On the K-theory of Crossed Products by Automorphic Semigroup Actions

LetP be a semigroup that admits an embedding into a groupG.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,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 R R× for a Dedekind domainR, theK-theory of the left and right regular semigroup C*-algebras C∗ λ(P ) and C∗ ρ(P ) coincide, although the structure of these algebras can be very different.