On Arens regularity of projective tensor product of Schatten spaces

Abstract

We discuss Arens regularity of the projective tensor product of Banach algebras consisting of Schatten class operators. More precisely, exploiting Ülger’s biregularity criterion, we prove that for any Hilbert space $\mathcal{\mathscr{H}}$, the Banach algebras $S_p\left(\mathcal{\mathscr{H}}\right){\otimes }^{\gamma }{S}_q\left(\mathcal{\mathscr{H}}\right)$ and $B\left(S_2\left(\mathcal{\mathscr{H}}\right)\right){\otimes }^{\gamma }{S}_2\left(\mathcal{\mathscr{H}}\right)$ are not Arens regular for every pair $1\le p,q\le 2$; whereas, when $\mathcal{\mathscr{H}}$ is separable and $S_2\left(\mathcal{\mathscr{H}}\right)$ is equipped with the Schur product, we show that $S_2\left(\mathcal{\mathscr{H}}\right)$ is completely continuous and that $S_2\left(\mathcal{\mathscr{H}}\right){\otimes }^{\gamma }{S}_2\left(\mathcal{\mathscr{H}}\right)$ is Arens regular.