  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.
