Biprojectivity and biflatness of amalgamated duplication of Banach algebras

Abstract

Let [Formula: see text] and [Formula: see text] be two Banach algebras such that [Formula: see text] is a Banach [Formula: see text]-bimodule with the left and right compatible action of [Formula: see text] on [Formula: see text]. Let [Formula: see text] be a strongly splitting Banach algebra extension of [Formula: see text] by [Formula: see text]. We show that (super) amenability of [Formula: see text] implies (super) module amenability of [Formula: see text] and (super) amenability [Formula: see text]. We investigate biprojectivity and biflatness of [Formula: see text] in the some especial cases. We also give some results related to module biprojectivity and module biflatness of [Formula: see text], when [Formula: see text] is biprojective or biflat.

Characterization of homological properties of θ-Lau product of Banach algebras

“…As a consequence, in the case where B has a bounded approximate identity, we conclude that the biflatness of A × θ B is equivalent to the amenability of A and B. This result gives a negative answer to an open question whether biflatness of A and X implies biflatness of the generalized module extension Banach algebra X A, which is arised in [6].…”

“…Our aim in this section is to study of biprojectivity of θ-Lau product of Banach algebras A × θ B, where A and B are arbitrary Banach algebras and θ ∈ σ(B). First, we introduce the generalized module extension Banach algebras which can be regarded as a generalization of θ-Lau product of Banach algebras, see [6], [7], [12] and [17], for more details. Recall that we shall use the notation, introduced in [6].…”

“…First, we introduce the generalized module extension Banach algebras which can be regarded as a generalization of θ-Lau product of Banach algebras, see [6], [7], [12] and [17], for more details. Recall that we shall use the notation, introduced in [6].…”

“…Suppose that A and X are Banach algebras such that X is a Banach A-bimodule. Following [6], we say that X is an algebraic Banach A-bimodule if the left and right actions of A on X are compatible, i.e. for all a ∈ A and x, x ∈ X, a…”

“…Remark 2.3. Recently, the authors in [6] considered biprojectivity of X A, for an essential Banach algebra X equipped with non-zero idempotent element p ∈ Z(X), where Z(X) denotes algebraic center of X. It seems that part of the proofs in [6, Theorem 3.1, Theorem 3.3] has a gap.…”