Periodicity of functionals and representations of normed algebras on reflexive spaces

Abstract

It is a well-known fact that any normed algebra can be represented isometrically as an algebra of operators with the operator norm. As might be expected from the very universality of this property, it is little used in the study of the structure of an algebra. Far more helpful are representations on Hilbert space, though these are correspondingly hard to come by: isometric representations on Hilbert space are not to be expected in general, and even continuous nontrivial representations may fail to exist. The purpose of this paper is to examine a class of representations intermediate in both availability and utility to those already mentioned—namely, representations on reflexive spaces. There certainly are normed algebras which admit isometric representations of the latter type but have not even faithful representations on Hilbert space: the most natural example is the algebra of all continuous linear operators on E where E = lp with 1 < p ≠ 2 < ∞, for Berkson and Porta proved in (2) that if E, F are taken from the spaces lp with 1 < p < ∞ and E ≠ F then the only continuous homomorphism from into is the zero mapping. On the other hand there are also algebras which have no continuous nontrivial representation on any reflexive space—for example the algebra of finite-rank operators on an irreflexive Banach space (see Berkson and Porta (2) or Barnes (1) or Theorem 3, Corollary 1 below).

Some geometric properties on the Fourier and Fourier-Stieltjes algebras of locally compact groups, Arens regularity and related problems

“…Indeed, if A is any Arens regular Banach algebra, then, as one can see very easily, A* c wap(A**). Now it is enough to remark that the operator algebra A = K(LP(G)) is Arens regular (see [72] or [75]) and A** = B(LP(G)), see [21,Chapter VIII]. n Corollary 8.5.…”

Weakly almost periodic functionals on the measure algebra

“…The following result was first shown by Young in [21], building upon [5], and was recast in terms of the real interpolation method by Kaiser in [12] (see also the similar arguments in [6]). …”

“…The connection between weakly almost periodic functionals and representations of Banach algebras goes back to Young, [21], and Kaiser, [12]. For L 1 (G), there is a correspondence between (non-degenerate) representations of L 1 (G) and representations of G. Using Young and Kaiser's work, it is easy to see that weakly almost periodic functionals on L 1 (G) correspond to coefficient functionals for representations of G on reflexive spaces.…”

Arens Regularity of the Algebra of Operators on a Banach Space

“…It is known that if B(E) is Arens regular, then E is reflexive (see Proposition 8 below). Thus there is some motivation for the belief that B(E) should be Arens regular for E = l p , 1 < p < ∞; indeed, this question was raised in [12]. We shall show that if E is a super-reflexive Banach space (defined below), then B(E) is Arens regular.…”