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.
