Operator Algebras and Dual Spaces

1. An isomorphism theorem. In this paper we discuss a particular method of introducing a convolution product into certain topological linear spaces so that they become algebras. The space in each case will be essentially the dual space of a linear space of continuous functions, and arises in the study of general group algebras and in the theory of distributions. Let G be a topological group, written additively but not necessarily abelian. Let C*(G) be the linear space of all complex-valued continuous functions on G. For xEG let Ux be the left translation operator which sends a function into the function Ux=x whose value at t is (x+t). Let X be a subspace of C*(G), topologized by a locally convex topology r. Let L(X) be the space of linear functionals on X which are r continuous on all r bounded sets of X. Let B(X) be the algebra of linear operators on X which are r bounded and r continuous on the t bounded sets of X. These contain the usual dual space X' and endomorphism algebra E(X) and in certain important cases coincide with them, for example if r is a norm topology. The operators Ux form a group G0 of operators on C*(G) anti-isomorphic to G. If X is invariant under Go and r suitable, we may regard Go as part of B(X). In this case, let B°(X) be the set of all T in B(X) which commute with Go. B°(X) is evidently a subalgebra of B(X). Our first theorem shows that under certain conditions L(X) and B°(X) are isomorphic as linear spaces, thus permitting us to introduce an algebraic structure into L(X). We first define two fundamental mappings between operators and functionals. For any operator T in B(X), let £ be a functional on X defined by (i) £(*) = r(*)(0).