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The structure of crossed products of irrational rotation algebras by finite subgroups of SL_2 (Z)

Siegfried Echterhoff·Wolfgang Lueck·N. Christopher Phillips·Samuel Walters
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Abstract

Let F be a finite subgroup of SL_2 (Z) (necessarily isomorphic to one of Z/2Z, Z/3Z, Z/4Z, or Z/6Z), and let F act on the irrational rotational algebra A_θ via the restriction of the canonical action of SL_2 (Z). Then the crossed product of A_θ by F, and the fixed point algebra for the action of F on A_θ, are AF algebras. The same is true for the crossed product and fixed point algebra of the flip action of Z/2Z on any simple d-dimensional noncommutative torus A_Θ. Along the way, we prove a number of general results which should have useful applications in other situations.

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Siegfried Echterhoff, Wolfgang Lueck, N. Christopher Phillips, Samuel Walters. 2006-09-28. The structure of crossed products of irrational rotation algebras by finite subgroups of SL_2 (Z). https://arxiv.org/abs/math/0609784

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