Crossed products of $C\sp{\ast}$-dynamical systems with ground states

Let ${C^*}(A,{\mathbf{R}})$ be the crossed product associated with a ${C^*}$-dynamical system $(A,{\mathbf{R}},\alpha )$. We show that if there is a ground state for $A$. then there is an ideal $I$ in ${C^*}(A,{\mathbf{R}})$ such that if ${\lambda _1} < {\lambda _2}$, then ${\hat \alpha _{{\lambda _1}}}(I)\mathop \subset \limits_ \ne {\hat \alpha _{{\lambda _2}}}(I)$. where $\hat \alpha$ is the dual automorphism group on ${C^*}(A,R)$, and such that ${C^*}(A,{\mathbf{R}}) = { \cup _{\lambda \geqslant 0}}{\hat \alpha _\lambda }(I)$. Conversely, we show that if $A$ is unital, and if ${C^*}(A,{\mathbf{R}})$ has such a decomposition for some proper ideal $I$. then $A$ has a ground state.