Enveloping $\sigma$-$C^*$-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 $\alpha$ 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.