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

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Given an m-tempered strongly continuous action α of by continuous *-automorphisms of a Frechet *-algebra A, it is shown that the enveloping σ-C*-algebra E(S(,A∞,α)) of the smooth Schwartz crossed product S(,A∞,α) of the Frechet algebra A∞ of C∞-elements of A is isomorphic to the σ-C*-crossed product C*(,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(,A∞,α)) = K*(C*(,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.