Continuous orbit equivalence of topological Markov shifts and Cuntz-Krieger algebras

By Kengo Matsumoto. 2007

128 citations recorded by the source corpus.

Let A,B be square irreducible matrices with entries in {0,1}. We will show that if the one-sided topological Markov shifts (X_A,\sigma_A) and (X_B,\sigma_B) are continuously orbit equivalent, then the two-sided topological Markov shifts (\bar X_A,\bar\sigma_A) and (\bar X_B,\bar\sigma_B) are flow equivalent, and hence det(id-A)=det(id-B). As a result, the one-sided topological Markov shifts (X_A,\sigma_A) and (X_B,\sigma_B) are continuously orbit equivalent if and only if the Cuntz-Krieger algebras O_A and O_B are isomorphic and det(id-A)=det(id-B).

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1aef9daa0e51478f814cd643ea2aa7efed18d88f

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Citations in the corpus

Equivalence and stable isomorphism of groupoids, and diagonal-preserving stable isomorphisms of graph C*-algebras and Leavitt path algebras (2016, methodology): “Recall [10, 19] that graphs E and F are orbit equivalent if there exist a homeomorphism h : ∂E → ∂F and continuous functions k, l : ∂E → N and k, l : ∂F → N such that σ k(x) F (h(σE(x))) = σ l(x) F (h(x)) and σ k(y) E (h (σF (y))) = σ l(y) E (h (y)) for all x ∈ ∂E and y ∈ ∂F.”
Reconstruction of groupoids and C*-rigidity of dynamical systems (2017, methodology): “Renault’s machinery, and techniques from groupoid homology, underpinned Matsumoto and Matui’s remarkable recent results [30] that irreducible (two-sided) shifts of finite type are flow equivalent if and only if there is a diagonal-preserving isomorphism of the stabilisations of the associated Cuntz–”
Homology and topological full groups of étale groupoids on totally disconnected spaces (2009, unknown): “When G is the étale groupoid arising from a subshift of finite type, G and its connection with C∗-algebras were studied by Matsumoto [20].”
Topological full groups of one-sided shifts of finite type (2012, background): “Matsumoto [26, 27] obtained the following classification theorem (see also [31, Theorem 5.”
Graph algebras and orbit equivalence (2014, methodology): “However, because the crossed product C∗-algebras are classifiable, Giordano, Putnam and Skau’s result means that orbit equivalence can be determined using K-theory.”
Etale groupoids arising from products of shifts of finite type (2015, background): “Matsumoto and the author [25] completely determined when these [[G]] are mutually isomorphic.”
\'Etale groupoids and their $C^*$-algebras (2017, background): “19 is a beautiful theorem of Matsumoto and Matui [28].”
Continuous orbit equivalence rigidity (2015, result): “This can be used to generalize and conceptually explain recent results [19] on orbit equivalence and Cartan isomorphism for shifts of finite type.”