More on the Arens regularity of B(X)

TL;DR

This paper investigates the conditions under which the algebra of bounded operators on a Banach space is Arens regular, establishing a precise equivalence with the space being ultrareflexive.

Contribution

It proves that B(X) is Arens regular if and only if the Banach space X is ultrareflexive, clarifying a long-standing question in functional analysis.

Findings

B(X) is Arens regular iff X is ultrareflexive

Provides a characterization of Arens regularity for operator algebras

Answers a question posed by Daws in 2004

Abstract

We focus on a question raised by Daws [Arens regularity of the algebra of operators on a Banach space, Bull. Lond. Math. Soc. 36 (2004), 493-503] concerning the Arens regularity of B(X), the algebra of operators on a Banach space X. Among other things, we show that B(X) is Arens regular if and only if X is ultrareflexive .