Fourier Spaces and Completely Isometric Representations of Arens Product Algebras

Motivated 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.

Complementation of the subspace of G-invariant vectors

Let [Formula: see text] be an isometric representation of a group [Formula: see text] in a Banach space [Formula: see text] over a normalizing non-discrete absolute valued division ring [Formula: see text]. If [Formula: see text] and [Formula: see text] are supportive and [Formula: see text] verifies the separation property, then [Formula: see text] is 1-complemented in [Formula: see text] along [Formula: see text]. As an immediate consequence, in an isometric representation of a group in a smooth Banach space whose dual is also smooth, the subspace of [Formula: see text]-invariant vectors is [Formula: see text]-complemented.

ON ISOMETRIC REPRESENTATION SUBSETS OF BANACH SPACES

Let $X,Y$ be two Banach spaces and $B_{X}$ the closed unit ball of $X$. We prove that if there is an isometry $f:B_{X}\rightarrow Y$ with $f(0)=0$, then there exists an isometry $F:X\rightarrow Y^{\ast \ast }$. If, in addition, $Y$ is weakly nearly strictly convex, then there is an isometry $F:X\rightarrow Y$. Making use of these results, we show that if $Y$ is weakly nearly strictly convex and there is an isometry $f:B_{X}\rightarrow Y$ with $f(0)=0$, then there exists a linear isometry $S:X\rightarrow Y$.

Representing Multipliers of the Fourier Algebra on Non-Commutative Lp Spaces

We show that the multiplier algebra of the Fourier algebra on a locally compact group G can be isometrically represented on a direct sum on non-commutative Lp spaces associated with the right von Neumann algebra of G. The resulting image is the idealiser of the image of the Fourier algebra. If these spaces are given their canonical operator space structure, then we get a completely isometric representation of the completely bounded multiplier algebra. We make a careful study of the noncommutative Lp spaces we construct and show that they are completely isometric to those considered recently by Forrest, Lee, and Samei. We improve a result of theirs about module homomorphisms. We suggest a definition of a Figa-Talamanca–Herz algebra built out of these non-commutative Lp spaces, say . It is shown that is isometric to L1(G), generalising the abelian situation.

A REPRESENTATION THEOREM FOR LOCALLY COMPACT QUANTUM GROUPS

Recently, Neufang, Ruan and Spronk proved a completely isometric representation theorem for the measure algebra M(G) and for the completely bounded (Herz–Schur) multiplier algebra McbA(G) on [Formula: see text], where G is a locally compact group. We unify and generalize both results by extending the representation to arbitrary locally compact quantum groups 𝔾 = (M, Γ, φ, ψ). More precisely, we introduce the algebra [Formula: see text] of completely bounded right multipliers on L1(𝔾) and we show that [Formula: see text] can be identified with the algebra of normal completely bounded [Formula: see text]-bimodule maps on [Formula: see text] which leave the subalgebra M invariant. From this representation theorem, we deduce that every completely bounded right centralizer of L1(𝔾) is in fact implemented by an element of [Formula: see text]. We also show that our representation framework allows us to express quantum group "Pontryagin" duality purely as a commutation relation.