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


Author: Masaharu Kusuda
Journal: Proc. Amer. Math. Soc. 89 (1983), 273-278
MSC: Primary 46L55; Secondary 46L40
DOI: https://doi.org/10.1090/S0002-9939-1983-0712636-9
MathSciNet review: 712636
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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)\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.


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DOI: https://doi.org/10.1090/S0002-9939-1983-0712636-9
Article copyright: © Copyright 1983 American Mathematical Society