We introduce and study "norm-multiplicative" homomorphisms ϕ : L 1 (F ) → M r (G) between group and measure algebras, and ϕ : L 1 (ω F ) → M(ω G ) between Beurling group and measure algebras, where F and G are locally compact groups with continuous weights ω F and ω 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 ϕ : L 1 (F ) → M r (G). We state versions of our results that describe a variety of (possibly unbounded) homomorphisms ϕ : CF → CG for (discrete) groups F and G.
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