Is there a noncommutative version of von Neumann's ergodic theorem.

Question

The two most celebrated ergodic theorems are Birkhoff's ergodic theorem and von Neumann's ergodic theorem.

E. C. Lance in his remarkable work (Ergodic Theorems for Convex Sets and Operator Algebras) formulated that can be considered to be the noncommutative version of Birkhoff's ergodic theorem, for a von Neumann algebra, $T*$-automorphism and a faithful $T$-invariant normal state.

I would like to know whether someone has done the same for von Neumann's ergodic theorem. In other words, is there a noncommutative version of von Neumann's ergodic theorem ?

I think it is more likely that you receive answers if you post this question in mathoverflow. — Eduardo Longa, Jun 22 '19 at 21:04
On MO at https://mathoverflow.net/questions/334592/is-there-a-noncommutative-version-of-von-neumanns-ergodic-theorem — Nate Eldredge, Jun 22 '19 at 23:08

score 8 · Accepted Answer · answered Jun 22 '19 at 21:15

8

Yes, and non-commutative ergodic theory is a big chunk of the industry of von Neumann algebras these days. Perhaps the starting point should be this paper by Yeadon:

F. J. Yeadon, Ergodic Theorems for Semifinite Von Neumann Algebras—I, Journal of the London Mathematical Society, s2-16, Issue 2 (1977), 326–332.

It gives you exactly what you seek. Part II of the paper as well as papers citing it is something you may be interested in.

answered Jun 22 '19 at 21:15

Tomasz Kania

16,996

Thank you so much. I am gonna take a look at the mentioned articles. – Neil hawking Jun 23 '19 at 11:26

Is there a noncommutative version of von Neumann's ergodic theorem.

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