Algebraic and strong splittings of extensions of Banach algebras

Bade, William G.; Dales, H. G.; Lykova, Zinaida A.

doi:10.1090/memo/0656

Cited by 39 publications

(62 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Every extension which splits strongly is evidently admissible and splits algebraically, and the admissibility of (1.1) is equivalent to ker ϕ being complemented in A as a Banach space. Motivated by Bade, Dales, and Lykova's comprehensive study [1] of extensions of Banach algebras, the second-and third-named authors [6] investigated the interrelationship among the above properties for extensions of Banach algebras of the form B(E), that is, all bounded operators acting on a Banach space E, showing that there exist Banach spaces E 1 and E 2 such that:…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

A singular, admissible extension which splits algebraically, but not strongly, of the algebra of bounded operators on a Banach space

Kania

Laustsen

Skillicorn

2016

Journal of Functional Analysis

View full text Add to dashboard Cite

show abstract

Section: Introduction and The Main Resultsmentioning

confidence: 99%

A singular, admissible extension which splits algebraically, but not strongly, of the algebra of bounded operators on a Banach space

Kania

Laustsen

Skillicorn

2016

Journal of Functional Analysis

View full text Add to dashboard Cite

show abstract

“…1 , and S. BAROOTKOOB 2 We are ready to prove one of our main result characterizing (2n+1)−amenability of A ⊲⊳ X.…”

Section: (2n + 1)−weak Amenability Of a ⊲⊳ Xmentioning

confidence: 94%

“…• (The classical module extension algebras) The (classical) module extension algebra A ⋉ X, as introduced in [2], is the ℓ 1 −direct sum of A by the Banach A−module X equipped with the multiplication (a, x)(b, y) = (ab, ay + xb) (a, b ∈ A, x, y ∈ X).…”

Section: Introduction and Some Preliminariesmentioning

confidence: 99%

Generalized module extension Banach algebras: Derivations and weak amenability

Ramezanpour

Barootkoob

2017

Quaestiones Mathematicae

View full text Add to dashboard Cite

Abstract. Let A and X be Banach algebras and let X be an algebraic Banach A−module. Then the ℓ 1 −direct sum A × X equipped with the multiplicationis a Banach algebra, denoted by A ⊲⊳ X, which will be called "a generalized module extension Banach algebra". Module extension algebras, Lau product and also the direct sum of Banach algebras are the main examples satisfying this framework. We characterize the structure of n−dual valued (n ∈ N)) derivations on A ⊲⊳ X from which we investigate the n−weak amenability for the algebra A ⊲⊳ X. We apply the results and the techniques of proofs for presenting some older results with simple direct proofs.

show abstract

“…Some aspects of algebras of this form have been discussed in [2] and [10]. We choose this class of Banach algebras to investigate for the preceding question because this class is neither too small nor is it too large; it contains permanently weakly amenable Banach algebras (see Section 6), and it contains no amenable Banach algebras due to [8,Lemma 2.7], since X is a complemented nilpotent ideal in the algebra.…”

Section: D(ab) = D(a)b + Ad(b) (A B ∈ A)mentioning

confidence: 99%