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 π a continuous representation of A on Y . We call π subordinate to X if each coordinate function π y,λ ∈ X, for all y ∈ Y, λ ∈ 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 W AP (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 LU C(A) are all coordinate functions of some norm continuous representation of A on a dual Banach space.

Fourier Spaces and Completely Isometric Representations of Arens Product Algebras

“…Note that a homogeneous subspace of A * will often fail to be closed in A * . When • S is the dual (operator space matrix) subspace norm • A * on a closed subspace S(A * ) of A * , our definition of a left introverted homogeneous space agrees with the the usual definition of a left introverted subspace of A * [8,11]. The statement of the next proposition includes the introduction of some notation and terminology.…”

“…Many of the most well-studied and basic objects associated with a locally compact group G -more generally a locally compact quantum group -are introverted subspaces of L 1 (G) * = L ∞ (G) and their dual spaces under an Arens product: examples include the introverted space of continuous functions vanishing at infinity, C 0 (G) (its dual with Arens product is the measure algebra M (G) with convolution product); the introverted space of continuous almost periodic functions on G, AP (G); the introverted space of continuous Eberlein functions on G, E(G); the introverted space of continuous weakly almost periodic functions on G, W AP (G); the left introverted space of left uniformly continuous functions on G, LU C(G); and L ∞ (G). A small sample of papers in which the duals of these and other spaces are studied as left Arens product algebras is [1,8,11,13,15,16,18,19,22].…”

Almost periodic functionals and finite-dimensional representations

“…It is easy to see that W is a neighborhood of 0 in H. In fact, since ξ is algebraically cyclic, we have π(A)ξ = H and hence the map Γ : It remains to show that W is relatively norm compact in H. This is equivalent to showing that for every sequence {a n } in B A , the sequence {π(a n )ξ} in W has a cluster point in H. Identifying {a n } with its natural image in A * * , let Φ ∈ A * * be a w * -cluster point of {a n }, and {a α } be a subnet such that a α → Φ in the w * -topology of A * * . Let π be the normal representation of A * * on H extending π (Filali, Neufang, and Monfared [14,Theorem 3.3]). We shall show that for a suitable subnet…”

Tame functionals on Banach algebras

“…Weakly almost periodic (WAP) functionals play a major role in the theory of Banach algebras. See for example Young [29], Lau [19], Filali, Neufang and Monfared [3] and references therein. Recall that a functional λ ∈ A * on a Banach algebra A is said to be WAP (and write λ ∈ WAP(A)) if the natural linear operator…”