Linear preservers on idempotents of Fourier algebras

TL;DR

This paper characterizes bounded linear operators on Fourier algebras that preserve idempotents, showing they are induced by group homomorphisms or anti-homomorphisms, especially in totally disconnected groups.

Contribution

It provides a new representation of idempotent-preserving operators on Fourier algebras and links their structure to group homomorphisms and anti-homomorphisms.

Findings

Operators preserving idempotents are induced by group homomorphisms or anti-homomorphisms.

Positive or contractive operators are characterized by group structure.

In totally disconnected groups, bounded homomorphisms correspond to idempotent-preserving operators.

Abstract

In this article, we give a representation of bounded complex linear operators which preserve idempotent elements on the Fourier algebra of a locally compact group. When such an operator is moreover positive or contractive, we show that the operator is induced by either a continuous group homomorphism or a continuous group anti-homomorphism. If the groups are totally disconnected, bounded homomorphisms on the Fourier algebra can be realised by the idempotent preserving operators.