Ebrahim Samei, Nico Spronk, Ross Stokke
We investigate generalized amenability and biflatness properties of various (operator) Segal algebras in both the group algebra, L1(G), and the Fourier algebra, A(G), of a locally compact group, G.
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Let $\mathcal A$ be a Banach algebra. Using the concept of module biflatness, we show that the module amenability of the second dual $\mathcal A^{**}$ (with the first Arens product) necessitates the module amenability of $\mathcal A$. We give some examples of Banach algebras $\mathcal A$ such that $\mathcal A^{**}$ are module biflat, but which are not themselves module biflat.
For a locally compact group G, the convolution product on the space N(L^p(G)) of nuclear operators was defined by Neufang. We study homological properties of the convolution algebra N(L^p(G)) and relate them with some properties of the group G, such as compactness, finiteness, discreteness, and amenability.
Given a Banach algebra $ \mathcal{A} $ and a continuous homomorphism $σ$ on it, the notion of $σ$-biflatness for $ \mathcal{A} $ is introduced. This is a generalization of biflatness and it is shown that they are distinct. The relations between $σ$-biflatness and some other close concepts such as $σ$-biprojectivity and $σ$-amenability are studied. The $σ$-biflatness of tensor product of Banach algebras are also discussed.
In this paper we are going to investigate the approximate biprojectivity and the $ϕ$-biflatness of some Banach algebras related to the locally compact groups. We show that a Segal algebra $S(G)$ is approximate biprojective if and only if $G$ is compact. Also for a continuous weight $w\geq 1$, we show that $L^{1}(G,w)$ is a approximate biprojective if and only if $G$ is compact. We study $ϕ$-biflatness of some Banach algebras, where $ϕ:A\rightarrow \mathbb{C}$ is a multiplicative linear functional. We show that if $S(G)$ is $ϕ$-biflat, then $G$ is amenable group. Also we show that the $ϕ$-bifla…
We prove that every proper Segal algebra of a SIN group is not approximately amenable.
In this paper, we study the notion of $ϕ$-biflatness for some Banach algebras, where $ϕ$ is a non-zero multiplicative linear functional. We show that the Segal algebra $S(G)$ is left $ϕ$-biflat if and only if $G$ is amenable. Also, we characterize left $ϕ$-biflatness of semigroup algebra $\ell^{1}(S)$ in the term of biflatness, where $S$ is a Clifford semigroup.
In this paper we first show that for a locally compact amenable group $G$, every proper abstract Segal algebra of the Fourier algebra on $G$ is not approximately amenable; consequently, every proper Segal algebra on a locally compact abelian group is not approximately amenable. Then using the hypergroup generated by the dual of a compact group, it is shown that all proper Segal algebras of a class of compact groups including the $2\times 2$ special unitary group, SU(2), are not approximately amenable.
We prove that the $L^1$-algebra of any non-Kac type compact quantum group does not satisfy operator biflatness. Since operator amenability implies operator biflatness, this result shows that any co-amenable, non-Kac type compact quantum group gives a counter example to the conjecture that $L^1(\Gb)$ is operator amenable if and only if $\Gb$ is amenable and co-amenable for any locally compact quantum group $\Gb$. The result also implies that the $L^1$-algebra of a locally compact quantum group is operator biprojective if and only if $\Gb$ is compact and of Kac type.
Let T be a homomorphism from a Banach algebra B to a Banach algebra A.The Cartesian product space A * B with T-Lau multiplication and l^1-norm becomes a new Banach algebra A *_T B. We investigate the notions such as approximate amenability, pseudo amenability, phi-pseudo amenability,phi-biflatness and phi-biprojectivity for Banach algebra A *_T B. We also present an example to show that approximate amenability of A and B is not stable for A *_T B. Finally we characterize the double centralizer algebra of A *_T B and present an application of this characterization.
In this paper, we countinue our work in \cite{11}. We show that $L^{1}(G,w)$ is $ϕ_{0}$-biprojective if and only if $G$ is compact, where $ϕ_{0}$ is the augmentation character. We introduce the notions of character Johnson amenability and character Johnson contractibility for Banach algebras. We show that $\ell^{1}(S)$ is pseudo-amenable if and only if $\ell^{1}(S)$ is character Johnson-amenable, provided that $S$ is a uniformly locally finite band semigroup. We give some conditions whether $ϕ$-biprojectivity ($ϕ$-biflatness) of $\ell^{1}(S)$ implies the finiteness (amenability) of $S$, respec…
In this paper we introduce the notion of $Δ$-weak character amenable Banach algebras and investigate $Δ$-weak character amenability of certain Banach algebras such as projective tensor product $A\widehat{\otimes}B$, Lau product $A\times_TB$ for every Banach algebra $ A$ and $B$, where $T$ be a homomorphism from $B$ into $A$, abstract Segal algebras and module extension Banach algebras.
In recent work of the authors the notion of a derivation being approximately semi-inner arose as a tool for investigating (approximate) amenability questions for Banach algebras. Here we investigate this property in its own right, together with the consequent one of approximately semi-amenability. Under certain hypotheses regarding approximate identities this new notion is the same as approximate amenability, but more generally it covers some important classes of algebras which are not approximately amenable, in particular Segal algebras on amenable SIN-groups.
We provide a survey of the similarities and differences between Banach and Frechet algebras including some known results and examples. We also collect some important generalizations in abstract harmonic analysis; for example, the generalization of the concepts of vector-valued Lipschitz algebras, abstract Segal algebras, Arens regularity, amenability, weak amenability, ideal amenability, etc.
Let G be a locally compact group, and let A(G) and B(G) denote its Fourier and Fourier-Stieltjes algebras. These algebras are dual objects of the group and measure algebras, L^1(G) and M(G), in a sense which generalizes the Pontryagin duality theorem on abelian groups. We wish to consider the amenability properties of A(G) and B(G) and compare them to such properties for L^1(G) and M(G). For us, ``amenability properties'' refers to amenability, weak amenability, and biflatness, as well as some properties which are more suited to special settings, such as the hyper-Tauberian property for semisi…
Amenability of any of the algebras described in the title is known to force them to be finite-dimensional. The analogous problems for \emph{approximate} amenability have been open for some years now. In this article we give a complete solution for the first two classes, using a new criterion for showing that certain Banach algebras without bounded approximate identities cannot be approximately amenable. The method also provides a unified approach to existing non-approximate amenability results, and is applied to the study of certain commutative Segal algebras. Using different techniques, we pr…
Let $\mathfrak{A}$ be a Banach algebra, and $\mathcal{X}$ a Banach $\mathfrak{A}$-bimodule. A bounded linear mapping $\mathcal{D}:\mathfrak{A}\rightarrow \mathcal{X}$ is approximately semi-inner derivation if there eixist nets $(ξ_α)_α$ and $(μ_α)_α$ in $\mathcal{X}$ such that, for each $a\in\mathfrak{A}$, $\mathcal{D}(a)=\lim_α(a.ξ_α-μ_α.a)$. $\mathfrak{A}$ is called approximately semi-amenable if for every Banach $\mathfrak{A}$-bimodule $\mathcal{X}$, every $\mathcal{D}\in\mathcal{Z}^{1}(\mathfrak{A},\mathcal{X}^{*})$ is approximtely semi-inner. There are some Banach algebras which are appro…
We continue the investigation of notions of approximate amenability that were introduced in work of the second and third authors. It is shown that every boundedly approximately contractible Banach algebra has a bounded approximate identity. Among our other results, it is shown that the Fourier algebra of the free group on two generators is not approximately amenable. Further examples are obtained of ${\ell}^1$-semigroup algebras which are approximately amenable but not amenable; using these, we show that bounded approximate amenability need not imply sequential approximate amenability. Results…
We study various operator homological properties of the Fourier algebra $A(G)$ of a locally compact group $G$. Establishing the converse of two results of Ruan and Xu, we show that $A(G)$ is relatively operator 1-projective if and only if $G$ is IN, and that $A(G)$ is relatively operator 1-flat if and only if $G$ is inner amenable. We also exhibit the first known class of groups for which $A(G)$ is not relatively operator $C$-flat for any $C\geq1$. As applications of our techniques, we establish a hereditary property of inner amenability, answer an open question of Lau and Paterson, and answer…
In this paper we will study the projetivity of various natural modules associated to operator Segal algebras of the Fourier algebra of a locally compact group. In particular, we will focus on the question of identifying when such modules will be projective in the category of operator spaces. Projectivity often implies that the underlying group is discrete or even finite. We will also look at the projectivity for modules of $A_{cb}(G)$, the closure of $A(G)$ in the space of its completely bounded mutipliers. Here we give an evidence to show that weak amenability of $G$ plays an important role.
Let $G$ be a locally compact group and $(Φ, Ψ)$ be a complementary pair of $N$-functions. In this paper, using the powerful tool of porosity, it is proved that when $G$ is an amenable group, then the Figà-Talamanca-Herz-Orlicz algebra ${\A}_Φ(G)$ is a Banach algebra under convolution product if and only if $G$ is compact. Then it is shown that ${\A}_Φ(G)$ is a Segal algebra, and as a consequence, the amenability of ${\A}_Φ(G)$ and the existence of a bounded approximate identity for ${\A}_Φ(G)$ under the convolution product is discussed. Furthermore, it is shown that for a compact abelian group…
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