Amenability and topological centres of the second duals of Banach algebras

Abstract

Let  be a Banach algebra and let ** be the second dual algebra of  endowed with the first or the second Arens product. We investigate relations between amenability of ** and Arens regularity of  and the rôle topological centres in amenability of **. We also find conditions under which weak amenability of ** implies weak amenability of .

Arens regularity of module actions

“…In [12], Ghahramani et al also gave examples which are different from ours to answer these questions.…”

“…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.…”

“…In Section 5, we retake an example used by Ghahramani et al in [12], and we provide in Theorem 5.1 a class of Banach algebras which are left strongly Arens irregular but not right strongly Arens irregular. This class includes the recent example given by Dales and Lau in [5,Example 4.5].…”

Weighted convolution algebras on subsemigroups of the real line

“…The two topological centres of the space A were first systematically studied by Lau andÜlger in 1996 [53], where they are denoted by Z 1 and Z 2 , respectively; some questions that were raised in [53] were answered in [24] and [28]. Two recent preprints that present the theory of topological centres in an abstract setting are those of Hu, Neufang, and Ruan [46,47].…”

Weak amenability of the second dual of a Banach algebra

“…Another by-product is that if the topological centre of A * * is weakly amenable and each derivation D : A → A * is weakly compact, then A is weakly amenable. That the amenability of the topological centre of A * * implies that of A without any extra condition is proved in [17].…”