Similarity problems, Folner sets and isometric representations of amenable semigroups

TL;DR

This paper explores the role of Folner sets in similarity problems for bounded linear operators, extending Sz.-Nagy's criteria and discussing analogs of the Dixmier-Day theorem for amenable groups and semigroups.

Contribution

It introduces new criteria for operator similarity and highlights the significance of Folner sets in the context of amenable groups and semigroups.

Findings

New criteria for operator similarity to isometries and unitaries

Extension of Dixmier-Day theorem to semigroups with Folner sets

Role of Folner sets in similarity problems for amenable structures

Abstract

We revisit Sz.-Nagy's criteria for similarity of Hilbert space bounded linear operators to isometries or unitaries and present new ones. We also discuss counterparts of the Dixmier-Day theorem concerning bounded representations of amenable groups and semigroups. We highlight the role of Folner sets in similarity problems in both settings of unimodular, $\sigma$-compact, amenable groups and in discrete semigroups possessing the Strong Folner condition (SFC).