Crossed products of 𝐶*-dynamical systems with ground states

Kusuda, Masaharu

doi:10.1090/S0002-9939-1983-0712636-9

Crossed products of $C^{\ast }$-dynamical systems with ground states
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by Masaharu Kusuda PDF

Proc. Amer. Math. Soc. 89 (1983), 273-278 Request permission

Abstract:

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 $\hat {\alpha }_{\lambda _1}(I) \subsetneqq \hat {\alpha }_{\lambda _2}(I)$. 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.

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