Cuntz–Krieger algebras and one-sided conjugacy of shifts of finite type and their groupoids

Kevin Aguyar Brix; Toke Meier Carlsen

doi:10.1017/S1446788719000168

Cuntz–Krieger algebras and one-sided conjugacy of shifts of finite type and their groupoids

Kevin Aguyar Brix, Toke Meier Carlsen

Náttúruvísindadeildin - Faculty of Science and Technology

Research output: Contribution to journal › Article › peer-review

11 Citations (Scopus)

Abstract

A one-sided shift of finite type (X_A,σ_A) determines on the one hand a Cuntz–Krieger algebra O_A with a distinguished abelian subalgebra D_A and a certain completely positive map τ_A on O_A . On the other hand, (X_A,σ_A) determines a groupoid G_A together with a certain homomorphism ϵ_A on G_A . We show that each of these two sets of data completely characterizes the one-sided conjugacy class of X_A . This strengthens a result of Cuntz and Krieger. We also exhibit an example of two irreducible shifts of finite type which are eventually conjugate but not conjugate. This provides a negative answer to a question of Matsumoto of whether eventual conjugacy implies conjugacy.

Original language	English
Pages (from-to)	289-298
Number of pages	10
Journal	Journal of the Australian Mathematical Society
Volume	109
DOIs	https://doi.org/10.1017/S1446788719000168
Publication status	Published - 21 Dec 2020

Keywords

Shifts of finite type
groupoids
Cuntz-Krieger algebras

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10.1017/S1446788719000168

https://arxiv.org/abs/1712.00179

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N2 - A one-sided shift of finite type (XA,σA) determines on the one hand a Cuntz–Krieger algebra OA with a distinguished abelian subalgebra DA and a certain completely positive map τA on OA . On the other hand, (XA,σA) determines a groupoid GA together with a certain homomorphism ϵA on GA . We show that each of these two sets of data completely characterizes the one-sided conjugacy class of XA . This strengthens a result of Cuntz and Krieger. We also exhibit an example of two irreducible shifts of finite type which are eventually conjugate but not conjugate. This provides a negative answer to a question of Matsumoto of whether eventual conjugacy implies conjugacy.

AB - A one-sided shift of finite type (XA,σA) determines on the one hand a Cuntz–Krieger algebra OA with a distinguished abelian subalgebra DA and a certain completely positive map τA on OA . On the other hand, (XA,σA) determines a groupoid GA together with a certain homomorphism ϵA on GA . We show that each of these two sets of data completely characterizes the one-sided conjugacy class of XA . This strengthens a result of Cuntz and Krieger. We also exhibit an example of two irreducible shifts of finite type which are eventually conjugate but not conjugate. This provides a negative answer to a question of Matsumoto of whether eventual conjugacy implies conjugacy.

KW - Shifts of finite type

KW - groupoids

KW - Cuntz-Krieger algebras

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EP - 298

JO - Journal of the Australian Mathematical Society

JF - Journal of the Australian Mathematical Society

ER -

Cuntz–Krieger algebras and one-sided conjugacy of shifts of finite type and their groupoids

Abstract

Keywords

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