On a family of C^*-subalgebras of Cuntz-Krieger algebras
aa r X i v : . [ m a t h . OA ] J a n On a family of C ∗ -subalgebras of Cuntz–Krieger algebras Kengo MatsumotoDepartment of MathematicsJoetsu University of EducationJoetsu, 943-8512, JapanJanuary 8, 2021
Abstract
In this paper, we study a family of C ∗ -subalgebras defined by fixed points ofgeneralized gauge actions of a Cuntz–Krieger algebra, by introducing a family of ´etalegroupoids whose associated C ∗ -algebras are these C ∗ -subalgebras. We know thattopological conjugacy classes of one-sided topological Markov shifts are characterizedin terms of the isomorphism classes of these ´etale groupoids. Mathematics Subject Classification : Primary 37A55; Secondary 46L35, 37B10.
Keywords and phrases : Topological Markov shifts, Cuntz–Krieger algebras, AF-algebra,´etale groupoid, gauge action
In this paper, we will study a family of C ∗ -subalgebras defined by fixed points of generalizedgauge actions of a Cuntz–Krieger algebra O A from a groupoid view point. Each of the C ∗ -subalgebras contains the canonical maximal abelian C ∗ -subalgebra of O A . They are gener-alization of the canonical AF subalgebra of a Cuntz–Krieger algebra. Let A = [ A ( i, j )] Ni,j =1 be an irreducible non permutation matrix with entries in { , } . The Cuntz–Kriegeralgebra O A is defined by N partial isometries S , . . . , S N satisfying the operator re-lations: 1 = P Nj =1 S j S ∗ j , S ∗ i S i = P Nj =1 A ( i, j ) S j S ∗ j , i = 1 , . . . , N ([3]). The algebrasare closely related to a class of symbolic dynamical systems called topological Markovshifts. The one-sided topological Markov shift ( X A , σ A ) for the matrix A is defined byits shift space X A consisting of right one-sided sequences ( x n ) n ∈ N ∈ { , , . . . , N } N sat-isfying A ( x n , x n +1 ) = 1 , n ∈ N with its shift transformation σ A : X A −→ X A definedby σ A (( x n ) n ∈ N ) = ( x n +1 ) n ∈ N , where N denotes the set of positive integers. The topol-ogy of X A is endowed with the relative topology of the infinite product topology on { , , . . . , N } N for the discrete set { , , . . . , N } . Hence the shift space X A is a compactHausdorff space homeomorphic to a Cantor discontinuum and the shift transformation σ A : X A −→ X A is a continuous surjection. Let us denote by B k ( X A ) the set of ad-missible words { ( x , . . . , x k ) ∈ { , . . . , N } k | ( x n ) n ∈ N ∈ X A } of X A with its length k .Let U µ be the cylinder set { ( x n ) n ∈ N ∈ X A | x = µ , . . . , µ m = x m } for the word µ = ( µ , . . . , µ m ) ∈ B m ( X A ). Let χ U µ be the characteristic function of U µ on X A . It1s well-known that the corresondence χ U µ −→ S µ · · · S µ m S ∗ µ m · · · S ∗ µ gives rise to an iso-morphism from the commutative C ∗ -algebra C ( X A ) of complex valued continuous func-tions on X A to the C ∗ -subalgebra D A of O A generated by the projections of the form S µ · · · S µ m S ∗ µ m · · · S ∗ µ . The gauge action written ρ A of T to the automorphism group of O A is defined by the one-parameter family of automorphisms ρ At , t ∈ T defined by the cor-respondence S i −→ exp(2 π √− t ) S i , t ∈ R / Z = T , i = 1 , . . . , N . The fixed point algebraof O A under ρ A is an AF algebra defined by the matrix A ([3]). It is called the standardAF-subalgebra of O A denoted by F A . Let us denote by C ( X A , Z ) the set of integer valuedcontinuous functions on X A . A continuous function f ∈ C ( X A , Z ) gives rise to an elementof C ( X A ) and hence of D A . Since exp(2 π √− tf ) is a unitary in D A for each t ∈ T , thecorrespondence S i −→ exp(2 π √− tf ) S i , i = 1 , , . . . , N, t ∈ T yields an automorphismof O A written ρ A,ft . The family of automorphisms ρ A,ft , t ∈ T defines an action of T onthe C ∗ -algebra O A . It is called the gauge action with potential f , or a generalized gaugeaction. For f ≡
1, the action ρ A, coincides with the gauge action ρ A that is in particularcalled the standard gauge action on O A .In the first half of the paper, we will study a family F A,f , f ∈ C ( X A , Z ) of C ∗ -subalgebras of O A introduced in [12]. They are defined in the following way. Definition 1.1 ([12, Definition 2.5]) . For f ∈ C ( X A , Z ), define a C ∗ -subalgebra F A,f of O A by the fixed point subalgebra of O A under the action ρ A,f F A,f := { X ∈ O A | ρ A,ft ( X ) = X for all t ∈ T } . (1.1)We call the C ∗ -algebra F A,f the cocycle algebra for f .For constant functions f ≡ f ≡
1, we know that F A, = O A and F A, = F A ,respectively. Hence the family F A,f , f ∈ C ( X A , Z ) of C ∗ -subalgebras of O A generalizeboth O A and F A . In [12], the following result was proved. Theorem 1.2 ([12, Therem 1.4], cf. [10]) . Let A and B be irreducible non permutationmatrices with entries in { , } . Then the one-sided topological Markov shifts ( X A , σ A ) and ( X B , σ B ) are topologically conjugate if and only if there exists an isomorphism Φ : O A −→O B of C ∗ -algebras such that Φ( D A ) = D B and Φ( F A,f ) = F B, Φ( f ) for all f ∈ C ( X A , Z ) , (1.2) where Φ( f ) ∈ C ( X B , Z ) for f ∈ C ( X A , Z ) . Since F A, = O A , F A, = F A and ∩ f ∈ C ( X A , Z ) F A,f = D A (Proposition 2.4), the isomor-phisms classes of the family of cocycle algebras F A,f for f ∈ C ( X A , Z ) completely deter-mine the topological conjugacy class of the one-sided topological Markov shift ( X A , σ A ).Hence it seems to deserve to study the family of cocycle algebras F A,f for f ∈ C ( X A , Z ).In this paper, we will investigate the family of the C ∗ -subalgebras.Let us denote by Z + the set of nonnegative integers. It is well-known that the algebra O A is realized as the C ∗ -algebra C ∗ ( G A ) of an amenable ´etale groupoid G A defied by G A := { ( x, n, z ) ∈ X A × Z × X A | there exist k, l ∈ Z + such that n = k − l, σ kA ( x ) = σ lA ( z ) } G (0) A = { ( x, , x ) ∈ G A | x ∈ X A } . Define the maps s ( x, n, z ) = ( z, , z ) ∈ G A , r ( x, n, z ) = ( x, , x ) ∈ G A . The product and inverse operation are defined by( x, n, z ) · ( z, m, w ) = ( x, n + m, w ) , ( x, n, z ) − = ( z, − n, x ) . The unit space G (0) A is naturally homeomorphic to the shift space X A . The subgroupoid G AF A = { ( x, , z ) ∈ G A | x, z, ∈ X A } is an AF-groupoid whose C ∗ -algebra is isomorphic to the standard AF-algebra F A (cf. [3],[19]).For f ∈ C ( X A , Z ) and n ∈ N , we define f n ∈ C ( X A , Z ) by setting f n ( x ) = n − X i =0 f ( σ iA ( x )) , x ∈ X A . (1.3)For n = 0, we put f ≡
0. It is straightforward to see that the identity f n + k ( x ) = f n ( x ) + f k ( σ nA ( x )) , x ∈ X A , n, k ∈ Z + holds. Definition 1.3.
For f ∈ C ( X A , Z ), define an ´etale subgroupoid G A,f of G A by G A,f := { ( x, n, z ) ∈ X A × Z × X A | there exist k, l ∈ Z + such that n = k − l, σ kA ( x ) = σ lA ( z ) , f k ( x ) = f l ( z ) } . It is called the cocycle groupoid for f .We will see the following result. Theorem 1.4 (Proposition 2.12, Theorem 2.14) . The groupoid G A,f is an essentiallyprincipal amenable clopen ´etale subgroupoid of G A such that there exists an isomorphism Φ : C ∗ ( G A ) −→ O A of C ∗ -algebra such that Φ ( C ∗ ( G A,f )) = F A,f and Φ ( C ( G (0) A )) = D A . Simplicity condition of the C ∗ -algebra F A,f is obtained in Proposition 2.17. We willthen see that there exists an isomorphism Φ : O A −→ O B of C ∗ -algebras such thatΦ( D A ) = D B and Φ( F A,f ) = F B,g if and only if there exists an isomorphism ϕ : G A −→ G B of ´etale groupoids such that ϕ ( G A,f ) = G B,g (Proposition 2.18). Therefore we will knowthe following characterization of topological conjugacy of one-sided topological Markovshifts in terms of these ´etale groupoids in the following way.
Theorem 1.5 (Corollary 2.20) . One-sided topological Markov shifts ( X A , σ A ) and ( X B , σ B ) are topologically conjugate if and only if there exists an isomorphism ϕ : G A −→ G B of´etale groupoids such that ϕ ( G A,g ◦ h ) = G B,g for all g ∈ C ( X B , Z ) , where h : X A −→ X B isa homeomorphism defined by the restriction of ϕ to its unit space G (0) A under the identifi-cation between G (0) A and X A , and G (0) B and X B , respectively.
3n the second half of the paper, we will study the following three subcalsses of cocyclealgebras.
Definition 1.6. (i) For a subset H ⊂ { , , . . . , N } , let χ H ∈ C ( X A , Z ) be the function defined by χ H (( x n ) n ∈ N ) = ( x ∈ H, χ H gives rise to an element of C ( X A , Z ). We set F A,H := F A,χ H the cocycle algebra for χ H . It is called the support algebra for H .(ii) For a function b ∈ C ( X A , Z ), put 1 b = 1 − b + b ◦ σ A ∈ C ( X A , Z ) . We write F bA := F A, b the cocycle algebra for 1 b . It is called the coboundary algebra for b .(iii) For a positive integer valued function f ∈ C ( X A , N ), the cocycle algebra F A,f iscalled the suspension algebra for f .We will finally study isomorphism classes, stable isomorohism classes of these cocyclealgebras belonging to the above three classes from different view points. A subset H ⊂{ , , . . . , N } is said to be saturated if any periodic word of B ∗ ( X A ) contains a symbolbelonging to H . We thus finally obtain the following results. Theorem 1.7 (Theorem 3.14, Theorem 3.21 and Theorem 3.26) . Let A be an irreduciblenon permutation matrix with entries in { , } . (i) For a saturated subset H ⊂ { , , . . . , N } , the support algebra F A,H is a unital AF-algebra defined by a certain inclusion matrix A H defined in (3.5) . Furthermore, if H ⊂ { , , . . . , N } is primitive in the sense of Definition 3.13, the AF-algebra F A,H is simple. (ii)
Assume that A is primitive. For b ∈ C ( X A , Z ) , the coboundary algebra F bA is aunital simple AF-algebra stably isomorphic to the standard AF-algebra F A of O A . (iii) Assume that A is primitive. For f ∈ C ( X A , N ) , the suspension algebra F A,f is aunital simple AF-algebra stably isomorphic to the standard AF-algebra defined by thesuspended matrix of the K -higher block matrix A [ K ] of A by the ceiling function f for some K . The proofs are given by three different ways for each.Throughout the paper, S , . . . , S N denote the canonical generating partial isometriesof O A satisfying P Nj =1 S j S ∗ j = 1 , S ∗ i S i = P Nj =1 A ( i, j ) S j S ∗ j , i = 1 , . . . , N . For a word µ = ( µ , . . . , µ m ) ∈ B m ( X A ), the partial isometry S µ is defined by S µ = S µ · · · S µ m in O A .The contents of the present paper is the following:1. Introduction and Prelimary2. Cocycle algebras and cocycle groupoids2.1. Basic properties of cocycle algebras4.2. Basic properties of cocycle groupoids2.3. The C ∗ -algebra C ∗ ( G A,f )2.4. Continuous orbit equivalence3. Three classes of cocycle algebras3.1. Support algebras3.2. Coboundary algebras3.3. Suspension algebras
In this subsection, we present several basic lemmas to study cocycle algebras. Take and fixan irreducible non permutation matrix A and a continuous function f ∈ C ( X A , Z ). It iseasy to see that D A ⊂ F A,f for any f ∈ C ( X A , Z ). For a word µ = ( µ , . . . , µ m ) ∈ B m ( X A ),let us denote by | µ | the length m of µ . We first note the following lemma. Lemma 2.1 ([7, Lemma 3.1]) . For µ ∈ B ∗ ( X A ) , the identity ρ A,ft ( S µ ) = exp(2 π √− f | µ | t ) S µ , t ∈ T (2.1) holds, where f | µ | ∈ C ( X A , Z ) is defined by (1.3) . The following is basic to analyze the structure of the algebra F A,f . Lemma 2.2.
For µ, ν ∈ B ∗ ( X A ) , the partial isometry S µ S ∗ ν belongs to F A,f if and only if f | µ | S µ S ∗ ν = S µ S ∗ ν f | ν | .Proof. By (2.1), we know that ρ A,ft ( S µ S ∗ ν ) =exp(2 π √− f | µ | t ) S µ S ∗ ν exp( − π √− f | ν | t )=exp(2 π √− f | µ | t ) S µ S ∗ ν exp( − π √− f | ν | t ) S ν S ∗ µ S µ S ∗ ν =exp(2 π √− f | µ | S µ S ∗ ν S ν S ∗ µ − S µ S ∗ ν f | ν | S ν S ∗ µ ) t ) S µ S ∗ ν . Hence S µ S ∗ ν ∈ F A,f if and only if f | µ | S µ S ∗ ν S ν S ∗ µ − S µ S ∗ ν f | ν | S ν S ∗ µ = 0 . (2.2)The equality (2.2) is equivalent to the equality f | µ | S µ S ∗ ν = S µ S ∗ ν f | ν | .For b ∈ C ( X A , Z ), we define 1 b ∈ C ( X A , Z ) by 1 b ( x ) = 1 − b ( x ) + b ( σ A ( x )) , x ∈ X A . Lemma 2.3.
For µ, ν ∈ B ∗ ( X A ) , the partial isometry S µ S ∗ ν belongs to F A, b if and onlyif ( | µ | − b ) S µ S ∗ ν = S µ S ∗ ν ( | ν | − b ) .Proof. By Lemma 2.2, we know that S µ S ∗ ν ∈ F A,f if and only if f | µ | S µ S ∗ ν = S µ S ∗ ν f | ν | . Put m = | µ | , n = | ν | . Now f = 1 − b + b ◦ σ A so that f m = m − b + b ◦ σ mA . Hence the equality f | µ | S µ S ∗ ν = S µ S ∗ ν f | ν | is equivalent to the equality( m − b + b ◦ σ mA ) S µ S ∗ ν = S µ S ∗ ν ( n − b + b ◦ σ nA ) . C ( X A ) and D A , we see that b ◦ σ mA = P ξ ∈ B m ( X A ) S ξ bS ∗ ξ ,so that b ◦ σ mA S µ S ∗ ν = X ξ ∈ B m ( X A ) S ξ bS ∗ ξ S µ S ∗ ν = S µ bS ∗ µ S µ S ∗ ν = S µ bS ∗ ν and similarly S µ S ∗ ν b ◦ σ nA = S µ bS ∗ ν . Hence we have b ◦ σ mA S µ S ∗ ν = S µ S ∗ ν b ◦ σ nA . Therefore f | µ | S µ S ∗ ν = S µ S ∗ ν f | ν | is equivalent to the equality( m − b ) S µ S ∗ ν = S µ S ∗ ν ( n − b ) . (2.3)We note that if in particular m = n , the equality (2.3) goes to b S µ S ∗ ν = S µ S ∗ ν b, that means S µ S ∗ ν ∈ { b } ′ ∩ F A for S µ S ∗ ν ∈ F A, b with | µ | = | ν | . Proposition 2.4. \ f ∈ C ( X A , Z ) F A,f = \ b ∈ C ( X A , Z ) F A, b = D A . Proof. As F A,f ⊃ D A for all f ∈ C ( X A , Z ), we will show the inclusion T b ∈ C ( X A , Z ) F A, b ⊂D A . By the preceding lemma, we see that F A, b ∩ F A = { X ∈ F A | bX = Xb } . For b ≡
0, we have F A, b = F A . For µ ∈ B ∗ ( X A ), let b = χ U µ that is identified with S µ S ∗ µ . Hence we have { b } ′ ∩ F A = { S µ S ∗ µ } ′ ∩ F A , Since the projections S µ S ∗ µ , µ ∈ B ∗ ( X A )generate D A , we know that \ µ ∈ B ∗ ( X A ) { S µ S ∗ µ } ′ ∩ F A = D A ′ ∩ F A = D A , because D A is maximal abelian in F A . We conclude that T b ∈ C ( X A , Z ) F A, b = D A andhence T f ∈ C ( X A , Z ) F A,f = D A . Lemma 2.5.
For µ, ν ∈ B ∗ ( X A ) , the partial isometry S µ S ∗ ν belongs to F A,f if and only if S µi S ∗ νi belongs to F A,f for all i ∈ { , , . . . , N } satisfying µi, νi ∈ B ∗ ( X A ) .Proof. Since the identity S µ S ∗ ν = P Ni =1 S µi S ∗ νi holds, it suffices to prove the only if partof the assertion. For µ, ν ∈ B ∗ ( X A ), put m = | µ | , n = | ν | . Suppose that the partialisometry S µ S ∗ ν belongs to F A,f . By Lemma 2.2, the equality f m S µ S ∗ ν = S µ S ∗ ν f n holds.Since S µ S ∗ ν = P Ni =1 S µi S ∗ νi , we know that S µ S ∗ ν · S νi S ∗ νi = S µi S ∗ νi S νi S ∗ νi = S µi S ∗ νi . As f n commutes with S νi S ∗ νi , we see the equality f m S µi S ∗ νi = f m S µ S ∗ ν · S νi S ∗ νi = S µ S ∗ ν f n · S νi S ∗ νi = S µi S ∗ νi f n . (2.4)6e also have f ◦ σ mA · S µi S ∗ νi = X ξ ∈ B m ( X A ) S ξ f S ∗ ξ S µi S ∗ νi = S µ f S ∗ µ S µ S i S ∗ i S ∗ ν = S µ f S i S ∗ i S ∗ ν and S µi S ∗ νi · f ◦ σ nA = S µi S ∗ νi X η ∈ B n ( X A ) S η f S ∗ η = S µ S i S ∗ i S ∗ ν S ν f S ∗ ν = S µ f S i S ∗ i S ∗ ν so that we have f ◦ σ mA · S µi S ∗ νi = S µi S ∗ νi · f ◦ σ nA . (2.5)As f m +1 = f m + f ◦ σ mA and f n +1 = f n + f ◦ σ nA , we have the equality f m +1 · S µi S ∗ νi = S µi S ∗ νi · f n +1 by (2.4) and (2.5). Hence S µi S ∗ νi belongs to F A,f .Define an expectation E A,f : O A −→ F A,f by E A,f ( X ) = Z T ρ A,ft ( X ) dt for X ∈ O A , where dt stands for the normalized Lebesgue measure on T . Let P A be the ∗ -algebraalgebraically generated by S , . . . , S N . For S µ S ∗ ν ∈ P A , we have E A,f ( S µ S ∗ ν ) = Z T exp(2 π √− f | µ | t ) S µ S ∗ ν exp( − π √− f | ν | t ) dt = Z T exp(2 π √− f | µ | − S µ S ∗ ν f | ν | S ν S ∗ µ ) t ) dt S µ S ∗ ν . Put f µ,ν = f | µ | − S µ S ∗ ν f | ν | S ν S ∗ µ ∈ D A and P µ,ν = Z T exp(2 π √− f µ,ν t ) dt ∈ D A , that is a projection in D A . We thus obtain the following lemma. Lemma 2.6. E A,f ( S µ S ∗ ν ) = P µ,ν S µ S ∗ ν . Hence we have
Lemma 2.7. P A ∩ F A,f is dense in F A,f .Proof.
As Lemma 2.6, the equality E A,f ( S µ S ∗ ν ) = P µ,ν S µ S ∗ ν holds. Since P µ,ν is a pro-jection in D A , it belongs to P A ∩ F A,f , so that E A,f ( S µ S ∗ ν ) ∈ P A ∩ F A,f . This means E A,f ( P A ) ⊂ P A ∩ F A,f and hence E A,f ( P A ) = P A ∩ F A,f . To show that P A ∩ F A,f is densein F A,f , it suffices to prove that E A,f ( P A ) is dense in F A,f . Take an arbitrary X ∈ F A,f .Since P A is dense in O A , there exists a sequence X n ∈ P A such that lim n →∞ k X n − X k = 0 . As we have k E A,f ( X n ) − X k = k E A,f ( X n − X ) k ≤ k X n − X k and E A,f ( X n ) belongs to P A ∩ F A,f , we know that P A ∩ F A,f is dense in F A,f .Therefore we have the following proposition.
Proposition 2.8.
The C ∗ -algebra F A,f is generated by the partial isometries of the form S µ S ∗ ν satisfying f | µ | S µ S ∗ ν = S µ S ∗ ν f | ν | , µ, ν ∈ B ∗ ( X A ) . .2 Basic properties of cocycle groupoids In this subsection, we will study a family G A,f , f ∈ C ( X A , Z ) of ´etale subgroupoids of G A .They are called the cocycle groupoid for f . For f ∈ C ( X A , Z ), define the ´etale subgroupoid G A,f by setting G A,f := { ( x, n, z ) ∈ X A × Z × X A | there exist k, l ∈ Z + such that n = k − l, σ kA ( x ) = σ lA ( z ) , f k ( x ) = f l ( z ) } . Put the unit space G (0) A,f = { ( x, , x ) ∈ G A,f | x ∈ X A } . The product and the inverseoperation on G A,f are inherited from G A . Lemma 2.9. (i) If ( x, n, z ) , ( z, m, w ) ∈ G A,f , then ( x, n + m, w ) ∈ G A,f . (ii) If ( x, n, z ) ∈ G A,f , then ( z, − n, x ) ∈ G A,f .Proof.
For ( x, n, z ) , ( z, m, w ) ∈ G A,f , take k, l, p, q ∈ Z + such that n = k − l, σ kA ( x ) = σ lA ( z ) , f k ( x ) = f l ( z ) ,m = p − q, σ pA ( z ) = σ qA ( w ) , f p ( z ) = f q ( w ) . We then have σ p + kA ( x ) = σ pA ( σ kA ( x )) = σ pA ( σ lA ( z )) = σ lA ( σ pA ( z )) = σ lA ( σ qA ( w )) = σ l + qA ( w ) . Since σ kA ( x ) = σ lA ( z ) and σ pA ( z ) = σ qA ( w ), we have f p + k ( x ) = f k ( x ) + f p ( σ kA ( x )) = f l ( z ) + f p ( σ lA ( z )) = f l + p ( z )= f p ( z ) + f l ( σ pA ( z )) = f q ( w ) + f l ( σ qA ( w )) = f l + q ( w ) . The assertion (ii) is obvious.Recall that the topology on the groupoid G A is defined in the following way. For opensubsets U, V ⊂ X A and k, l ∈ Z + such that both σ kA : U −→ σ kA ( U ) and σ lA : V −→ σ lA ( V )are injective and hence homeomorphisms, an open neighborhood basis of G A is defined by U ( U, k, l, V ) = { ( y, k − l, w ) ∈ G A | y ∈ U, w ∈ V, σ kA ( y ) = σ lA ( w ) } As G A,f ⊂ G A , we endow G A,f with the relative topology from G A , so that G A,f is atopological subgroupoid of G A such that the unit space G (0) A,f is homeomorphic to X A under the natural correspondence ( x, , x ) ∈ G (0) A,f −→ x ∈ X A . We henceforth identify G (0) A,f with X A through the natural identification. Lemma 2.10. G A,f is open and closed in G A . Hence G A,f is a clopen subgroupoid of G A .Proof. For ( x, n, z ) ∈ G A,f , there exist k, l ∈ Z + such that n = k − l, σ kA ( x ) = σ lA ( z ) and f k ( x ) = f l ( z ) . Both f k and f l are continuous so that there exist p, q ∈ N with p ≥ k, q ≥ l such that f k ( x ) = f k ( x ′ ) for all x ′ ∈ U ( x ,...,x p ) and f l ( z ) = f l ( z ′ ) for all z ′ ∈ U ( z ,...,z q ) . (2.6)8ence we have ( x, n, z ) ∈ U ( U ( x ,...,x p ) , k, l, U ( z ,...,z q ) )and σ kA ( x ′ ) = σ lA ( z ′ ) and f k ( x ′ ) = f l ( z ′ )for all ( x ′ , n, z ′ ) ∈ U ( U ( x ,...,x p ) , k, l, U ( z ,...,z q ) ) . This means that U ( U ( x ,...,x p ) , k, l, U ( z ,...,z q ) )is an open neighborhood of ( x, k − l, z ) in G A,f , proving that G A,f is open in G A .We will next show that G A,f is closed in G A . Let g α = ( x α , n α , z α ) ∈ G A,f , α ∈ Λ be anet converging to g = ( x, n, z ) ∈ G A . Take k, l ∈ Z + such that n = k − l, σ kA ( x ) = σ lA ( z ).Since both f k , f l are continuous on X A one may find p, q ∈ N with p ≥ k, q ≥ l satisfying(2.6), respectively. As ( x, n, z ) ∈ U ( U ( x ,...,x p ) , k, l, U ( z ,...,z q ) ) , there exists α ∈ Λ such that g α ∈ U ( U ( x ,...,x p ) , k, l, U ( z ,...,z q ) ) , so that n α = k − l, x α ∈ U ( x ,...,x p ) , z α ∈ U ( z ,...,z q ) , σ kA ( x α ) = σ lA ( z α ) (2.7)and hence n α = n . By x α ∈ U ( x ,...,x p ) , z α ∈ U ( z ,...,z q ) as in (2.7), we have by (2.6) f k ( x ) = f k ( x α ) , f l ( z ) = f l ( z α ) . (2.8)On the other hand, as g α = ( x α , n α , z α ) ∈ G A,f , there exist k α , l α ∈ Z + such that n α = k α − l α , σ k α A ( x α ) = σ l α A ( z α ) , f k α ( x α ) = f l α ( z α ) . (2.9)We note that for any p α > k α and q α > l α such that p α − q α = n α we have σ p α A ( x α ) = σ q α A ( z α ) , f p α ( x α ) = f q α ( z α ) . We indeed see that σ p α A ( x α ) = σ p α − k α A ( σ k α A ( x α )) = σ q α − l α A ( σ l α A ( z α )) = σ q α A ( z α ) ,f p α ( x α ) = f k α ( x α ) + f p α − k α ( σ k α A ( x α )) = f l α ( z α ) + f q α − l α ( σ l α A ( z α )) = f q α ( z α ) . Hence we may assume that k α > k and l α > l . As we have f k α ( x α ) = f k ( x α ) + f k α − k ( σ kA ( x α )) , f l α ( z α ) = f l ( z α ) + f l α − l ( σ lA ( z α ))together with f k α ( x α ) = f l α ( z α ), σ kA ( x α ) = σ lA ( z α ) and k α − k = l α − l by (2.7) and (2.9),we see that f k ( x α ) = f l ( z α ) . (2.10)By (2.8) and (2.10), we conclude f k ( x ) = f l ( z ) so that g = ( x, n, z ) belongs to G A,f ,proving G A,f is closed in G A .We note that if f ≡
0, then G A,f = G A , and if f ≡
1, then G A,f = G AF A . Hence the´etale groupoid G A,f is a generalization of both G A and G AF A . Lemma 2.11.
Keep the above notation. (i) G A,f is an ´etale groupoid. G A,f is essentially principal. (iii) G A,f is amenable.Proof. (i) It suffices to show that the map r : G A,f −→ G A,f is a local homeomorphism.Take ( x, n, z ) ∈ G A,f . There exists k, l ∈ Z + such that σ kA ( x ) = σ lA ( z ) and n = k − l, f k ( x ) = f l ( z ) . Consider the following open neighborhood U of ( x, n, z ) in G A,f U = U ( U ( x ,...,x k ) , k, l, U ( z ,...,z l ) )= { ( y, k − l, w ) ∈ G A,f | y ∈ U ( x ,...,x k ) , w ∈ U ( z ,...,z l ) , σ kA ( y ) = σ lA ( w ) , f k ( y ) = f l ( w ) } . It is straightforward to see that r : U −→ r ( U ) is injective and hence homeomorphic bydefinition of the topology on G A,f , so that the groupoid G A,f is an ´etale groupoid.(ii) Put the isotropy bundles G ′ A = { ( x, n, x ) ∈ G A | x ∈ X A } and G ′ A,f = { ( x, n, x ) ∈ G A,f | x ∈ X A } , so that G (0) A,f ⊂ G ′ A,f ⊂ G ′ A . To prove that G A,f is essentially principal, we will show thatthe set Int( G ′ A,f ) of interiers of G ′ A,f coincides with G (0) A,f . As G A,f is ´etale, the unit space G (0) A,f is open in G A,f . Since G A is essentially principal, Int( G ′ A ) = G (0) A so that we have G (0) A,f = Int( G (0) A,f ) ⊂ Int( G ′ A,f ) ⊂ Int( G ′ A ) = G (0) A = G (0) A,f , proving that G A,f is essentially principal.(iii) Since G A is ´etale and amenable together with G A,f being clopen in G A , we seethat G A,f is amenable (see for example [25, Proposition 4.1.14]).Hence we have the following proposition.
Proposition 2.12.
Let A be an irreducible, non permutation matrix with entries in { , } .For any f ∈ C ( X A , Z ) , the cocycle groupoid G A,f is an essentially principal amenableclopen ´etale subgroupoid of G A . C ∗ -algebra C ∗ ( G A,f ) In this subsection we will study the groupoid C ∗ -algebra C ∗ ( G A,f ). Let C c ( G ) standfor the ∗ -algebra of complex valued compactly supported continuous functions on G . Itsproduct structure and ∗ -involution in C c ( G ) are defined by( ξ ∗ η )( t ) = X s ∈ G,r ( s )= r ( t ) ξ ( s ) η ( s − t ) , ξ ∗ ( t ) = ξ ( t − )for ξ, η ∈ C c ( G ) , t ∈ G . Define the C ∗ -algebra C ∗ ( G ) by the completion of C c ( G ) bythe universal C ∗ -norm on C c ( G ). For the amenable ´etale groupoid G , the C ∗ -algebracoincides with the reduced C ∗ -algebra C ∗ r ( G ) of G (cf. [20], [25]). For the groupoid G A ,it is well-known that there exists an isomorphism Φ : C ∗ ( G A ) −→ O A of C ∗ -algebras suchthat Φ( C ( G (0) A )) = D A . emma 2.13. Let f ∈ C ( X A , Z ) . There exists a natural embedding ι f : C ∗ ( G A,f ) ֒ → C ∗ ( G A ) such that ι f ( C ( G (0) A,f )) = C ( G (0) A ) . Proof.
Since G A,f is clopen in G A , the ∗ -algebra C c ( G A,f ) is naturally embedded into C c ( G A ). By the universality of the C ∗ -norm on C ∗ ( G A,f ) , there exists a ∗ -homomorphism ι f : C ∗ ( G A,f ) −→ C ∗ ( G A ) . As it is identical on C ( G (0) A,f ), the ∗ -homomorphism ι f : C ∗ ( G A,f ) −→ C ∗ ( G A ) is actually injective (cf. [25, Theorem 4.2.9]). Hence the naturalembedding C c ( G A,f ) ֒ → C c ( G A ) induces the embedding ι f : C ∗ ( G A,f ) ֒ → C ∗ ( G A ) as C ∗ -subalgebra such that ι f ( C ( G (0) A,f )) = C ( G (0) A ) . We will show the following theorem.
Theorem 2.14.
Assume that A is an irreducible, non permutation matrix with entriesin { , } . Let f ∈ C ( X A , Z ) . Then there exsits an isomorphism Φ : C ∗ ( G A ) −→ O A of C ∗ -algebra such that Φ ( C ∗ ( G A,f )) = F A,f and Φ ( C ( G (0) A,f )) = D A .Proof. By Lemma 2.13, we may regard C ∗ ( G A,f ) as a C ∗ -subalgebra of C ∗ ( G A ) . For µ, ν ∈ B ∗ ( X A ) with | µ | = k, | ν | = l , put U µ,ν = { ( x, k − l, y ) ∈ G A | x ∈ U µ , y ∈ U ν , σ kA ( x ) = σ lA ( y ) } that is a clopen set in G A . Let χ U µ,ν be the characteristic function of U µ,ν on G A . It iswell-known that the correspondence Φ : χ U µ,ν ∈ C ∗ ( G A ) −→ S µ S ∗ ν ∈ O A gives rise to anisomorphism from C ∗ ( G A ) onto O A . Put V µ,ν = { ( x, | µ | − | ν | , z ) ∈ G A | x ∈ U µ , z ∈ U ν , σ | µ | A ( x ) = σ | ν | A ( z ) , f | µ | ( x ) = f ν | ( z ) } By noticing that the continuous functions f | µ | , f | ν | on X A are regarded as elements of C c ( G A,f ) through the natural identification between X A an G (0) A,f so that f | µ | ( x, n, z ) = ( f | µ | ( x ) if n = 0 , z = x, f | ν | ( x, n, z ) = ( f | ν | ( x ) if n = 0 , z = x, f | µ | ∗ χ V µ,ν )( x, n, z ) = f | µ | ( x ) χ V µ,ν ( x, n, z ) , ( χ V µ,ν ∗ f | ν | )( x, n, z ) = χ V µ,ν ( x, n, z ) f | ν | ( z ) . Hence we have f | µ | ∗ χ V µ,ν = χ V µ,ν ∗ f | ν | . The C ∗ -algebra C ∗ ( G A,f ) is generated by the characteristic functions χ V µ,ν of V µ,ν , µ, ν ∈ B ∗ ( X A ). By using Proposition 2.8 together with Lemma 2.13, we have Φ : C ∗ ( G A ) −→ O A satisfies Φ ( C ∗ ( G A,f )) = F A,f and Φ ( C ( G (0) A,f )) = D A .We will state a simplicity condition of the C ∗ -algebra C ∗ ( G A,f ) in terms of the function f in the following way. Following [15], an ´etale groupoid G is said to be minimal if forany z ∈ G (0) , the set { r ( g ) ∈ G (0) | s ( g ) = z } is dense in G (0) . efinition 2.15. A function f ∈ C ( X A , Z ) is said to be minimal if for z ∈ X A and µ ∈ B ∗ ( X A ), there exist x ∈ U µ and k, l ∈ Z + such that σ kA ( x ) = σ lA ( z ) , f k ( x ) = f l ( z ) . (2.11)By the above definition of the minimality of the function f , the next lemma is imme-diate. Lemma 2.16.
A function f ∈ C ( X A , Z ) is minimal if and only if the groupoid G A,f isminimal.
We note that the ´etale groupoid G A,f is essentially principal and amenable. By [15],an essentially principal amenable ´etale groupoid G is minimal if and only if the C ∗ -algebra C ∗ ( G ) of the groupoid is simple. Therefore we have the following proposition. Proposition 2.17.
A function f ∈ ( X A , Z ) is minimal if and only if the C ∗ -algebra F A,f is simple.
Let
A, B be irreducible non permutation matrices with entries in { , } . It is well-knownthat the groupoids G A and G B are isomorphic as ´etale groupoids if and only if thereexists an isomorphism Φ : O A −→ O B of C ∗ -algebras such that Φ( D A ) = D B (cf. [13],[15], [16], [22], [23]). In [5], a notion called continuous orbit equivalence generalizingtopological conjugacy in one-sided topological Markov shifts was introduced. One-sidedtopological Mrkov shifts ( X A , σ A ) and ( X B , σ B ) are said to be continuous orbit equivalent if there exist k , l ∈ C ( X A , Z + ) and k , l ∈ C ( X B , Z + ) such that σ k ( x ) B ( h ( σ A ( x ))) = σ l ( x ) B ( h ( x )) , x ∈ X A and σ k ( y ) A ( h − ( σ B ( y ))) = σ l ( y ) A ( h − ( y )) , y ∈ X B . As in [5], it wasshown that ( X A , σ A ) and ( X B , σ B ) are continuous orbit equivalent if and only if thereexists an isomorphism Φ : O A −→ O B of C ∗ -algebras such that Φ( D A ) = D B . Henceone-sided topological Mrkov shifts ( X A , σ A ) and ( X B , σ B ) are continuous orbit equivalentif and only if the ´etale groupoids G A and G B are isomorphic ([13]). In our situation, wesee the following proposition. Proposition 2.18.
Let f ∈ C ( X A , Z ) and g ∈ C ( X B , Z ) . There exists an isomorphism Φ : O A −→ O B of C ∗ -algebras such that Φ( D A ) = D B and Φ( F A,f ) = F B,g if and only ifthere exists an isomorphism ϕ : G A −→ G B of ´etale groupoids such that ϕ ( G A,f ) = G B,g .Proof.
We note that F A,f = C ∗ ( G A,f ) and F B,g = C ∗ ( G B,g ). Hence by the proof of [22,Proposition 4.1] (see also [15, Theorem 5.17]), we know the desired assertion. We in factsee that the condition that there exists an isomorphism ϕ : G A −→ G B of ´etale groupoidssuch that ϕ ( G A,f ) = G B,g implies an isomorphism Φ : O A −→ O B of C ∗ -algebras such thatΦ( D A ) = D B and Φ( F A,f ) = F B,g . Conversely, suppose that there exists an isomorphismΦ : O A −→ O B of C ∗ -algebras such that Φ( D A ) = D B and Φ( F A,f ) = F B,g . By consid-ering of germs of the normalizers of the subalgebras D A ⊂ O A and D B ⊂ O B , we knowthat the additional condition Φ( F A,f ) = F B,g implies the restriction of the isomorphism ϕ : G A −→ G B of ´etale groupoids yields the equality ϕ ( G A,f ) = G B,g . Corollary 2.19.
Suppose that there exists a homeomorphism h : X A −→ X B that givesrise to a continuous orbit equivalence between ( X A , σ A ) and ( X B , σ B ) . There exists an isomorphism ϕ h : G A −→ G B of ´etale groupoids such that ϕ h ( G A, Ψ h ( g ) ) = G B,g for all g ∈ C ( X B , Z ) , where Ψ h ( g ) ∈ C ( X A , Z ) is defined by Ψ h ( g )( x ) = l ( x ) X i =0 g ( σ iB ( h ( x ))) − k ( x ) X j =0 g ( σ jB ( h ( σ A ( x )))) , x ∈ X A . (2.12) In particular, the ´etale groupoids G A, Ψ h ( g ) and G B,g are isomorphic. (ii)
There exists an isomorphism
Φ : O A −→ O B of C ∗ -algebras such that Φ( D A ) = D B and Φ( F A, Ψ h ( g ) ) = F B,g for all g ∈ C ( X B , Z ) .Proof. (i) Assume that a homeomorphism h : X A −→ X B gives rise to a continuousorbit equivalence between ( X A , σ A ) and ( X B , σ B ). By [7, Theorem 3.2], there exists anisomorphism Φ : O A −→ O B of C ∗ -algebras such that Φ( D A ) = D B and Φ ◦ ρ A, Ψ h ( g ) t = ρ B,gt , t ∈ T . Hence we see that Φ( F A, Ψ h ( g ) ) = F B,g for all g ∈ C ( X B , Z ) so that byProposition 2.18, there exists an isomorphism ϕ h : G A −→ G B of ´etale groupoids suchthat ϕ h ( G A, Ψ h ( g ) ) = G B,g for all g ∈ C ( X B , Z ) . (ii) The assertion follows from (i) together with Proposition 2.18. Corollary 2.20.
One-sided topological Markov shifts ( X A , σ A ) and ( X B , σ B ) are topologi-cally conjugate if and only if there exists an isomorphism ϕ : G A −→ G B of ´etale groupoidssuch that ϕ ( G A,g ◦ h ) = G B,g for all g ∈ C ( X B , Z ) , where h : X A −→ X B is a homeomor-phism defined by the restriction of ϕ to its unit space G (0) A under the identification between G (0) A and X A , and G (0) B and X B , respectively.Proof. By Theorem 1.2, ( X A , σ A ) and ( X B , σ B ) are topologically conjugate if and onlyif there exists an isomorphism Φ : O A −→ O B of C ∗ -algebras such that Φ( D A ) = D B and Φ( F A,g ◦ h ) = F B,g for all g ∈ C ( X B , Z ) . Hence the assertion follows from Proposition2.18.
We fix an irreducible non permutation matrix A with entries in { , } . For b ∈ C ( X A , Z )and H ⊂ { , , . . . , N } , define continuous functions 1 b , χ H ∈ C ( X A , Z ) by1 b ( x ) = 1 − b ( x ) + b ( σ A ( x )) , χ H ( x ) = ( x ∈ H, x H, for x = ( x n ) n ∈ N ∈ X A . For x ∈ X A and k ∈ N , we write x [ k, ∞ ) = σ k − A ( x ) ∈ X A . Lemma 3.1.
For a nonempty subset H ⊂ { , , . . . , N } , a continuous function b ∈ C ( X A , Z ) and a positive integer valued function f ∈ C ( X A , N ) , we have (i) If χ H = 1 b , then H = { , , . . . , N } and b is a constant. (ii) If b = f , then b is a constant and f ≡ . (iii) If f = χ H , then f ≡ and H = { , , . . . , N } . roof. (i) Suppose that χ H = 1 b . For x = ( x n ) n ∈ N ∈ X A and k ∈ N , we have b ( x [ k, ∞ ) ) = ( b ( x [ k +1 , ∞ ) ) if x k ∈ H, b ( x [ k +1 , ∞ ) ) if x k H, so that for n ∈ N b ( x ) = N H c ( x , . . . , x n ) + b ( x [ n +1 , ∞ ) ) , x ∈ X A , holds where N H c ( x , . . . , x n ) = |{ i | x i H, i = 1 , , . . . , n }| . Since b : X A −→ Z iscontinuous and hence bounded, so that N H c ( x , . . . , x n ) must be bounded for all x ∈ X A and n ∈ N . As A is irreducible, it does not occur unless H = { , , . . . , N } . Therefore weconclude that H = { , , . . . , N } , so that b is a constant.(ii) Suppose that 1 b = f for some f ∈ C ( X A , N ). Since f ( x ) ≥ x ∈ X A , wehave 1 − b ( x ) + b ( σ A ( x )) ≥ b ( σ A ( x )) ≥ b ( x ) for all x ∈ X A . (3.1)Assume that b is not constant so that there exist y, z ∈ X A such that b ( y ) > b ( z ) . (3.2)Since b ∈ C ( X A , Z ), one may find K ∈ N such that b = X ν ∈ B K ( X A ) b ν χ U ν for some b ν ∈ Z for ν ∈ B K ( X A ) . Hence there exist µ y , µ z ∈ B K ( X A ) such that y ∈ U µ y , z ∈ U µ z and b ( y ) = b µ y , b ( z ) = b µ z . As the matrix A is irreducible, there exists ξ ∈ B M ( X A ) such that µ y ξµ z ∈ B K + M ( X A ),so that w := µ y ξµ z z [ k +1 , ∞ ) ∈ X A . As w ∈ U µ y , we have b ( w ) = b µ y = b ( y ). On the otherhand, we have by (3.1) b ( z ) = b ( µ z z [ K +1 , ∞ ) ) = b ( σ K + MA ( w )) ≥ b ( w ) = b ( y ) , (3.3)a contradiction to (3.2). We thus conclude that b is a constant and hence f ≡ χ H ( x ) = f ( x ) ≥ x ∈ X A . This implies that H = { , , . . . , N } . In this subsection, we will study the first class of cocycle algebras called support algebras.For H ⊂ { , , . . . , N } , let us denote by ρ A,H the gauge action ρ A,χ H with potentialfunction χ H . Definition 3.2.
Define the C ∗ -subalgebra F A,H of O A by the cocycle algebra F A,χ H forthe funtion χ H , that is defined by the fixed point subalgebra of O A under the action ρ A,H F A,H := { X ∈ O A | ρ A,Ht ( X ) = X for all t ∈ T } . (3.4)14he algebra F A,H is called the support algebra for H . If H = { , , . . . , N } , then ρ A,Ht = ρ At so that F A,H = F A . If H = ∅ , then ρ A,Ht = id so that F A,H = O A .For a word µ = ( µ , . . . , µ m ) ∈ B m ( X A ), let N H ( µ ) := |{ i ∈ { , . . . , m } | µ i ∈ H }| the cardinal number of symbols in { µ , . . . , µ m } contained in H . Lemma 3.3.
For µ, ν ∈ B ∗ ( X A ) such that S µ S ∗ ν = 0 , we have S µ S ∗ ν ∈ F A,H if and onlyif N H ( µ ) = N H ( ν ) .Proof. Since χ H is identified with P j ∈ H S j S ∗ j , for µ = ( µ , . . . , µ m ) ∈ B m ( X A ) we have χ H S µ = ( S µ if µ ∈ H, S µ χ H S ∗ µ S µ = S µ χ H S µ ··· µ m = ( S µ if µ ∈ H, ≤ n ≤ m − S µ ··· µ n χ H S ∗ µ ··· µ n S µ = S µ ··· µ n χ H S µ n +1 ··· µ m = ( S µ if µ n +1 ∈ H, χ | µ | H S µ = ( χ H + S µ χ H S ∗ µ + S µ µ χ H S ∗ µ µ + · · · + S µ ··· µ m − χ H S ∗ µ ··· µ m − ) S µ , we have χ | µ | H S µ = N H ( µ ) S µ . Similarly we have S ∗ ν χ | ν | H = ( χ | ν | H S ν ) ∗ = N H ( ν ) S ∗ ν . We thus have χ | µ | H S µ S ∗ ν = S µ S ∗ ν χ | ν | H if andonly if N H ( µ ) S µ S ∗ ν = N H ( ν ) S µ S ∗ ν . Assume that S µ S ∗ ν = 0. By Lemma 2.2, S µ S ∗ ν ∈ F A,H if and only if N H ( µ ) = N H ( ν ) . By Proposition 2.8 and Lemma 3.3, we have the following proposition.
Proposition 3.4.
For H ⊂ { , , . . . , N } , the support algebra F A,H is the C ∗ -subalgebra C ∗ ( S µ S ∗ ν | N H ( µ ) = N H ( ν ) , µ, ν ∈ B ∗ ( X A )) of O A generated by partial isometries S µ S ∗ ν satisfying N H ( µ ) = N H ( ν ) , µ, ν ∈ B ∗ ( X A ) . We will next study a family of ´etale subgroupoids G H and its C ∗ -algebras C ∗ ( G H ). Definition 3.5.
For H ⊂ { , , . . . , N } , define an ´etale subgroupoid G H of G A by setting G H := { ( x, n, z ) ∈ X A × Z × X A | there exist k, l ∈ Z + such that n = k − l, σ kA ( x ) = σ lA ( z ) , N H ( x , . . . , x k ) = N H ( z , . . . , z l ) } . Put the unit space G (0) H = { ( x, , x ) ∈ G H | x ∈ X A } . The product and the inverseoperation are inherited from G A . 15 emma 3.6. G A,H = G A,χ H the cocycle groupoid for χ H .Proof. For x = ( x n ) n ∈ N ∈ X A , we have χ kH ( x ) = χ H ( x ) + χ H ( σ A ( x )) + · · · + χ H ( σ k − A ( x )) = N H ( x , x , . . . , x k ) . Hence for x = ( x n ) n ∈ N , z = ( z n ) n ∈ N ∈ X A we have χ kH ( x ) = χ lH ( z ) if and only if N H ( x , . . . , x k ) = N H ( z , . . . , z l ) . Therefore we conclude that G A,χ H = G A,H .Hence by Proposition 2.12 and Theorem 2.14, we know that G H is an essentiallyprincipal amenable ´etale subgroupoid of G A such that C ∗ ( G H ) = F A,H .Let G A = ( V A , E A ) be the directed graph associated with the matrix A . It is definedin the following way. The vertex set V A is defined by { , , . . . , N } , and the edge set E A is defined by the set of edge ( i, j ) ∈ V A × V A satisfying A ( i, j ) = 1 whose source vertex isthe vertex i and terminal vertex is the vertex j . A path ( µ , . . . , µ m ) ∈ B m ( X A ) in thegraph G A is called a cycle if µ = µ m . It is equivalent to say that the word ( µ , . . . , µ m − )is a periodic word in B ∗ ( X A ).We henceforth assume that a subset H ⊂ { , , . . . , N } is not empty. Definition 3.7.
A subset H ⊂ { , , . . . , N } is said to be saturated if any cycle in thegraph G A has a vertex in V H . This means that if a word ( µ , . . . , µ m ) ∈ B m ( X A ) satisfies µ = µ m , then there exists i ∈ { , , . . . , m − } such that µ i ∈ H .We then have the following lemma. Lemma 3.8.
Let H ⊂ { , , . . . , N } be a nonempty subset. The following assertions areequivalent. (i) H is saturated. (ii) The cardinality |{ µ ∈ B ∗ ( X A ) | N H ( µ ) = n }| is finite for all n ∈ N . (iii) The cardinality |{ µ ∈ B ∗ ( X A ) | N H ( µ ) = n }| is finite for some n ∈ N .Proof. (i) = ⇒ (ii): Assume that H is saturated in { , , . . . , N } . Suppose that thereexists n ∈ N such that the cardinality |{ µ ∈ B ∗ ( X A ) | N H ( µ ) = n }| is infinite. Hence forany p ∈ N , there exists q ∈ N with q > p such that there exists ν ∈ B q ( X A ) satisfying N H ( ν ) = n . Since there exists at least one cycle in a word of length N + 1, there existsmore than n + 1 cycles in a word of length longer than ( N + 1)( n + 1). Now H is saturatedso that for p > ( N + 1)( n + 1) and q > p , any word ν ∈ B q ( X A ) satisfies N H ( ν ) ≥ n + 1,a contradiction.(ii) = ⇒ (iii): The implication is clear.(iii) = ⇒ (i): Assume the assertion (iii). Suppose that H is not saturated in { , , . . . , N } .There exists a periodic word w = ( w , . . . , w p ) ∈ B ∗ ( X A ) with A ( w p , w ) = 1 such that w i H for all i = 1 , , . . . , p , so that N H ( w ) = 0. Since A is irreducible and H = ∅ ,one may find a word ν ∈ B ∗ ( X A ) such that N H ( ν ) = n and νw ∈ B ∗ ( X A ). Put ν ( m ) = ν m times z }| { w · · · w ∈ B ∗ ( X A ) so that N H ( ν ( m )) = N H ( ν ) = n because N H ( w ) = 0.The family { ν ( m ) | m = 1 , , . . . } is infinite, a contradiction.16nder the assumption that H is saturated, for each n ∈ Z + the set { S µ S ∗ ν | N H ( µ ) = N H ( ν ) = n } is finite because of Lemma 3.8. Let us denote by F nA,H the linear span of { S µ S ∗ ν | N H ( µ ) = N H ( ν ) = n } . Lemma 3.9.
For S µ S ∗ ν , S ξ S ∗ η ∈ F nA,H , we have S µ S ∗ ν · S ξ S ∗ η ∈ F nA,H . Hence F nA,H is afinite dimensional C ∗ -subalgebra of O A if H is saturated.Proof. For S µ S ∗ ν , S ξ S ∗ η ∈ F nA,H , we may assume that S µ S ∗ ν · S ξ S ∗ η = 0 and | ν | ≤ | ξ | . Since S ∗ ν S ξ = 0, we have ξ = ν ¯ ξ for some ¯ ξ ∈ B ∗ ( X A ). As N H ( ξ ) = N H ( ν ) = n , we have N H ( ¯ ξ ) = 0. By putting ν = ( ν , . . . , ν p ) and ¯ ξ = ( ¯ ξ , . . . , ¯ ξ q ) , we have S ∗ ν S ξ = S ∗ ν S ν S ¯ ξ = S ∗ ν p S ν p S ¯ ξ = N X j =1 A ( ν p , j ) S j S ∗ j S ¯ ξ = S ¯ ξ so that S µ S ∗ ν · S ξ S ∗ η = S µ S ¯ ξ S ∗ η = S µ ¯ ξ S ∗ η . As N H ( µ ¯ ξ ) = N H ( µ ) + N H ( ¯ ξ ) = N H ( µ ) = n = N H ( η ), we see that S µ S ∗ ν · S ξ S ∗ η belongsto F nA,H . If H is aturated, the linear span of { S µ S ∗ ν | N H ( µ ) = N H ( ν ) = n } is finitedimensional and closed under both multiplication and ∗ -operation, so that it is a finitedimensional C ∗ -algebra. Lemma 3.10.
Assume that H is saturated. For each i ∈ { , , . . . , N } , there exists afinite family { ω i ( j ) } p i j =1 of words ω i ( j ) = ( ω i ( j ) , . . . , ω iℓ ( j ) ( j )) ∈ B ℓ ( j ) ( X A ) , j = 1 , , . . . , p i such that for each j = 1 , , . . . , p i , ω i ( j ) = i, ω iℓ ( j ) ( j ) ∈ H, N H ( ω i ( j )) = 1 ,U i = ∪ p i j =1 U ω i ( j ) , U ω i ( j ) ∩ U ω i ( j ′ ) = ∅ for j = j ′ . Proof. If i ∈ H , then p i = 1 and take ω i (1) as i . If i H , take the set of words( ω i ( j ) , . . . , ω iℓ ( j ) ( j )) starting with ω i ( j ) = i and ending with ω iℓ ( j ) ( j ) in H such that ω iq ( j ) H for q = 1 , , . . . , ℓ ( j ) −
1. Since H is saturated, the family satisfies U i = ∪ p i j =1 U ω i ( j ) . Lemma 3.11.
Assume that H is saturated. (i) F nA,H ⊂ F n +1 A,H , n ∈ N . (ii) 1 ∈ F A,H and hence ∈ F nA,H for all n ∈ N , where is the unit of the C ∗ -algebra O A .Proof. Take the finite family { ω i ( j ) } p i j =1 of words as in Lemma 3.10, so that we have U i = ∪ p i j =1 U ω i ( j ) , U ω i ( j ) ∩ U ω i ( j ′ ) = ∅ for j = j ′ . This means that the equality S i S ∗ i = P p i j =1 S ω i ( j ) S ∗ ω i ( j ) holds for i = 1 , , . . . , N . Hencewe have 1 = N X i =1 S i S ∗ i = N X i =1 p i X j =1 S ω i ( j ) S ∗ ω i ( j ) . N H ( ω i ( j )) = 1, we have S ω i ( j ) S ∗ ω i ( j ) ∈ F A,H so that 1 ∈ F A,H .For µ, ν ∈ B ∗ ( X A ) such that N H ( µ ) = N H ( ν ) = n , we have S µ S ∗ ν ∈ F nA,H and S µ S ∗ ν = N X i =1 p i X j =1 S µ S ω i ( j ) S ∗ ω i ( j ) S ∗ ν = N X i =1 p i X j =1 S µω i ( j ) S ∗ νω i ( j ) . As N H ( µω i ( j )) = N H ( µ ) + N H ( ω i ( j )) = n + 1 and similarly N H ( νω i ( j )) = n + 1 , we knowthat S µ S ∗ ν ∈ F n +1 A,H .Assume that H is saturated in { , , . . . , N } . Let us denote by Σ iH the set { ω i ( j ) } p i j =1 in Lemma 3.10. We set Σ H = ∪ Ni =1 Σ iH = ∪ Ni =1 { ω i (1) , . . . , ω i ( p i ) } . Put M = P Ni =1 p i and denote by { ω (1) , . . . , ω ( M ) } the set Σ H . Write each word ω ( m ) as ω ( m ) = ( ω ( m ) , . . . , ω ℓ ( m ) ( m )) ∈ B ℓ ( m ) ( X A ) , m = 1 , , . . . , M. We then have for m = 1 , , . . . , Mω ℓ ( m ) ( m ) ∈ H, ω k ( m ) H for k = 1 , , . . . , ℓ ( m ) − ,X A = M [ m =1 U ω ( m ) : disjoint union . Define an M × M matrix A H with entries in { , } by setting for m, n ∈ { , , . . . , M } A H ( m, n ) = A ( ω ℓ ( m ) ( m ) , ω ( n )) = ( ω ( m ) ω ( n ) ∈ B ∗ ( X A ) , Proposition 3.12.
Assume that H is saturated. The C ∗ -algebra F A,H is a unital AF-algebra defined by the inclusion matrix A H .Proof. The subalgebras F nA,H , n ∈ N are increasing sequence of finite dimensional C ∗ -algebras with the common unit 1 of O A . Since F A,H = ∪ ∞ n =1 F nA,H , one knows that F A,H isa unital AF-algebra. As in the proof of Lemma 3.11, the inclusion F nA,H ⊂ F n +1 A,H is givenby the matrix A H . Definition 3.13.
A saturated subset H ⊂ { , , . . . , N } is said to be primitive if the M × M matrix A H is primitive, that is, there exists K ∈ N such that A KH ( m, n ) ≥ m, n = 1 , . . . , M .It is well-known that an AF-algebra defined by an inclusion matrix is simple if andonly if the matrix is primitive. We thus have the following theorem by Proposition 3.12. Theorem 3.14.
Assume that H is saturated in { , , . . . , N } . The C ∗ -algebra F A,H is aunital AF-algebra defined by the inclusion matrix A H . Furthermore, if H is primitive, the C ∗ -algebra F A,H is simple. F A,H . A = (cid:20) (cid:21) , H = { } ⊂ { , } . It is easy to see that H is saturated. Put α = 1 , α = 21 so that Σ H = { α , α } . Then we have A H = (cid:20) (cid:21) and hence H is primitive. We may regard X A H ⊂ { α , α } N . Let S , S be the canonical generatingpartial isometries of O A . The we have F A,H = C ∗ ( S ξ S ∗ η | ξ, η ∈ B ∗ ( X A H ); | ξ | = | η | ) = F A H . Hence the C ∗ -algebra F A,H is isomorphic to the UHF algebra M ∞ of type 2 ∞ . A = , H = { , } ⊂ { , , } . It is easy to se that H is saturated.Put α = 1 , α = 2 , α = 31 , α = 32, so that Σ H = { α , α , α , α } . Then we have A H = and hence H is primitive. We may regard X A H ⊂ { α , α , α , α } N .Let S , S , S be the canonical generating partial isometries of O A . Then we have F A,H = C ∗ ( S ξ S ∗ η | ξ, η ∈ B ∗ ( X A H ) , | ξ | = | η | ) = F A H Hence the C ∗ -algebra F A,H is isomorphicto the simple AF-algebra F A H . A = (cid:20) (cid:21) , that is denoted by [2], and H = { } ⊂ { , } . In the one-sidedtopological Markov shift ( X [2] , σ [2] ), z = 2 ∞ = (2 , , , . . . ) ∈ X [2] is the fixed point, butthe symbol 2 does not belong to H , so that H is not saturated in { , } . It is easy to seethat { z } is a G H -invariant subset, so that the point corresponds to a closed ideal of the C ∗ -algebra C ∗ ( G H ). It is a proper ideal of C ∗ ( G H ). Hence the C ∗ -algebra F [2] , { } is notsimple. In this subsection, we will study the second class of cocycle algebras called coboundaryalgebras from a view point of ´etale groupoids.
Lemma 3.15.
Suppose that f = 1 − b + b ◦ σ A for some b ∈ C ( X A , Z ) . For ( x, n, z ) ∈ G A ,we have ( x, n, z ) ∈ G A,f if and only if n = b ( x ) − b ( z ) .Proof. Suppose that ( x, n, z ) ∈ G A,f . Take k, l ∈ Z + such that n = k − l, σ kA ( x ) = σ lA ( z )and f k ( x ) = f l ( z ). Since f k ( x ) = k − b ( x ) + b ( σ kA ( x )) and f l ( z ) = l − b ( z ) + b ( σ lA ( z )) , wehave k − l = b ( x ) − b ( z ).Conversely, suppose that n = b ( x ) − b ( z ). Since ( x, n, z ) ∈ G A , there exist k, l ∈ Z + such that n = k − l, σ kA ( x ) = σ lA ( z ) . As f k ( x ) = k − b ( x ) + b ( σ kA ( x )) and f l ( z ) = l − b ( z ) + b ( σ lA ( z )) , the condition n = b ( x ) − b ( z ) implies that f k ( x ) = f l ( z ) and hence( x, n, z ) ∈ G A,f . 19 efinition 3.16.
For b ∈ C ( X A , Z ), define an ´etale subgroupoid G bA of G A by setting G bA := { ( x, n, z ) ∈ X A × Z × X A | there exist k, l ∈ Z + such that n = k − l = b ( x ) − b ( z ) , σ kA ( x ) = σ lA ( z ) } . The unit space ( G bA ) (0) is defined by ( G bA ) (0) = { ( x, , x ) ∈ G bA | x ∈ X A } . The productand the inverse operation are inherited from G A . Since by putting 1 b = 1 − b + b ◦ σ A ∈ C ( X A , Z ), under the condition σ kA ( x ) = σ lA ( z ) , we have 1 kb ( x ) = 1 lb ( z ) if and only if k − l = b ( x ) − b ( z ). Hence we see Lemma 3.17. G bA = G A, b the cocycle groupoid for b . The cocycle groupoid G bA is called the coboundary groupoid for b . By Proposition2.12, Theorem 2.14 and Lemma 3.17, we know that the coboundary groupoid G bA is anessentially principal amenable ´etale clopen subgroupoid of G A . Lemma 3.18.
Assume that A is primitive. Then the function b for each b ∈ C ( X A , Z ) is minimal. Hence the coboundary groupoid G bA is minimal.Proof. Take arbitrary z ∈ X A and µ = ( µ , . . . , µ m ) ∈ B m ( X A ). Since b ∈ C ( X A , Z ), onemay find K ∈ N with K ≥ m such that b = X ν ∈ B K ( X A ) b ν χ U ν for some b ν ∈ Z for ν ∈ B K ( X A ) . Since X A = ∪ ν ∈ B K ( X A ) U ν , one may find ν z ∈ B K ( X A ) such that z ∈ U ν z so that b ( z ) = b ν z .Since | µ | = m ≤ K , there exists ν = ( ν , ν , . . . , ν K ) ∈ B K ( X A ) such that U ν ⊂ U µ .We see that b ( x ) = b ν for all x ∈ U ν . Now A is primitive, so that there exists L ∈ N such that A L ( i, j ) ≥ i, j = 1 , , . . . , N .We have three cases.Case 1, b ν = b ν z : We have b ( x ) = b ( z ) . Since A L ( ν K , z K + L ) ≥ , one finds a word( ν K +1 , ν K +2 , . . . , ν K + L − ) ∈ B L − ( X A ) such that A ( ν K , ν K +1 ) = A ( ν K + L − , z K + L ) = 1 . Put x [1 ,K ] = ν , x [ K +1 ,K + L − = ( ν K +1 , . . . , ν K + L − ) , x [ K + L, ∞ ) = z [ K + L, ∞ ) so that we have x = ( x n ) n ∈ N ∈ U ν ⊂ X A and hence b ( x ) = b ν . We then have σ K + L − A ( x ) = σ K + L − A ( z ) = z [ K + L, ∞ ) . By putting k = l = K + L −
1, we have x ∈ U µ and σ kA ( x ) = σ lA ( z ) , k − l = b ( x ) − b ( z )(= 0) . Case 2, b ν > b ν z : We put M = b ( x ) − b ( z )(= b ν − b ν z ) > . Take ν ′ = ( ν ′ , . . . , ν ′ M ) ∈ B M ( X A ) and ν ′′ = ( ν ′′ K +1 , . . . , ν ′′ K + L − ) ∈ B L − ( X A ) such that A ( ν K , ν ′ ) = A ( ν ′ M , ν ′′ K +1 ) = A ( ν ′′ K + L − , z K + L ) = 1 . Put x [1 ,K ] = ν , x [ K +1 ,K + M ] = ( ν ′ , . . . , ν ′ M ) , x [ K + M +1 ,K + M + L − = ( ν ′′ K +1 , . . . , ν ′′ K + L − ) ,x [ K + M + L, ∞ ) = z [ K + L, ∞ )
20o that we have x = ( x n ) n ∈ N ∈ U ν ⊂ X A and σ K + L + M − A ( x ) = x [ K + M + L, ∞ ) = z [ K + L, ∞ ) = σ K + L − A ( z ) . By putting k = K + L + M − l = K + L −
1, we have x ∈ U µ and σ kA ( x ) = σ lA ( z ) , k − l = M = b ( x ) − b ( z ) . Case 3, b ν < b ν z : We put M ′ = b ( z ) − b ( x )(= b ν z − b ν ) > . One may find( ξ , . . . , ξ L − ) ∈ B L − ( X A ) such that A ( ν K , ξ ) = A ( ξ L − , z K + M ′ + L ) = 1 . Put x [1 ,K ] = ν , x [ K +1 ,K + L − = ( ξ , . . . , ξ L − ) , x [ K + L, ∞ ) = z [ K + M ′ + L, ∞ ) so that we have x = ( x n ) n ∈ N ∈ U ν ⊂ X A and σ K + L − A ( x ) = x [ K + L, ∞ ) = z [ K + M ′ + L, ∞ ) = σ K + M ′ + L − A ( z ) . By putting k = K + L − l = K + M ′ + L −
1, we have σ kA ( x ) = σ lA ( z ) , k − l = − M ′ = b ( x ) − b ( z ) . Therefore the function 1 b = 1 − b + b ◦ σ A is minimal. Definition 3.19.
For b ∈ C ( X A , Z ), the coboundary algebra F bA is defined by the cocyclealgebra F A, b for the function 1 b .By Lemma 3.17, F bA is isomorphic to the C ∗ -algebra C ∗ ( G bA ) of the coboundarygroupoid G bA . Proposition 3.20.
Assume that A is primitive. For any b ∈ C ( X A , Z ) , the coboundaryalgebra F bA is simple.Proof. By the previous lemma, the coboundary groupoid G bA is minimal, so that the C ∗ algebra C ∗ ( G bA ) that is F bA is simple. Theorem 3.21.
Assume that A is primitive. For any b ∈ C ( X A , Z ) , the coboundaryalgebra F bA is a unital simple AF-algebra that is stably isomorphic to the standard AF-algebra F A .Proof. The function 1 b = 1 − b + b ◦ σ A is cohomologous to 1, so that by putting aone-parameter unitary group u t = exp(2 π √− bt ) ∈ D A , t ∈ T , we have by [7, Lemma 2.3] ρ A, b t = Ad( u t ) ◦ ρ At , t ∈ T . This means that the actions ρ A, b and ρ A are cocycle conjugate. By general theory of C ∗ -crossed products, we know that their crossed products O A ⋊ ρ A, b T and O A ⋊ ρ A T are isomorphic. By [2], we know that O A ⋊ ρ A T is stably isomorphic to F A . Now A isprimitive, so that the AF-algebra F A is simple and hence so is O A ⋊ ρ A T . Hence thecrossed product O A ⋊ ρ A, B T is simple. By [24], there exists a projection P b ∈ O A ⋊ ρ A, b T such that ( O A ) ρ A, b = P b ( O A ⋊ ρ A, b T ) P b . As the algebra F bA is the fixed point algebra( O A ) ρ A, b under the action ρ A, b , it is a full corner of O A ⋊ ρ A, b T . Hence F bA is stablyisomorphic to O A ⋊ ρ A, b T and hence to the standard AF-algebra F A .21n [5], the continuous full group Γ A , writtten as [ σ A ] c in [5], for a topological Markovshift ( X A , σ A ) plays a crucial role to study continuous orbit equivalence in one-sided topo-logical Markov shifts (see also [14], [16], etc.). The group Γ A consists of homeomorphisms τ on X A such that there exist k τ , l τ ∈ C ( X A , Z + ) satisfying σ k τ ( x ) A ( τ ( x )) = σ l τ ( x ) A ( x ) for x ∈ X A . The function d τ = l τ − k τ ∈ C ( X A , Z ) is called the cocycle functin of τ . Since thehomeomorphism τ gives rise to a continuous orbit equivalence on X A , one may considerthe function Ψ τ (1) ∈ C ( X A , Z ) for 1 by the formula (2.12). It is straightforward to seethat the formula Ψ τ (1)( x ) = 1 − d τ ( x ) + d τ ◦ σ A ( x ) , x ∈ X A (3.6)holds (cf. [11]). We then know the following proposition. Proposition 3.22.
Let A be an irreducible, non permutation matrix with entries in { , } .The coboundary algebra F d τ A for the cocycle function d τ = l τ − k τ of an element τ of thecontinuous full group Γ A is isomorphic to the standard AF-algebra F A .Proof. Take an arbitrary τ ∈ Γ A . By Corollary 2.19 (ii), there exists an isomorphismΦ τ : F A, Ψ τ (1) −→ F A, of C ∗ -algebras. As Ψ τ (1) = 1 − d τ + d τ ◦ σ A by (3.6), we see that F d τ A is isomorphic to F A .There are lots of examples of homeomorphisms in the group Γ A as in [6], [14] and [16]. In this subsection, we will study the third class of cocycle algebras called suspesion alge-bras from a view point of suspension of two-sided subshifts. We fix an irreducible, nonpermutation matrix A = A [ i, j ] Ni,j =1 with entries in { , } . Let us denote by ( ¯ X A , ¯ σ A ) thetwo-sided topological Markov shift for the matrix A that is defined by the shift space¯ X A = { ( x n ) n ∈ Z ∈ { , , . . . , N } Z | A ( x n , x n +1 ) = 1 for all n ∈ Z } and the shift homeomorphism ¯ σ A (( x n ) n ∈ Z ) = ( x n +1 ) n ∈ Z on ¯ X A . Let f : X A −→ N be acontinuous function on X A . By taking a higher block representation of ( X A , σ A ) (cf. [4]),we may assume that the function f depends only on the first coordinate on X A . Thismeans that there exists a finte family f , . . . , f N of positive integers such that f ( x ) = f x for x = ( x n ) n ∈ N . Put m j = f j − ∈ Z + , j = 1 , . . . , N. Let ¯ f be the continuous functionon ¯ X A defined by ¯ f (( x n ) n ∈ Z ) = f (( x n ) n ∈ N ) for ( x n ) n ∈ Z ∈ ¯ X A .Let us denote by ( ¯ X fA , S fA ) the discrete suspension of ( ¯ X A , ¯ σ A ) by the ceiling function¯ f , that is defined in the following way (cf. [1], [18]). The space ¯ X fA is defined by¯ X fA = { ( x, n ) ∈ ¯ X A × Z + | ≤ n < ¯ f ( x ) } , and the transformation S fA on ¯ X fA is defined by for ( x, n ) ∈ ¯ X fA ,S fA ( x, n ) = ( ( x, n + 1) if n < ¯ f ( x ) − , (¯ σ A ( x ) ,
0) if n = ¯ f ( x ) − . G A = ( V A , E A ) denotes the directed graph defined by the matrix A such thatits vertex set V A = { , , . . . , N } and its edge set E A consists of the ordered pair ( i, j )of vertices such that A ( i, j ) = 1. The source s ( i, j ) of ( i, j ) is i , and the terminal t ( i, j )of ( i, j ) is j . As in [1], [9], [18], let us construct a new directed graph G A f = ( V A f , E A f )with its transition matrix A f in the following way . Let V A f = ∪ Nj =1 { j , j , . . . , j m j } . For j, k ∈ { , , . . . , N } with A ( j, k ) = 1, we define A f ( j , j ) = A f ( j , j ) = · · · = A f ( j m j − , j m j ) = A f ( j m j , k ) = 1 . We define A f ( j m , k n ) = 0 for other pair ( j m , k n ) ∈ V A f × V A f . We call the matrix A f the suspended matrix of A by f .Denote by ¯ X A f the two-sided shift space defined by the suspended matrix A f . LetΣ A f = V A f . Hence we have¯ X A f = { ( x fi ) i ∈ Z ∈ Σ Z A f | x fi ∈ Σ A f , A f ( x fi , x fi +1 ) = 1 for all i ∈ Z } . As x fi = j ( i ) n ( i ) for some j ( i ) ∈ { , , . . . , N } and n ( i ) ∈ { , , . . . , m j ( i ) } . Let x = j (1).Let x be the first return time of j (1) n (1) to the j in ( x fi ) i ∈ Z . Similarly we have a sequence( x k ) k ∈ Z ∈ ¯ X A of the first return time in both forward and backward in ( x fi ) i ∈ Z . We thenhave a correspondence η : ¯ X A f −→ ¯ X fA such that η (( x fi ) i ∈ Z ) = (( x k ) k ∈ Z , n (1)) ∈ ¯ X fA . The following lemma is well-known.
Lemma 3.23 (cf. [1], [9], [17], [18]) . For f ∈ C ( X A , N ) , the map η : ¯ X A f −→ ¯ X fA is ahomeomorphism satisfying η ◦ ¯ σ A = S fA ◦ η . Hence the discrete suspension ( ¯ X fA , S fA ) is identified with the two-sided topologicalMarkov shift ( ¯ X A f , ¯ σ A f ) defined by the suspended matrix A f .We will next study a crucial relation between these two C ∗ -algebras F A f and F A,f . Letus denote by ˜ S j , ˜ S j , . . . , ˜ S j mj , j = 1 , , . . . , N the canonical generating partial isometriesof the Cuntz–Krieger algebra O A f so that N X j =1 ( ˜ S j ˜ S ∗ j + ˜ S j ˜ S ∗ j + · · · + ˜ S j mj ˜ S ∗ j mj ) = 1 , ˜ S ∗ j n ˜ S j n = ( ˜ S j n +1 ˜ S ∗ j n +1 for n = 0 , , . . . , m j − , P Nk =1 A ( j, k ) ˜ S k ˜ S ∗ k for n = m j . Let us denote by ρ A f t , t ∈ T the standard gauge action on O A f that is defined by ρ A f t ( ˜ S j n ) = exp(2 π √− t ) ˜ S j n , n = 0 , , . . . , m j , j = 1 , , . . . , N. Let F A f be the standard AF-algebra inside O A f that is realized as the fixed point algebraof O A f under the gauge action ρ A f . Define partial isometries s , . . . , s N and projection P A by s j := ˜ S j ˜ S j · · · ˜ S j mj , P A := N X j =1 ˜ S j ˜ S ∗ j in O A f . The following lemma is similarly shown to [9, Section 6].23 emma 3.24. Keep the above notation. (i) P Nj =1 s j s ∗ j = P A , s ∗ i s i = P Nk =1 A ( j, k ) s k s ∗ k , i = 1 , , . . . , N (ii) P A O A f P A = C ∗ ( s , . . . , s N ) the C ∗ -subalgebra of O A f generated by s , . . . , s N . Hence P A O A f P A is isomorphic to O A . (iii) ρ A f t ( P A ) = P A , t ∈ T . (iv) Let Φ A : P A O A f P A −→ O A = C ∗ ( s , . . . , s N ) be the identification Φ A ( ˜ S j ˜ S j · · · ˜ S j mj ) = s j in (ii) between P A O A f P A and O A . Then we have Φ A ◦ ρ A f t | P A O Af P A = ρ A,ft ◦ Φ A , t ∈ T . Let us denote by K the C ∗ -algebra of compact operators on the separable infinitedimensional Hilbert space ℓ ( N ), and C its maximal abelian C ∗ -subalgebra of K consistingof diagonal operators on ℓ ( N ). By the above lemma, we see the following proposition. Proposition 3.25. (i) P A F A f P A = F A,f . (ii) Assume that A is primitive. Then P A is a full projection in F A f , so that the cocyclealgebra F A,f is stably isomorphic to F A f . More exactly, there exists an isomorphism Φ f : F A f ⊗ K −→ F A,f ⊗ K of C ∗ -algebras such that Φ f ( D A f ⊗ C ) = D A ⊗ C . Proof. (i) Let S , . . . , S N be the canonical generating partial isometries of O A . Sincethe correspondence Φ A : P A O A f P A −→ O A defined by Φ A ( ˜ S j ˜ S j · · · ˜ S j mj ) = S j is anisomorphism satisfying Φ A ◦ ρ A f t = ρ A,ft ◦ Φ A , we see that ( P A O A f P A ) ρ Af = ( O A ) ρ A,f . As ρ A f t ( P A ) = P A , we know that P A ( O A f ) ρ Af P A = ( O A ) ρ A,f and hence P A F A f P A = F A,f .(ii) Now the matrix A is primitive, so is the suspended matrix A f . Hence the AF-algebra F A f is simple. As any nonzero projection of a simple C ∗ -algebra is full, theprojection P A is a full projection in F A f . It is easy to see that P A D A f P A = D A , so thatthe pair ( F A,f , D A ) is a relative full corner of the pair ( F A f , D A f ) in the sense of [8]. Hencethe pairs ( F A f ⊗ K , D A f ⊗ C ) and ( F A,f ⊗ K , D A ⊗ C ) are relative Morita equivalence inthe sense of [8], so that there exists an isomorphismΦ f : F A f ⊗ K −→ F A,f ⊗ K of C ∗ -algebras such that Φ f ( D A f ⊗ C ) = D A ⊗ C by [8].For a continuous function f ∈ C ( X A , N ), there exists K ∈ N such that f = P ν ∈ B K ( X A ) f ν χ U ν for some f ν ∈ N for each ν ∈ B K ( X A ). By taking the K -higher block matrix A [ K ] of A as in [4], the function f is regarded as the one depending on only the first coordinate of X A [ K ] . We thus obtain the following theorem. Theorem 3.26.
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