Orthogonal ℓ1-sets and extreme non-Arens regularity of preduals of von Neumann algebras

Abstract

A Banach algebra is Arens-regular when all its continuous functionals are weakly almost periodic, in symbols when ⁎ . To identify the opposite behaviour, Granirer called a Banach algebra extremely non-Arens regular (enAr, for short) when the quotient ⁎ contains a closed subspace that has ⁎ as a quotient. In this paper we propose a simplification and a quantification of this concept. We say that a Banach algebra is r-enAr, with , when there is an isomorphism with distortion r of ⁎ into ⁎ . When , we obtain an isometric isomorphism and we say that is isometrically enAr. We then identify sufficient conditions for the predual ⁎ of a von Neumann algebra to be r-enAr or isometrically enAr. With the aid of these conditions, the following algebras are shown to be r-enAr: (i) the weighted semigroup algebra of any weakly cancellative discrete semigroup, when the weight is diagonally bounded with diagonal bound . When the weight is multiplicative, i.e., when , the algebra is isometrically enAr,

(ii) the weighted group algebra of any non-discrete locally compact infinite group and for any weight,

(iii) the weighted measure algebra of any locally compact infinite group, when the weight is diagonally bounded with diagonal bound . When the weight is multiplicative, i.e., when , the algebra is isometrically enAr.

The Fourier algebra of a locally compact infinite group G is shown to be isometrically enAr provided that (1) the local weight of G is greater or equal than its compact covering number, or (2) G is countable and contains an infinite amenable subgroup.