Fourier spaces and completely isometric representations of Arens product algebras

Abstract

Motivated by the definition of a semigroup compactification of a locally compact group and a large collection of examples, we introduce the notion of an (operator) "homogeneous left dual Banach algebra" (HLDBA) over a (completely contractive) Banach algebra $A$. We prove a Gelfand-type representation theorem showing that every HLDBA over $A$ has a concrete realization as an (operator) homogeneous left Arens product algebra: the dual of a subspace of $A^*$ with a compatible (matrix) norm and a type of left Arens product ${\scriptscriptstyle \square}$. Examples include all left Arens product algebras over $A$, but also -- when $A$ is the group algebra of a locally compact group -- the dual of its Fourier algebra. Beginning with any (completely) contractive (operator) $A$-module action $Q$ on a space $X$, we introduce the (operator) Fourier space $({\cal F}_Q(A^*), \| \cdot \|_Q)$ and prove that $({\cal F}_Q(A^*)^*, {\scriptscriptstyle \square})$ is the unique (operator) HLDBA over $A$ for which there is a weak$^*$-continuous completely isometric representation as completely bounded operators on $X^*$ extending the dual module representation. Applying our theory to several examples of (completely contractive) Banach algebras $A$ and module operations, we provide new characterizations of familiar HLDBAs over $A$ and we recover -- and often extend -- some (completely) isometric representation theorems concerning these HLDBAs.