Norm-multiplicative homomorphisms of Beurling algebras

Abstract

We introduce and study "norm-multiplicative" homomorphisms $\phi :{\mathcal{L}}^1(F)\to {\mathcal{M}}_r(G)$ between group and measure algebras, and $\phi :{\mathcal{L}}^1({\omega }_F)\to \mathcal{M}({\omega }_G)$ 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 $\phi :{\mathcal{L}}^1(F)\to {\mathcal{M}}_r(G)$. We state versions of our results that describe a variety of (possibly unbounded) homomorphisms $\phi :\mathbb{C}F\to \mathbb{C}G$ for (discrete) groups $F$ and $G$.

26 pages