Amenable partial actions

TL;DR

This paper develops the theory of amenability for partial group actions and representations, establishing connections between amenability of actions, representations, and induced structures in various mathematical contexts.

Contribution

It introduces new notions of amenability for partial actions and representations, and explores their properties and relationships, including induction and measure-theoretic aspects.

Findings

Amenability of partial actions is characterized via Koopman representations.

A partial action on a measure space is amenable iff its Koopman representation is amenable.

Induction preserves amenability properties in partial representations.

Abstract

We introduce and study various notions of amenability continuous (Borel) partial actions of locally compact (Borel) groups $G$ on topological (standard Borel) spaces. We also study amenability of partial representations of a locally compact group in a Banach space and show that a partial action on a measure space is amenable iff the corresponding Koopman partial representation on the corresponding $L^2$-space is amenable. We introduce the notion of induced partial representation from a closed subgroup and explore perseverance of amenability type properties under induction.