Vol. 38, No. 3, 1971

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On algebra actions on a group algebra

J. E. Kerlin

Vol. 38 (1971), No. 3, 669–680

A complete description of all algebra actions of the group algebra L1(K) on the group algebra L1(G)[M(G)] for locally compact Abelian groups K and G is presented. A fundamental algebra action of L1(K) on L1(G) is that induced by a continuous homomorphism 𝜃 : K G via a generalized convolution; such actions have been considered by Gelbaum in characterizing topological tensor products of group algebras. It is shown in this paper that conversely every algebra action of L1(K) on L1(G)[M(G)] is induced by a necessarily continuous homomorphism of K into the quotient of G by a compact subgroup. The analysis is based on a representation theorem for algebra actions on L1(G) for general locally compact group G. Namely, every algebra action of a Banach algebra C on L1(G) is the composition of a necessarily continuous cetral homomorphism Ψ of C into M(G) and convolution in M(G): c a = Ψ(c) a for all c C and a L1(G). Applications to lopological tensor products of group algebras are announced.

Mathematical Subject Classification 2000
Primary: 43A20
Received: 28 September 1970
Published: 1 September 1971
J. E. Kerlin