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

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

2006-07-27

arxiv.org/abs/math/0607701v1

Mathematics

Operator Algebras

13 pages, 3 figures

Scientific paper

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.

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