Isometric Jordan isomorphisms of group algebras

TL;DR

This paper characterizes contractive Jordan isomorphisms between group algebras as either isometric isomorphisms or anti-isomorphisms, and explores their applications to zero product preservers and automorphisms.

Contribution

It establishes a complete classification of contractive Jordan isomorphisms of group algebras and applies this to study related automorphisms and zero product preservers.

Findings

Contractive Jordan isomorphisms are either isometric isomorphisms or anti-isomorphisms.

Application to zero product preservers on group algebras.

Analysis of local and approximately local isometric automorphisms.

Abstract

Let $G$ and $H$ be locally compact groups. We will show that each contractive Jordan isomorphism $\Phi\colon L^1(G)\to L^1(H)$ is either an isometric isomorphism or an isometric anti-isomorphism. We will apply this result to study isometric two-sided zero product preservers on group algebras and, further, to study local and approximately local isometric automorphisms of group algebras.