Vol. 246, No. 1, 2010

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Orbit equivalence of topological Markov shifts and Cuntz–Krieger algebras

Kengo Matsumoto

Vol. 246 (2010), No. 1, 199–225
Abstract

We prove that one-sided topological Markov shifts (XAA) and (XBB) 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 𝒪A and 𝒪B keeping their commutative C-subalgebras C(XA) and C(XB). The “if” part (and hence the “only if” part) above is equivalent to the condition that there exists a homeomorphism from XA to XB intertwining their topological full groups. We will also study structure of the automorphisms of 𝒪A preserving the commutative C-algebra C(XA).

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
Authors
Kengo Matsumoto
Department of Mathematics
Joetsu University of Education
943-8512 Joetsu
Japan