Homomorphisms of convolution algebras

We establish an explicit, algebraic, one-to-one correspondence between the ⁎-homomorphisms, φ:L1(F)→M(G), of group and measure algebras over locally compact groups F and G, and group homomorphisms, ϕ:F→Mϕ, where Mϕ is a semi-topological subgroup of (M(G),w⁎). We show how to extend any such ⁎-homomorphism to a larger convolution algebra to obtain nicer continuity properties. We augment Greenleafʼs characterization of the contractive subgroups of M(G) (Greenleaf, 1965 [17], ) by completing the description of their topological structures. We show that not every contractive homomorphism has the dual form of Cohenʼs factorization in the abelian case, thus answering a question posed by Kerlin and Pepe (1975) in [27]. We obtain an alternative factorization of any contractive homomorphism φ:L1(F)→M(G) into four homomorphisms, where each of the four factors is one of the natural types appearing in the Cohen factorization.