Abstract

A Banach algebra \(\mathscr{A}\) is Arens-regular when all its continuous functionals are weakly almost periodic, in symbols when \(\mathscr{A^{∗}} = \mathscr{WAP}(\mathscr{A})\). To identify the opposite behaviour, Granirer called a Banach algebra extremely non-Arens regular (enAr, for short) when the quotient \(\mathscr{A^{∗}} /\mathscr{WAP}(\mathscr{A})\) contains a closed subspace that has \(\mathscr{A^{∗}}\) as a quotient. In this paper we propose a simplification and a quantification of this concept. We say that a Banach algebra \(\mathscr{A}\) is r-enAr, with \(r ≥ 1\), when there is an isomorphism with distortion \(r\) of \(\mathscr{A^{∗}}\) into \(\mathscr{A^{∗}} /\mathscr{WAP}(\mathscr{A})\). When \(r = 1\), we obtain an isometric isomorphism and we say that \(\mathscr{A}\) is isometrically enAr. We then identify sufficient conditions for the predual \(\mathfrak{V_{∗}}\) of a von Neumann algebra \(\mathfrak{V}\) 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 \(c ≥ r\). When the weight is multiplicative, i.e., when \(c = 1\), 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 \(c ≥ r\). When the weight is multiplicative, i.e., when \(c = 1\), the algebra is isometrically enAr.

The Fourier algebra \(A(G)\) 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.