On strong Arens irregularity of projective tensor product of Hilbert-Schmidt space

Abstract

It was shown in [16] that the Banach algebra $A:=S_2({\ell }^2){\otimes }^{\gamma }{S}_2({\ell }^2)$ is not Arens regular, where $S_2({\ell }^2)$ denotes the Banach algebra of the Hilbert-Schmidt operators on ${\ell }^2$. In this article, employing the notion of limits along ultrafilters, we prove that the irregularity of $S_2({\ell }^2){\otimes }^{\gamma }{S}_2({\ell }^2)$ is not strong. Along the way, we provide a class of functionals in $A^{\ast \ast }$ which lie in the topological center but are not in $A$; and, as a consequence, we deduce that $A^{\ast \ast }$ is not an annihilator Banach algebra with respect to any of the two Arens products.

Major revision undertaken keeping in focus the analysis of strong Arens irregularity of the projective tensor product of the Hibert-Schmidt space