On the k-Theory of crossed products by automorphic semigroup actions

Cuntz, J., Echterhoff, S. and Li, X. (2013) On the k-Theory of crossed products by automorphic semigroup actions. Quarterly Journal of Mathematics, 64(3), pp. 747-784. (doi: 10.1093/qmath/hat021)

Abstract

Let P be a semigroup that admits an embedding into a group G. Assume that the embedding satisfies the Toeplitz condition recently introduced by the third named author and that the Baum–Connes conjecture holds for G. We prove a formula describing the K-theory of the reduced crossed product A⋊α, rP 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 R⋊ R× for a Dedekind domain R, the K-theory of the left and right regular semigroup C*-algebras Cλ*(P) and Cρ*(P) coincide, although the structure of these algebras can be very different.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Li, Professor Xin
• Authors: Cuntz, J., Echterhoff, S., and Li, X.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Quarterly Journal of Mathematics
• Publisher: Oxford University Press
• ISSN: 0033-5606
• ISSN (Online): 1464-3847
• Published Online: 04 June 2013