Orthogonal $\ell_1$-sets and extreme non-Arens regularity of preduals of von Neumann algebras

Abstract: We propose a new definition for a Banach algebra $\mathfrak{A}$ to be extremely non-Arens regular, namely that the quotient $\mathfrak{A}^\ast/\mathscr{WAP}(\mathfrak{A})$ of $\mathfrak{A}^\ast$ with the space of its weakly almost periodic elements contains an isomorphic copy of $\mathfrak{A}^\ast.$ This definition is simpler and formally stronger than the original one introduced by Granirer in the nineties. We then identify sufficient conditions for the predual $\mathfrak{V}\ast$ of a von Neumann algebra $\mathfrak{V}$ to be extremely non-Arens regular in this new sense. These conditions are obtained with the help of orthogonal $\ell_1$-sets of $\mathfrak{V}\ast.$ We show that some of the main algebras in Harmonic Analysis satisfy these conditions. Among them,there is ${\small \bullet}$ the weighted semigroup algebra of any weakly cancellative discrete semigroup, for any diagonally bounded weight, ${\small \bullet}$ the weighted group algebra of any non-discrete locally compact infinite group and for any weight, ${\small \bullet}$ the weighted measure algebra of any locally compact infinite group, for any diagonally bounded weight, ${\small \bullet}$ the Fourier algebra of any locally compact infinite group having its local weight greater or equal than its compact covering number, ${\small \bullet}$ the Fourier algebra of any countable discrete group containing an infinite amenable subgroup.