Extreme non-Arens regularity of the group algebra

Abstract

Abstract The Banach algebras of Harmonic Analysis are usually far from being Arens regular and often turn out to be as irregular as possible. This utmost irregularity has been studied by means of two notions: strong Arens irregularity, in the sense of Dales and Lau, and extreme non-Arens regularity, in the sense of Granirer. Lau and Losert proved in 1988 that the convolution algebra L 1 ⁢ ( G ) {L^{1}(G)} is strongly Arens irregular for any infinite locally compact group. In the present paper, we prove that L 1 ⁢ ( G ) {L^{1}(G)} is extremely non-Arens regular for any infinite locally compact group. We actually prove the stronger result that for any non-discrete locally compact group G, there is a linear isometry from L ∞ ⁢ ( G ) {L^{\infty}(G)} into the quotient space L ∞ ⁢ ( G ) / ℱ ⁢ ( G ) {L^{\infty}(G)/\mathcal{F}(G)} , with ℱ ⁢ ( G ) {\mathcal{F}(G)} being any closed subspace of L ∞ ⁢ ( G ) {L^{\infty}(G)} made of continuous bounded functions. This, together with the known fact that ℓ ∞ ⁢ ( G ) / 𝒲 ⁢ 𝒜 ⁢ 𝒫 ⁢ ( G ) {\ell^{\infty}(G)/\mathscr{W\!A\!P}(G)} always contains a linearly isometric copy of ℓ ∞ ⁢ ( G ) {\ell^{\infty}(G)} , proves that L 1 ⁢ ( G ) {L^{1}(G)} is extremely non-Arens regular for every infinite locally compact group.