Norm-multiplicative homomorphisms of Beurling algebras

TL;DR

This paper introduces and analyzes norm-multiplicative homomorphisms between Beurling and group measure algebras, providing a unified framework that generalizes and strengthens existing results, including a description of positive homomorphisms.

Contribution

It offers a comprehensive study of norm-multiplicative homomorphisms in Beurling algebras, including the first characterization of positive homomorphisms and extensions to discrete groups.

Findings

Unified approach to homomorphisms between Beurling algebras

First description of positive homomorphisms between group algebras

Generalizations to unbounded homomorphisms for discrete groups

Abstract

We introduce and study "norm-multiplicative" homomorphisms $\varphi: {\cal L}^1(F) \rightarrow {\cal M}_r(G)$ between group and measure algebras, and $\varphi: {\cal L}^1(\omega_F) \rightarrow {\cal M}(\omega_G)$ between Beurling group and measure algebras, where $F$ and $G$ are locally compact groups with continuous weights $\omega_F$ and $\omega_G$. Through a unified approach we recover, and sometimes strengthen, many of the main known results concerning homomorphisms and isomorphisms between these (Beurling) group and measure algebras. We provide a first description of all positive homomorphisms $\varphi: {\cal L}^1(F) \rightarrow {\cal M}_r(G)$. We state versions of our results that describe a variety of (possibly unbounded) homomorphisms $\varphi: \mathbb{C} F \rightarrow \mathbb{C} G$ for (discrete) groups $F$ and $G$.