Let be a Banach algebra. is called ideally amenable if for every closed ideal of , the first cohomology group of with coefficients in is zero, i.e. . Some examples show that ideal amenability is different from weak amenability and amenability. Also for , is called -ideally amenable if for every closed ideal of , . In this paper we find the necessary and sufficient conditions for a module extension Banach algebra to be 2-ideally amenable.
It is known that a Banach algebra inherits amenability from its second Banach dual **. No example is yet known whether this fails if one considers the weak amenability instead, but the property is known to hold for the group algebra L¹(G), the Fourier algebra A(G) when G is amenable, the Banach algebras which are left ideals in **, the dual Banach algebras, and the Banach algebras which are Arens regular and have every derivation from into * weakly compact. In this paper, we extend this class of...
We study the Arens regularity of module actions of Banach left or right modules over Banach algebras. We prove that if has a brai (blai), then the right (left) module action of on * is Arens regular if and only if is reflexive. We find that Arens regularity is implied by the factorization of * or ** when is a left or a right ideal in **. The Arens regularity and strong irregularity of are related to those of the module actions of on the nth dual of . Banach algebras for which Z( **) = but are...
Let be a dual Banach algebra. We investigate the first weak-continuous cohomology group of with coefficients in . Hence, we obtain conditions on for which
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