$ C^* $-algebras associated with asymptotic equivalence relations defined by hyperbolic toral automorphisms

Kengo Matsumoto; Kengo Matsumoto

doi:10.3934/era.2021006

Electronic Research Archive

2021, Volume 29, Issue 4: 2645-2656. doi: 10.3934/era.2021006

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$ C^* $-algebras associated with asymptotic equivalence relations defined by hyperbolic toral automorphisms

Kengo Matsumoto

Department of Mathematics, Joetsu University of Education, Joetsu, 943-8512, Japan

Received: 01 December 2020 Published: 11 January 2021
Primary: 37D20, 37A55; Secondary: 46L35

We study the $ C^* $-algebras of the étale groupoids defined by the asymptotic equivalence relations for hyperbolic automorphisms on the two-dimensional torus. The algebras are proved to be isomorphic to four-dimensional non-commutative tori by an explicit numerical computation. The ranges of the unique tracial states of its $ K_0 $-groups of the $ C^* $-algebras are described in terms of the hyperbolic matrices of the automorphisms on the torus.
- Hyperbolic toral automorphisms,
- Smale space,
- asymptotic equivalence relation,
- étale groupoid,
- non-commutative tori
Citation: Kengo Matsumoto. $ C^* $-algebras associated with asymptotic equivalence relations defined by hyperbolic toral automorphisms[J]. Electronic Research Archive, 2021, 29(4): 2645-2656. doi: 10.3934/era.2021006

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Abstract

We study the $ C^* $-algebras of the étale groupoids defined by the asymptotic equivalence relations for hyperbolic automorphisms on the two-dimensional torus. The algebras are proved to be isomorphic to four-dimensional non-commutative tori by an explicit numerical computation. The ranges of the unique tracial states of its $ K_0 $-groups of the $ C^* $-algebras are described in terms of the hyperbolic matrices of the automorphisms on the torus.

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