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

TL;DR

This paper investigates the Arens irregularity of the projective tensor product of Hilbert-Schmidt spaces, showing it is not strongly irregular and exploring the structure of its bidual.

Contribution

It proves that the tensor product is not strongly Arens irregular and identifies functionals in the bidual that lie in the topological center but are outside the algebra.

Findings

The tensor product is not strongly Arens irregular.

Existence of functionals in the bidual within the topological center but outside the algebra.

The bidual is not an annihilator Banach algebra under any Arens product.

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 $A^{**}$ which lie in the topological center but are not in $A$; and, as a consequence, we deduce that $A^{**}$ is not an annihilator Banach algebra with respect to any of the two Arens products.