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

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.