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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 ?

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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:

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.

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  • $\begingroup$ Thank you so much. I am gonna take a look at the mentioned articles. $\endgroup$ – Neil hawking Jun 23 at 11:26

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