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Noncommutative Maximal Inequality and Ergodic Theorem for Actions of Amenable Groups

Simons Semester on Dynamics via YouTube

Overview

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Watch a 58-minute mathematics lecture exploring the noncommutative maximal inequality and ergodic theorem for actions of amenable groups. Delve into the generalization of Birkhoff's pointwise theorem, examining both amenable group actions and noncommutative measure spaces. Learn how tools from Calderon-Zygmund theory are adapted to amenable groups, following a progression from basic concepts through advanced applications. Explore topics including maximal operators, noncommutative ergodic theory, martingale maximal inequalities, dyadic systems, and the geometry of local estimates. Understand the collaborative research between Léonard Cadilhac of Sorbonne Université and Simeng Wang of Harbin, which extends classical ergodic theory into more general mathematical frameworks.

Syllabus

Intro
The starting point
Question: ergodic theorem for group actions
Precising the question: amenable groups
Possible solutions to the problem
Maximal operators
To the noncommutative world
Examples
Noncommutative ergodic theorem: early days
Maximal norms in L
Martingale maximal inequality
Dyadic martingale and averages
Construction of dyadic systems
Difference operator in amenable groups
Steps of the proof
Geometry of local estimates

Taught by

Simons Semester on Dynamics

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