Evanescent affine isometric actions and weak identity excluding groups

TL;DR

This paper explores the properties of evanescent affine isometric actions on Hilbert spaces, their decomposition, and their relation to identity excluding groups and von Neumann algebras.

Contribution

It introduces the concept of evanescence for affine actions, analyzes their unique decomposition, and connects these properties to group theory and operator algebras.

Findings

Evanescent actions are opposite to irreducible actions.

Decomposition into evanescent and irreducible parts can be unique.

Almost fixed point actions are often evanescent.

Abstract

We investiguate a property of affine isometric actions on Hilbert spaces called evanescence. Evanescent actions are the extreme opposite of irreducible actions. Every affine isometric action decomposes naturally into an evanescent part and an irreducible part. We study when this decomposition is unique. We also study when an action that has almost fixed points is automatically evanescent. We relate these questions to the identity excluding property for groups. We also relate them to the finiteness of the von Neumann algebras generated by the linear part of the action.