Cuntz-Krieger algebras, topological Markov shifts and groupoids

Abstract: It is well-known that there is a strong connection between Cuntz-Krieger algebras and a certain type of shifts of finite type called topological Markov shifts. Recently, it has been discovered that topological Markov shifts can be recovered up to different kinds of equivalence from the corresponding Cuntz-Krieger algebras.

I will give an overview of these results and explain how groupoids can be used to prove and generalise them.

The talk will primarily be based on the following papers.

K. Matsumoto: "Orbit equivalence of topological Markov shifts and Cuntz-Krieger algebras".

K. Matsumoto: "Continuous orbit equivalence, flow equivalence of Markov shifts and circle actions on Cuntz–Krieger algebras".

K. Matsumoto and H. Matui: "Continuous orbit equivalence of topological Markov shifts and Cuntz–Krieger algebras".

T. M. Carlsen, S. Eilers, E. Ortega, and G. Restorff: "Flow equivalence and orbit equivalence for shifts of finite type and isomorphism of their groupoids".

T. M. Carlsen and J. Rout: "Diagonal-preserving gauge-invariant isomorphisms of graph C*-algebras".

T. M. Carlsen, E. Ruiz, A. Sims, and M. Tomforde: "Reconstruction of groupoids and C*-rigidity of dynamical systems".

geometric topologynumber theoryoperator algebrasrepresentation theory

Audience: researchers in the topic

( slides | video )

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Noncommutative geometry in NYC

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