On arens regularity of projective tensor product of Schatten p-class operators

TL;DR

This paper investigates the Arens regularity of the projective tensor product of Schatten p-class operators, showing it is not regular in most cases and highlighting the role of biregularity conditions.

Contribution

It demonstrates the non-Arens regularity of certain tensor products of Schatten classes and emphasizes the importance of biregularity conditions in such analyses.

Findings

$S_p(\

ext{tensor}^ ext{gamma} S_q( ext{Hilbert space})$ is not Arens regular.

$B(S_2( ext{Hilbert space})) ext{tensor}^ ext{gamma} S_2( ext{Hilbert space})$ is not Arens regular with usual multiplication but regular with Schur product.

Abstract

In this paper we discuss the Arens regularity of projective tensor product of Schatten p-class operators. We use the biregularity condition given by \"Ulger to prove that $S_p(\mathcal H)\otimes^\gamma S_q(\mathcal H)$ is not Arens regular. We further prove that $B(S_2(\mathcal H))\otimes^\gamma S_2(\mathcal H)$ is not Arens regular(with respect to usual multiplication) while it is regular with respect to Schur product. Thus we demonstrate the importance of biregularity condition given in \cite{Ulger} and the convenience of its use to prove Arens regularity or irregularity through some concrete examples.