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

Lav Kumar Singh

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 Ülger to prove that $S_p(\mathcal H)\otimes^γS_q(\mathcal H)$ is not Arens regular. We further prove that $B(S_2(\mathcal H))\otimes^γS_2(\mathcal H)$ 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.

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

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 Ülger to prove that is not Arens regular. We further prove that 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.

Paper Structure

This paper contains 9 sections, 14 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Schatten spaces
  4. Arens regularity
  5. Projective tensor product and its Arens reguality
  6. Arens regularity of some bilinear maps
  7. Arens regularity of projective tensor product of Schatten spaces
  8. Arens regularity of tensor product of Hilbert-Schmit spaces with Schur product
  9. Acknowledgement

Key Result

Theorem 2.1

Ulger Let $A$ and $B$ be Banach algebras. Then, their projective tensor product $A \otimes^{\gamma} B$ is Arens regular if and only if every bilinear form $m: A \times B \rightarrow \mathbb C$ is biregular.

Theorems & Definitions (29)

  • Theorem 2.1
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Theorem 4.1
  • proof
  • Corollary 4.2
  • proof
  • Corollary 4.3
  • ...and 19 more