Individual ergodic theorems on von Neumann algebras

Abstract: Birkhoff’s celebrated individual ergodic theorem asserts that for a measure-preserving ergodic transformation on a measure space, the time average is equal to the space average almost everywhere. Since the theory of von Neumann algebras is a quantum analogue of the classical measure theory, it is natural to study similar individual ergodic theorems in the setting of von Neumann algebras. The study was exactly initiated by Lance in 1970s, and witnessed fruitful progress in recent decades with the help of modern tools from the operator space theory, such as the noncommutative vector-valued $L^p$-spaces studied by Pisier, Junge and Xu. This talk aims to give a gentle introduction to the aforementioned topic, and if time permits, we may also present some recent results in this direction, in particular ergodic theorems for some group actions on von Neumann algebras and for positive contractions on $L^p$-spaces, which is joint work with Guixiang Hong, Ben Liao and Samya Ray.

functional analysisgroup theoryoperator algebras

Audience: researchers in the topic

Groups, Operators, and Banach Algebras Webinar

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