Approximately Biprojective Banach Algebras and Nilpotent Ideals

Abstract

By introducing a new notion of approximate biprojectivity we show that nilpotent ideals in approximately amenable or pseudo-amenable Banach algebras, and nilpotent ideals with the nilpotency degree larger than two in biflat Banach algebras cannot have the special property which we call 'property (B)' (Definition 5.2 below) and hence, as a consequence, they cannot be boundedly approximately complemented in those Banach algebras.2010 Mathematics subject classification: primary 46H20; secondary 46B28.

Bounded pseudo-amenability and contractibility of certain Banach algebras

“…Let {ρ α } be a net satisfying Definition 5.2. As in Proposition 3.5 of[Pou1], there are subnets {e β i } of {e β } and {ρ α i } of {ρ α } such that m i := ρ α i (e β i ) is an approximate diagonal for A. We show that {m i } is a multiplier-bounded approximate diagonal.…”

“…Consider the Banach algebra A constructed in [GhR] which is boundedly approximately amenable but not boundedly approximately contractible. Then it follows from [CGZ,Proposition 2.4] that A # is boundedly approximately amenable and so A # is pseudo-amenable by [Pou1,Corollary 3.7]. Using Proposition 2.1 and the fact that A is not boundedly approximately contractible we conclude that A # is not boundedly pseudo-amenable.…”

“…Suppose that A is the algebra introduced in Proposition 2.2. Approximate amenability of A # implies its approximate biprojectivity [Pou1,Proposition 3.4]. On the other hand, A # is not boundedly pseudo-amenable and so by Proposition 5.4 is not multiplier-boundedly approximately biprojective.…”

“…Therefore it is boundedly pseudo contractible and consequently, by Proposition 5.3, multiplier-boundedly approximately biprojective. However, it is known that 2 (S) is not boundedly approximately biprojective (see [Pou1,Example 4.1]).…”

Some homological and cohomological notions on $T$-Lau product of Banach algebras

“…for all a ∈ A, will be called an approximate A-bimodule morphism from X into Y . Recently, a new notion of approximate biprojectivity introduced and studied by Pourmahmoud-Aghababa [20], which is based on approximate A-bimodule morphisms. Precisely, in the sense of [20], a Banach algebra A is called approximately biprojective if there exists an approximate A-bimodule morphism (ρ A α ) from A into A ⊗A such that π A • ρ A α (a) −→ a for all a ∈ A. Lemma 2.1.…”