Enveloping σ-C^*-algebra of a smooth Frechet algebra crossed product by R,K-theory and differential structure in C^*-algebras

Abstract

Given an m-tempered strongly continuous action α of R by continuous ^*-automorphisms of a Frechet^*-algebra A, it is shown that the enveloping σ-C^*-algebra E(S(R, A^∞, α)) of the smooth Schwartz crossed product S(R, A^∞, α) of the Frechet algebra A^∞ of C^∞-elements of A is isomorphic to the σ-C^*-crossed product C^*(R,E (A), α) of the enveloping σ-C^*-algebra E(A) of A by the induced action. When A is a hermitian Q-algebra, one gets K-theory isomorphism RK_*(S (R, A^∞, α)) =K _*(C^*(R, E(A), α) for the representable K-theory of Frechet algebras. An application to the differential structure of a C^*-algebra defined by densely defined differential seminorms is given.