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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}$.

References [Enhancements On Off] (What's this?)

  • [1] O. Bratelli and D. W. Robinson, Operator algebras and quantum statistical mechanics. I, Springer-Verlag, Berlin and New York, 1979.
  • [2] M. S. Goldstein, Theorems on almost everywhere convergence in von Neumann algebras, J. Operator Theory 6 (1981), 233-311. (Russian) MR 643693 (84g:46096)
  • [3] E. C. Lance, Ergodic theorems for convex sets and operator algebras, Invent. Math. 37 (1976), 201-214. MR 0428060 (55:1089)
  • [4] A. Paszkiewicz, Convergences in $ {W^*}$-algebras, J. Funct. Anal. (to appear). MR 865218 (87m:46130)
  • [5] D. Petz, Quasi-uniform ergodic theorems in von Neumann algebras, Bull. London Math. Soc. 16 (1984), 151-156. MR 737243 (85c:46066)
  • [6] C. Radin, Automorphism of von Neumann algebras as point transformations, Proc. Amer. Math. Soc. 39 (1973), 343-346. MR 0313829 (47:2383)
  • [7] -, Pointwise ergodic theory on operator algebras, J. Math. Phys. 19 (1978), 1983-1985. MR 0495989 (58:14621)
  • [8] M. Takesaki, Theory of operator algebras. I, Springer-Verlag, Berlin and New York, 1979. MR 548728 (81e:46038)

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Additional Information

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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