Abstract
AbstractMotivated by the definition of a semigroup compactication 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
$A^{\ast }$
with a compatible (matrix) norm and a type of left Arens product
$\Box$
. 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
$({\mathcal{F}}_{Q}(A^{\ast }),\Vert \cdot \Vert _{Q})$
and prove that
$({\mathcal{F}}_{Q}(A^{\ast })^{\ast },\Box )$
is the unique (operator) HLDBA over
$A$
for which there is a weak
$^{\ast }$
-continuous completely isometric representation as completely bounded operators on
$X^{\ast }$
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.
Publisher
Canadian Mathematical Society