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Quasi‐Uniform Ergodic Theorems in von Neumann Algebras

Quasi‐Uniform Ergodic Theorems in von Neumann Algebras QUASI-UNIFORM ERGODIC THEOREMS IN VON NEUMANN ALGEBRAS DENES PETZ The first "almost everywhere" ergodic theorem in the von Neumann algebra context was proved by E. C. Lance [7] and by Ja. G. Sinai and V. V. Anselevic [16]. Namely, if J / is a von Neumann algebra with a faithful normal state # and a is an n - 1 l l automorphism of si such that <j> is a-invariant then the means s {a) = n~ £ oc (A) t = 0 converge in some sense. The notion of "pointwise" convergence used here is the following. A bounded sequence {A ) <= si converges to 0 ^-almost uniformly if for every 6 > 0 there is a projection E e si such that on the one hand $(/ — £ ) < e and on the other hand \\A E\\ -> 0 as n -> oo. When si is commutative, si = L°°(X, pi) with a probability space {X,y,\ this convergence coincides with almost everywhere pointwise convergence via Egorov's theorem. Almost uniform convergence has proved to be useful and several classical theorems have been generalized in a von Neumann algebra setting by means of it (see, for example, http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Quasi‐Uniform Ergodic Theorems in von Neumann Algebras

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/16.2.151
Publisher site
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Abstract

QUASI-UNIFORM ERGODIC THEOREMS IN VON NEUMANN ALGEBRAS DENES PETZ The first "almost everywhere" ergodic theorem in the von Neumann algebra context was proved by E. C. Lance [7] and by Ja. G. Sinai and V. V. Anselevic [16]. Namely, if J / is a von Neumann algebra with a faithful normal state # and a is an n - 1 l l automorphism of si such that <j> is a-invariant then the means s {a) = n~ £ oc (A) t = 0 converge in some sense. The notion of "pointwise" convergence used here is the following. A bounded sequence {A ) <= si converges to 0 ^-almost uniformly if for every 6 > 0 there is a projection E e si such that on the one hand $(/ — £ ) < e and on the other hand \\A E\\ -> 0 as n -> oo. When si is commutative, si = L°°(X, pi) with a probability space {X,y,\ this convergence coincides with almost everywhere pointwise convergence via Egorov's theorem. Almost uniform convergence has proved to be useful and several classical theorems have been generalized in a von Neumann algebra setting by means of it (see, for example,

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Mar 1, 1984

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