Invariant states

Herman, Richard H.

doi:10.1090/S0002-9947-1971-0281013-6

Invariant states
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by Richard H. Herman

Trans. Amer. Math. Soc. 158 (1971), 503-512

DOI: https://doi.org/10.1090/S0002-9947-1971-0281013-6

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Abstract:

States of a ${C^ \ast }$-algebra invariant under the action of a group of automorphisms of the ${C^ \ast }$-algebra are considered. It is shown that “clustering” states in the same part are equal and thus the same is true of extremal invariant states under suitable conditions. The central decomposition of an invariant state is considered and it is shown that the central measure is mixing if and only if the state satisfies a strong notion of clustering. Under transitivity of the central measure and some reasonable restrictions, the central decomposition is the ergodic decomposition of the state with respect to the isotropy subgroup.

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