Reversibility and Banach Algebras

TL;DR

This paper reviews the structure of groups with reversible elements and explores related questions in the context of Banach algebras, highlighting the interplay between algebraic properties and functional analysis.

Contribution

It provides a survey of known results on reversible elements in groups and introduces new questions about their role in Banach algebra groups.

Findings

Reversible elements are conjugate to their inverses in groups.

The paper discusses structural properties involving reversible elements.

Open questions are posed regarding groups associated with Banach algebras.

Abstract

An element $g$ of a group is called {\em reversible} if it is conjugate in the group to its inverse. In this paper we review some results about the structure of groups involving the reversible elements and we pose some questions about groups associated to a Banach algebra.