Vol. 15, No. 4, 1965

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Norm decreasing homomorphisms of group algebras

Frederick Paul Greenleaf

Vol. 15 (1965), No. 4, 1187–1219

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).

Mathematical Subject Classification
Primary: 46.80
Secondary: 42.56
Received: 24 April 1964
Revised: 1 August 1964
Published: 1 December 1965
Frederick Paul Greenleaf