Approximate and weak amenability of certain Banach algebras

P. Bharucha; R. J. Loy

Displaying similar documents to “Approximate and weak amenability of certain Banach algebras”

Some notions of amenability for certain products of Banach algebras

Eghbal Ghaderi, Rasoul Nasr-Isfahani, Mehdi Nemati (2013)

Colloquium Mathematicae

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For two Banach algebras and ℬ, an interesting product $\times_{θ} ℬ$ , called the θ-Lau product, was recently introduced and studied for some nonzero characters θ on ℬ. Here, we characterize some notions of amenability as approximate amenability, essential amenability, n-weak amenability and cyclic amenability between and ℬ and their θ-Lau product.

Weak amenability of the second dual of a Banach algebra

M. Eshaghi Gordji, M. Filali (2007)

Studia Mathematica

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It is known that a Banach algebra inherits amenability from its second Banach dual **. No example is yet known whether this fails if one considers the weak amenability instead, but the property is known to hold for the group algebra L¹(G), the Fourier algebra A(G) when G is amenable, the Banach algebras which are left ideals in **, the dual Banach algebras, and the Banach algebras which are Arens regular and have every derivation from into * weakly compact. In this paper, we extend this...

Existence theorem for the Hammerstein integral equation

Mieczysław Cichoń, Ireneusz Kubiaczyk (1996)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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In this paper we prove an existence theorem for the Hammerstein integral equation $x (t) = p (t) + λ \int_{I} K (t, s) f (s, x (s)) d s$ , where the integral is taken in the sense of Pettis. In this theorem continuity assumptions for f are replaced by weak sequential continuity and the compactness condition is expressed in terms of the measures of weak noncompactness. Our equation is considered in general Banach spaces.

Weak amenability of weighted group algebras on some discrete groups

Varvara Shepelska (2015)

Studia Mathematica

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Weak amenability of ℓ¹(G,ω) for commutative groups G was completely characterized by N. Gronbaek in 1989. In this paper, we study weak amenability of ℓ¹(G,ω) for two important non-commutative locally compact groups G: the free group ₂, which is non-amenable, and the amenable (ax + b)-group. We show that the condition that characterizes weak amenability of ℓ¹(G,ω) for commutative groups G remains necessary for the non-commutative case, but it is sufficient neither for ℓ¹(₂,ω) nor for...

Approximate amenability for Banach sequence algebras

H. G. Dales, R. J. Loy, Y. Zhang (2006)

Studia Mathematica

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We consider when certain Banach sequence algebras A on the set ℕ are approximately amenable. Some general results are obtained, and we resolve the special cases where $A = ℓ^{p}$ for 1 ≤ p < ∞, showing that these algebras are not approximately amenable. The same result holds for the weighted algebras $ℓ^{p} (ω)$ .

A generalized notion of $n$ -weak amenability

Abasalt Bodaghi, Behrouz Shojaee (2014)

Mathematica Bohemica

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In the current work, a new notion of $n$ -weak amenability of Banach algebras using homomorphisms, namely $(ϕ, ψ)$ - $n$ -weak amenability is introduced. Among many other things, some relations between $(ϕ, ψ)$ - $n$ -weak amenability of a Banach algebra $𝒜$ and $M_{m} (𝒜)$ , the Banach algebra of $m \times m$ matrices with entries from $𝒜$ , are studied. Also, the relation of this new concept of amenability of a Banach algebra and its unitization is investigated. As an example, it is shown that the group algebra $L^{1} (G)$ is ( $ϕ, ψ$ )- $n$ -weakly amenable...

Proper cocycles and weak forms of amenability

Paul Jolissaint (2015)

Colloquium Mathematicae

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Let G and H be locally compact, second countable groups. Assume that G acts in a measure class preserving way on a standard space (X,μ) such that $L^{\infty} (X, μ)$ has an invariant mean and that there is a Borel cocycle α: G × X → H which is proper in the sense of Jolissaint (2000) and Knudby (2014). We show that if H has one of the three properties: Haagerup property (a-T-menability), weak amenability or weak Haagerup property, then so does G. In particular, we show that if Γ and Δ are measure equivalent...

Character Connes amenability of dual Banach algebras

Mohammad Ramezanpour (2018)

Czechoslovak Mathematical Journal

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We study the notion of character Connes amenability of dual Banach algebras and show that if $A$ is an Arens regular Banach algebra, then $A^{* *}$ is character Connes amenable if and only if $A$ is character amenable, which will resolve positively Runde’s problem for this concept of amenability. We then characterize character Connes amenability of various dual Banach algebras related to locally compact groups. We also investigate character Connes amenability of Lau product and module extension...

3-Weak amenability of (2n)th duals of Banach algebras

Mina Ettefagh (2012)

Colloquium Mathematicae

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We show that under some conditions, 3-weak amenability of the (2n)th dual of a Banach algebra A for some n ≥ 1 implies 3-weak amenability of A.

Ideal amenability of module extensions of Banach algebras

Eshaghi M. Gordji, F. Habibian, B. Hayati (2007)

Archivum Mathematicum

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Let $𝒜$ be a Banach algebra. $𝒜$ is called ideally amenable if for every closed ideal $I$ of $𝒜$ , the first cohomology group of $𝒜$ with coefficients in $I^{*}$ is zero, i.e. $H^{1} (𝒜, I^{*}) = {0}$ . Some examples show that ideal amenability is different from weak amenability and amenability. Also for $n \in N$ , $𝒜$ is called $n$ -ideally amenable if for every closed ideal $I$ of $𝒜$ , $H^{1} (𝒜, I^{(n)}) = {0}$ . In this paper we find the necessary and sufficient conditions for a module extension Banach algebra to be 2-ideally amenable.

Constructions preserving $n$ -weak amenability of Banach algebras

A. Jabbari, Mohammad Sal Moslehian, H. R. E. Vishki (2009)

Mathematica Bohemica

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A surjective bounded homomorphism fails to preserve $n$ -weak amenability, in general. We however show that it preserves the property if the involved homomorphism enjoys a right inverse. We examine this fact for certain homomorphisms on several Banach algebras.

Weak Distances between Random Subproportional Quotients of $ℓ ₁^{m}$

Piotr Mankiewicz (2012)

Bulletin of the Polish Academy of Sciences. Mathematics

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Lower estimates for weak distances between finite-dimensional Banach spaces of the same dimension are investigated. It is proved that the weak distance between a random pair of n-dimensional quotients of $ℓ ₁^{n ²}$ is greater than or equal to c√(n/log³n).

Metric spaces admitting only trivial weak contractions

Richárd Balka (2013)

Fundamenta Mathematicae

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If (X,d) is a metric space then a map f: X → X is defined to be a weak contraction if d(f(x),f(y)) < d(x,y) for all x,y ∈ X, x ≠ y. We determine the simplest non-closed sets X ⊆ ℝⁿ in the sense of descriptive set-theoretic complexity such that every weak contraction f: X → X is constant. In order to do so, we prove that there exists a non-closed $F_{σ}$ set F ⊆ ℝ such that every weak contraction f: F → F is constant. Similarly, there exists a non-closed $G_{δ}$ set G ⊆ ℝ such that every weak...

Weak dimensions and Gorenstein weak dimensions of group rings

Yueming Xiang (2021)

Czechoslovak Mathematical Journal

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Let $K$ be a field, and let $G$ be a group. In the present paper, we investigate when the group ring $K [G]$ has finite weak dimension and finite Gorenstein weak dimension. We give some analogous versions of Serre’s theorem for the weak dimension and the Gorenstein weak dimension.

Weak Baire measurability of the balls in a Banach space

José Rodríguez (2008)

Studia Mathematica

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Let X be a Banach space. The property (∗) “the unit ball of X belongs to Baire(X, weak)” holds whenever the unit ball of X* is weak*-separable; on the other hand, it is also known that the validity of (∗) ensures that X* is weak*-separable. In this paper we use suitable renormings of $ℓ^{\infty} (ℕ)$ and the Johnson-Lindenstrauss spaces to show that (∗) lies strictly between the weak*-separability of X* and that of its unit ball. As an application, we provide a negative answer to a question raised...