Orbit equivalence of topological Markov shifts and Cuntz-Krieger algebras

2007/07/14 by Matsumoto, Kengo
#46L55 #Dynamical Systems (math.DS) #FOS: Mathematics #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.0707.2114

Abstract

We will prove that one-sided topological Markov shifts (X_A,σ_A) and (X_B,σ_B) for matrices A and B with entries in \0,1\ are topologically orbit equivalent if and only if there exists an isomorphism between the Cuntz-Krieger algebras \Cal O_A and \Cal O_B keeping their commutative C^*-subalgerbas C(X_A) and C(X_B). It is also equivalent to the condition that there exists a homeomorphism from X_A to X_B intertwining their topological full groups. We will also study structure of the automorphisms of \Cal O_A keeping the commutative C^*-algebra C(X_A).

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