More on the Arens regularity of B(X)

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 .