The first cohomology group of module extension Banach algebras

Medghalchi, Alireza; Pourmahmood-Aghababa, Hasan

doi:10.1216/rmj-2011-41-5-1639

Cited by 10 publications

(16 citation statements)

References 8 publications

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“…Proposition 5.18 is the same as Theorem 2.5 of [19]. So it can be said that Theorem 4.4 is a generalization of Theorem 2.5 in [19]. (i) If U is a closed ideal of a Banach algebra A such that U 2 = U, then for any continuous A-bimodule homomorphism T : U → A with T (x)y + xT (y) = 0 (x, y ∈ U) we have T (xy) = 0.…”

Section: By This Proposition It Is Clear Ifmentioning

confidence: 85%

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Some Notes on Semidirect Products of Banach Algebras

Farhadi

Ghahramani

2019

Results Math

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Let A and U be Banach algebras such that U is also a Banach Abimodule with compatible algebra operations, module actions and norm. By defining an approprite action, we turn l 1 -direct product A × U into a Banach algebra such that A is closed subalgebra and U is a closed ideal of it. This algebra, is in fact semidirect product of A and U which we denote it by A ⋉ U and every semidirect products of Banach algebras can be represented as this form. In this paper we consider the Banach algebra A ⋉ U as mentioned and study the derivations on it. In fact we consider the automatic continuity of the derivations on A ⋉ U and obtain some results in this context and study its relation with the automatic continuity of the derivations on A and U . Also we calculate the first cohomology group of A ⋉ U in some different cases and establish relations among the first cohomology group of A ⋉ U and those of A and U . As applications of these contents, we present various results about the automatic continuity of derivations and the first cohomology group of direct products of Banach algebras, module extension Banach algebras and θ-Lau products of Banach algebras. MSC(2010): 16E40; 46H40; 46H25.

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Section: By This Proposition It Is Clear Ifmentioning

confidence: 85%

“…Also in this case, for any r a ∈ C A (U), since a ∈ Z(A), then r a = 0. Therefore if H 1 (A) = (0), in this case by Proposition 4.4 we have Various examples of the trivial extension of Banach algebras and computing their first cohomology group are given in [19].…”

Section: By This Proposition It Is Clear Ifmentioning

confidence: 99%

Some Notes on Semidirect Products of Banach Algebras

Farhadi

Ghahramani

2019

Results Math

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“…The set of all linear and bounded operators from X into X that are A-morphisms is denoted by Hom A (X). In light of [13,Proposition 2.2] (see also [6,Lemma 1]) we have the following result. LEMMA 2.1.…”

Section: Bse Property Onmentioning

confidence: 91%

“…These algebras were studied initially in [2,17]. Some cohomological results on module extension Banach algebras are given in [6,13]. In this paper, we assume that A is a commutative Banach algebra and X is a symmetric Banach A-bimodule.…”

Section: Introductionmentioning

confidence: 99%

The Bochner–schoenberg-Eberlein Property of Extensions of Banach Algebras and Banach Modules

Alizadeh¹,

Ebadian

Ostadbashi³

et al. 2021

Bull. Aust. Math. Soc.

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Let A be a Banach algebra and let X be a Banach A-bimodule. We consider the Banach algebra ${A\oplus _1 X}$ , where A is a commutative Banach algebra. We investigate the Bochner–Schoenberg–Eberlein (BSE) property and the BSE module property on $A\oplus _1 X$ . We show that the module extension Banach algebra $A\oplus _1 X$ is a BSE Banach algebra if and only if A is a BSE Banach algebra and $X=\{0\}$ . Furthermore, we consider $A\oplus _1 X$ as a Banach $A\oplus _1 X$ -module and characterise the BSE module property on $A\oplus _1 X$ . We show that $A\oplus _1 X$ is a BSE Banach $A\oplus _1 X$ -module if and only if A and X are BSE Banach A-modules.

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“…We begin with [13, Lemma 2.1] which describes derivations on trivial extensions (see also [12,Proposition 2.2]). We set [13, Lemma 2.1] using the following terminology:…”

Section: Derivations On Trivial Extension Algebrasmentioning

confidence: 99%

Derivations and the First Cohomology Group of Trivial Extension Algebras

Bennis

Fahid

2017

Mediterr. J. Math.

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Abstract. In this paper we investigate in details derivations on trivial extension algebras. We obtain generalizations of both known results on derivations on triangular matrix algebras and a known result on first cohomology group of trivial extension algebras. As a consequence we get the characterization of trivial extension algebras on which every derivation is inner. We show that, under some conditions, a trivial extension algebra on which every derivation is inner has necessarily a triangular matrix representation. The paper starts with detailed study (with examples) of the relation between the trivial extension algebras and the triangular matrix algebras.

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The first cohomology group of module extension Banach algebras

Cited by 10 publications

References 8 publications

Some Notes on Semidirect Products of Banach Algebras

Some Notes on Semidirect Products of Banach Algebras

The Bochner–schoenberg-Eberlein Property of Extensions of Banach Algebras and Banach Modules

Derivations and the First Cohomology Group of Trivial Extension Algebras

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