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

A Banach algebra is Arens-regular when all its continuous functionals are weakly almost periodic, in symbols A⁎=WAP(A). To identify the opposite behaviour, Granirer called a extremely non-Arens regular (enAr, for short) quotient A⁎/WAP(A) contains closed subspace that has A⁎ as quotient. In this paper we propose simplification and quantification of concept. We say r-enAr, with r≥1, there an isomorphism distortion r into A⁎/WAP(A). When r=1, obtain isometric isometrically enAr. then sufficient conditions predual V⁎ von Neumann V to be r-enAr or With aid these conditions, following algebras shown r-enAr: weighted semigroup any cancellative discrete semigroup, weight diagonally bounded diagonal bound c≥r. multiplicative, i.e., c=1, enAr, group non-discrete locally compact infinite weight, measure group,