Enveloping $\sigma$-$C^{\ast }$-algebra of a smooth Frechet algebra crossed product by $R$, $K$-theory and differential structure in $C^{\ast }$-algebras

Abstract

Given an $m$-tempered strongly continuous action $\alpha$ of $\mathbb{R}$ by continuous ${}^{\ast }$-automorphisms of a Frechet ${}^{\ast }$-algebra $A$, it is shown that the enveloping \hbox{$\sigma$-$C^{\ast }$-algebra} $E(S(\mathbb{R},A^{\infty },\alpha ))$ of the smooth Schwartz crossed product $S(\mathbb{R},A^{\infty },\alpha )$ of the Frechet algebra $A^{\infty }$ of $C^{\infty }$-elements of $A$ is isomorphic to the \hbox{$\sigma$-$C^{\ast }$-crossed} product $C^{\ast }(\mathbb{R},E(A),\alpha )$ of the enveloping $\sigma$-$C^{\ast }$-algebra $E(A)$ of $A$ by the induced action. When $A$ is a hermitian $Q$-algebra, one gets $K$-theory isomorphism $R{K}_{\ast }(S(\mathbb{R},A^{\infty },\alpha ))=K_{\ast }(C^{\ast }(\mathbb{R},E(A),\alpha )$ for the representable $K$-theory of Frechet algebras. An application to the differential structure of a $C^{\ast }$-algebra defined by densely defined differential seminorms is given.

arXiv comment13 pages, 3 figures