Norm-multiplicative homomorphisms of Beurling algebras

Abstract

We introduce and study “norm-multiplicative” homomorphisms φ:L1(F)→Mr(G) between group and measure algebras, and φ:L1(ωF)→M(ωG) between Beurling group and measure algebras, where F and G are locally compact groups with continuous weights ωF and ωG. 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 φ:L1(F)→Mr(G). We state versions of our results that describe a variety of (possibly unbounded) homomorphisms φ:CF→CG for (discrete) groups F and G.