Weak and cyclic amenability for non-commutative Banach algebras

Grønbæk, Niels

doi:10.1017/s0013091500005587

Cited by 24 publications

(18 citation statements)

References 9 publications

(11 reference statements)

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“…If every continuous cyclic derivation from A to A ′ is inner, then A is called cyclic amenable [11].…”

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confidence: 99%

On certain products of Banach algebras with applications to harmonic analysis

Monfared

2007

Studia Math.

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Abstract. Given Banach algebras A and B with spectrum σ(B) = ∅, and given θ ∈ σ(B), we define a product A × θ B, which is a strongly splitting Banach algebra extension of B by A. We obtain characterizations of bounded approximate identities, spectrum, topological center, minimal idempotents, and study the ideal structure of these products. By assuming B to be a Banach algebra in C 0 (X) whose spectrum can be identified with X, we apply our results to harmonic analysis, and study the question of spectral synthesis, and primary ideals.

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“…If every continuous cyclic derivation from A to A ′ is inner, then A is called cyclic amenable [11].…”

mentioning

confidence: 99%

On certain products of Banach algebras with applications to harmonic analysis

Monfared

2007

Studia Math.

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“…In particular, C is called contractible [respectively, amenable] if H 1 (C, X ) = 0 [respectively, H 1 (C, X (1) ) = 0] for every Banach C-bimodule X and for n ∈ N ∪ {0} it is called n-weakly amenable if H 1 (C, C (n) ) = 0. Moreover, recall from [13] that C is cyclic amenable if H 1 λ (C, C (1) ) = 0. For more information about these notions we refer the reader to the impressive references [5,13].…”

Section: Preliminariesmentioning

confidence: 99%

“…Moreover, recall from [13] that C is cyclic amenable if H 1 λ (C, C (1) ) = 0. For more information about these notions we refer the reader to the impressive references [5,13].…”

Section: Preliminariesmentioning

confidence: 99%

Amalgamated duplication of the Banach algebra 𝔄 along a 𝔄-bimodule 𝒜

Javanshiri

Nemati

2018

J. Algebra Appl.

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Let A and A be Banach algebras such that A is a Banach A-bimodule with compatible actions. We define the product A ⋊ A, which is a strongly splitting Banach algebra extension of A by A. After characterization of the multiplier algebra, topological centre, (maximal) ideals and spectrum of A ⋊ A, we restrict our investigation to the study of semisimplicity, regularity, Arens regularity of A ⋊ A in relation to that of the algebras A, A and the action of A on A. We also compute the first cohomology group H 1 (A ⋊ A, (A ⋊ A) (n) ) for all n ∈ N ∪ {0} as well as the first-order cyclic cohomology group H 1 λ (A ⋊ A, (A ⋊ A) (1) ), where (A ⋊ A) (n) is the n-th dual space of A ⋊ A when n ∈ N and A ⋊ A itself when n = 0. These results are not only of interest in their own right, but also they pave the way for obtaining some new results for Lau products and module extensions of Banach algebras as well as triangular Banach algebra. Finally, special attention is devoted to the cyclic and n-weak amenability of A ⋊ A.Mathematics Subject Classification (2010). 46H20, 46H25, 46H10, 47B48, 47B47, 16E40.

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“…We start this section with a definition which is analogous to [13 Thus, hD..x//; ai D h.x/ .x/; ai, for all a 2 .E/ and x 2 E. Therefore D is -inner.…”

Section: Weak Amenability Of Banach Modulesmentioning

confidence: 99%