Ergodic theory for quantum semigroups

TL;DR

This paper extends ergodic theory to quantum semigroups by generalizing the Jacobs-de Leeuw-Glicksberg splitting theorem within the framework of Hopf-von Neumann algebras, introducing new concepts of almost periodicity.

Contribution

It introduces a novel approach to ergodic theory for quantum semigroups, generalizing classical results to the setting of Hopf-von Neumann algebras.

Findings

Established a splitting theorem for quantum semigroup actions on von Neumann algebras.

Developed a new notion of almost periodic vectors and operators for quantum settings.

Extended classical ergodic results to non-commutative quantum frameworks.

Abstract

Recent results of L. Zsido, based on his previous work with C. P. Niculescu and A. Stroh, on actions of topological semigroups on von Neumann algebras, give a Jacobs-de Leeuw-Glicksberg splitting theorem at the von Neumann algebra (rather than Hilbert space) level. We generalize this to the framework of actions of quantum semigroups, namely Hopf-von Neumann algebras. To this end, we introduce and study a notion of almost periodic vectors and operators that is suitable for our setting.