The homomorphisms φ
of the group algebra L1(F) into the algebra M(G) of measures, where F
and G are locally compact groups, has been completely determined when
both groups are abelian by P. J. Cohen, and when G is compact and the
homomorphism is norm decreasing and order-preserving by Glicksberg. In this
paper the structure of norm decreasing homomorphisms φ is determined for
arbitrary locally compact F and G. As an application the special structure of
all norm decreasing monomorphisms is determined, along with the rather
elegant structure of all norm decreasing homomorphisms mapping L1(F) onto
L1(G).
The analysis is effected by finding all multiplicative subgroups of the unit ball of
measures on a locally compact group for, as we show, each φ extends to a norm
decreasing homomorphism φ : M(F) → M(G), and is determined by the image under
φ of the group of point masses on G, a multiplicative subgroup of the unit ball in
M(G).
