The Second Transpose of a Derivation

Abstract

Let A be a Banach algebra, and let D

Arens regularity of module actions

“…Arıkan proved in [3] that a unital Banach algebra A is reflexive if and only if every left module action of A is Arens regular.Ülger extended this theorem by proving in [19] that a unital Banach algebra A is reflexive if and only if the left module action of A on the dual A * of A is Arens regular. Dales, Rodríguez and Velasco obtained in [6] the same result when A has just a bounded left approximate identity (blai) but assumed that A is Arens regular. With a short and simpler proof, and without the condition that A is Arens regular, our first theorem includes all these results.…”

“…In Section 2, we prove that if A has a brai (blai), then the right (left) module action of A on A * is Arens regular if and only if A is reflexive. This includes results proved by Arıkan in [3], Ulger in [19], and Dales, Rodríguez and Velasco in [6].…”

“…First, we determine the first topological centre of B * * . As in [12] or [6], we see from (1) that (x ′′ , a ′′ ) ∈ Z(B * * ) if and only if (i) b ′′ → a ′′ b ′′ : A * * → A * * is weak * -weak * continuous, (ii) y ′′ → a ′′ y ′′ : X * * → X * * is weak * -weak * continuous, (iii) b ′′ → x ′′ b ′′ : A * * → X * * is weak * -weak * continuous.…”

“…We define the product on A := X by ab = f, b a. Then A is clearly a Banach algebra with a right identity x (see [6]). It is straightforward to check that the left action of A on A * is given by aa ′ = f, a a ′ , and so a ′′ a ′′′ = a ′′ , f a ′′′ .…”

“…In [6,Section 5], it was proved that for a Banach algebra A, A * is an A * * -submodule of A * * * if and only if A is Arens regular. Below we show that in fact A * is always a Z A (A * * )-submodule of A * * * .…”