Orbit equivalence of topological Markov shifts and Cuntz-Krieger algebras

Details Orbit equivalence of topological Markov shifts and Cuntz-Krieger algebras Orbit equivalence of topological Markov shifts and Cuntz-Krieger algebras

2007-07-14

arxiv.org/abs/0707.2114v1

Mathematics

Operator Algebras

26 pages

Scientific paper

We will prove that one-sided topological Markov shifts $(X_A,\sigma_A)$ and $(X_B,\sigma_B)$ 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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