Representations of Banach algebras subordinate to topologically introverted spaces

Abstract

Let A A be a Banach algebra, X X a closed subspace of A ∗ A^* , Y Y a dual Banach space with predual Y ∗ Y_* , and π \pi a continuous representation of A A on Y Y . We call π \pi subordinate to X X if each coordinate function π y , λ ∈ X \pi _{y,\lambda }\in X , for all y ∈ Y , λ ∈ Y ∗ y\in Y, \lambda \in Y_* . If X X is topologically left (right) introverted and Y Y is reflexive, we show the existence of a natural bijection between continuous representations of A A on Y Y subordinate to X X , and normal representations of X ∗ X^* on Y Y . We show that if A A has a bounded approximate identity, then every weakly almost periodic functional on A A is a coordinate function of a continuous representation of A A subordinate to W A P ( A ) WAP(A) . We show that a function f f on a locally compact group G G is left uniformly continuous if and only if it is the coordinate function of the conjugate representation of L 1 ( G ) L^1(G) , associated to some unitary representation of G G . We generalize the latter result to an arbitrary Banach algebra with bounded right approximate identity. We prove the functionals in L U C ( A ) LUC(A) are all coordinate functions of some norm continuous representation of A A on a dual Banach space.