  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.
