Weak Amenability Research Articles

Pointwise Convergence of Noncommutative Fourier Series

This paper is devoted to the study of pointwise convergence of Fourier series for group von Neumann algebras and quantum groups. It is well-known that a number of approximation properties of groups can be interpreted as summation methods and mean convergence of the associated noncommutative Fourier series. Based on this framework, this paper studies the refined counterpart of pointwise convergence of these Fourier series. As a key ingredient, we develop a noncommutative bootstrap method and establish a general criterion of maximal inequalities for approximate identities of noncommutative Fourier multipliers. Based on this criterion, we prove that for any countable discrete amenable group, there exists a sequence of finitely supported positive definite functions tending to 1 1 pointwise, so that the associated Fourier multipliers on noncommutative L p L_p -spaces satisfy the pointwise convergence for all p > 1 p>1 . In a similar fashion, we also obtain results for a large subclass of groups (as well as quantum groups) with the Haagerup property and the weak amenability. We also consider the analogues of Fejér and Bochner-Riesz means in the noncommutative setting. Our approach heavily relies on the noncommutative ergodic theory in conjunction with abstract constructions of Markov semigroups, inspired by quantum probability and geometric group theory. Finally, we also obtain as a byproduct the dimension free bounds of the noncommutative Hardy-Littlewood maximal inequalities associated with convex bodies.

Weak amenability for dual Banach algebras

Abstract A suitable notion of weak amenability for dual Banach algebras, which we call weak Connes amenability, is defined and studied. Among other things, it is proved that the measure algebra M(G) of a locally compact group G is always weakly Connes amenable. It can be a complement to Johnson’s theorem that $L^1(G)$ is always weakly amenable [10].

Weak Amenability of Banach Algebra-Valued $$\ell _p$$-Sequence Algebras

Weak Amenability of Banach Algebra-Valued $$\ell _p$$-Sequence Algebras

On von Neumann equivalence and group approximation properties

The notion of von Neumann equivalence (vNE), which encapsulates both measure equivalence and W^* -equivalence, was introduced recently by Ishan, Peterson, and Ruth (2019). They have shown that many analytic properties, such as amenability, property (T), the Haagerup property, and proper proximality are preserved under von Neumann equivalence. In this article, we expand on the list of properties that are stable under von Neumann equivalence, and prove that weak amenability, weak Haagerup property, and the approximation property (AP) are von Neumann equivalence invariants. In particular, we get that AP is stable under measure equivalence. Furthermore, our techniques give an alternate proof for vNE-invariance of the Haagerup property.

Harmonic analysis of little [formula omitted]-Legendre polynomials

Many classes of orthogonal polynomials satisfy a specific linearization property giving rise to a polynomial hypergroup structure, which offers an elegant and fruitful link to Fourier analysis, harmonic analysis and functional analysis. From the opposite point of view, this allows regarding certain Banach algebras as L1-algebras, associated with underlying orthogonal polynomials. The individual behavior strongly depends on these underlying polynomials. We study the little q-Legendre polynomials, which are orthogonal with respect to a discrete measure. We will show that their L1-algebras have the property that every element can be approximated by linear combinations of idempotents. This particularly implies that these L1-algebras are weakly amenable (i.e., every bounded derivation into the dual module is an inner derivation), which is known to be shared by any L1-algebra of a locally compact group; in the polynomial hypergroup context, weak amenability is rarely satisfied and of particular interest because it corresponds to a certain property of the derivatives of the underlying polynomials and their (Fourier) expansions w.r.t. the polynomial basis. To our knowledge, the little q-Legendre polynomials yield the first example of a polynomial hypergroup whose L1-algebra is weakly amenable and right character amenable but fails to be amenable. As a crucial tool, we establish certain uniform boundedness properties of the characters. Our strategy relies on the Fourier transformation on hypergroups, the Plancherel isomorphism, continued fractions, character estimations and asymptotic behavior.

Amenability, weak amenability and bounded approximate identities in multipliers of the Fourier algebra

Amenability, weak amenability and bounded approximate identities in multipliers of the Fourier algebra

The first cohomology group and weak amenability of a Morita context Banach algebra

The first cohomology group and weak amenability of a Morita context Banach algebra

Amalgamated duplication of Banach algebras from homological point of view

In a paper from 2002, Zhang [Weak amenability of module extensions of Banach algebras, Trans. Amer. Math. Soc. 354 (2002) 4131–4151] used module extensions to construct an example of a weakly amenable Banach algebra which is not 3-weakly amenable. Following his approach, we show that for amalgamated duplication of dual Banach algebras, we are able to choose appropriate conditions for weak Connes amenability of those Banach algebras which extend the Zhang’s result in a much more general setting. Moreover, apart from the characterization of minimal idempotents, we restrict our investigation to the study of injectivity, projectivity and flatness of amalgamated duplication of Banach algebras. Particularly, it is worth mentioning that our result paves the way for studying of homological properties and weak Connes amenability of those dual Banach algebras that have a semidirect product-like structure including Lau algebras.

Weak amenability of free products of hyperbolic and amenable groups

AbstractWe show that if G is an amenable group and H is a hyperbolic group, then the free product $G\ast H$ is weakly amenable. A key ingredient in the proof is the fact that $G\ast H$ is orbit equivalent to $\mathbb{Z}\ast H$ .

On σ-amenability of TσA,σB

Let ?A and ?B be two homomorphisms on Banach algebras A and B, respectively. In this paper, we study ?-amenability, ?-weak amenability, ?-biflatness, and ?-biprojectivity of triangular Banach algebras of the form T?A,?B, where ? = ?A ? ?B.

CENTRE OF BANACH ALGEBRA VALUED BEURLING ALGEBRAS

AbstractWe prove that for a Banach algebra A having a bounded $\mathcal {Z}(A)$ -approximate identity and for every $\mathbf {[IN]}$ group G with a weight w which is either constant on conjugacy classes or satisfies $w \geq 1$ , $\mathcal {Z}(L^{1}_{w}(G) \otimes ^{\gamma } A) \cong \mathcal {Z}(L^{1}_{w}(G)) \otimes ^{\gamma } \mathcal {Z}(A)$ . As an application, we discuss the conditions under which $\mathcal {Z}(L^{1}_{\omega }(G,A))$ enjoys certain Banach algebraic properties, such as weak amenability or semisimplicity.

Turán’s inequality, nonnegative linearization and amenability properties for associated symmetric Pollaczek polynomials

An elegant and fruitful way to bring harmonic analysis into the theory of orthogonal polynomials and special functions, or to associate certain Banach algebras with orthogonal polynomials satisfying a specific but frequently satisfied nonnegative linearization property, is the concept of a polynomial hypergroup. Polynomial hypergroups (or the underlying polynomials, respectively) are accompanied by L1-algebras and a rich, well-developed and unified harmonic analysis. However, the individual behavior strongly depends on the underlying polynomials. We study the associated symmetric Pollaczek polynomials, which are a two-parameter generalization of the ultraspherical polynomials. Considering the associated L1-algebras, we will provide complete characterizations of weak amenability and point amenability by specifying the corresponding parameter regions. In particular, we shall see that there is a large parameter region for which none of these amenability properties hold (which is very different to L1-algebras of locally compact groups). Moreover, we will rule out right character amenability. The crucial underlying nonnegative linearization property will be established, too, which particularly establishes a conjecture of R. Lasser (1994). Furthermore, we shall prove Turán’s inequality for associated symmetric Pollaczek polynomials. Our strategy relies on chain sequences, asymptotic behavior, further Turán type inequalities and transformations into more convenient orthogonal polynomial systems.

A note on weak amenability of semigroup algebras

We study weak amenability of certain classes of commutative semigroup algebras. First, we present a class of commutative semigroups which semigroup algebras (like arbitrary group algebras) are always 2n-weakly amenable. Weak amenability of $$\ell ^1(S)$$ in terms of regular and irregular subsets of a commutative semigroup S is also studied.

Weak amenability for dynamical systems

Using the recently developed notion of a Herz-Schur multiplier of a C*-dynamical system we introduce weak amenability of C*- and W*-dynamical systems. As a special case we recover Haagerup's characterisation of weak amenability of a discrete group. We also consider a generalisation of the Fourier algebra and its multipliers to crossed products.

Addendum to “Amenability and weak amenability of second conjugate Banach algebras”

Addendum to “Amenability and weak amenability of second conjugate Banach algebras”

Ternary n-Weak Amenability of Certain Commutative Banach Algebras and $$\hbox {JBW}^*$$-Triples

We show that a commutative unital Banach *-algebra is ternary n-weakly amenable when it is n-weakly amenable. We apply this result for a wide variety of commutative n-weakly amenable algebras such as for a convolution group algebra on a discrete abelian group and for a commutative unital $$\hbox {C}^*$$ -algebra. We also show that every commutative $$\hbox {JBW}^*$$ -triple is ternary n-weakly amenable. These results present a somehow unified extension of the previous ternary weak amenability results in the category of triple systems and n-weak amenability results in the category of Banach algebras.

On the fourier algebra of certain hypergroups

We consider certain classes of hypergroups and address some problems regarding their Fourier algebras, including zero product preserving maps, local derivations and weak amenability. The key tool in our approach is to show that the Fourier algebra of such hypergroups has the so called property (𝔹). This property has already been shown to be very effective in solving problems related to zero product preserving maps and local derivations provided that some other conditions are satisfied. Some examples of groups and hypergroups, the Fourier algebras of which have the property (𝔹) satisfying all those conditions are given. Our examples are the class of almost abelian groups and double coset hypergroups of Gelfand pairs. We also prove that if H is the double coset hypergroup of a Gelfand pair, then A(H) is weakly amenable. We provide some results on the direct product of commutative (regular) Fourier hypergroups.

Point derivations and cohomologies of Lipschitz algebras

AbstractFor a compact metric space (K, d), LipK denotes the Banach algebra of all complex-valued Lipschitz functions on (K, d). We show that the continuous Hochschild cohomology Hn(LipK, (LipK)*) and Hn(LipK, ℂe) are both infinite-dimensional vector spaces for each n ≥ 1 if the space K contains a certain infinite sequence which converges to a point e ∈ K. Here (LipK)* is the dual module of LipK and ℂe denotes the complex numbers with a LipK-bimodule structure defined by evaluations of LipK-functions at e. Examples of such metric spaces include all compact Riemannian manifolds, compact geodesic metric spaces and infinite compact subsets of ℝ. In particular, the (small) global homological dimension of LipK is infinite for every such space. Our proof uses the description of point derivations by Sherbert [‘The structure of ideals and point derivations in Banach algebras of Lipschitz functions’, Trans. Amer. Math. Soc.111 (1964), 240–272] and directly constructs non-trivial cocycles with the help of alternating cocycles of Johnson [‘Higher-dimensional weak amenability’, Studia Math.123 (1997), 117–134]. An alternating construction of cocycles on the basis of the idea of Kleshchev [‘Homological dimension of Banach algebras of smooth functions is equal to infinity’, Vest. Math. Mosk. Univ. Ser. 1. Mat. Mech.6 (1988), 57–60] is also discussed.

A Unified Approach to Various Notions of Derivation

The main goal of this article is introducing a fairly general framework to treat several types of derivations simultaneously. Let $${{\mathcal{A}}}$$ and $${\mathcal{B}}$$ be Banach algebras, $$\alpha$$ and $$\beta$$ be homomorphisms from $${{\mathcal{A}}}$$ onto $${\mathcal{B}}$$ , and $${\mathcal{X}}$$ be a Banach $${\mathcal{B}}$$ -bimodule. A map $$D\in {{\mathcal {B}}}({{\mathcal {A}}},{{\mathcal {X}}})$$ is called an $$(\alpha , \beta )$$ -derivation if $$D(ab) =\alpha (a)D(b)+D(a)\beta (b)$$ . All homomorphisms, ordinary derivations, skew derivations, and point derivations are certain types of $$(\alpha , \beta )$$ -derivations. We define $$(\alpha , \beta )$$ -analog of notions of amenability and weak amenability. Amenability modulo a closed ideal in the sense of Rahimi (Acta Math Hungarica 144:407–415, 2014) and generalized weak amenability in the sense of Bodaghi et al. (Banach J Math Anal 3:131–142, 2009) are special cases of our concepts. We provide several characterizations of $$(\alpha ,\beta )$$ -weak amenability under some mild conditions. Moreover $$({\alpha }^{**},{\beta }^{**})$$ -weak amenability of $${{\mathcal{A}}}^{**}$$ implies $$(\alpha ,\beta )$$ -weak amenability of $${{\mathcal{A}}}$$ .

Weak module amenability of triangular banach algebras II

Abstract Let A and B be Banach 𝔄-bimodule and Banach 𝔅-bimodule algebras, respectively. Also let M be a Banach A, B-module and Banach 𝔄, 𝔅-module with compatible actions. In the case of 𝔄 = 𝔅, the author along with Pourabbas [5] have studied the weak 𝔄-module amenability of triangular Banach algebra T = A M B $\begin{array}{} \displaystyle \mathcal{T}=\left[\begin{array}{rr} A & M \\ & B \end{array} \right] \end{array}$ and showed that 𝓣 is weakly 𝔄-module amenable if and only if the corner Banach algebras A and B are weakly 𝔄-module amenable, where A, B and M are unital. In this paper we investigate a special structure of 𝔄 ⊕ 𝔅-bimodule derivation from 𝓣 into 𝓣∗ and show that 𝓣 is weakly 𝔄 ⊕ 𝔅-bimodule amenable if and only if the corner Banach algebras A and B are weakly 𝔄-module amenable and weakly 𝔅-module amenable, respectively, where A, B and M are essential and not necessary unital.