Representations of Banach algebras subordinate to topologically introverted spaces

Abstract:

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