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arXiv · 1205.5412

On the K-theory of crossed products by automorphic semigroup actions

Joachim Cuntz·Siegfried Echterhoff·Xin Li
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Abstract

Let P be a semigroup that admits an embedding into a group G. Assume that the embedding satisfies a certain Toeplitz condition and that the Baum-Connes conjecture holds for G. We prove a formula describing the K- theory of the reduced crossed product A \rtimesα,r P by any automorphic action of P. This formula is obtained as a consequence of a result on the K-theory of crossed products for special actions of G on totally disconnected spaces. We apply our result to various examples including left Ore semigroups and quasi-lattice ordered semigroups. We also use the results to show that for certain semigroups P, including the ax + b-semigroup for a Dedekind domain R, the K-theory of the left and right regular semigroup C*-algebras of P coincide, although the structure of these algebras can be very different.

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Joachim Cuntz, Siegfried Echterhoff, Xin Li. 2012-05-24. On the K-theory of crossed products by automorphic semigroup actions. https://arxiv.org/abs/1205.5412

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