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 Lp-spaces studied by Pisier, Junge and Xu. This talk aims to give a gentle introduction to the aforementioned topic, and present some recent results on ergodic theorems for actions on von Neumann algebras by amenable groups. In particular, we established a quantum analogue of Lindenstrauss’s pointwise ergodic theorem. Our methods rely essentially on geometric constructions of martingales based on the Ornstein-Weiss quasi-tilings and harmonic analytic estimates coming from noncommutative Calderón-Zygmund theory. The talk is based on joint papers with Guixiang Hong, Ben Liao and Léonard Cadilhac.

会议密码:2022