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

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 at 21:04
• – Nate Eldredge Jun 22 at 23:08