Given an $m$-tempered strongly continuous action $\alpha$ of $\R$ by continuous $^{ *}$-automorphisms of a Frechet $^{ *}$-algebra $A$, it is shown that the enveloping \hbox{ $\sigma$-$C^{ *}$-algebra} $E(S(\R,A^{ \infty},\alpha))$ of the smooth Schwartz crossed product $S(\R,A^{ \infty},\alpha)$ of the Frechet algebra $A^{ \infty}$ of $C^{ \infty}$-elements of $A$ is isomorphic to the \hbox{ $\sigma$-$C^{ *}$-crossed} product $C^{ *}(\R,E(A),\alpha)$ of the enveloping $\sigma$-$C^{ *}$-algebra $E(A)$ of $A$ by the induced action. When $A$ is a hermitian $Q$-algebra, one gets $K$-theory isomorphism $RK_{ *}(S(\R,A^{ \infty},\alpha)) = K_{ *}(C^{ *}(\R,E(A),\alpha)$ 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.
Categories: math.OA