Involutions and Trivolutions in Algebras Related to Second Duals of Group Algebras

TL;DR

This paper introduces the concept of trivolutions in complex algebras, explores their natural occurrence in Banach and group algebras, and investigates conditions for their existence on dual spaces.

Contribution

It defines trivolutions, characterizes their properties, and examines their presence in duals of topologically introverted spaces, especially in relation to group algebras.

Findings

Trivolutions naturally occur in many Banach algebras.

Conditions for the existence of involutions on dual spaces are established.

Examples demonstrate when trivolutions can or cannot exist on duals.

Abstract

We define a trivolution on a complex algebra $A$ as a non-zero conjugate-linear, anti-homomorphism $\tau$ on $A$, which is a generalized inverse of itself, that is, $\tau^3=\tau$. We give several characterizations of trivolutions and show with examples that they appear naturally on many Banach algebras, particularly those arising from group algebras. We give several results on the existence or non-existence of involutions on the dual of a topologically introverted space. We investigate conditions under which the dual of a topologically introverted space admits trivolutions.