Weak module amenability of triangular Banach algebras

Abstract

ABSTRACT. Let A and B be unital Banach algebras and let M be a unital Banach A, have studied the weak amenability of triangular Banach algebra T = A M B and showed that T is weakly amenable if and only if the corner algebras A and B are weakly amenable. When A is a Banach algebra and A and B are Banach A-module with compatible actions, and M is a commutative left Banach A-A-module and right Banach A-B-module, we show that A and B are weakly A-module amenable if and only if triangular Banach algebra T is weakly T-module amenable, where Amini in [1] developed the concept of module amenability for a class of Banach algebras which is in fact a generalization of the Johnson's amenability. For example for every inverse semigroup S with subsemigroup E of idempotents, he showed that the 1 (E)-module amenability of 1 (S) is equivalent to amenability of S. Duncan and Namioka in [4] for E-unitary semigroup S have shown that 1 (S) is not amenable, whenever E = E S , the set of idempotent elements in S, is finite. For the general case of this result see [3, Chapter 10] But there are many amenable inverse semigroups including the bicyclic semigroup and Clifford semigroups with an infinite set of idempotents. Amini and Bagha in [2] introduced the concept of weak module amenability and showed that if S is commutative, 1 (S) is always weak 1 (E)-module amenable. Note that in the group case Johnson [7] showed that a group G is amenable if and only if L 1 (G) is amenable and in [8] he showed that L 1 (G) is always weakly amenable.On the other hand, Forrest and Marcoux in [6] have studied the weak amenability of triangular Banach algebra

Pseudo-amenability and pseudo-contractibility for certain products of Banach algebras

“…We then have proved that if A × θ B is (2n)-weakly amenable for some n ∈ N, then so are A and B; furthermore, if A and B are (2n + 1)-weakly amenable, then A × θ B is (2n + 1)-weakly amenable. We finally have shown that cyclic amenability of A and B is equivalent to cyclic amenability of their θ-Lau product; see also [21].…”