The third dual of a Banach algebra

Abstract

Let A be a Banach algebra and ( A ″, □) be its second dual with first Arens product. The third dual of A can be regarded as dual of A ″ ( A ‴ = ( A ″)′) or as the second dual of A ′ ( A ‴ = ( A ′)″), so there are two ( A ″, □)-bimodule structures on A ‴ that are not always equal. This paper determines the conditions that make these structures equal. As a consequence, there are some relations between weak amenability of A and ( A ″, □).

m−WEAK AMENABILITY OF (2n)−TH DUALS OF BANACH ALGEBRAS

“…In each case we will find conditions to make two different actions equal. These are generalizations of the methods in [9]. In Section 4 we consider continuous derivations D : A → A ′ and D : A → A ′′ .…”

Weak amenability for the second dual of Banach modules

“…Then X 000 can have two A 00 -bimodule structures (see [9]). First we regard X 000 , as the dual space of X 00 .…”

3-Weak amenability of (2n)th duals of Banach algebras

“…In Section 2 we investigate two A -bimodule structures on A (5) given by A (5) = (((A ) ) ) and A (5) = (((A ) ) ) , and also two A (4) -bimodule structures on A (7) = ((((A ) ) ) ) and A (7) = ((((A ) ) ) ) . In a similar work [6] we investigated two A -bimodule structures on A (3) = (A ) and…”

“…Now we are ready to consider two A (4) -bimodule structures on A (7) . Take a (7) ∈ A (7) , a (6) ∈ A (6) and a (4) ∈ A (4) with bounded nets (a (5) β ) ⊂ A (5) , (a (4) i ) ⊂ A (4) and (a α ) ⊂ A such that a (7) = w * -lim β a (5) β , a (6) = w * -lim i a (4) i and a (4) = w * -lim α a α .…”

“…For the A (4) -bimodule structure on A (7) = ((((A ) ) ) ) we can write a (4) .a (7) , a (6) = lim α lim β a (6) , a α .a…”