Orbit equivalence of topological Markov shifts and Cuntz–Krieger algebras

Kengo Matsumoto

Vol. 246 (2010), No. 1, 199–225

DOI: 10.2140/pjm.2010.246.199

Abstract

We 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 continuously orbit equivalent if and only if there exists an isomorphism between the Cuntz–Krieger algebras 𝒪_A and 𝒪_B keeping their commutative C^∗-subalgebras C(X_A) and C(X_B). The “if” part (and hence the “only if” part) above is 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 𝒪_A preserving the commutative C^∗-algebra C(X_A).

Keywords

topological Markov shifts, orbit equivalence, full group, Cuntz–Krieger algebra

Mathematical Subject Classification 2000

Primary: 46L55

Secondary: 46L35, 37B10

Milestones

Received: 5 December 2007

Accepted: 16 January 2010

Published: 1 May 2010