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Transformations induced in the state space of a $ C\sp \ast$-algebra and related ergodic theorems


Author: A. Łuczak
Journal: Proc. Amer. Math. Soc. 96 (1986), 617-625
MSC: Primary 46L50
DOI: https://doi.org/10.1090/S0002-9939-1986-0826491-2
MathSciNet review: 826491
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Abstract: Let $ A$ be a norm-separable $ {C^*}$-algebra with unit 1, $ \sigma $-weakly dense in a $ {W^*}$-algebra $ M$, and let $ \alpha $ be a positive linear mapping of $ M$ into itself leaving 1 invariant. We show that $ \alpha $ induces a transformation $ \tilde \alpha $ defined "almost everywhere" on the state space $ \sigma $ of $ A$ with values in $ \sigma $. If $ \alpha $ is a $ ^*$-automorphism of $ M$, then there exists

$\displaystyle \mathop {\lim }\limits_{N \to \infty } \frac{1}{N}\sum\limits_{n = 0}^{N - 1} {{{\tilde \alpha }_n}\psi } $

for "almost all" states $ \psi $ of $ A$, where $ {\tilde \alpha _n}$ are transformations on $ \sigma $ induced by $ {\alpha ^n}$.

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DOI: https://doi.org/10.1090/S0002-9939-1986-0826491-2
Keywords: Convergence in von Neumann algebras, state space of a $ {C^*}$-algebra, ergodic theorems
Article copyright: © Copyright 1986 American Mathematical Society

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