}} = ρ_{2t}^{A} on O_{A} and ρ_{t}^{Z}|_{P_{B}O_{Z}P_{B}} = ρ_{2t}^{B} on O_{B} for t ∈ 𝕋.

In this case, we say that (O_{A}, D_{A}, ρ^{A}) and (O_{B}, D_{B}, ρ^{B}) are strong Morita equivalent in 1-step via (O_{Z}, D_{Z}, ρ^{Z}). If two Cuntz–Krieger triplets (O_{A}, D_{A}, ρ^{A}) and (O_{B}, D_{B}, ρ^{B}) are connected through n-chains of strong Morita equivalences in 1-step, (O_{A}, D_{A}, ρ^{A}) and (O_{B}, D_{B}, ρ^{B}) are said to be strong Morita equivalent in n-step, or simply, strong Morita equivalent. The one-sided topological Markov shifts (X_{A}, σ_{A}) and (X_{B}, σ_{B}) are said to be eventually conjugate if there exist a homeomorphism h : X_{A} → X_{B} and a nonnegative integer K such that

σ_{B}^{K}(h(σ_{A}(x))) =σ_{B}^{K}(h(x)),  x ∈ X_{A},
σ_{A}^{K}(h^{−1}(σ_{B}(y))) =σ_{A}^{K}(h^{−1}(y)),  y ∈ X_{B}.

It has been shown that there exists an isomorphism Φ : O_{A} ⟶ O_{B} satisfying Φ(D_{A}) = D_{B} and Φ ∘ ρ_{t}^{A} = ρ_{t}^{B} ∘ Φ, t ∈ 𝕋 if and only if the one-sided topological Markov shifts (X_{A}, σ_{A}) and (X_{B}, σ_{B}) are eventually conjugate ([15, Corollary 3.5]). The latter condition implies that their two-sided topological Markov shifts (X̄_{A}, σ̄_{A}) and (X̄_{B}, σ̄_{B}) are topologically conjugate by [14, Theorem 5.5] (cf. [14, Theorem 6.7]). Hence an isomorphic Cuntz–Krieger triplets (O_{A}, D_{A}, ρ^{A}) and (O_{B}, D_{B}, ρ^{B}) yields a strong shift equivalence between the underlying matrices A and B.

Proposition 2.2. If A and B are elementary equivalent, then their Cuntz– Krieger triplets (O_{A}, D_{A}, ρ^{A}) and (O_{B}, D_{B}, ρ^{B}) are strong Morita equivalent in 1-steps,

Proof. Let A and B be elementary equivalent matrices, and choose matrices C

and D satisfying A = CD, B = DC. Let Z be the square matrix Z = [\begin{matrix}0 & C \\ D & 0 \end{matrix}] .

By the above discussions, there exist projections P_{C}, P_{D} in D_{Z} satisfying P_{C} + P_{D} = 1 and (2.3) (2.4).

The main purpose of this section is to study the converse implication of Proposition 2.2. We henceforth assume that (O_{A}, D_{A}, ρ^{A}) and (O_{B}, D_{B}, ρ^{B}) are strong Morita equivalent in 1-step via (O_{Z}, D_{Z}, ρ^{Z}) for some matrix Z. We may take two projections P_{A}, P_{B} in D_{Z} having the properties (1), (2), (3) and (4) in Definition 2.1. Let us denote by G_{Z} = (V_{Z}, E_{Z}) the directed graph for the matrix Z. The Cuntz–Krieger algebra O_{Z} is then generated by partial isometries S_{γ}, γ ∈ E_{Z} satisfying the relations:

∑_{η∈E_{Z}} S_{η}S_{η}^{∗} = 1,  S_{γ}^{∗}S_{γ} = ∑_{η∈E_{Z}} Z^{G}(γ, η)S_{η}S_{η}^{∗} for γ ∈ E_{Z}  (2.5)

where Z^{G}(γ, η) = 1 if t(γ) = s(η), and 0 otherwise. We have the following lemmas.

Lemma 2.3. Let S_{γ}, γ ∈ E_{Z} be the generating partial isometries of O_{Z} satisfying (2.5). Then we have

(i) P_{A}S_{γ}P_{A} = P_{B}S_{γ}P_{B} = 0.

(ii) S_{γ} = P_{A}S_{γ}P_{B} + P_{B}S_{γ}P_{A}.

(iii) P_{A}S_{γ} = S_{γ}P_{B} and P_{B}S_{γ} = S_{γ}P_{A}.

Proof. By the equality P_{A} + P_{B} = 1, we have

S_{γ} = P_{A}S_{γ}P_{A} + P_{A}S_{γ}P_{B} + P_{B}S_{γ}P_{A} + P_{B}S_{γ}P_{B}.

Since P_{A}S_{γ}P_{A} belongs to P_{A}O_{Z}P_{A} which is identified with O_{A}, the condition (4) of Definition 2.1 gives rise to the equality

ρ_{t}^{Z}(P_{A}S_{γ}P_{A}) = ρ_{2t}^{A}(P_{A}S_{γ}P_{A}).  (2.6)

As ρ_{t}^{Z}|_{D_{Z}} = id and P_{A}, P_{B} ∈ D_{Z}, the left hand side for t = \frac{1}{2} of (2.6) equals

P_{A}ρZ_{\frac{1}{2}}(S_{γ})P_{A} = −P_{A}S_{γ}P_{A}.

As ρ_{1}^{A} = id, the right hand side for t = \frac{1}{2} equals P_{A}S_{γ}P_{A}. Hence we have P_{A}S_{γ}P_{A} = 0 and similarly P_{B}S_{γ}P_{B} = 0. Therefore we know (i) and hence (ii). The assertion (iii) follows from (ii) since P_{A} and P_{B} are mutually orthogonal projections.

Lemma 2.4.

∑_{γ∈E_{Z}} S_{γ}P_{A}S_{γ}^{∗} = P_{B},  ∑_{γ∈E_{Z}} S_{γ}P_{B}S_{γ}^{∗} = P_{A}.  (2.7)

Proof. By Lemma 2.3, we know S_{γ}P_{A} = P_{B}S_{γ} so that

∑_{γ∈E_{Z}} S_{γ}P_{A}S_{γ}^{∗} = ∑_{γ∈E_{Z}} P_{B}S_{γ}S_{γ}^{∗} = P_{B}.  (2.8)

Similarly we see that ∑_{γ∈E_{Z}} S_{γ}P_{B}S_{γ}^{∗} = P_{A}.

We notice the identities in the following lemma which immediately come from Lemma 2.3 (iii).

Lemma 2.5. For γ_{1}, γ_{2} ∈ E_{Z}, we have the following identities.

(i) S_{γ_{1}}S_{γ_{2}}P_{A} = P_{A}S_{γ_{1}}S_{γ_{2}} ∈ O_{A} and S_{γ_{1}}S_{γ_{2}}P_{B} = P_{B}S_{γ_{1}}S_{γ_{2}} ∈ O_{B}.

(ii) S_{γ_{1}}P_{B}S_{γ_{2}} = P_{A}S_{γ_{1}}P_{B}S_{γ_{2}}P_{A} ∈ O_{A} and S_{γ_{1}}P_{A}S_{γ_{2}} = P_{B}S_{γ_{1}}P_{A}S_{γ_{2}}P_{B} ∈ O_{B}.

Lemma 2.6. Let γ_{1}, γ_{2} ∈ E_{Z}. Then P_{A}S_{γ_{1}} ≠ 0, P_{B}S_{γ_{2}} ≠ 0 and Z^{G}(γ_{1}, γ_{2}) = 1 if and only if P_{A}S_{γ_{1}}S_{γ_{2}} ≠ 0.

Proof. Since the identity

P_{A}S_{γ_{1}}S_{γ_{2}} = P_{A}S_{γ_{1}}P_{B}S_{γ_{2}} = S_{γ_{1}}S_{γ_{2}}P_{A} = Z(γ_{1}, γ_{2})S_{γ_{1}}S_{γ_{2}}P_{A}  (2.9)

holds, the if part is obvious. It suffices to show the only if part. By the identity (2.9), we have

(P_{A}S_{γ_{1}}S_{γ_{2}})^{∗}(P_{A}S_{γ_{1}}S_{γ_{2}}) = (S_{γ_{1}}S_{γ_{2}}P_{A})^{∗}S_{γ_{1}}S_{γ_{2}}P_{A}

= P_{A}S_{γ_{2}}^{∗}S_{γ_{1}}^{∗}S_{γ_{1}}S_{γ_{2}}P_{A}

= ∑_{η_{1}∈E_{Z}} Z^{G}(γ_{1}, η_{1})P_{A}S_{γ_{2}}^{∗}S_{η_{1}}S_{η_{1}}^{∗}S_{γ_{2}}P_{A}

= Z^{G}(γ_{1}, γ_{2})P_{A}S_{γ_{2}}^{∗}S_{γ_{2}}P_{A}

= Z^{G}(γ_{1}, γ_{2})(P_{B}S_{γ_{2}})^{∗}(P_{B}S_{γ_{2}}).

The above equalities ensure the only if part.

Lemma 2.7. Let γ_{1}, γ_{2}, η_{1}, η_{2} ∈ E_{Z}. Then S_{γ_{1}}S_{γ_{2}} ≠ 0, S_{γ_{2}}S_{η_{1}} ≠ 0, P_{A}S_{η_{1}}S_{η_{2}} ≠ 0 if and only if P_{A}S_{γ_{1}}S_{γ_{2}}S_{η_{1}}S_{η_{2}} ≠ 0.

Proof. Since P_{A}S_{γ_{1}}S_{γ_{2}}S_{η_{1}}S_{η_{2}} = S_{γ_{1}}S_{γ_{2}}P_{A}S_{η_{1}}S_{η_{2}}, the if part is obvious. It suffices to show the only if part. We have

(P_{A}S_{γ_{1}}S_{γ_{2}}S_{η_{1}}S_{η_{2}})^{∗}(P_{A}S_{γ_{1}}S_{γ_{2}}S_{η_{1}}S_{η_{2}})
=P_{A}S_{η_{2}}^{∗}S_{η_{1}}^{∗}S_{γ_{2}}^{∗}S_{γ_{1}}^{∗}S_{γ_{1}}S_{γ_{2}}S_{η_{1}}S_{η_{2}}P_{A}
= ∑_{ζ_{1}∈E_{Z}} Z^{G}(γ_{1}, ζ_{1})P_{A}S_{η_{2}}^{∗}S_{η_{1}}^{∗}S_{γ_{2}}^{∗}S_{ζ_{1}}S_{ζ_{1}}^{∗}S_{γ_{2}}S_{η_{1}}S_{η_{2}}P_{A}
=Z^{G}(γ_{1}, γ_{2}) ∑_{ζ_{2}∈E_{Z}} Z^{G}(γ_{2}, ζ_{2})P_{A}S_{η_{2}}^{∗}S_{η_{1}}^{∗}S_{ζ_{2}}S_{ζ_{2}}^{∗}S_{η_{1}}S_{η_{2}}P_{A}
=Z^{G}(γ_{1}, γ_{2})Z^{G}(γ_{2}, η_{1}) ∑_{ζ_{3}∈E_{Z}} Z^{G}(η_{1}, ζ_{3})P_{A}S_{η_{2}}^{∗}S_{ζ_{3}}S_{ζ_{3}}^{∗}S_{η_{2}}P_{A}
=Z^{G}(γ_{1}, γ_{2})Z^{G}(γ_{2}, η_{1})Z^{G}(η_{1}, η_{2})P_{A}S_{η_{2}}^{∗}S_{η_{2}}P_{A}.

The above equalities ensure the only if part.

Now we are assuming that the Cuntz–Krieger triplets (O_{A}, D_{A}, ρ^{A}) and (O_{B}, D_{B}, ρ^{B}) are strong Morita equivalent in 1-step via (O_{Z}, D_{Z}, ρ^{Z}). Recall that B_{k}(X_{A}) denotes the set of admissible words γ_{1} ⋯ γ_{k} of X_{Z} with length k. For k = 2, we see

B_{2}(X_{A}) = {γ_{1}γ_{2} | Z^{G}(γ_{1}, γ_{2}) = 1}.

Let us note that for γ_{1}, γ_{2} ∈ E_{Z}, the word γ_{1}γ_{2} belongs to B_{2}(X_{A}) if and only if S_{γ_{1}}S_{γ_{2}} ≠ 0.

We introduce several directed graphs in this situation. Define edge sets EÃ, EB̃, EC̃, ED̃ by setting

EÃ = {(A, γ_{1}γ_{2}) ∈ {A} × B_{2}(X_{Z}) | P_{A}S_{γ_{1}}S_{γ_{2}} ≠ 0},
EB̃ = {(B, γ_{1}γ_{2}) ∈ {B} × B_{2}(X_{Z}) | P_{B}S_{γ_{1}}S_{γ_{2}} ≠ 0},
EC̃ = {(A, γ_{1}) ∈ {A} × E_{Z} | P_{A}S_{γ_{1}} ≠ 0},
ED̃ = {(B, γ_{1}) ∈ {B} × E_{Z} | P_{B}S_{γ_{1}} ≠ 0}

and vertex sets V_{Ãs}, V_{Ãt}, V_{B̃s}, V_{B̃t}, V_{C̃s}, V_{C̃t}, V_{D̃s}, V_{D̃t} by setting

V_{Ãs} = {(A, s(γ_{1})) ∈ {A} × V_{Z} | (A, γ_{1}γ_{2}) ∈ EÃ},
V_{Ãt} = {(A, t(γ_{2})) ∈ {A} × V_{Z} | (A, γ_{1}γ_{2}) ∈ EÃ},
V_{B̃s} = {(B, s(γ_{1})) ∈ {B} × V_{Z} | (B, γ_{1}γ_{2}) ∈ EB̃},
V_{B̃t} = {(B, t(γ_{2})) ∈ {B} × V_{Z} | (B, γ_{1}γ_{2}) ∈ EB̃},
V_{C̃s} = {(A, s(γ_{1})) ∈ {A} × V_{Z} | (A, γ_{1}) ∈ EC̃},
V_{C̃t} = {(B, t(γ_{1})) ∈ {A} × V_{Z} | (A, γ_{1}) ∈ EC̃},
V_{D̃s} = {(B, s(γ_{1})) ∈ {B} × V_{Z} | (B, γ_{1}) ∈ ED̃},
V_{D̃t} = {(A, t(γ_{1})) ∈ {B} × V_{Z} | (B, γ_{1}) ∈ ED̃}.

Lemma 2.8. Keep the above notation. We have

(i) V_{Ãs} = V_{Ãt} = V_{C̃s} = V_{D̃t}.

(ii) V_{B̃s} = V_{B̃t} = V_{D̃s} = V_{C̃t}.

Proof. (i) We will first show the equality V_{Ãs} = V_{Ãt}. Take an arbitrary vertex (A, s(γ_{1})) ∈ V_{Ãs} and γ_{2} ∈ E_{Z} with P_{A}S_{γ_{1}}S_{γ_{2}} ≠ 0, so that t(γ_{1}) = s(γ_{2}). We may find η_{1}, η_{2} ∈ E_{Z} such that S_{η_{1}}S_{η_{2}} ≠ 0 and t(η_{2}) = s(γ_{1}). By Lemma 2.7, we have S_{η_{1}}S_{η_{2}}S_{γ_{1}}S_{γ_{2}}P_{A} ≠ 0. Since S_{η_{1}}S_{η_{2}}S_{γ_{1}}S_{γ_{2}}P_{A} = P_{A}S_{η_{1}}S_{η_{2}}S_{γ_{1}}S_{γ_{2}}, we have P_{A}S_{η_{1}}S_{η_{2}} ≠ 0 so that (A, t(η_{2})) ∈ V_{Ãt} and hence (A, s(γ_{1}) ∈ V_{Ãt}. This shows that the inclusion relation V_{Ãs} ⊂ V_{Ãt} holds. Similarly we obtain that V_{Ãt} ⊂ V_{Ãs} so that V_{Ãs} = V_{Ãt}. We will second show the equality V_{C̃s} = V_{D̃t}. Take an arbitrary vertex (A, s(γ_{1})) ∈ V_{C̃s}. We see that P_{A}S_{γ_{1}} ≠ 0 and hence S_{γ_{1}}P_{B} ≠ 0. Now both matrices A and B are assumed to be irreducible and not any permutations, so that Z and hence Z^{G} are irreducible and not any permutations. This implies that ∑_{γ′∈E_{Z}} S_{γ′}^{∗}S_{γ′} ≥ 1. Hence we may find γ_{2} ∈ E_{Z} such that S_{γ_{2}}S_{γ_{1}}P_{B} ≠ 0 so that t(γ_{2}) = s(γ_{1}). Since S_{γ_{2}}S_{γ_{1}}P_{B} = P_{B}S_{γ_{2}}S_{γ_{1}}, we have P_{B}S_{γ_{2}} ≠ 0. This implies that (B, γ_{2}) ∈ ED̃ and (A, t(γ_{2})) ∈ V_{D̃t}. As t(γ_{2}) = s(γ_{1}), we obtain that (A, s(γ_{1})) ∈ V_{D̃t} so that V_{C̃s} ⊂ V_{D̃t}. We see V_{D̃t} ⊂ V_{C̃s} similarly so that V_{C̃s} = V_{D̃t}. We will finally show that V_{Ãs} = V_{C̃s}. Since the condition P_{A}S_{γ_{1}}S_{γ_{2}} ≠ 0 implies P_{A}S_{γ_{1}} ≠ 0, we have V_{Ãs} ⊂ V_{C̃s}. Conversely, for (A, s(γ_{1})) ∈ V_{C̃s}, we have P_{A}S_{γ_{1}} ≠ 0 so that S_{γ_{1}}P_{B} ≠ 0. Since P_{B} = ∑_{γ′∈E_{Z}} S_{γ′}P_{A}S_{γ′}^{∗}, we may find γ_{2} ∈ E_{Z} such that S_{γ_{1}}S_{γ_{2}}P_{A} ≠ 0. Hence we see that P_{A}S_{γ_{1}}S_{γ_{2}} ≠ 0 so that (A, s(γ_{1})) ∈ V_{Ãs}. This shows that V_{Ãs} = V_{C̃s}. Therefore (i) has been shown. (ii) is shown similarly.

Let us denote by VÃ and by VB̃ the first four vertex sets and the second four vertex sets in Lemma 2.8, respectively. Namely we put

VÃ := V_{Ãs} = V_{Ãt} = V_{C̃s} = V_{D̃t},
VB̃ := V_{B̃s} = V_{B̃t} = V_{D̃s} = V_{C̃t}.

For an edge (A, γ_{1}γ_{2}) ∈ EÃ, define its source and terminal vertices by

s(A, γ_{1}γ_{2}) = (A, s(γ_{1})) ∈ V_{Ãs}, t(A, γ_{1}γ_{2}) = (A, t(γ_{2})) ∈ V_{Ãt}.

We then have a directed graph (VÃ, EÃ) denoted by GÃ. We have a directed graph GB̃ = (VB̃, EB̃) similarly. From an edge (A, γ_{1}) ∈ EC̃, define its source and terminal vertices by

s(A, γ_{1}) = (A, s(γ_{1})) ∈ V_{C̃s}, t(A, γ_{1}) = (A, t(γ_{1})) ∈ V_{C̃t}.

We have a directed graph GC̃ = (VÃ ⟶^{EC̃} VB̃) and similarly GD̃ = (VB̃ ⟶^{ED̃} VÃ). Let Ã be the vertex transition matrix Ã : VÃ ×VÃ ⟶ ℤ_{+} of the directed graph GÃ which is defined by

Ã((A, u), (A, v)) = |{(A, γ_{1}γ_{2}) ∈ EÃ | s(γ_{1}) = u, t(γ_{2}) = v}|

for (A, u), (A, v) ∈ VÃ. The edge transition matrix Ã^{G} : EÃ × EÃ ⟶ {0, 1} of GÃ is defined by

Ã^{G}(γ_{1}γ_{2}, η_{1}η_{2}) = \begin{cases}1 & if t(A, γ_{1}γ_{2}) = s(A, η_{1}η_{2}), \\ 0 & otherwise \end{cases}

for (A, γ_{1}γ_{2}), (A, η_{1}η_{2}) ∈ EÃ. We similarly have the vertex transition matrices B̃, C̃, D̃ and the edge transition matrices B̃^{G}, C̃^{G}, D̃^{G} of the directed graphs GB̃, GC̃, GD̃, respectively.

Proposition 2.9. The matrices Ã and B̃ are elementary equivalent such that

Ã = C̃D̃ and B̃ = D̃C̃.

Hence the two-sided topological Markov shifts (X̄Ã, σ̄Ã) and (X̄B̃, σ̄B̃) are topologically conjugate.

Proof. For (A, γ_{1}γ_{2}) with γ_{1}, γ_{2} ∈ E_{Z}, Lemma 2.6 ensures that (A, γ_{1}) ∈ EC̃, (B, γ_{2}) ∈ ED̃, Z^{G}(γ_{1}, γ_{2}) = 1 if and only if (A, γ_{1}γ_{2}) ∈ EÃ. Since t(A, γ_{1}) = s(B, γ_{2}) if and only if Z^{G}(γ_{1}, γ_{2}) = 1, we know that Ã = C̃D̃, and B̃ = D̃C̃ similarly.

A directed graph G = (V, E) with vertex set V and edge set E is said to be bipartite if V and E may be decomposed into disjoint unions V = V_{1} ⊔ V_{2} and E = E_{12} ⊔ E_{21} such that

V_{1} ={s(γ) ∈ V | γ ∈ E_{12}} = {t(γ) ∈ V | γ ∈ E_{21}},
V_{2} ={s(γ) ∈ V | γ ∈ E_{21}} = {t(γ) ∈ V | γ ∈ E_{12}}.

Let EZ̃ = EC̃ ∪ED̃ and VZ̃ = VÃ∪VB̃. It is now obvious that the directed graph GZ̃ = (VZ̃, EZ̃) is bipartite. Let us denote by Z̃ and Z̃^{G} the vertex transition matrix and the edge transition matrix of the directed graph GZ̃, respectively. Since GZ̃ is bipartite, by the above proposition, we have

Z̃ = [\begin{matrix}0 & C̃ \\ D̃ & 0 \end{matrix}] ,  Z̃^{2} = [\begin{matrix}Ã & 0 \\ 0 & B̃ \end{matrix}] .

We will study the relationship between the two matrices Z̃ and Z. For γ ∈ E_{Z}, denote by S_{(A,γ)}, S_{(B,γ)} the partial isometries P_{A}S_{γ}, P_{B}S_{γ}, respectively, so that S_{γ} = S_{(A,γ)} + S_{(B,γ)}.

Lemma 2.10. Let γ_{1}, γ_{2} ∈ E_{Z} satisfy Z^{G}(γ_{1}, γ_{2}) = 1.

(i) S_{(B,γ_{2})} ≠ 0 implies S_{(A,γ_{1})} ≠ 0.

(ii) S_{(A,γ_{2})} ≠ 0 implies S_{(B,γ_{1})} ≠ 0.

Proof. (i) Since S_{(A,γ_{1})}S_{(B,γ_{2})} = P_{A}S_{γ_{1}}P_{B}S_{γ_{2}} = S_{γ_{1}}S_{γ_{2}}P_{A}, we have

(S_{(A,γ_{1})}S_{(B,γ_{2})})^{∗}(S_{(A,γ_{1})}S_{(B,γ_{2})}) = P_{A}S_{γ_{2}}^{∗}S_{γ_{1}}^{∗}S_{γ_{1}}S_{γ_{2}}P_{A}

= ∑_{η_{1}∈E_{Z}} Z^{G}(γ_{1}, η_{1})P_{A}S_{γ_{2}}^{∗}S_{η_{1}}S_{η_{1}}^{∗}S_{γ_{2}}P_{A}

= Z^{G}(γ_{1}, γ_{2})S_{(B,γ_{2})}^{∗}S_{(B,γ_{2})}.

The above equality ensures the assertion. (ii) is shown similarly.

Lemma 2.11. Either of the following two situations occurs:

(1) Both S_{(A,γ)} and S_{(B,γ)} are not zeros for all γ ∈ E_{Z}. In this case we have

C̃^{G} = D̃^{G} = Z^{G} so that Ã = B̃ and Z̃ = [\begin{matrix}0 & Z \\ Z & 0 \end{matrix}]

(2) For each γ ∈ E_{Z}, either S_{(A,γ)} = 0, S_{(B,γ)} ≠ 0 or S_{(A,γ)} ≠ 0, S_{(B,γ)} = 0 holds. In this case we have Z̃ = Z.

Proof. Suppose that there exists γ_{0} ∈ E_{Z} such that both conditions S_{(A,γ_{0})} ≠ 0 and S_{(B,γ_{0})} ≠ 0 hold. Since the directed graph G_{Z} = (V_{Z}, E_{Z}) is irreducible, for any edge γ ∈ E_{Z}, there exists a finite sequence of edges γ_{1}, …, γ_{n} in E_{Z} such that

Z^{G}(γ, γ_{1}) = Z^{G}(γ_{1}, γ_{2}) = ⋯ = Z^{G}(γ_{n}, γ_{0}) = 1.

By the preceding lemma, any edge η ∈ E_{Z} satisfying Z^{G}(η, γ_{0}) = 1 forces that S_{(A,η)} ≠ 0 and S_{(B,η)} ≠ 0. By using this argument repeatedly, we see that S_{(A,γ)} ≠ 0 and S_{(B,γ)} ≠ 0. Hence either of the following two cases occurs:

(1) Both S_{(A,γ)} and S_{(B,γ)} are not zeros for all γ ∈ E_{Z}.

(2) For each γ ∈ E_{Z}, either S_{(A,γ)} = 0 or S_{(B,γ)} = 0.

Case (1): We have the following equalities.

S_{γ}^{∗}S_{γ} =(S_{(A,γ)}^{∗} + S_{(B,γ)}^{∗})(S_{(A,γ)} + S_{(B,γ)})
=S_{(A,γ)}^{∗}S_{(A,γ)} + S_{(B,γ)}^{∗}S_{(B,γ)}
= ∑_{(B,η)∈ED̃} C̃^{G}((A, γ), (B, η))S_{(B,η)}S_{(B,η)}^{∗}

+ ∑_{(A,η)∈EC̃} D̃^{G}((B, γ), (A, η))S_{(A,η)}S_{(A,η)}^{∗}.

On the other hand, we have

S_{γ}^{∗}S_{γ} = ∑_{η∈E_{Z}} Z^{G}(γ, η)S_{η}S_{η}^{∗}

= ∑_{η∈E_{Z}} Z^{G}(γ, η)(P_{B}S_{η}S_{η}^{∗}P_{B} + P_{A}S_{η}S_{η}^{∗}P_{A})

= ∑_{η∈E_{Z}} Z^{G}(γ, η)S_{(B,η)}S_{(B,η)}^{∗} + ∑_{η∈E_{Z}} Z^{G}(γ, η)S_{(A,η)}S_{(A,η)}^{∗}.

Since both S_{(A,γ)} ≠ 0 and S_{(B,γ)} ≠ 0 for all γ ∈ E_{Z}, we have

C̃^{G}((A, γ), (B, η)) = Z^{G}(γ, η),  D̃^{G}((B, γ), (A, η)) = Z^{G}(γ, η)

for all γ, η ∈ E_{Z}. Hence we have C̃^{G} = D̃^{G} = Z^{G} so that Ã^{G} = B̃^{G} and hence

Ã = B̃. As Z̃ = [\begin{matrix}0 & C̃ \\ D̃ & 0 \end{matrix}] , we have Z̃^{G} = [\begin{matrix}0 & Z^{G} \\ Z^{G} & 0 \end{matrix}] and hence Z̃ = [\begin{matrix}0 & Z \\ Z & 0 \end{matrix}] .

Case (2): Suppose that for each γ ∈ E_{Z}, either S_{(A,γ)} ≠ 0 or S_{(B,γ)} ≠ 0 occurs. Since the identity

S_{γ}^{∗}S_{γ} = S_{(A,γ)}^{∗}S_{(A,γ)} + S_{(B,γ)}^{∗}S_{(B,γ)}

always holds, the situation S_{(A,γ)} ≠ 0 or S_{(B,γ)} ≠ 0 occurs. Hence in this case we see that for each γ ∈ E_{Z}, either S_{(A,γ)} = 0, S_{(B,γ)} ≠ 0 or S_{(A,γ)} ≠ 0, S_{(B,γ)} = 0 occurs. This implies that the edge set E_{Z} is a disjoint union E_{Z} = EC̃ ⊔ ED̃. As S_{(A,γ_{1})}S_{(A,γ_{2})} = 0, S_{(B,γ_{1})}S_{(B,γ_{2})} = 0 for all γ_{1}, γ_{2} ∈ E_{Z},

we have Z = [\begin{matrix}0 & C̃ \\ D̃ & 0 \end{matrix}] so that Z̃ = Z.

We thus see the following lemma and proposition.

Lemma 2.12. We have a natural identification between the Cuntz–Krieger triplets (OZ̃, DZ̃, ρZ̃) and (O_{Z}, D_{Z}, ρ^{Z}).

Proof. For γ ∈ E_{Z}, we have that S_{γ} = P_{A}S_{γ} + P_{B}S_{γ}. If P_{A}S_{γ} ≠ 0, then (A, γ) ∈ EC̃. If P_{B}S_{γ} ≠ 0, then (B, γ) ∈ ED̃. Hence S_{γ} belongs to the C^{∗}-algebra C^{∗}(S_{(A,γ)}, S_{(B,γ′)} | (A, γ) ∈ EC̃, (B, γ′) ∈ ED̃) generated by S_{(A,γ)}, S_{(B,γ′)} with (A, γ) ∈ EC̃, (B, γ′) ∈ ED̃. Hence we have

C^{∗}(S_{(A,γ)}, S_{(B,γ′)} | (A, γ) ∈ EC̃, (B, γ′) ∈ ED̃) = O_{Z}.

Since EZ̃ = EC̃ ∪ ED̃ and VZ̃ = VC̃ ∪ VD̃, the algebra C^{∗}(S_{(A,γ)}, S_{(B,γ′)} | (A, γ) ∈ EC̃, (B, γ′) ∈ ED̃) is nothing but OZ̃, so that OZ̃ is identified with O_{Z} through the correspondence between S_{(A,γ)} + S_{(B,γ)} ∈ OZ̃ and S_{γ} ∈ O_{Z}. We then have

S_{γ}S_{γ}^{∗} =(S_{(A,γ)} + S_{(B,γ)})(S_{(A,γ)} + S_{(B,γ)})^{∗}
=(P_{A}S_{γ} + P_{B}S_{γ})(P_{A}S_{γ} + P_{B}S_{γ})^{∗}
=P_{A}S_{γ}S_{γ}^{∗}P_{A} + P_{B}S_{γ}S_{γ}^{∗}P_{A} + P_{A}S_{γ}S_{γ}^{∗}P_{B} + P_{B}S_{γ}S_{γ}^{∗}P_{B}
=P_{A}S_{γ}S_{γ}^{∗}P_{A} + P_{B}S_{γ}S_{γ}^{∗}P_{B}.

Similarly, by a routine calculation, we have the equalities

S_{γ_{1}}S_{γ_{2}} ⋯ S_{γ_{n}}S_{γ_{n}}^{∗} ⋯ S_{γ_{2}}^{∗}S_{γ_{1}}^{∗} =P_{A}S_{γ_{1}}P_{B}S_{γ_{2}} ⋯ S_{γ_{n}}S_{γ_{n}}^{∗} ⋯ S_{γ_{2}}^{∗}P_{B}S_{γ_{1}}^{∗}P_{A}
+P_{B}S_{γ_{1}}P_{A}S_{γ_{2}} ⋯ S_{γ_{n}}S_{γ_{n}}^{∗} ⋯ S_{γ_{2}}^{∗}P_{A}S_{γ_{1}}^{∗}P_{B}

and

P_{A}S_{γ_{1}}S_{γ_{2}} ⋯ S_{γ_{n}}S_{γ_{n}}^{∗} ⋯ S_{γ_{2}}^{∗}S_{γ_{1}}^{∗}P_{A} =P_{A}S_{γ_{1}}P_{B}S_{γ_{2}} ⋯ S_{γ_{n}}S_{γ_{n}}^{∗} ⋯ S_{γ_{2}}^{∗}P_{B}S_{γ_{1}}^{∗}P_{A},
P_{B}S_{γ_{1}}S_{γ_{2}} ⋯ S_{γ_{n}}S_{γ_{n}}^{∗} ⋯ S_{γ_{2}}^{∗}S_{γ_{1}}^{∗}P_{B} =P_{B}S_{γ_{1}}P_{A}S_{γ_{2}} ⋯ S_{γ_{n}}S_{γ_{n}}^{∗} ⋯ S_{γ_{2}}^{∗}P_{A}S_{γ_{1}}^{∗}P_{B}.

These equalities give us a natural identification between DZ̃ and D_{Z}. For t ∈ 𝕋, we have

ρ_{t}^{Z}(S_{γ}) =ρ_{t}^{Z}(P_{A}S_{γ} + P_{B}S_{γ})
=P_{A}ρ_{t}^{Z}(S_{γ}) + P_{B}ρ_{t}^{Z}(S_{γ})
=exp(2π\sqrt{−1}t)P_{A}S_{γ} + exp(2π\sqrt{−1}t)P_{B}S_{γ}
=ρZ̃ ; t(S_{(A,γ)}) + ρZ̃ ; t(S_{(B,γ)})
=ρZ̃ ; t(S_{(A,γ)} + S_{(B,γ)}).

Therefore the Cuntz–Krieger triplets (OZ̃, DZ̃, ρZ̃) and (O_{Z}, D_{Z}, ρ^{Z}) are naturally identified with each other.

Proposition 2.13. Z̃ = Z.

Proof. By Lemma 2.11, we know that the either of the following two cases occurs:

(1) Z̃ = [\begin{matrix}0 & Z \\ Z & 0 \end{matrix}] , (2) Z̃ = Z.

We assume the first case Z̃ = [\begin{matrix}0 & Z \\ Z & 0 \end{matrix}] . Let I_{Z} denote the identity matrix

whose size is the same as that of Z. By the unitary U = \frac{1}{\sqrt{2}} [\begin{matrix}I_{Z} & I_{Z} \\ I_{Z} & −I_{Z} \end{matrix}] ,

we have UZ̃U^{∗} = [\begin{matrix}Z & 0 \\ 0 & −Z \end{matrix}] , so that Sp^{×}(Z̃) = Sp^{×}(Z) ∪ (−Sp^{×}(Z)), where

Sp^{×}(Z) denotes the set of non zero spectra of Z. By Lemma 2.12, the Cuntz– Krieger triplets (OZ̃, DZ̃, ρZ̃) and (O_{Z}, D_{Z}, ρ^{Z}) are isomorphic. Hence, as we noted in the paragraph before Proposition 2.2, the two-sided topological Markov shifts (X̄Z̃, σ̄Z̃) and (X̄_{Z}, σ̄_{Z}) become topologically conjugate, so that Sp^{×}(Z̃) = Sp^{×}(Z) by a general theory of symbolic dynamics (cf. [10]). This is a contradiction, and the case (1) does not occur.

We will next study the bipartite graph GÃ from the C^{∗}-algebraic view point. For (A, γ_{1}γ_{2}) ∈ EÃ, define the partial isometry

S_{(A,γ_{1}γ_{2})} = P_{A}S_{γ_{1}}S_{γ_{2}}.

Lemma 2.14. The C^{∗}-subalgebra C^{∗}(S_{(A,γ_{1}γ_{2})}; (A, γ_{1}γ_{2}) ∈ EÃ) of O_{Z} is isomorphic to the Cuntz–Krieger algebra OÃ for the matrix Ã.

Proof. We first notice the identity

∑_{(A,γ_{1}γ_{2})∈EÃ} S_{(A,γ_{1}γ_{2})}S_{(A,γ_{1}γ_{2})}^{∗} = ∑_{γ_{1},γ_{2}∈E_{Z}} P_{A}S_{γ_{1}}S_{γ_{2}}S_{γ_{2}}^{∗}S_{γ_{1}}^{∗}P_{A} = P_{A}

holds. We also have

S_{(A,γ_{1}γ_{2})}^{∗}S_{(A,γ_{1}γ_{2})}
=P_{A}S_{γ_{2}}^{∗}S_{γ_{1}}^{∗}S_{γ_{1}}S_{γ_{2}}P_{A}
= ∑_{ζ_{1}∈E_{Z}} Z^{G}(γ_{1}, ζ_{1})P_{A}S_{γ_{2}}^{∗}S_{ζ_{1}}S_{ζ_{1}}^{∗}S_{γ_{2}}P_{A}
= ∑_{η_{1}∈E_{Z}} Z^{G}(γ_{1}, γ_{2})Z^{G}(γ_{2}, η_{1})P_{A}S_{η_{1}}S_{η_{1}}^{∗}P_{A}
= ∑_{η_{1},η_{2}∈E_{Z}} Z^{G}(γ_{1}, γ_{2})Z^{G}(γ_{2}, η_{1})Z^{G}(η_{1}, η_{2})P_{A}S_{η_{1}}S_{η_{2}}S_{η_{2}}^{∗}S_{η_{1}}^{∗}P_{A}.

For (A, γ_{1}γ_{2}), (A, η_{1}η_{2}) ∈ EÃ, the condition t(A, γ_{1}γ_{2}) = s(A, η_{1}η_{2}) holds if and only if Z^{G}(γ_{2}, η_{1}) = 1. Hence we know

Z^{G}(γ_{1}, γ_{2})Z^{G}(γ_{2}, η_{1})Z^{G}(η_{1}, η_{2}) = Ã^{G}(γ_{1}γ_{2}, η_{1}η_{2}).

By the above equalities, we have

S_{(A,γ_{1}γ_{2})}^{∗}S_{(A,γ_{1}γ_{2})} = ∑_{(A,η_{1}η_{2})∈EÃ} Ã^{G}(γ_{1}γ_{2}, η_{1}η_{2})S_{(A,η_{1}η_{2})}S_{(A,η_{1}η_{2})}^{∗},

thus proving that the C^{∗}-subalgebra C^{∗}(S_{(A,γ_{1}γ_{2})}; (A, γ_{1}γ_{2}) ∈ EÃ) of O_{Z} is isomorphic to the Cuntz–Krieger algebra OÃ for the matrix Ã.

Lemma 2.15. The C^{∗}-subalgebra C^{∗}(S_{(A,γ_{1}γ_{2})}; (A, γ_{1}γ_{2}) ∈ EÃ) of O_{Z} is nothing but P_{A}O_{Z}P_{A}. Hence the Cuntz–Krieger algebra OÃ is isomorphic to O_{A}.

Proof. Since S_{(A,γ_{1}γ_{2})} = P_{A}S_{γ_{1}}S_{γ_{2}}P_{A} for (A, γ_{1}γ_{2}) ∈ EÃ, we have C^{∗}(S_{(A,γ_{1}γ_{2})}; (A, γ_{1}γ_{2}) ∈ EÃ) ⊂ P_{A}O_{Z}P_{A}. We will show the converse inclusion relation. Take an arbitrary fixed X ∈ O_{Z} with P_{A}XP_{A} ≠ 0. Let P_{Z} be the dense ∗-subalgebra of O_{Z} algebraically generated by S_{γ}, γ ∈ E_{Z}. We may find X_{n} ∈ P_{Z} such that ∥X −X_{n}∥ → 0. Since ∥P_{A}XP_{A}−P_{A}X_{n}P_{A}∥ ≤ ∥X −X_{n}∥ → 0, it suffices to show that P_{A}X_{n}P_{A} belongs to C^{∗}(S_{(A,γ_{1}γ_{2})}; (A, γ_{1}γ_{2}) ∈ EÃ). By [7, Lemma 2.2], any element of the subalgebra P_{Z} is a finite linear combination of elements of the form S_{μ}S_{i}S_{i}^{∗}S_{ν}^{∗} for some admissible words μ = (μ_{1}, …, μ_{m}), ν = (ν_{1}, …, ν_{n}) in X_{Z}. Assume that P_{A}S_{μ}S_{i}S_{i}^{∗}S_{ν}^{∗}P_{A} ≠ 0. Since P_{A}S_{j} = S_{j}P_{B}, we have

P_{A}S_{μ} = P_{A}S_{μ_{1}} ⋯ S_{μ_{m}} = \begin{cases}S_{μ_{1}} ⋯ S_{μ_{m}}P_{A} & if m is even, \\ S_{μ_{1}} ⋯ S_{μ_{m}}P_{B} & if m is odd. \end{cases}  (2.10)

The assumption P_{A}S_{μ}S_{i}S_{i}^{∗}S_{ν}^{∗}P_{A} ≠ 0 forces the numbers m, n to be both even, or both odd. Case 1: m, n are both even. We have

P_{A}S_{μ}S_{i}S_{i}^{∗}S_{ν}^{∗}P_{A}

=P_{A}S_{μ_{1}}S_{μ_{2}}P_{A}S_{μ_{3}}S_{μ_{4}}P_{A} ⋯ P_{A}S_{μ_{m−1}}S_{μ_{m}}P_{A}S_{i}S_{i}^{∗}P_{A}

· S_{ν_{n}}^{∗}S_{ν_{n−1}}^{∗}P_{A} ⋯ S_{ν_{4}}^{∗}S_{ν_{3}}^{∗}P_{A}S_{ν_{2}}^{∗}S_{ν_{1}}^{∗}P_{A}

=S_{(A,μ_{1}μ_{2})}S_{(A,μ_{3}μ_{4})} ⋯ S_{(A,μ_{m−1}μ_{m})}P_{A}S_{i}S_{i}^{∗}P_{A}S_{(A,ν_{n−1}ν_{n})}^{∗} ⋯ S_{(A,ν_{3}ν_{4})}^{∗}S_{(A,ν_{1}ν_{2})}^{∗}.

Now we have

P_{A}S_{i}S_{i}^{∗}P_{A} = ∑_{j∈E_{Z}} P_{A}S_{i}S_{j}S_{j}^{∗}S_{i}^{∗}P_{A} = ∑_{j∈E_{Z}} S_{(A,ij)}S_{(A,ij)}^{∗}

so that P_{A}S_{μ}S_{i}S_{i}^{∗}S_{ν}^{∗}P_{A} is a finite linear combination of products of the elements S_{(A,γ_{1}γ_{2})}, S_{(A,γ_{1}γ_{2})}^{∗} for (A, γ_{1}γ_{2}) ∈ EÃ and hence it belongs to C^{∗}(S_{(A,γ_{1}γ_{2})}; (A, γ_{1}γ_{2}) ∈ EÃ). Case 2: m, n are both odd.

Similarly to Case 1, we have

P_{A}S_{μ}S_{i}S_{i}^{∗}S_{ν}^{∗}P_{A}
=S_{(A,μ_{1}μ_{2})} ⋯ S_{(A,μ_{m−2}μ_{m−1})}S_{(A,μ_{m}i)}S_{(A,ν_{n}i)}^{∗}S_{(A,ν_{n−2}ν_{n−1})}^{∗} ⋯ S_{(A,ν_{1}ν_{2})}^{∗}

so that P_{A}S_{μ}S_{i}S_{i}^{∗}S_{ν}^{∗}P_{A} belongs to C^{∗}(S_{(A,γ_{1}γ_{2})}; (A, γ_{1}γ_{2}) ∈ EÃ).

Proposition 2.16. The Cuntz–Krieger triplet (OÃ, DÃ, ρÃ) for the matrix Ã is isomorphic to (O_{A}, D_{A}, ρ^{A}).

Proof. By Lemma 2.14 and Lemma 2.15, we know that

OÃ = C^{∗}(S_{(A,γ_{1}γ_{2})}; (A, γ_{1}γ_{2}) ∈ EÃ) = P_{A}O_{Z}P_{A} = O_{A}.  (2.11)

Under the identification between C^{∗}(S_{(A,γ_{1}γ_{2})}; (A, γ_{1}γ_{2}) ∈ EÃ) and P_{A}O_{Z}P_{A} in Lemma 2.15, the C^{∗}-subalgebra

C^{∗}(S_{(A,γ_{1}γ_{2})} ⋯S_{(A,γ_{n−1}γ_{n})}S_{(A,γ_{n−1}γ_{n})}^{∗} ⋯ S_{(A,γ_{1}γ_{2})}^{∗};
(A, γ_{1}γ_{2}), …, (A, γ_{n−1}γ_{n}) ∈ EÃ)

of C^{∗}(S_{(A,γ_{1}γ_{2})}; (A, γ_{1}γ_{2}) ∈ EÃ) generated by the projections

S_{(A,γ_{1}γ_{2})} ⋯ S_{(A,γ_{n−1}γ_{n})}S_{(A,γ_{n−1}γ_{n})}^{∗} ⋯ S_{(A,γ_{1}γ_{2})}^{∗}

for (A, γ_{1}γ_{2}), …, (A, γ_{n−1}γ_{n}) ∈ EÃ is naturally identified with the C^{∗}- subalgebra P_{A}D_{Z}P_{A} of D_{Z}, so that DÃ = D_{A}. By regarding the generating partial isometry S_{(A,γ_{1}γ_{2})} for (A, γ_{1}γ_{2}) ∈ EÃ as an element of P_{A}O_{Z}P_{A} = O_{A}, we have

ρÃ ; 2t(S_{(A,γ_{1}γ_{2})}) =e^{2π\sqrt{−1}2t}S_{(A,γ_{1}γ_{2})}
=P_{A}e^{2π\sqrt{−1}t}S_{γ_{1}}e^{2π\sqrt{−1}t}S_{γ_{2}}
=P_{A}ρ_{t}^{Z}(S_{γ_{1}})ρ_{t}^{Z}(S_{γ_{2}})
=ρ_{t}^{Z}(P_{A}S_{γ_{1}}S_{γ_{2}}).

Since P_{A}S_{γ_{1}}S_{γ_{2}} ∈ P_{A}O_{Z}P_{A} = O_{A} and ρ_{t}^{Z}|_{P_{A}O_{Z}P_{A}} = ρ_{2t}^{A} on O_{A}, we have

ρ_{t}^{Z}(P_{A}S_{γ_{1}}S_{γ_{2}}) = ρ_{2t}^{A}(P_{A}S_{γ_{1}}S_{γ_{2}}) = ρ_{2t}^{A}(S_{(A,γ_{1}γ_{2})})

so that ρÃ ; 2t = ρ_{2t}^{A} for all t ∈ 𝕋 and hence ρÃ = ρ^{A}.

We thus have

Proposition 2.17. Suppose that the Cuntz–Krieger triplets (O_{A}, D_{A}, ρ^{A}) and (O_{B}, D_{B}, ρ^{B}) are strong Morita equivalent in 1-step. Then the two-sided topological Markov shifts (X̄_{A}, σ̄_{A}) and (X̄_{B}, σ̄_{B}) are topologically conjugate.

Proof. Assume that the Cuntz–Krieger triplets (O_{A}, D_{A}, ρ^{A}) and (O_{B}, D_{B}, ρ^{B}) are strong Morita equivalent in 1-step. By Proposition 2.9, the matrices Ã, B̃ are elementary equivalent so that their two-sided topological Markov shifts (X̄Ã, σ̄Ã) and (X̄B̃, σ̄B̃) are topologically conjugate. Proposition 2.16 with [15, Corollary 3.5] ensures that the ons-sided topological Markov shifts (XÃ, σÃ) and (X_{A}, σ_{A}) are eventually conjugate and hence strongly continuous orbit equivalent in the sense of [15]. Since the latter property yields topological conjugacy of their two-sided topological Markov shifts, the two-sided topological Markov shifts (X̄Ã, σ̄Ã) and (X̄_{A}, σ̄_{A}) are topologically conjugate. Similarly we know that the two-sided topological Markov shifts (X̄B̃, σ̄B̃) and (X̄_{A}, σ̄_{B}) are topologically conjugate. Therefore we get the assertion.

Now we reach one of the main results of the paper.

Theorem 2.18. Let A, B be irreducible non-permutation matrices. The Cuntz– Krieger triplets (O_{A}, D_{A}, ρ^{A}) and (O_{B}, D_{B}, ρ^{B}) are strong Morita equivalent if and only if their two-sided topological Markov shifts (X̄_{A}, σ̄_{A}) and (X̄_{B}, σ̄_{B}) are topologically conjugate.

Proof. The if part comes from Proposition 2.2. The only if part follows from Proposition 2.17.

By the Williams’s fundamental theorem on topological Markov shifts which states that two irreducible matrices A and B are strong shift equivalent if and only if their two-sided topological Markov shifts (X̄_{A}, σ̄_{A}) and (X̄_{B}, σ̄_{B}) are topologically conjugate ([25]), we obtain the following corollary.

Corollary 2.19. Let A, B be irreducible non-permutation matrices. The Cuntz–Krieger triplets (O_{A}, D_{A}, ρ^{A}) and (O_{B}, D_{B}, ρ^{B}) are strong Morita equivalent if and only if the matrices A and B are strong shift equivalent.

3 Strong shift equivalence and circle actions on O_{A}

It is well-known that two unital C^{∗}-algebras A and B are strong Morita equivalent if and only if their stabilizations A ⊗ K and B ⊗ K are isomorphic by Brown–Green–Rieffel Theorem [3, Theorem 1.2] (cf. [2], [3], [4], [20]). We will next study relationships between stabilized Cuntz–Krieger algebras with their gauge actions and strong shift equivalence matrices. We will investigate stabilizations of generalized gauge actions from a view point of flow equivalence. Recall that for a function f ∈ C(X_{A}, ℤ) and t ∈ 𝕋, an automorphism ρ_{t}^{A,f} ∈ Aut(O_{A}) is defined by ρ_{t}^{A,f}(S_{i}) = U_{t}(f)S_{i}, i = 1, …, N, t ∈ 𝕋 for the unitary U_{t}(f) = exp(2π\sqrt{−1}tf) ∈ D_{A} as in (1.5). It is easy to see that the automorphisms ρ_{t}^{A,f}, t ∈ 𝕋 yield an action of 𝕋 to O_{A} such that ρ_{t}^{A,f}(a) = a for all a ∈ D_{A}. For f ∈ C(X_{A}, ℤ) and n ∈ ℤ_{+}, let us denote by f^{n} the function f^{n}(x) = ∑_{i=0}^{n−1} f(σ_{A}^{i}(x)), x ∈ X_{A}. We know that the identity

ρ_{t}^{A,f}(S_{μ}) = U_{t}(f^{n})S_{μ}  (3.1)

for f ∈ C(X_{A}, ℤ), μ = (μ_{1}, …, μ_{n}) ∈ B_{n}(X_{A}), t ∈ 𝕋 holds (cf. [15, Lemma 3.1]).

For a C^{∗}-algebra A without unit, let M(A) stand for its multiplier C^{∗}-algebra defined by

M(A) = {a ∈ A^{∗∗} | aA ⊂ A, Aa ⊂ A}

where A^{∗∗} denotes the second dual (A^{∗})^{∗} of the C^{∗}-algebra A. An action α of 𝕋 to A extends to M(A) and is still denoted by α. For an action α of 𝕋 to A, a unitary one-cocycle u_{t}, t ∈ 𝕋 relative to α is a strongly continuous map t ∈ 𝕋 → u_{t} ∈ U(M(A)) to the unitary group U(M(A)) satisfying u_{t+s} = u_{s}α_{s}(u_{t}), s, t ∈ 𝕋. The following proposition has been proved in [15].

Proposition 3.1 ([15, Proposition 4.3]). Let A and B be elementary equivalent matrices, and choose matrices C and D satisfying A = CD and B = DC. Then there exist an isomorphism Φ : O_{A} ⊗ K → O_{B} ⊗ K satisfying Φ(D_{A} ⊗ C) = D_{B} ⊗ C and a homomorphism φ : C(X_{A}, ℤ) → C(X_{B}, ℤ) of ordered groups such that for each function f ∈ C(X_{A}, ℤ) there exists a unitary one-cocycle u_{t}^{f} ∈ U(M(O_{A} ⊗ K)) relative to ρ^{A,f} ⊗ id such that

Φ ∘ Ad(u_{t}^{f}) ∘ (ρ_{t}^{A,f} ⊗ id) = (ρ_{t}^{B,φ(f)} ⊗ id) ∘ Φ for t ∈ 𝕋.  (3.2)

In this section, we will first review the proof in [15] of the above proposition to investigate the K-theoretic behavior of the above isomorphism Φ : O_{A} ⊗ K → O_{B} ⊗ K. The proof of the above proposition is based on the the proof of [11, Proposition 4.1], in which Morita equivalence of C^{∗}-algebras has been used (cf. [2], [3], [4], [5], [8], [12], [16], [18], [24]). Let A and B be elementary equivalent matrices, and choose matrices C and D satisfying A = CD and B = DC. As in the previous section, the equality A(i, j) = ∑_{k=1}^{N_{B}} C(i, k)D(k, j) for i, j = 1, …, N_{A} forces that the cardinal numbers of the two sets {a ∈ E_{A} | s(a) = v_{i}^{A}, t(a) = v_{j}^{A}} and {(c, d) ∈ E_{C} × E_{D} | s(c) = v_{i}^{A}, t(c) = s(d), t(d) = v_{j}^{A}} coincide. Hence we may take a bijection from E_{A} to the above subset of E_{C} × E_{D}. We fix it and write it as φ_{A,CD}. By the other equality B = DC, one may take a bijection

written φ_{B,DC} from E_{B} to a subset of E_{D} ×E_{C} similarly. We set Z = [\begin{matrix}0 & C \\ D & 0 \end{matrix}]

as a block matrix, and use the same notation as in the previous sections. For an arbitrary fixed function f ∈ C(X_{A}, ℤ), we may regard it as an element of D_{A} and hence of D_{Z} by identifying it with f ⊕ 0 in D_{A} ⊕ D_{B} = D_{Z}. As

exp(2π\sqrt{−1}t(f ⊕ 0)) = exp(2π\sqrt{−1}tf) ⊕ P_{D} ∈ U(D_{Z}),

the automorphism ρ_{t}^{Z,f⊕0} of O_{Z} for t ∈ 𝕋 defined by (1.5) satisfies

ρ_{t}^{Z,f⊕0}(S_{c}) = exp(2π\sqrt{−1}tf)S_{c} for c ∈ E_{C},  (3.3)
ρ_{t}^{Z,f⊕0}(S_{d}) = S_{d} for d ∈ E_{D}.  (3.4)

Fix c ∈ E_{C} and d ∈ E_{D} such that t(c) = s(d), and let a ∈ E_{A} be the unique edge satisfying φ_{A,CD}(a) = cd. Let b ∈ E_{B} be the unique edge in E_{B} satisfying φ_{B,DC}(b) = dc, in a similar way. The equalities (3.3), (3.4) imply

ρ_{t}^{Z,f⊕0}(S_{c}S_{d}) = exp(2π\sqrt{−1}tf)S_{c}S_{d} = ρ_{t}^{A,f}(S_{a}),
ρ_{t}^{Z,f⊕0}(S_{d}S_{c}) = S_{d}exp(2π\sqrt{−1}tf)S_{c} = S_{d}exp(2π\sqrt{−1}tf)S_{d}^{∗}S_{b}.

We set φ(f) = ∑_{d∈E_{D}} S_{d}fS_{d}^{∗} ∈ D_{Z}. As P_{D}φ(f)P_{D} = φ(f), we see that φ(f) ∈ D_{B} and hence φ(f) ∈ C(X_{B}, ℤ) satisfies

∑_{d∈E_{D}} S_{d}exp(2π\sqrt{−1}tf)S_{d}^{∗} = exp(2π\sqrt{−1}tφ(f)) ∈ U(D_{B}).

We similarly set ψ(g) = ∑_{c∈E_{C}} S_{c}gS_{c}^{∗} ∈ C(X_{A}, ℤ) for g ∈ C(X_{B}, ℤ). We thus see the following lemma.

Lemma 3.2 ([15, Lemma 4.1]). For f ∈ C(X_{A}, ℤ), g ∈ C(X_{B}, ℤ) and t ∈ 𝕋, we have

ρ_{t}^{Z,f⊕0}(S_{c}S_{d}) = ρ_{t}^{A,f}(S_{a}),  ρ_{t}^{Z,f⊕0}(S_{d}S_{c}) = ρ_{t}^{B,φ(f)}(S_{b}),  (3.5)
ρ_{t}^{Z,0⊕g}(S_{d}S_{c}) = ρ_{t}^{B,g}(S_{b}),  ρ_{t}^{Z,0⊕g}(S_{c}S_{d}) = ρ_{t}^{A,ψ(g)}(S_{a})  (3.6)

where a ∈ E_{A}, b ∈ E_{B} and c ∈ E_{C}, d ∈ E_{D} satisfy φ_{A,CD}(a) = cd and φ_{B,DC}(b) = dc, respectively.

We note that the homomorphisms φ : C(X_{A}, ℤ) → C(X_{B}, ℤ) and ψ : C(X_{B}, ℤ) → C(X_{A}, ℤ) satisfy the equalities

(ψ ∘ φ)(f) = f ∘ σ_{A}, (φ ∘ ψ)(g) = g ∘ σ_{B}

for f ∈ C(X_{A}, ℤ) and g ∈ C(X_{B}, ℤ) ([15, Lemma 4.2]). By [11, Proposition 4.1], one may find partial isometries v_{A}, v_{B} ∈ M(O_{Z} ⊗ K) such that

v_{A}^{∗}v_{A} = v_{B}^{∗}v_{B} = 1 ⊗ 1,  v_{A}v_{A}^{∗} = P_{C} ⊗ 1,  v_{B}v_{B}^{∗} = P_{D} ⊗ 1.  (3.7)

Since

Ad(v_{A}^{∗}) : O_{A} ⊗ K → O_{Z} ⊗ K and Ad(v_{B}^{∗}) : O_{B} ⊗ K → O_{Z} ⊗ K (3.8)

are isomorphisms satisfying

Ad(v_{A}^{∗})(D_{A} ⊗ C) = D_{Z} ⊗ C and Ad(v_{B}^{∗})(D_{B} ⊗ C) = D_{Z} ⊗ C.

By putting

w = v_{B}v_{A}^{∗} ∈ M(O_{Z} ⊗ K),  (3.9)
Φ = Ad(w) : O_{A} ⊗ K → O_{B} ⊗ K,  (3.10)
u_{t}^{A,f} = w^{∗}(ρ_{t}^{Z,f⊕0} ⊗ id)(w) for f ∈ C(X_{A}, ℤ),  (3.11)
u_{t}^{B,g} = w(ρ_{t}^{Z,0⊕g} ⊗ id)(w^{∗}) for g ∈ C(X_{B}, ℤ),  (3.12)

they satisfy Φ(D_{A} ⊗ C) = D_{B} ⊗ C and the equalities

Φ ∘ Ad(u_{t}^{A,f}) ∘ (ρ_{t}^{A,f} ⊗ id) = (ρ_{t}^{B,φ(f)} ⊗ id) ∘ Φ for f ∈ C(X_{A}, ℤ),
Φ ∘ (ρ_{t}^{A,ψ(g)} ⊗ id) = Ad(u_{t}^{B,g}) ∘ (ρ_{t}^{B,g} ⊗ id) ∘ Φ for g ∈ C(X_{B}, ℤ).

The above discussion is a sketch of the proof of Proposition 3.1 given in [15]. In what follows, we will reconstruct partial isometries v_{A}, v_{B} satisfying (3.7) to investigate the K-theoretic behavior of the map Φ : O_{A} ⊗ K → O_{B} ⊗ K in the following section. The idea of the reconstruction is due to the proof of [2, Lemma 2.5] (cf. [11, Proposition 4.1]). We are assuming that A = CD, B = DC. Keep the notation as in the preceding section. Put E_{C} = {c_{1}, …, c_{N_{C}} } and E_{D} = {d_{1}, …, d_{N_{D}}} for the matrices C and D respectively. For k = 1, …, N_{D}, take c(k) ∈ E_{C} such that c(k)d_{k} ∈ B_{2}(X_{Z}) so that we have

S_{c(k)}^{∗}S_{c(k)} ≥ S_{d_{k}}S_{d_{k}}^{∗}.

Similarly for l = 1, …, N_{C}, take d(l) ∈ E_{D} such that d(l)c_{l} ∈ B_{2}(X_{Z}) so that we have

S_{d(l)}^{∗}S_{d(l)} ≥ S_{c_{l}}S_{c_{l}}^{∗}.

Put

U_{0} = P_{C},  U_{k} = S_{c(k)}S_{d_{k}}S_{d_{k}}^{∗} for k = 1, …, N_{D},  (3.13)
T_{0} = P_{D},  T_{l} = S_{d(l)}S_{c_{l}}S_{c_{l}}^{∗} for l = 1, …, N_{C}.  (3.14)

We then have

∑_{k=1}^{N_{D}} U_{k}^{∗}U_{k} = ∑_{k=1}^{N_{D}} S_{d_{k}}S_{d_{k}}^{∗}S_{c(k)}^{∗}S_{c(k)}S_{d_{k}}S_{d_{k}}^{∗} = ∑_{k=1}^{N_{D}} S_{d_{k}}S_{d_{k}}^{∗} = P_{D},

∑_{k=1}^{N_{C}} T_{l}^{∗}T_{l} = ∑_{l=1}^{N_{C}} S_{c_{l}}S_{c_{l}}^{∗}S_{d(l)}^{∗}S_{d(l)}S_{c_{l}}S_{c_{l}}^{∗} = ∑_{l=1}^{N_{C}} S_{c_{l}}S_{c_{l}}^{∗} = P_{C}.

We decompose the set ℕ of natural numbers into disjoint infinite subsets ℕ = ∪_{j=1}^{∞}ℕ_{j}, and decompose ℕ_{j} for each j once again into disjoint infinite sets ℕ_{j} = ∪_{k=0}^{∞}ℕ_{j_{k}}. Let {e_{i,j}}_{i,j∈ℕ} be a set of matrix units which generate the algebra K = K(ℓ^{2}(ℕ)). Put the projections f_{j} = ∑_{i∈ℕ_{j}} e_{i,i} and f_{j_{k}} = ∑_{i∈ℕ_{j_{k}}} e_{i,i}, both of which converge in the strong operator topology on ℓ^{2}(ℕ). Take a partial isometry s_{j_{k},j} such that s_{j_{k},j}^{∗}s_{j_{k},j} = f_{j}, s_{j_{k},j}s_{j_{k},j}^{∗} = f_{j_{k}} and put s_{j,j_{k}} = s_{j_{k},j}^{∗}. We set for n = 1, 2, …,

u_{n} = ∑_{k=1}^{N_{D}} U_{k} ⊗ s_{n_{k},n},  w_{n} = P_{C} ⊗ s_{n_{0},n} + u_{n},

t_{n} = ∑_{l=1}^{N_{C}} T_{l} ⊗ s_{n_{l},n},  z_{n} = P_{D} ⊗ s_{n_{0},n} + t_{n}.

Markov Shifts and Cuntz–Krieger Algebras 895

Lemma 3.3. Keep the above notations.

(i) w_{n}^{∗}w_{n} = 1 ⊗ f_{n} and w_{n}w_{n}^{∗} ≤ P_{C} ⊗ f_{n}. (ii) z_{n}^{∗}z_{n} = 1 ⊗ f_{n} and z_{n}z_{n}^{∗} ≤ P_{D} ⊗ f_{n}. Proof. (i) Since u_{n}^{∗}u_{n} = P_{D} ⊗ f_{n}, we have

w_{n}^{∗}w_{n} = P_{C} ⊗ f_{n} + u_{n}^{∗}u_{n} = P_{C} ⊗ f_{n} + P_{D} ⊗ f_{n} = 1 ⊗ f_{n}.

On the other hand, we know that u_{n}(P_{C} ⊗ s_{n,n_{0}}) = (P_{C} ⊗ s_{n,n_{0}})u_{n}^{∗} = 0 so that we have

w_{n}w_{n}^{∗} = P_{C} ⊗ f_{n_{0}} + u_{n}u_{n}^{∗} = P_{C} ⊗ f_{n_{0}} + ∑_{k=1}^{N_{D}} S_{c(k)}S_{d_{k}}S_{d_{k}}^{∗}S_{c(k)}^{∗} ⊗ f_{n_{k}}.

As f_{n_{0}}, f_{n_{k}} ≤ f_{n}, we have

w_{n}w_{n}^{∗} ≤ P_{C} ⊗ f_{n}.

(ii) is shown similarly. We will reconstruct and study the isometry v_{A} in (3.7). Let f_{n,m} be a partial isometry satisfying f_{n,m}^{∗}f_{n,m} = f_{m}, f_{n,m}f_{n,m}^{∗} = f_{n}. We put

v_{1} = w_{1} = P_{C} ⊗ s_{1_{0},1} + u_{1},
v_{2n} = (P_{C} ⊗ f_{n} − v_{2n−1}v_{2n−1}^{∗})(P_{C} ⊗ f_{n,n+1}) for 1 ≤ n ∈ ℕ,
v_{2n−1} = w_{n}(1 ⊗ f_{n} − v_{2n−2}^{∗}v_{2n−2}) for 2 ≤ n ∈ ℕ.

Lemma 3.4. Keep the above notation.

(i) v_{2n−2}^{∗}v_{2n−2} + v_{2n−1}^{∗}v_{2n−1} = 1 ⊗ f_{n}. (ii) v_{2n−1}v_{2n−1}^{∗} + v_{2n}v_{2n}^{∗} = P_{C} ⊗ f_{n}. Proof. (i) As w_{n}^{∗}w_{n} = 1 ⊗ f_{n}, we have

v_{2n−2}^{∗}v_{2n−2} + v_{2n−1}^{∗}v_{2n−1}

=v_{2n−2}^{∗}v_{2n−2} + (1 ⊗ f_{n} − v_{2n−2}^{∗}v_{2n−2})w_{n}^{∗}w_{n}(1 ⊗ f_{n} − v_{2n−2}^{∗}v_{2n−2})
=v_{2n−2}^{∗}v_{2n−2} + 1 ⊗ f_{n} − v_{2n−2}^{∗}v_{2n−2}
=1 ⊗ f_{n}.

(ii) We have v_{2n−1}v_{2n−1}^{∗} + v_{2n}v_{2n}^{∗}

=v_{2n−1}v_{2n−1}^{∗} + (P_{C} ⊗ f_{n} − v_{2n−1}v_{2n−1}^{∗})(P_{C} ⊗ f_{n})(P_{C} ⊗ f_{n} − v_{2n−1}v_{2n−1}^{∗})
=v_{2n−1}v_{2n−1}^{∗} + P_{C} ⊗ f_{n} − v_{2n−1}v_{2n−1}^{∗}
=P_{C} ⊗ f_{n}.

By the above lemma, one may see that the summations ∑_{n=1}^{∞} v_{2n−2} and ∑_{n=1}^{∞} v_{2n−1} converge in M(O_{Z} ⊗ K) to certain partial isometries written v_{ev} and v_{od} respectively in the strict topology of the multiplier algebra of O_{Z} ⊗ K. Similarly we obtain a partial isometry v_{A} = ∑_{n=1}^{∞} v_{n} in M(O_{Z} ⊗ K) in the strict topology. Therefore we have the next lemma.

Lemma 3.5. The partial isometries v_{ev}, v_{od} and v_{A} defined above satisfy the following relations:

(i) v_{A} = v_{od} + v_{ev}.

(ii) v_{od}^{∗}v_{od} + v_{ev}^{∗}v_{ev} = 1 ⊗ 1.

(iii) v_{od}v_{od}^{∗} + v_{ev}v_{ev}^{∗} = P_{C} ⊗ 1.

(iv) v_{A}^{∗}v_{A} = 1 ⊗ 1 and v_{A}v_{A}^{∗} = P_{C} ⊗ 1.

We put

q_{od}^{C} = ∑_{n=1}^{∞} v_{2n−1}(P_{C} ⊗ 1)v_{2n−1}^{∗}, q_{od}^{D} = ∑_{n=1}^{∞} v_{2n−1}(P_{D} ⊗ 1)v_{2n−1}^{∗}

so that

q_{od}^{C} + q_{od}^{D} = v_{od}v_{od}^{∗} and hence q_{od}^{C} + q_{od}^{D} + v_{ev}v_{ev}^{∗} = P_{C} ⊗ 1.

We will show the following lemma.

Lemma 3.6. v_{A}(ρ_{t}^{Z,f⊕0} ⊗ id)(v_{A}^{∗}) = q_{od}^{C} + (U_{t}(−f) ⊗ 1)q_{od}^{D} + v_{ev}v_{ev}^{∗}.

Proof. We notice that ρ_{t}^{Z,f⊕0}(S_{c}) = U_{t}(f)S_{c} for c ∈ E_{C} and ρ_{t}^{Z,f⊕0}(S_{d}) = S_{d} for d ∈ E_{D}. As v_{2n−1}v_{2n−1}^{∗} ∈ D_{Z} ⊗ C, we have (ρ_{t}^{Z,f⊕0} ⊗ id)(v_{2n−1}v_{2n−1}^{∗}) = v_{2n−1}v_{2n−1}^{∗} and hence (ρ_{t}^{Z,f⊕0} ⊗ id)(v_{ev}) = v_{ev}. We then have

v_{A}(ρ_{t}^{Z,f⊕0} ⊗ id)(v_{A}^{∗}) = v_{od}(ρ_{t}^{Z,f⊕0} ⊗ id)(v_{od}^{∗}) + v_{ev}(ρ_{t}^{Z,f⊕0} ⊗ id)(v_{ev}^{∗})

= ∑_{n=1}^{∞} v_{2n−1}(ρ_{t}^{Z,f⊕0} ⊗ id)(v_{2n−1}^{∗}) + v_{ev}v_{ev}^{∗}.

Since

v_{1}(P_{C} ⊗ 1) = P_{C} ⊗ s_{1_{0},1} and v_{1}(P_{D} ⊗ 1) = ∑_{k=1}^{N_{D}} S_{c(k)}S_{d_{k}}S_{d_{k}}^{∗} ⊗ s_{1_{k},1},

Markov Shifts and Cuntz–Krieger Algebras 897

we have

(ρ_{t}^{Z,f⊕0} ⊗ id)(v_{1}^{∗}) = (P_{C} ⊗ 1)v_{1}^{∗} + (ρ_{t}^{Z,f⊕0} ⊗ id)((P_{D} ⊗ 1)v_{1}^{∗})

= (P_{C} ⊗ 1)v_{1}^{∗} + ∑_{k=1}^{N_{D}} S_{d_{k}}S_{d_{k}}^{∗}ρ_{t}^{Z,f⊕0}(S_{c(k)}^{∗}) ⊗ s_{1_{k},1}^{∗}

= (P_{C} ⊗ 1)v_{1}^{∗} + ∑_{k=1}^{N_{D}} S_{d_{k}}S_{d_{k}}^{∗}S_{c(k)}^{∗}U_{t}(−f) ⊗ s_{1_{k},1}^{∗}

= (P_{C} ⊗ 1)v_{1}^{∗} + (P_{D} ⊗ 1)v_{1}^{∗}(U_{t}(−f) ⊗ 1),

so that

v_{1}(ρ_{t}^{Z,f⊕0} ⊗ id)(v_{1}^{∗}) = v_{1}(P_{C} ⊗ 1)v_{1}^{∗} + v_{1}(P_{D} ⊗ 1)v_{1}^{∗}(U_{t}(−f) ⊗ 1)
= v_{1}(P_{C} ⊗ 1)v_{1}^{∗} + (U_{t}(−f) ⊗ 1)v_{1}(P_{D} ⊗ 1)v_{1}^{∗}.

For 2 ≤ n ∈ ℕ, we have

v_{2n−1}(P_{C} ⊗ 1) = (P_{C} ⊗ s_{n_{0},n})(1 ⊗ f_{n} − v_{2n−2}^{∗}v_{2n−2}),
v_{2n−1}(P_{D} ⊗ 1) = ∑_{k=1}^{N_{D}}(S_{c(k)}S_{d_{k}}S_{d_{k}}^{∗} ⊗ s_{n_{k},n})(1 ⊗ f_{n} − v_{2n−2}^{∗}v_{2n−2}),

and hence

(ρ_{t}^{Z,f⊕0} ⊗ id)((P_{D} ⊗ 1)v_{2n−1}^{∗})
=(1 ⊗ f_{n} − v_{2n−2}^{∗}v_{2n−2}) ∑_{k=1}^{N_{D}} S_{d_{k}}S_{d_{k}}^{∗}ρ_{t}^{Z,f⊕0}(S_{c(k)}^{∗}) ⊗ s_{n_{k},n}^{∗}
=(1 ⊗ f_{n} − v_{2n−2}^{∗}v_{2n−2}) ∑_{k=1}^{N_{D}} S_{d_{k}}S_{d_{k}}^{∗}S_{c(k)}^{∗}U_{t}(−f) ⊗ s_{n_{k},n}^{∗}
=(P_{D} ⊗ 1)v_{2n−1}^{∗}(U_{t}(−f) ⊗ 1)

so that

v_{2n−1}(ρ_{t}^{Z,f⊕0} ⊗ id)(v_{2n−1}^{∗})
=v_{2n−1}(P_{C} ⊗ 1)v_{2n−1}^{∗} + v_{2n−1}(P_{D} ⊗ 1)v_{2n−1}^{∗}(U_{t}(−f) ⊗ 1)
=v_{2n−1}(P_{C} ⊗ 1)v_{2n−1}^{∗} + (U_{t}(−f) ⊗ 1)v_{2n−1}(P_{D} ⊗ 1)v_{2n−1}^{∗}.

Therefore we have

v_{od}(ρ_{t}^{Z,f⊕0} ⊗ id)(v_{od}^{∗}) = q_{od}^{C} + (U_{t}(−f) ⊗ 1)q_{od}^{D}

and hence v_{A}(ρ_{t}^{Z,f⊕0} ⊗ id)(v_{A}^{∗}) = q_{od}^{C} + (U_{t}(−f) ⊗ 1)q_{od}^{D} + v_{ev}v_{ev}^{∗}.

By using t_{n}, z_{n} instead of u_{n}, w_{n} respectively, we similarly obtain a partial isometry v_{B} in M(O_{Z} ⊗ K) in the strict topology. We then have the following lemmas.

Lemma 3.7.

(i) The partial isometry v_{A}(ρ_{t}^{Z,f⊕0} ⊗ id)(v_{A}^{∗}) for f ∈ C(X_{A}, ℤ), t ∈ 𝕋 belongs to M(D_{A} ⊗ C) and satisfies

v_{A}(ρ_{t}^{Z,(f_{1}+f_{2})⊕0} ⊗ id)(v_{A}^{∗}) = v_{A}(ρ_{t}^{Z,f_{1}⊕0} ⊗ id)(v_{A}^{∗})v_{A}(ρ_{t}^{Z,f_{2}⊕0} ⊗ id)(v_{A}^{∗})

for f_{1}, f_{2} ∈ C(X_{A}, ℤ), t ∈ 𝕋.

(ii) The partial isometry v_{B}(ρ_{t}^{Z,0⊕g} ⊗ id)(v_{B}^{∗}) for g ∈ C(X_{B}, ℤ), t ∈ 𝕋 belongs to M(D_{B} ⊗ C) and satisfies

v_{B}(ρ_{t}^{Z,0⊕(g_{1}+g_{2})} ⊗ id)(v_{B}^{∗}) = v_{B}(ρ_{t}^{Z,0⊕g_{1}} ⊗ id)(v_{B}^{∗})v_{B}(ρ_{t}^{Z,0⊕g_{2}} ⊗ id)(v_{B}^{∗})

for g_{1}, g_{2} ∈ C(X_{B}, ℤ), t ∈ 𝕋.

Proof. (i) Since the projections q_{od}^{C}, q_{od}^{D}, v_{ev}v_{ev}^{∗} all belong to the multiplier algebra M(D_{A}⊗C) of D_{A}⊗C, the preceding lemma ensures that the partial isometry v_{A}(ρ^{Z,f⊕0} ⊗ id)(v_{A}^{∗}) belongs to M(D_{A} ⊗ C). As U_{t}(f_{1} + f_{2}) = U_{t}(f_{1})U_{t}(f_{2}), the desired equality follows. (ii) is shown similarly.

Lemma 3.8.

(i) (ρ_{t}^{Z,0⊕g} ⊗ id)(v_{A}) = v_{A} for g ∈ C(X_{B}, ℤ), t ∈ 𝕋.

(ii) (ρ_{t}^{Z,f⊕0} ⊗ id)(v_{B}) = v_{B} for f ∈ C(X_{A}, ℤ), t ∈ 𝕋.

Proof. (i) Since ρ_{t}^{Z,0⊕g}(S_{c}) = S_{c}, ρ_{t}^{Z,0⊕g}(S_{d}) = e^{2π\sqrt{−1}tg}S_{d}, we have

ρ_{t}^{Z,0⊕g}(U_{k}) =ρ_{t}^{Z,0⊕g}(S_{c(k)}S_{d_{k}}S_{d_{k}}^{∗})
=S_{c(k)}e^{2π\sqrt{−1}tg}S_{d_{k}}S_{d_{k}}^{∗}e^{−2π\sqrt{−1}tg}
=S_{c(k)}S_{d_{k}}S_{d_{k}}^{∗} = U_{k}.

Hence (ρ_{t}^{Z,0⊕g} ⊗ id)(u_{n}) = u_{n} so that (ρ_{t}^{Z,0⊕g} ⊗ id)(w_{n}) = w_{n}. We then have

(ρ_{t}^{Z,0⊕g} ⊗ id)(v_{1}) = (ρ_{t}^{Z,0⊕g} ⊗ id)(P_{C} ⊗ s_{1_{0},1} + u_{1}) = P_{C} ⊗ s_{1_{0},1} + u_{1} = v_{1}.

Since v_{2n−1}v_{2n−1}^{∗}, v_{2n−2}^{∗}v_{2n−2} ∈ D_{Z} ⊗ C and the restriction of ρ_{t}^{Z,0⊕g} ⊗ id to D_{Z} ⊗ C is the identity, we easily know that

(ρ_{t}^{Z,0⊕g} ⊗ id)(v_{2n}) = v_{2n}, (ρ_{t}^{Z,0⊕g} ⊗ id)(v_{2n−1}) = v_{2n−1} for n ∈ ℕ.

We thus have (ρ_{t}^{Z,0⊕g} ⊗id)(v_{n}) = v_{n} for all n ∈ ℕ and hence (ρ_{t}^{Z,0⊕g} ⊗id)(v_{A}) = v_{A}. (ii) is shown similarly.

We put

w = v_{B}v_{A}^{∗} ∈ M(O_{Z} ⊗ K),
u_{t}^{A,f} = w^{∗}(ρ_{t}^{Z,f⊕0} ⊗ id)(w) for f ∈ C(X_{A}, ℤ),
u_{t}^{B,g} = w(ρ_{t}^{Z,0⊕g} ⊗ id)(w^{∗}) for g ∈ C(X_{B}, ℤ).

By Lemma 3.8, we have

u_{t}^{A,f} = v_{A}v_{B}^{∗}(ρ_{t}^{Z,f⊕0} ⊗ id)(v_{B})(ρ_{t}^{Z,f⊕0} ⊗ id)(v_{A}^{∗}) = v_{A}(ρ_{t}^{Z,f⊕0} ⊗ id)(v_{A}^{∗}) (3.15)

and similarly u_{t}^{B,g} = v_{B}(ρ_{t}^{Z,0⊕g} ⊗ id)(v_{B}^{∗}).

Lemma 3.9.

(i) For each f ∈ C(X_{A}, ℤ), the unitaries u_{t}^{A,f}, t ∈ 𝕋 give rise to a unitary representation of 𝕋 in M(D_{A} ⊗ C) which satisfies u_{t}^{A,f_{1}+f_{2}} = u_{t}^{A,f_{1}}u_{t}^{A,f_{2}} for f_{1}, f_{2} ∈ C(X_{A}, ℤ).

(ii) For each g ∈ C(X_{B}, ℤ), the unitaries u_{t}^{B,g}, t ∈ 𝕋 give rise to a unitary representation of 𝕋 in M(D_{B} ⊗ C) which satisfies u_{t}^{B,g_{1}+g_{2}} = u_{t}^{B,g_{1}}u_{t}^{B,g_{2}} for g_{1}, g_{2} ∈ C(X_{B}, ℤ).

Proof. (i) By Lemma 3.6 and (3.15), we have

u_{t}^{A,f}u_{s}^{A,f} =v_{A}(ρ_{t}^{Z,f⊕0} ⊗ id)(v_{A}^{∗})v_{A}(ρ_{s}^{Z,f⊕0} ⊗ id)(v_{A}^{∗})
=(q_{od}^{C} + (U_{t}(−f) ⊗ 1)q_{od}^{D} + v_{ev}v_{ev}^{∗})(q_{od}^{C} + (U_{s}(−f) ⊗ 1)q_{od}^{D} + v_{ev}v_{ev}^{∗})
=q_{od}^{C} + (U_{t+s}(−f) ⊗ 1)q_{od}^{D} + v_{ev}v_{ev}^{∗} = u_{t+s}^{A,f}.

The equality u_{t}^{A,f_{1}+f_{2}} = u_{t}^{A,f_{1}}u_{t}^{A,f_{2}} immediately follows from Lemma 3.7. (ii) is shown similarly.

We thus have

Proposition 3.10. Let A, B be nonnegative irreducible and non-permutation matrices. Suppose that they are elementary equivalent, and choose matrices C and D satisfying A = CD, B = DC. Then there exist an isomorphism Φ : O_{A} ⊗K → O_{B} ⊗K satisfying Φ(D_{A} ⊗C) = D_{B} ⊗C, and unitary representations t ∈ 𝕋 → u_{t}^{A,f} ∈ M(D_{A} ⊗ C) for each f ∈ C(X_{A}, ℤ) and t ∈ 𝕋 → u_{t}^{B,g} ∈ M(D_{B} ⊗ C) for each g ∈ C(X_{B}, ℤ) such that

Φ ∘ Ad(u_{t}^{A,f}) ∘ (ρ_{t}^{A,f} ⊗ id) = (ρ_{t}^{B,φ(f)} ⊗ id) ∘ Φ for f ∈ C(X_{A}, ℤ),
Φ ∘ (ρ_{t}^{A,ψ(g)} ⊗ id) = Ad(u_{t}^{B,g}) ∘ (ρ_{t}^{B,g} ⊗ id) ∘ Φ for g ∈ C(X_{B}, ℤ).

Proof. As in the proof of [15, Proposition 4.3], the map Φ = Ad(w) where w = v_{B}v_{A}^{∗} gives rise to an isomorphism Φ : O_{A} ⊗ K → O_{B} ⊗ K such that Φ(D_{A} ⊗ C) = D_{B} ⊗ C and

Φ ∘ Ad(u_{t}^{A,f}) ∘ (ρ_{t}^{A,f} ⊗ id) = (ρ_{t}^{B,φ(f)} ⊗ id) ∘ Φ.

The other equality is shown similarly.

Since both homomorphisms φ : C(X_{A}, ℤ) → C(X_{B}, ℤ) and ψ : C(X_{B}, ℤ) → C(X_{A}, ℤ) satisfy φ(1) = 1, ψ(1) = 1, we have the following corollary.

Corollary 3.11 (cf. [7, Theorem 3.8], [6, Theorem 2.3]). Let A, B be irreducible non-permutation matrices. Suppose that the two-sided topological Markov shifts (X̄_{A}, σ̄_{A}) and (X̄_{B}, σ̄_{B}) are topologically conjugate. Then there exist an isomorphism Φ : O_{A} ⊗ K → O_{B} ⊗ K of C^{∗}-algebras satisfying Φ(D_{A} ⊗ C) = D_{B} ⊗ C, and unitary representations t ∈ 𝕋 → v_{t}^{A} ∈ M(D_{A} ⊗ C) and t ∈ 𝕋 → v_{t}^{B} ∈ M(D_{B} ⊗ C) such that

Φ ∘ Ad(v_{t}^{A}) ∘ (ρ_{t}^{A} ⊗ id) = (ρ_{t}^{B} ⊗ id) ∘ Φ,
Φ ∘ (ρ_{t}^{A} ⊗ id) = Ad(v_{t}^{B}) ∘ (ρ_{t}^{B} ⊗ id) ∘ Φ

where ρ_{t}^{A} and ρ_{t}^{B} are the gauge actions on O_{A} and O_{B}, respectively.

Remark 3.12. We must emphasize that Cuntz–Krieger in [7, Theorem 3.8] and Cuntz in [6, Theorem 2.3] have shown that the stabilized Cuntz–Krieger triplet (O_{A} ⊗ K, D_{A} ⊗ C, ρ^{A} ⊗ id) is invariant under topological conjugacy of the two-sided topological Markov shifts (X̄_{A}, σ̄_{A}). Hence the above corollary is weaker than their result.

Before ending this section, we will introduce a notion of strong Morita equivalence in the stabilized Cuntz–Krieger triplets. The triplet (O_{A}⊗K, D_{A}⊗C, ρ^{A}⊗ id) is called the stabilized Cuntz–Krieger triplet. Two stabilized Cuntz–Krieger triplets (O_{A} ⊗ K, D_{A} ⊗ C, ρ^{A} ⊗ id) and (O_{B} ⊗ K, D_{B} ⊗ C, ρ^{B} ⊗ id) are said to be strong Morita equivalent in 1-step if there exist a stabilized Cuntz–Krieger triplet (O_{Z} ⊗ K, D_{Z} ⊗ C, ρ^{Z} ⊗ id) and isomorphisms of C^{∗}-algebras

Φ_{A} : O_{Z} ⊗ K ⟶ O_{A} ⊗ K, Φ_{B} : O_{Z} ⊗ K ⟶ O_{B} ⊗ K

satisfying

Φ_{A}(D_{Z}⊗C) = D_{A} ⊗ C,  Φ_{B}(D_{Z} ⊗ C) = D_{B} ⊗ C,
ρ_{t}^{Z} ⊗ id =(Φ_{B}^{−1} ∘ ρ_{t}^{B} ⊗ id ∘ Φ_{B}) ∘ (Φ_{A}^{−1} ∘ ρ_{t}^{A} ⊗ id ∘ Φ_{A})
=(Φ_{A}^{−1} ∘ ρ_{t}^{A} ⊗ id ∘ Φ_{A}) ∘ (Φ_{B}^{−1} ∘ ρ_{t}^{B} ⊗ id ∘ Φ_{B}).

If two stabilized Cuntz–Krieger triplets (O_{A} ⊗ K, D_{A} ⊗ C, ρ^{A} ⊗ id) and (O_{B} ⊗ K, D_{B} ⊗ C, ρ^{B} ⊗ id) are connected by n-chains of strong Morita equivalences in 1-step, they are said to be strong Morita equivalent in n-step, or simply strong Morita equivalent.

Proposition 3.13. Let A and B be irreducible and not any permutation matrices. Suppose that A, B are elementary equivalent. Then the stabilized Cuntz– Krieger triplets (O_{A} ⊗ K, D_{A} ⊗ C, ρ^{A} ⊗ id) and (O_{B} ⊗ K, D_{B} ⊗ C, ρ^{B} ⊗ id) are strong Morita equivalent in 1-step.

Proof. Let Z = [\begin{matrix}0 & C \\ D & 0 \end{matrix}] . Take isometries v_{A}, v_{B} ∈ M(O_{Z} ⊗K) satisfying (3.7).

By Lemma 3.8, the following identities hold

(ρ_{t}^{Z,0⊕1} ⊗ id)(v_{A}) = v_{A}, (ρ_{t}^{Z,1⊕0} ⊗ id)(v_{B}) = v_{B}.

Define Φ_{A} = Ad(v_{A}), Φ_{B} = Ad(v_{B}). As in (3.8), they give rise to isomorphisms

Φ_{A} : O_{Z} ⊗ K ⟶ O_{A} ⊗ K, Φ_{B} : O_{Z} ⊗ K ⟶ O_{B} ⊗ K

satisfying

Φ_{A}(D_{Z} ⊗ C) = D_{A} ⊗ C, Φ_{B}(D_{Z} ⊗ C) = D_{B} ⊗ C.

Since we see

ρ_{t}^{Z,0⊕1}(S_{c}) = S_{c}, ρ_{t}^{Z,0⊕1}(S_{d}) = e^{2π\sqrt{−1}t}S_{d},
ρ_{t}^{Z,1⊕0}(S_{c}) = e^{2π\sqrt{−1}t}S_{c}, ρ_{t}^{Z,1⊕0}(S_{d}) = S_{d}

for c ∈ C, d ∈ D, we have for x ⊗ K ∈ O_{Z} ⊗ K

((ρ_{t}^{A} ⊗ id) ∘ Φ_{A})(x ⊗ K) =(ρ_{t}^{Z,0⊕1} ⊗ id)(v_{A}(x ⊗ K)v_{A}^{∗})
=v_{A}(ρ_{t}^{Z,0⊕1} ⊗ id)(x ⊗ K)v_{A}^{∗}
=Φ_{A} ∘ (ρ_{t}^{Z,0⊕1} ⊗ id)(x ⊗ K).

Hence we have (ρ_{t}^{A} ⊗id)∘Φ_{A} = Φ_{A} ∘(ρ_{t}^{Z,0⊕1} ⊗id) and similarly (ρ_{t}^{B} ⊗id)∘Φ_{B} = Φ_{B} ∘ (ρ_{t}^{Z,1⊕0} ⊗ id). Since

ρ_{t}^{Z} ⊗ id = (ρ_{t}^{Z,1⊕0} ⊗ id) ∘ (ρ_{t}^{Z,0⊕1} ⊗ id) = (ρ_{t}^{Z,0⊕1} ⊗ id) ∘ (ρ_{t}^{Z,1⊕0} ⊗ id),

we know the assertion.

Therefore we have the following corollary.

Corollary 3.14. If A, B are strong shift equivalent, then the stabilized Cuntz– Krieger triplets (O_{A} ⊗ K, D_{A} ⊗ C, ρ^{A} ⊗ id) and (O_{B} ⊗ K, D_{B} ⊗ C, ρ^{B} ⊗ id) are strong Morita equivalent.

4 Behavior on K-theory

In this section we will study the behavior of the isomorphism Φ : O_{A} ⊗ K → O_{B} ⊗ K in Proposition 3.10 on their K-groups Φ_{∗} : K_{0}(O_{A}) → K_{0}(O_{B}) under the condition A = CD, B = DC. Recall that A = [A(i, j)]_{i,j=1}^{N} is an N × N matrix with entries in nonnegative integers. Then the associated graph G_{A} = (V_{A}, E_{A}) consists of the vertex set V_{A} = {v_{1}^{A}, …, v_{N}^{A}} of N vertices and edge set E_{A} = {a_{1}, …, a_{N_{A}}}, where there are A(i, j) edges from v_{i}^{A} to v_{j}^{A}. Denote by t(a_{i}), s(a_{i}) the terminal vertex of a_{i}, the source vertex of a_{i}, respectively. The graph G_{A} has the N_{A} × N_{A} transition matrix A^{G} = [A^{G}(i, j)]_{i,j=1}^{N_{A}} of edges defined by (2.1). The Cuntz–Krieger algebra O_{A} is defined as the Cuntz–Krieger algebra O_{A^{G}} for the matrix A^{G} which is the universal C^{∗}-algebra generated by partial isometries S_{a_{i}}, i = 1, …, N_{A} subject to the relations (2.2). We similarly consider the N_{B} × N_{B} matrix B^{G} with entries in {0, 1} for the graph G_{B} = (V_{B}, E_{B}) of the matrix B with vertex set V_{B} = {v_{1}^{B}, …, v_{M}^{B}} and edge set E_{B} = {b_{1}, …, b_{N_{B}}}, so that we have the other Cuntz–Krieger algebra O_{B^{G}} for the matrix B^{G} which is denoted by O_{B}. Now we are assuming that A = CD and B = DC for some nonnegative rectangular matrices C and D. Both A and B are also assumed to be irreducible and not any permutations. Since A = CD, the edge set E_{A} is regarded as a subset of the product E_{C} × E_{D} of those of E_{C} and E_{D}. As in Section 2, we may take a bijection φ_{A,CD} from E_{A} to a subset of E_{C} × E_{D}. For any a_{i} ∈ E_{A}, there uniquely exist c(a_{i}) ∈ E_{C} and d(a_{i}) ∈ E_{D} such that φ_{A,CD}(a_{i}) = c(a_{i})d(a_{i}). We write it simply as a_{i} = c(a_{i})d(a_{i}). Similarly, for any edge b_{l} ∈ E_{B}, there uniquely exist d(b_{l}) ∈ E_{D} and c(b_{l}) ∈ E_{C} such that φ_{B,DC}(b_{l}) = d(b_{l})c(b_{l}), simply written b_{l} = d(b_{l})c(b_{l}). We define the N_{A}×N_{B} matrix D̂ = [D̂(i, l)]_{i=1,...,N_{A}}^{l=1,...,N_{B}} by

D̂(i, l) = \begin{cases}1 & if d(a_{i}) = d(b_{l}), \\ 0 & otherwise. \end{cases}  (4.1)

Lemma 4.1. The matrix D̂^{t} : ℤ^{N_{A}} → ℤ^{N_{B}} induces a homomorphism from ℤ^{N_{A}}/(id − (A^{G})^{t})ℤ^{N_{A}} to ℤ^{N_{B}}/(id − (B^{G})^{t})ℤ^{N_{B}} as abelian groups.

Proof. For i = 1, …, N_{A} and l = 1, …, N_{B}, we know that both

[A^{G}D̂](i, l) = ∑_{j=1}^{N_{A}} A^{G}(i, j)D̂(j, l) and [D̂B^{G}](i, l) = ∑_{k=1}^{N_{B}} D̂(i, k)B^{G}(k, l)

are the cardinal number of the set {c ∈ E_{C} | d(a_{i})cd(b_{l}) ∈ B_{3}(X_{Z})}. Hence we have A^{G}D̂ = D̂B^{G}. We then have that D̂^{t}(id − (A^{G})^{t})ℤ^{N_{A}} ⊂ (id − (B^{G})^{t})ℤ^{N_{B}} so that D̂^{t} induces a desired homomorphism.

We denote by ΦD̂^{t} the above homomorphism from ℤ^{N_{A}}/(id − (A^{G})^{t})ℤ^{N_{A}} to ℤ^{N_{B}}/(id − (B^{G})^{t})ℤ^{N_{B}} induced by D̂^{t}.

Let us denote by [e_{i}^{N_{A}}] the class of the vector e_{i}^{N_{A}} = (0, …, 0, 1^{i}, 0, …, 0) ∈ ℤ^{N_{A}} in ℤ^{N_{A}}/(id − (A^{G})^{t})ℤ^{N_{A}}. It was shown in [6] that the correspondence ϵ_{A^{G}} : K_{0}(O_{A^{G}}) → ℤ^{N_{A}}/(id − (A^{G})^{t})ℤ^{N_{A}} defined by ϵ_{A^{G}}([S_{a_{i}}S_{a_{i}}^{∗}]) = [e_{i}^{N_{A}}] yields an isomorphism of abelian groups. We then have

Proposition 4.2. Suppose that A = CD, B = DC. Let Φ : O_{A} ⊗ K → O_{B} ⊗ K be the isomorphism constructed in the proof of Proposition 3.10 such that Φ = Ad(w) with w = v_{B}v_{A}^{∗} for the isometry v_{A} as well as v_{B} defined before Lemma 3.5. Then the diagram

K_{0}(O_{A^{G}})  ⟶^{Φ_{∗}}  K_{0}(O_{B^{G}})
ϵ_{A^{G}} ↓  ↓^{ϵ_{B^{G}}}
ℤ^{N_{A}}/(id − (A^{G})^{t})ℤ^{N_{A}} ⟶^{ΦD̂^{t}} ℤ^{N_{B}}/(id − (B^{G})^{t})ℤ^{N_{B}}

is commutative.

Proof. We note that K = K(ℓ^{2}(ℕ)) has a countable basis and ℕ is decomposed such as ℕ = ∪_{j=1}^{∞}ℕ_{j} where ℕ_{j} is also a disjoint infinite set such as ℕ_{j} = ∪_{k=0}^{∞}ℕ_{j_{k}} with disjoint infinite sets ℕ_{j_{k}} for every k = 0, 1, 2, …. We write ℕ_{j_{k}} as ℕ_{j_{k}} = {j_{k}(0), j_{k}(1), j_{k}(2), … }. In particular for j = 1, k = 0, we denote by n̄ = 1_{0}(n) for n = 0, 1, 2, … so that ℕ_{1_{0}} = {¯0, ¯1, ¯2, … }. Let pn̄, n = 0, 1, 2, … be the sequence of projections of rank one in K such that ∑_{n=0}^{∞} pn̄ = f_{1_{0}}. By [6, Proposition 3.1], the group K_{0}(O_{A^{G}}) is generated by the projections of the form

S_{a_{i}}S_{a_{i}}^{∗} ⊗ p¯0, i = 1, …, N_{A}.

Denote by 1_{A} the unit of O_{A^{G}} so that [1_{A}] = ∑_{i=1}^{N_{A}}[S_{a_{i}}S_{a_{i}}^{∗}⊗p¯0] in K_{0}(O_{A^{G}}). Let Φ = Ad(w) : O_{A^{G}} ⊗K → O_{B^{G}} ⊗K be the isomorphism constructed in the proof of Proposition 3.10. Hence Φ_{∗} : K_{0}(O_{A^{G}}) → K_{0}(O_{B^{G}}) satisfies Φ_{∗}([S_{a_{i}}S_{a_{i}}^{∗} ⊗ p¯0]) = [w(S_{a_{i}}S_{a_{i}}^{∗}⊗p¯0)^{∗}w^{∗}]. To complete the proof of the proposition, we provide the following two lemmas.

Let l(i) be the number l = 1, …, N_{C} satisfying c_{l} = c(a_{i}) so that d(l(i)) ∈ E_{D} satisfies T_{l(i)} = S_{d(l(i))}S_{c(a_{i})}S_{c(a_{i})}^{∗} in (3.14). We put s_{1_{l(i)},1_{0}} = s_{1_{l(i)},1}s_{1,1_{0}} and s_{1_{0},1_{l(i)}} = s_{1_{l(i)},1_{0}}^{∗}.

Lemma 4.3. Keep the above notation.

(i) w(S_{a_{i}}S_{a_{i}}^{∗} ⊗ p¯0)w^{∗} = v_{B}(S_{a_{i}}S_{a_{i}}^{∗} ⊗ s_{1,1_{0}}p¯0s_{1_{0},1})v_{B}^{∗}.

(ii) v_{B}(S_{a_{i}}S_{a_{i}}^{∗} ⊗ s_{1,1_{0}}p¯0s_{1_{0},1})v_{B}^{∗} = S_{d(l(i))}S_{c(a_{i})}S_{d(a_{i})}S_{d(a_{i})}^{∗}S_{c(a_{i})}^{∗}S_{d(l(i))}^{∗} ⊗ s_{1_{l(i)},1_{0}}p¯0s_{1_{0},1_{l(i)}}.

Proof. (i) The unitary w is given by w = v_{B}v_{A}^{∗}. We know v_{A} = ∑_{n=1}^{∞} v_{n} and v_{1} = P_{C} ⊗ s_{1_{0},1} + ∑_{k=1}^{N_{D}} U_{k} ⊗ s_{1_{k},1}. As p¯0s_{1_{k},1} = 0 for k = 1, …, N_{D}, we have

v_{A}^{∗}(S_{a_{i}}S_{a_{i}}^{∗} ⊗ p¯0)v_{A} = v_{1}^{∗}(S_{a_{i}}S_{a_{i}}^{∗} ⊗ p¯0)v_{1}
= (P_{C} ⊗ s_{1_{0},1})^{∗}(S_{a_{i}}S_{a_{i}}^{∗} ⊗ p¯0)(P_{C} ⊗ s_{1_{0},1})
= S_{a_{i}}S_{a_{i}}^{∗} ⊗ s_{1,1_{0}}p¯0s_{1_{0},1}.

(ii) For c_{l} ∈ E_{C} = {c_{1}, …, c_{N_{C}} } and a_{i} ∈ E_{A}, we note that S_{c_{l}}^{∗}S_{a_{i}} = S_{c_{l}}^{∗}S_{c(a_{i})}S_{d(a_{i})} if c_{l} = c(a_{i}), otherwise zero. Hence we have

v_{B}(S_{a_{i}}S_{a_{i}}^{∗} ⊗ s_{1,1_{0}}p¯0s_{1_{0},1})v_{B}^{∗}
= (∑_{l=1}^{N_{C}} T_{l} ⊗ s_{1_{l},1}) (S_{a_{i}}S_{a_{i}}^{∗} ⊗ s_{1,1_{0}}p¯0s_{1_{0},1}) (∑_{l′=1}^{N_{C}} T_{l′} ⊗ s_{1_{l′} ,1})^{∗}

= ∑_{l=1}^{N_{C}} S_{d(l)}S_{c_{l}}S_{c_{l}}^{∗}S_{a_{i}}S_{a_{i}}^{∗}S_{c_{l}}S_{c_{l}}^{∗}S_{d(l)}^{∗} ⊗ s_{1_{l},1}s_{1,1_{0}}p¯0s_{1_{0},1}s_{1_{l},1}^{∗}
=S_{d(l(i))}S_{c(a_{i})}S_{d(a_{i})}S_{d(a_{i})}^{∗}S_{c(a_{i})}^{∗}S_{d(l(i))}^{∗} ⊗ s_{1_{l(i)},1_{0}}p¯0s_{1_{0},1_{l(i)}}.

Lemma 4.4. S_{d(a_{i})}S_{d(a_{i})}^{∗} = ∑_{l=1}^{N_{B}} D̂(i, l)S_{b_{l}}S_{b_{l}}^{∗}.

Proof. In the algebra O_{B^{G}}, we have ∑_{l=1}^{N_{B}} S_{b_{l}}S_{b_{l}}^{∗} = 1. As b_{l} = d(b_{l})c(b_{l}), it implies that ∑_{l=1}^{N_{B}} S_{d(b_{l})}S_{c(b_{l})}S_{c(b_{l})}^{∗}S_{d(b_{l})}^{∗} = P_{D} in O_{Z}. By multiplying S_{d(a_{i})}S_{d(a_{i})}^{∗} to the equality we have

∑_{l=1}^{N_{B}} S_{d(a_{i})}S_{d(a_{i})}^{∗}S_{d(b_{l})}S_{c(b_{l})}S_{c(b_{l})}^{∗}S_{d(b_{l})}^{∗}S_{d(a_{i})}S_{d(a_{i})}^{∗} = S_{d(a_{i})}S_{d(a_{i})}^{∗}.

Since

S_{d(a_{i})}S_{d(a_{i})}^{∗}S_{d(b_{l})} = D̂(i, l)S_{d(b_{l})},

we have

∑_{l=1}^{N_{B}} D̂(i, l)S_{d(b_{l})}S_{c(b_{l})}S_{c(b_{l})}^{∗}S_{d(b_{l})}^{∗} = S_{d(a_{i})}S_{d(a_{i})}^{∗}.

As S_{b_{l}} = S_{d(b_{l})}S_{c(b_{l})}, we get the desired equality.

Proof of Proposition 4.2: By using Lemma 4.3, we have the equalities in K_{0}(O_{B^{G}}):

Φ_{∗}([S_{a_{i}}S_{a_{i}}^{∗} ⊗ p¯0]) = [S_{d(l(i))}S_{c(a_{i})}S_{d(a_{i})}S_{d(a_{i})}^{∗}S_{c(a_{i})}^{∗}S_{d(l(i))}^{∗} ⊗ s_{1_{l(i)},1_{0}}p¯0s_{1_{0},1_{l(i)}}].

Since

[S_{d(l(i))}S_{c(a_{i})}S_{d(a_{i})}S_{d(a_{i})}^{∗}S_{c(a_{i})}^{∗}S_{d(l(i))}^{∗} ⊗ s_{1_{l(i)},1_{0}}p¯0s_{1_{0},1_{l(i)}}]
=[S_{d(a_{i})}S_{d(a_{i})}^{∗} ⊗ f_{1_{0}}p¯0f_{1_{0}}] in K_{0}(O_{B^{G}}),

and f_{1_{0}}p¯0f_{1_{0}} = p¯0, we have

Φ_{∗}([S_{a_{i}}S_{a_{i}}^{∗} ⊗ p¯0]) = [S_{d(a_{i})}S_{d(a_{i})}^{∗} ⊗ p¯0].

As ϵ_{A^{G}}([S_{a_{i}}S_{a_{i}}^{∗} ⊗p¯0]) = [e_{i}^{N_{A}}] and ϵ_{B^{G}}([S_{b_{l}}S_{b_{l}}^{∗} ⊗p¯0]) = [e_{l}^{N_{B}}], By using Lemma 4.4, we complete the proof of Proposition 4.2.

Let S_{A} and R_{A} be the N_{A} × N matrix and N × N_{A} matrix defined by

S_{A}(i, j) = \begin{cases}1 & if t(a_{i}) = v_{j}^{A}, \\ 0 & otherwise, \end{cases} R_{A}(j, i) = \begin{cases}1 & if v_{j}^{A} = s(a_{i}), \\ 0 & otherwise, \end{cases}

for i = 1, …, N_{A} and j = 1, …, N, respectively. We then have A = R_{A}S_{A} and A^{G} = S_{A}R_{A}. We similarly have the matrices S_{B}, R_{B} for the other matrix B such that B = R_{B}S_{B} and B^{G} = S_{B}R_{B}. The matrix S_{A}^{t} : ℤ^{N_{A}} → ℤ^{N} induces a homomorphism ℤ^{N_{A}}/(id − (A^{G})^{t})ℤ^{N_{A}} → ℤ^{N}/(id − A^{t})ℤ^{N} of abelian groups which is actually an isomorphism since its inverse is given by a homomorphism induced by R_{A}^{t}. The above isomorphism is denoted by Φ_{S_{A}^{t}}. We have an

isomorphism Φ_{S_{B}^{t}} : ℤ^{N_{B}}/(id − (B^{G})^{t})ℤ^{N_{B}} → ℤ^{M}/(id − B^{t})ℤ^{M} in a similar way. Now we are assuming that A = CD, B = DC so that AC = CB and hence C^{t}A^{t} = B^{t}C^{t}. The matrix C^{t} : ℤ^{N} → ℤ^{M} induces a homomorphism from ℤ^{N}/(id − A^{t})ℤ^{N} to ℤ^{M}/(id − B^{t})ℤ^{M} as abelian groups, which is denoted by Φ_{C^{t}}. It is actually an isomorphism with Φ_{D^{t}} as its inverse. We notice the following lemma. The second assertion (ii) is pointed out by Hiroki Matui. The author thanks him for his advice.

Lemma 4.5. (i) The diagram

ℤ^{N_{A}}/(id − (A^{G})^{t})ℤ^{N_{A}} ⟶^{ΦD̂^{t}} ℤ^{N_{B}}/(id − (B^{G})^{t})ℤ^{N_{B}}
Φ_{S^{t}_{A}} ↓  ↓^{Φ_{S^{t}_{B}}}
ℤ^{N}/(id − A^{t})ℤ^{N}  ⟶^{Φ_{C^{t}}} ℤ^{M}/(id − B^{t})ℤ^{M}

is commutative.

(ii) Φ_{S_{A}^{t}}([(1, 1, …, 1)]) = [(1, 1, …, 1)].

Proof. (i) Since ΦD̂ is induced by the matrix D̂^{t}, it suffices to prove the equality D̂S_{B} = S_{A}C. Let (i, j) be i = 1, …, N_{A} and j = 1, …, M so that a_{i} ∈ E_{A} and v_{j}^{B} ∈ V_{B}. Let k be such that t(a_{i}) = v_{k}^{A}. Hence we have

[S_{A}C](i, j) = ∑_{n=1}^{N} S_{A}(i, n)C(n, j) = C(k, j)

which is the number of edges of E_{C} leaving v_{k}^{A} and terminating at v_{j}^{B}. On the other hand,

[D̂S_{B}](i, j) = ∑_{l=1}^{M_{B}} D̂(i, l)S_{B}(l, j).

It is easy to see that the above number is also C(k, j).

(ii) Since A = R_{A}S_{A}, for each k = 1, …, N_{A} with a_{k} ∈ E_{A} there exists a unique i = 1, …, N such that s(a_{k}) = v_{i}^{A}. Hence ∑_{i=1}^{N} R_{A}(i, k) = 1 so that we have for each j = 1, …, N

∑_{i=1}^{N} A^{t}(j, i) = ∑_{i=1}^{N} ∑_{k=1}^{N_{A}} R_{A}(i, k)S_{A}(k, j)

= ∑_{k=1}^{N_{A}} (∑_{i=1}^{N} R_{A}(i, k)) S_{A}(k, j)

= ∑_{k=1}^{N_{A}} S_{A}^{t}(j, k).

We then see

Φ_{S_{A}^{t}}([(1, 1, …, 1)]) = [(∑_{k=1}^{N_{A}} S_{A}(k, 1), ∑_{k=1}^{N_{A}} S_{A}(k, 2), …, ∑_{k=1}^{N_{A}} S_{A}(k, N))]

= [(∑_{i=1}^{N} A^{t}(1, i), ∑_{i=1}^{N} A^{t}(2, i), …, ∑_{i=1}^{N} A^{t}(N, i))]

= [(1, 1, …, 1)] in ℤ^{N}/(id − A^{t})ℤ^{N}.

Put

ϵ_{A} = Φ_{S_{A}^{t}} ∘ ϵ_{A^{G}} : K_{0}(O_{A}) → ℤ^{N}/(id − A^{t})ℤ^{N},  (4.2)

which is an isomorphism of groups such that ϵ_{A}([1_{A}]) = [(1, 1, …, 1)]. We thus reach the following theorem:

Theorem 4.6. Suppose that two nonnegative irreducible matrices A, B satisfy A = CD, B = DC for some nonnegative rectangular matrices C, D. Let Φ : O_{A} ⊗ K → O_{B} ⊗ K be the isomorphism constructed in the proof of Proposition 3.10 such that Φ = Ad(w) with w = v_{B}v_{A}^{∗} for the isometry v_{A} as well as v_{B} defined before Lemma 3.5. Then the diagram

K_{0}(O_{A})  ⟶^{Φ_{∗}}  K_{0}(O_{B})
ϵ_{A}↓  ↓^{ϵ_{B}}
ℤ^{N}/(id − A^{t})ℤ^{N} ⟶^{Φ_{C^{t}}} ℤ^{M}/(id − B^{t})ℤ^{M}

is commutative, where all maps are isomorphisms of abelian groups.

We write A ≈_{C,D} B if A = CD, B = DC. Recall that A, B are said to be strong

shift equivalent in n-step if there exist a finite sequence of square matrices A_{1}, …, A_{n−1} and two finite sequences of rectangular matrices C_{1}, …, C_{n} and D_{1}, …, D_{n} such that

A = A_{0} ≈_{C_{1},D_{1}} A_{1}, A_{1} ≈_{C_{2},D_{2}} A_{2}, …, A_{n−1} ≈_{C_{n},D_{n}} A_{n} = B.

This situation is written

A ≈_{C_{1},D_{1}} ⋯ ≈_{C_{n},D_{n}} B.  (4.3)

R. F. Williams proved that two-sided topological Markov shifts (X̄_{A}, σ̄_{A}) and (X̄_{B}, σ̄_{B}) are topologically conjugate if and only if A and B are strong shift equivalent in n-step for some n ([25]). Hence we have the following corollary.

Corollary 4.7. Suppose that two matrices A, B are strong shift equivalent in n-step for some two sequences of rectangular matrices C_{1}, …, C_{n} and D_{1}, …, D_{n} as in (4.3). Then there exist an isomorphism Φ : O_{A}⊗K → O_{B} ⊗K of C^{∗}-algebras and a unitary representation t ∈ 𝕋 → v_{t}^{A} ∈ M(D_{A} ⊗ C) such that

Φ(D_{A} ⊗ C) = D_{B} ⊗ C, Φ ∘ Ad(v_{t}^{A}) ∘ (ρ_{t}^{A} ⊗ id) = (ρ_{t}^{B} ⊗ id) ∘ Φ,

and the following diagram is commutative

K_{0}(O_{A})  ⟶^{Φ_{∗}}  K_{0}(O_{B})
ϵ_{A}↓  ↓^{ϵ_{B}}
ℤ^{N}/(id − A^{t})ℤ^{N} ⟶^{Φ_{_{(C_{1}C_{2}···Cn)}t}} ℤ^{M}/(id − B^{t})ℤ^{M}.

We note that the inverse of Φ_{(C_{1}C_{2}···C_{n})}^{t} : ℤ^{N}/(id − A^{t})ℤ^{N} → ℤ^{M}/(id − B^{t})ℤ^{M} is given by Φ_{(D_{n}···D_{2}D_{1})}^{t} : ℤ^{M}/(id − B^{t})ℤ^{M} → ℤ^{N}/(id − A^{t})ℤ^{N}.

5 Converse and Invariant

In this section, we will study the converse of Corollary 3.11 by using Corollary 4.7. We fix a projection p_{1} of rank one in K.

Proposition 5.1. The following assertions are equivalent.

(i) There exist an isomorphism Φ : O_{A} ⊗ K → O_{B} ⊗ K of C^{∗}-algebras and a unitary one-cocycle u_{t} ∈ M(O_{B} ⊗ K), t ∈ 𝕋 relative to ρ_{t}^{B} ⊗ id such that

Φ(D_{A} ⊗ C) = D_{B} ⊗ C, Φ ∘ (ρ_{t}^{A} ⊗ id) = Ad(u_{t}) ∘ (ρ_{t}^{B} ⊗ id) ∘ Φ, (5.1)
Φ_{∗}([1_{A} ⊗ p_{1}]) = [1_{B} ⊗ p_{1}] in K_{0}(O_{B}).  (5.2)

(ii) There exist an isomorphism φ : O_{A} → O_{B} and a unitary one-cocycle v_{t} ∈ U(O_{B}), t ∈ 𝕋 relative to ρ_{t}^{B} on O_{B} such that

φ(D_{A}) = D_{B} and φ ∘ ρ_{t}^{A} = Ad(v_{t}) ∘ ρ_{t}^{B} ∘ φ,  t ∈ 𝕋. (5.3)

Proof. The implication (ii) ⟹ (i) is obvious by putting Φ = φ ⊗ id and u_{t} = v_{t} ⊗ 1. We will show the implication (i) ⟹ (ii) in the following way. By [13, Proposition 3.13], the condition Φ_{∗}([1_{A} ⊗ p_{1}]) = [1_{B} ⊗ p_{1}] in K_{0}(O_{B}) ensures that there exists a partial isometry V ∈ O_{B} ⊗K satisfying the following conditions:

V (D_{B} ⊗ C)V^{∗} ⊂ D_{B} ⊗ C, V^{∗}(D_{B} ⊗ C)V ⊂ D_{B} ⊗ C,
V V^{∗} = 1_{B} ⊗ p_{1}, V^{∗}V = Φ(1_{A} ⊗ p_{1}).

Put Ψ = Ad(V) ∘ Φ : O_{A} ⊗ K → O_{B} ⊗ K. It is straightforward to see that

Ψ(O_{A} ⊗ ℂp_{1}) = O_{B} ⊗ ℂp_{1}, Ψ(D_{A} ⊗ ℂp_{1}) = D_{B} ⊗ ℂp_{1},
Ψ(1_{A} ⊗ p_{1}) = 1_{B} ⊗ p_{1}.

It is clear that Ψ_{∗} = Φ_{∗} : K_{0}(O_{A}) → K_{0}(O_{B}). We identify O_{B} ⊗ ℂp_{1} with O_{B}. Put the partial isometry v_{t} = V u_{t}(ρ_{t}^{B} ⊗ id)(V^{∗}) ∈ O_{B} ⊗ K. Since v_{t} = (1_{B} ⊗ p_{1})v_{t}(1_{B} ⊗ p_{1}), by this identification, v_{t} belongs to O_{B}. Define φ : O_{A} → O_{B} by setting φ(a) = Ψ(a ⊗ p_{1}) for a ∈ O_{A}. It then follows that

φ(ρ_{t}^{A}(a)) ⊗ p_{1} =V Φ(ρ_{t}^{A}(a) ⊗ p_{1})V^{∗}
=V (Ad(u_{t}) ∘ (ρ_{t}^{B} ⊗ id) ∘ Φ)(a ⊗ p_{1})V^{∗}
=V u_{t}(ρ_{t}^{B} ⊗ id)(V^{∗})(ρ_{t}^{B} ⊗ id)Φ(V (a ⊗ p_{1})V^{∗})(ρ_{t}^{B} ⊗ id)(V)u_{t}^{∗}V^{∗}
=v_{t}((ρ_{t}^{B} ⊗ id) ∘ Ψ)(a ⊗ p_{1})v_{t}^{∗}
=(Ad(v_{t}) ∘ (ρ_{t}^{B} ∘ φ)(a)) ⊗ p_{1}

so that we have φ(ρ_{t}^{A}(a)) = (Ad(v_{t}) ∘ ρ_{t}^{B} ∘ φ)(a). Since we have

(ρ_{t}^{B} ⊗ id)(Φ(1_{A} ⊗ p_{1})) =(Ad(u_{t}^{∗}) ∘ Φ ∘ (ρ_{t}^{A} ⊗ id))(1_{A} ⊗ p_{1})
=u_{t}^{∗}Φ(1_{A} ⊗ p_{1})u_{t} = u_{t}^{∗}V^{∗}V u_{t},

we have

v_{t}ρ_{t}^{B}(v_{s}) =V u_{t}(ρ_{t}^{B} ⊗ id)(V^{∗})(ρ_{t}^{B} ⊗ id)(V u_{s}(ρ_{s}^{B} ⊗ id)(V^{∗}))
=V u_{t}(ρ_{t}^{B} ⊗ id)(V^{∗}V)(ρ_{t}^{B} ⊗ id)(u_{s})(ρ_{t}^{B} ∘ ρ_{s}^{B} ⊗ id)(V^{∗})
=V u_{t}(ρ_{t}^{B} ⊗ id)(Φ(1_{A} ⊗ p_{1}))(ρ_{t}^{B} ⊗ id)(u_{s})(ρ_{t+s}^{B} ⊗ id)(V^{∗})
=V u_{t}u_{t}^{∗}V^{∗}V u_{t}(ρ_{t}^{B} ⊗ id)(u_{s})(ρ_{t+s}^{B} ⊗ id)(V^{∗})
=V u_{t}(ρ_{t}^{B} ⊗ id)(u_{s})(ρ_{t+s}^{B} ⊗ id)(V^{∗})
=V u_{t+s}(ρ_{t+s}^{B} ⊗ id)(V^{∗})
=v_{t+s}.

Hence v_{t}, t ∈ 𝕋 is a unitary one-cocycle relative to ρ^{B}.

Remark 5.2. Let v_{t} in O_{B} be a unitary one-cocycle relative to ρ_{t}^{B} satisfying (5.3). For a ∈ D_{A}, we see that φ(ρ_{t}^{A}(a)) = Ad(v_{t})(ρ_{t}^{B}(φ(a))). As ρ_{t}^{A}(a) = a and φ(a) belongs to D_{B} so that we have φ(a) = Ad(v_{t})(φ(a)). Hence v_{t} commutes with any element of D_{B}. This implies that v_{t} belongs to D_{B} and hence it is fixed by the action ρ^{B}. Therefore a unitary one-cocycle v_{t} in O_{B} relative to ρ_{t}^{B} satisfying (5.3) automatically belongs to D_{B} and yields a unitary representation t ∈ 𝕋 → v_{t} ∈ D_{B}. Since the unitary u_{t} in (5.1) is given by u_{t} = v_{t}⊗1 from the unitary v_{t} satisfying (5.3), the unitary one-cocycle u_{t} in the statement (i) of the above proposition can be taken as a unitary representation t ∈ 𝕋 → u_{t} ∈ M(D_{B} ⊗ C) which is fixed by the action ρ_{t}^{B} ⊗ id.

Corollary 5.3. If there exist an isomorphism Φ : O_{A} ⊗ K → O_{B} ⊗ K of C^{∗}-algebras and a unitary one-cocycle u_{t} in M(O_{B} ⊗ K) relative to ρ_{t}^{B} ⊗ id such that

Φ(D_{A} ⊗ C) = D_{B} ⊗ C, Φ ∘ (ρ_{t}^{A} ⊗ id) = Ad(u_{t}) ∘ (ρ_{t}^{B} ⊗ id) ∘ Φ,
Φ_{∗}([1_{A} ⊗ p_{1}]) = [1_{B} ⊗ p_{1}] in K_{0}(O_{B}),

then two-sided topological Markov shifts (X̄_{A}, σ̄_{A}) and (X̄_{B}, σ̄_{B}) are topologically conjugate.

Proof. Suppose that there exist an isomorphism Φ : O_{A} ⊗ K → O_{B} ⊗ K and a unitary one-cocycle u_{t} in M(O_{B} ⊗ K) relative to ρ_{t}^{B} ⊗ id satisfying the above equalities. Proposition 5.1 tells us that there exist an isomorphism φ : O_{A} → O_{B} and a unitary one-cocycle v_{t} ∈ U(O_{B}), t ∈ 𝕋 relative to ρ_{t}^{B} on O_{B} satisfying (5.3). Hence the one-sided topological Markov shifts (X_{A}, σ_{A}) and (X_{B}, σ_{B}) are strongly continuous orbit equivalent by [14, Theorem 6.7]. It also implies topological conjugacy of their two-sided topological Markov shifts (X̄_{A}, σ̄_{A}) and (X̄_{B}, σ̄_{B}) by [14, Theorem 5.5].

Definition 5.4. An isomorphism ξ : O_{B} ⊗ K → O_{A} ⊗ K of C^{∗}-algebras is said to be induced from strong shift equivalence if there exist a strong shift equivalence A ≈_{C_{1},D_{1}} ⋯ ≈_{C_{n},D_{n}} B and a unitary one-cocycle u_{t} in M(O_{A} ⊗ K)

relative to ρ_{t}^{A} ⊗ id such that

ξ(D_{B} ⊗ C) = D_{A} ⊗ C, ξ ∘ (ρ_{t}^{B} ⊗ id) = Ad(u_{t}) ∘ (ρ_{t}^{A} ⊗ id) ∘ ξ,
ξ_{∗} = ϵ_{A}^{−1} ∘ Φ_{(D_{n}···D_{2}D_{1})^{t}} ∘ ϵ_{B} : K_{0}(O_{B}) → K_{0}(O_{A}).

In this case, we say that ξ : O_{B} ⊗ K → O_{A} ⊗ K is induced from strong shift

equivalence A ≈_{C_{1},D_{1}} ⋯ ≈_{C_{n},D_{n}} B.

We will define the strong shift equivalence invariant subset K_{0}^{SSE}(O_{A}) of K_{0}(O_{A}) as follows.

Definition 5.5.

K_{0}^{SSE}(O_{A}) = {[p] ∈ K_{0}(O_{A}) | there exist a square matrix B and

an isomorphism ξ : O_{B} ⊗ K → O_{A} ⊗ K induced from strong shift equivalence such that ξ_{∗}([1_{B}]) = [p] in K_{0}(O_{A})}.

We note that the class [1_{A}] in K_{0}(O_{A}) of the unit 1_{A} of O_{A} always belongs to the set K_{0}^{SSE}(O_{A}), because we may take B = A and ξ = id.

Proposition 5.6. Suppose that there exists a topological conjugacy between (X̄_{A}, σ̄_{A}) and (X̄_{B}, σ̄_{B}). Then there exists an isomorphism η : K_{0}(O_{A}) → K_{0}(O_{B}) satisfying η(K_{0}^{SSE}(O_{A})) = K_{0}^{SSE}(O_{B}). Hence the pair (K_{0}(O_{A}), K_{0}^{SSE}(O_{A})) is an invariant under topological conjugacy of two-sided topological Markov shifts.

Proof. Suppose that (X̄_{A}, σ̄_{A}) and (X̄_{B}, σ̄_{B}) are topologically conjugate so that A ≈_{C_{1},D_{1}} ⋯ ≈_{C_{n},D_{n}} B for some nonnegative rectangular matrices

C_{1}, D_{1}, …, C_{n}, D_{n}. By Corollary 4.7, the strong shift equivalence induces an isomorphism ξ_{BA} : O_{A} ⊗ K → O_{B} ⊗ K and a unitary one-cocycle u_{t} in M(O_{B} ⊗ K) relative to ρ_{t}^{B} ⊗ id such that

ξ_{BA}(D_{A} ⊗ C) = D_{B} ⊗ C, ξ_{BA} ∘ (ρ_{t}^{A} ⊗ id) = Ad(u_{t}) ∘ (ρ_{t}^{B} ⊗ id) ∘ ξ_{BA}, ξ_{BA∗} = Φ_{(C_{1}···C_{n})^{t}} : K_{0}(O_{A}) → K_{0}(O_{B}).

Put η = ξ_{BA∗} : K_{0}(O_{A}) → K_{0}(O_{B}). Take an element [p] ∈ K_{0}^{SSE}(O_{A}). There exist a square nonnegative matrix A′ and an isomorphism ξ_{AA′} : O_{A′} ⊗ K → O_{A}⊗K of C^{∗}-algebras induced from strong shift equivalence A′ ≈_{C′_{1},D′_{1}} ⋯ ≈_{C′_{n′} ,D′_{n′}}

A such that ξ_{AA′∗}([1_{A′}]) = [p] in K_{0}(O_{A}). Then the isomorphism ξ_{BA} ∘ ξ_{AA′} : O_{A′} ⊗ K → O_{B} ⊗ K is induced from strong shift equivalence

A′ ≈_{C′_{1},D′_{1}} ⋯ ≈_{C′_{n′} ,D′_{n′}} A ≈_{C_{1},D_{1}} ⋯ ≈_{C_{n},D_{n}} B

such that η([p]) = (ξ_{BA} ∘ ξ_{AA′})_{∗}([1_{A′}]) in K_{0}(O_{B}) so that η([p]) ∈ K_{0}^{SSE}(O_{B}).

Suppose that two matrices A, B are strong shift equivalent in n-step such as (4.3). The matrix B in (4.3) is given by B = D_{n}C_{n} so that (4.3) is written as

A ≈_{C_{1},D_{1}} ⋯ ≈_{C_{n},D_{n}} D_{n}C_{n}.  (5.4)

We set the following sequence SSE_{n}(A), n = 1, 2, … of subsets of the group ℤ^{N}

SSE_{n}(A)
={v ∈ ℤ^{N} | v = D_{1}^{t} ⋯ D_{n−1}^{t}D_{n}^{t}[1, 1, …, 1]^{t}, A ≈_{C_{1},D_{1}} ⋯ ≈_{C_{n},D_{n}} D_{n}C_{n}},

where [1, 1, …, 1]^{t} denotes (the row size of D_{n}) × 1 matrix whose entries are all 1’s. We define the sequence K_{alg,n}^{SSE}(A), n = 1, 2, … of subsets of the group ℤ^{N}/(id − A^{t})ℤ^{N} by

K_{alg,n}^{SSE}(A) = {[v] ∈ ℤ^{N}/(id − A^{t})ℤ^{N} | v ∈ SSE_{n}(A)}, n = 1, 2, ….

We then define the subset K_{alg}^{SSE}(A) of ℤ^{N}/(id − A^{t})ℤ^{N} by

K_{alg}^{SSE}(A) = ∪_{n=1}^{∞} K_{alg,n}^{SSE}(A).

By Corollary 4.7, we have the following proposition

Proposition 5.7. Let ϵ_{A} : K_{0}(O_{A}) → ℤ^{N}/(id − A^{t})ℤ^{N} be the isomorphism defined in (4.2). Then we have

ϵ_{A}(K_{0}^{SSE}(O_{A})) = K_{alg}^{SSE}(A).

Proof. For [p] ∈ K_{0}^{SSE}(O_{A}), there exist a nonnegative square matrix B with a strong shift equivalence A ≈_{C_{1},D_{1}} ⋯ ≈_{C_{n},D_{n}} B, an isomorphism ξ : O_{B} ⊗ K →

O_{A} ⊗ K of C^{∗}-algebras and a unitary one-cocycle u_{t}, t ∈ 𝕋 relative to ρ^{A} ⊗ id such that

ξ(D_{B} ⊗ C) = D_{A} ⊗ C,  ξ ∘ (ρ_{t}^{B} ⊗ id) = Ad(u_{t}) ∘ (ρ_{t}^{A} ⊗ id) ∘ ξ,  (5.5)

ξ_{∗} = ϵ_{A}^{−1} ∘ Φ_{(D_{n}···D_{2}D_{1})}^{t} ∘ ϵ_{B} : K_{0}(O_{B}) → K_{0}(O_{A}) and ξ_{∗}([1_{B}]) = [p]. (5.6)

Since ϵ_{B}([1_{B}]) = [[1, 1, …, 1]^{t}] in ℤ^{M}/(id − B^{t})ℤ^{M}, we have

ϵ_{A}([p]) =ϵ_{A} ∘ ξ_{∗}([1_{B}])  (5.7)
=Φ_{(D_{n}···D_{2}D_{1})}^{t} ∘ ϵ_{B}([1_{B}]) = Φ_{(D_{n}···D_{2}D_{1})}^{t}([1, 1, …, 1]^{t})  (5.8)

so that ϵ_{A}([p]) ∈ K_{alg}^{SSE}(A) and hence ϵ_{A}(K_{0}^{SSE}(O_{A})) ⊂ K_{alg}^{SSE}(A). Conversely, take an arbitrary element [v] ∈ K_{alg}^{SSE}(A). We may find a strong shift

equivalence A ≈_{C_{1},D_{1}} ⋯ ≈_{C_{n},D_{n}} D_{n}C_{n} such that v = (D_{n} ⋯ D_{2}D_{1})^{t}[1, 1, …, 1]^{t}.

Put B = D_{n}C_{n}. By Corollary 4.7, there exist an isomorphism ξ : O_{B} ⊗ K → O_{A} ⊗ K of C^{∗}-algebras and a unitary one-cocycle u_{t}, t ∈ 𝕋 relative to ρ^{A} ⊗ id satisfying (5.5) and ξ_{∗} = ϵ_{A}^{−1} ∘ Φ_{(D_{n}···D_{2}D_{1})}^{t} ∘ ϵ_{B} : K_{0}(O_{B}) → K_{0}(O_{A}). Put [p] = ξ_{∗}([1_{B}]) which belongs to K_{0}^{SSE}(O_{A}). By the same equalities as (5.7), (5.8), we get ϵ_{A}([p]) = Φ_{(D_{n}···D_{2}D_{1})}^{t}([1, 1, …, 1]^{t}) which is the class of [v]. This shows that ϵ_{A}(K_{0}^{SSE}(O_{A})) ⊃ K_{alg}^{SSE}(A).

Theorem 5.8. Let A, B be nonnegative irreducible and non-permutation matrices. The following two assertions are equivalent.

(i) Two-sided topological Markov shifts (X̄_{A}, σ̄_{A}) and (X̄_{B}, σ̄_{B}) are topologically conjugate.

(ii) There exist an isomorphism Φ : O_{A} ⊗ K → O_{B} ⊗ K of C^{∗}-algebras and a unitary one-cocycle u_{t} in M(O_{B} ⊗ K) relative to ρ_{t}^{B} ⊗ id such that

Φ(D_{A} ⊗ C) = D_{B} ⊗ C, Φ ∘ (ρ_{t}^{A} ⊗ id) = Ad(u_{t}) ∘ (ρ_{t}^{B} ⊗ id) ∘ Φ,

Φ_{∗}(K_{0}^{SSE}(O_{A})) = K_{0}^{SSE}(O_{B}) in K_{0}(O_{B}).

Proof. (i) ⟹ (ii): The assertion follows from Corollary 3.11 and Proposition 5.6. (ii) ⟹ (i): Suppose that there exist an isomorphism Φ : O_{A} ⊗ K → O_{B} ⊗ K of C^{∗}-algebras and a unitary one-cocycle u_{t} in M(O_{B} ⊗ K) relative to ρ_{t}^{B} ⊗ id satisfying the conditions of (ii). Take a projection p_{1} of rank one in K. Put the projection p = Φ(1_{A} ⊗ p_{1}) ∈ O_{B} ⊗ K. As [1_{A}] ∈ K_{0}^{SSE}(O_{A}) and Φ_{∗}(K_{0}^{SSE}(O_{A})) = K_{0}^{SSE}(O_{B}), the class [p] = Φ_{∗}([1_{A}]) of p in K_{0}(O_{B}) belongs to K_{0}^{SSE}(O_{B}). One may take a nonnegative square matrix B′ and an isomorphism γ : O_{B} ⊗K → O_{B′} ⊗K with a unitary one-cocycle u′_{t} in M(O_{B′} ⊗K) relative to ρ_{t}^{B′} ⊗ id induced from strong shift equivalence B ≈_{C_{1},D_{1}} ⋯ ≈_{C_{n},D_{n}} B′ satisfying

γ(D_{B} ⊗ C) = D_{B′} ⊗ ℂ, γ ∘ (ρ_{t}^{B} ⊗ id) = Ad(u′_{t}) ∘ (ρ_{t}^{B′} ⊗ id) ∘ γ,
γ_{∗}([p]) = [1_{B′}] in K_{0}(O_{B′}).

Then the isomorphism γ ∘ Φ : O_{A} ⊗ K → O_{B′} ⊗ K satisfies the conditions

(γ ∘ Φ)(D_{A} ⊗ C) = D_{B′} ⊗ ℂ,
(γ ∘ Φ) ∘ (ρ_{t}^{A} ⊗ id) = Ad(γ(u_{t})u′_{t}) ∘ (ρ_{t}^{B′} ⊗ id) ∘ (γ ∘ Φ),
(γ ∘ Φ)_{∗}([1_{A}]) = [1_{B′}] in K_{0}(O_{B′}).

By Corollary 5.3, the two-sided topological Markov shifts (X̄_{A}, σ̄_{A}) and (X̄_{B′}, σ̄_{B′}) are topologically conjugate. Since (X̄_{B}, σ̄_{B}) and (X̄_{B′}, σ̄_{B′}) are topologically conjugate, so are (X̄_{A}, σ̄_{A}) and (X̄_{B}, σ̄_{B}).

Remark 5.9. The unitary one-cocycle u_{t} in M(O_{B} ⊗ K) in (ii) of the above theorem can be taken as a unitary representation t ∈ 𝕋 → u_{t} ∈ M(O_{B} ⊗ K) by Corollary 3.11.

Definition 5.10. A nonnegative square matrix A = [A(i, j)]_{i,j=1}^{N} is said to have full strong shift equivalent units in K_{0}-group if K_{alg}^{SSE}(A) = ℤ^{N}/(id − A^{t})ℤ^{N}. We simply call it that A has full units.

By Proposition 5.7, A has full units if and only if K_{0}^{SSE}(O_{A}) = K_{0}(O_{A}). Since the subset K_{0}^{SSE}(O_{A}) ⊂ K_{0}(O_{A}) is invariant under topological conjugacy of two-sided topological Markov shifts by Proposition 5.6, we have

Proposition 5.11. Suppose that two-sided topological Markov shifts (X̄_{A}, σ̄_{A}) and (X̄_{B}, σ̄_{B}) are topologically conjugate. Then A has full units if and only if B has full units.

As a consequence of Theorem 5.8, we have the following corollary.

Corollary 5.12. Suppose that both A and B have full units. Then the following two assertions are equivalent.

(i) Two-sided topological Markov shifts (X̄_{A}, σ̄_{A}) and (X̄_{B}, σ̄_{B}) are topologically conjugate.

(ii) There exist an isomorphism Φ : O_{A} ⊗ K → O_{B} ⊗ K of C^{∗}-algebras and a unitary one-cocycle u_{t} in M(O_{B} ⊗ K) relative to ρ_{t}^{B} ⊗ id such that

Φ(D_{A} ⊗ C) = D_{B} ⊗ C, Φ ∘ (ρ_{t}^{A} ⊗ id) = Ad(u_{t}) ∘ (ρ_{t}^{B} ⊗ id) ∘ Φ.

Example 5.13.

1. If K_{0}(O_{A}) = 0, then A has full units.

2. Let A be the 1 × 1 matrix [N] whose entry is N with 1 < N ∈ ℕ. Then the matrix A has full units. For any 0 ≤ k ≤ N − 1, let C be the 1 × (k + 1) matrix [1, …, 1, N −k] and D the (k+1)×1matrix (1, 1, …, 1)^{t}. Then A = CD and D^{t}[1, …, 1]^{t} = k + 1. Hence [k + 1] ∈ ℤ/(1 − N)ℤ so that K_{alg}^{SSE}(A) = ℤ/(1 − N)ℤ = K_{0}(O_{A}).

There is no known example of irreducible, non permutation matrix A such that A does not have full units.

Acknowledgments: The author is very grateful to the referee for his careful reading, valuable advices and useful suggestions in the presentation of the paper. This work was done during staying at the Institut Mittag-Leffler in the research program Classification of operator algebras: complexity, rigidity and dynamics, January–April 2016. The author is grateful to the Institut Mittag-Leffler for its wonderful research environment, hospitality and support. The author also thanks Hiroki Matui for his advices in Lemma 4.5 and suggestions.

Remark 5.14. After submitting the paper, there were several progress in the following papers related to this paper:

1. T. M. Carlsen and J. Rout, Diagonal-preserving gauge invariant isomorphisms of graph C^{∗}-algebras, preprint, arXiv: 1610.00692 [mathOA].

2. K. Matsumoto, State splitting, strong shift equivalence and stable isomorphism of Cuntz–Krieger algebras, preprint, arXiv: 1611.06627 [mathOA]. In the paper 1, the converse implication of [7, Theorem 3.8] was proved, In the paper 2, strong shift equivalence class of the matrix A was described in terms of (O_{A}, D_{A}, ρ^{A}) and (O_{A^{t}}, D_{A^{t}}, ρ^{A^{t}}).

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Kengo Matsumoto

Department of Mathematics Joetsu University of Education Joetsu, 943-8512, Japan kengo@juen.ac.jp 916

Documenta Mathematica 22 (2017)