Homomorphisms of convolution algebras

We establish an explicit, algebraic, one-to-one correspondence between the *-homomorphisms, (sic): L-1 (F) -> M (G), of group and measure algebras over locally compact groups F and G, and group homomorphisms, phi : F -> M-phi, where M-phi is a semi-topological subgroup of (M (G), omega*). We show how to extend any such *-homomorphism to a larger convolution algebra to obtain nicer continuity properties. We augment Greenleaf's 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 Cohen's 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 (sic): L1 (F) -> M (G) into four homomorphisms, where each of the four factors is one of the natural types appearing in the Cohen factorization. (C) 2011 Elsevier Inc. All rights reserved.