### Electronic Research Archive

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

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

• 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.

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

### Related Papers:

• 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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