Semicrossed Products, Dilations, and Jacobson Radicals

Abstract

We compute the C*-envelope of the isometric semicrossed product of a C*-algebra arising from number theory by the multiplicative semigroup of a number ring R, and prove that it is isomorphic to T[R], the left regular representation of the ax+b-semigroup of R. We do this by explicitly dilating an arbitrary representation of the isometric semicrossed product to a representation of T[R] and show that such representations are maximal.

We also study the Jacobson radical of the semicrossed product of a simple C*-algebra and either a subsemigroup of an abelian group or a free semigroup. A full characterization of the Jacobson radical is obtained for a large subset of these semicrossed products and we apply our results to a number of examples.