Abstract
In recent years, lots of papers have been published on module amenability. In this paper, our main aim is to study the homological properties of various module derivations and prove some results about module amenability. So this paper continous a line investigation in [3], [4] for Banach algebras.
Highlights
First relative cohomology group of A with coefficients in X∗
Throughout this paper, as in [2], A and U are Banach algebras such that A is a Banach U -bimodule with the compatible actions, as follows: α · = (α · a)b, · α = a(b · α) (α ∈ U, a, b ∈ A)
Let Φ : ZG(A, −) → Z(A, −) ⊕ F be a map of functors where F is the forgetful functor from the category of Banach A-U -modules to the category of Banach U -bimodules
Summary
Throughout this paper, as in [2], A and U are Banach algebras such that A is a Banach U -bimodule with the compatible actions, as follows:. When A is a commutative U -bimodule and acts on itself by algebra multiplication from both sides, it is a Banach A-U -module. If X is a (commutative) Banach A-U -module, so is X∗, where the actions of A and U on X∗ are defined as follows:. We consider the module projective tensor product A⊗U A which is a Banach A-U -module with canonical actions. (ii) If Y is an essential A-bimodule and φ : X∗ → Y ∗ is a bounded right A⊗U Aop-U -module homomorphism, φ is an A-U -module homomorphism
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