The norm decreasing
homomorphisms φ of L1(F) into M(G) for locally compact groups F and G have
been characterized by F. P. Greenleaf using an integral representation. In this note
the authors improve and unify some of the results and proofs of structure theorems in
the previous literature. Necessary and sufficient conditions that φ have a canonical
factorization of a general type are expressed in terms of the extensibility of a
φ-associated character on a φ-related closed normal subgroup. In particular, an
explicit factorization of φ can be obtained when either F or G is Abelian. Also
investigated is the structure of norm decreasing homomorphisms φ with range in
L1(G).
