M. Khosravi, M. S. Moslehian, A. N. Motlagh
In this paper, we define the first topological $(σ,τ)$-cohomology group and examine vanishing of the first $(σ,τ)$-cohomology groups of certain triangular Banach algebras. We apply our results to study the $(σ,τ)$-weak amenability and $(σ,τ)$-amenability of triangular Banach algebras.
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In this paper, we define some new notions of triangular Banach algebras and we investigate the derivations on these algebras.
Suppose that A and B are unital Banach algebras with units 1_A and 1_B, respectively, M is a unital Banach A-B-bimodule, T=Tri(A,M,B) is the triangular Banach algebra, X is a unital T-bimodule, X_{AA}=1_AX1_A, X_{BB}=1_BX1_B, X_{AB}=1_AX1_B and X_{BA}=1_BX1_A. Applying two nice long exact sequences related to A, B, T, X, X_{AA}, X_{BB}, X_{AB} and X_{BA} we establish some results on (co)homology of triangular Banach algebras.
In this paper we investigate Hirano invertibility of anti-triangular matrix over a Banach algebra. Let $a\in {\mathcal A}^H, b\in {\mathcal A}^{sD}.$ If $b^Da=0, bab^π=0,$ we prove that $\begin{pmatrix} a&1\\ b&0 \end{pmatrix}\in M_2(\mathcal A)^H.$ Moreover, we considered Hirano invertibility of anti-triangular matrices under commutative-like conditions. These provide new kind of operator matrices with tripotent and nilpotent decompositions.
Let B be a unital Banach algebra. A projection in B which is equivalent to the identitity may give rise to a matrix-like structure on any two-sided ideal A in B. In this set-up we prove a theorem to the effect that the bounded Hochschild cohomology H^n(A,A^*) vanishes for all n>=1. The hypothesis of this theorem involve (i) strong H-unitality of A, (ii) a growth condition on diagonal matrices in A, (iii) an extension of A in B with trivial bounded simplicial homology. As a corollary we show that if X is an infinite dimensional Banach space with the bounded approximation property, L_1(μ,Ω) is a…
We introduce the notions of approximate cohomology and approximate homotopy in Banach algebras and we study the relation between them. We show that the approximate homotopically equivalent cochain complexes give the same approximate cohomologies. As an special case, approximate Hochschild cohomology is introduced and studied.
Let $\mathcal A$, $\mathcal B$ be Banach $\mathfrak A$-modules with compatible actions and $\mathcal M$ be a left Banach $\mathcal A$-$\mathfrak A$-module and a right Banach $\mathcal B$-$\mathfrak A$-module. In the current paper, we study module amenability, $n$-weak module amenability and module Arens regularity of the triangular Banach algebra $\mathcal T=[ {cc} \mathcal A & \mathcal M & \mathcal B ]$ (as an $\mathfrak T:=\Big{[ {cc} α& & α ] | α\in\mathfrak A\Big}$-module). We employ these results to prove that for an inverse semigroup $S$ with subsemigroup $E$ of idempotents, the triangul…
Completion problem of upper triangular $3\times3$ operator matrices on arbitrary Banach spaces
本文研究了任意Banach空间上3×3上三角算子矩阵的补全问题，利用Harte的指数定理和Banach空间嵌入技术，推广了Han等人结论，并探讨了高维上三角算子的可逆性条件。
We prove vanishing of cohomology with coefficients in representations on a large class of Banach spaces for a group acting "nicely" on a simplicial complexes based on spectral properties of the 1-dimensional links of the simplicial complex.
Complementation in Continuous Cohomology with Coefficients in Banach Modules
本文引入弱唯一平稳表示概念，借此研究带有 Banach 模块系数的连续上同调中闭子空间的补全性，并以此扩展改进了幂零群、群乘积及格等设置下的上同调结果。
We revisit the old result that biflat Banach algebras have the same cyclic cohomology as $\mathbb C$, and obtain a quantitative variant (which is needed in forthcoming joint work of the author). Our approach does not rely on the Connes-Tsygan exact sequence, but is motivated strongly by its construction as found in [Connes,1985] and [Helemskii,1992].
Isomorphisms, Moduli, and Cohomological Dimension for Twisted Triangular Banach Algebras
本文提出扭曲三角 Banach 代数 T_sigma(A,B;X)，给出其完整同构分类，证明群代数情形下的新动力学不变量，并建立 Hochschild 上同调维数界与可和性判据。
The ${\ell}^1$-convolution algebra of a semilattice is known to have trivial cohom ology in degrees 1,2 and 3 whenever the coefficient bimodule is symmetric. We ex tend this result to all cohomology groups of degree $\geq 1$ with symmetric coef ficients. Our techniques prove a stronger splitting result, namely that the spli tting can be made natural with respect to the underlying semilattice.
A Banach algebra A is self-induced if the multiplication is an isomorphism from the A-balanced projective tensor-square of A to A. The class of self-induced Banach algebras is a natural generalization of unital Banach algebras, providing a fertile framework for developing homological aspects of unital Banach algebras. Elementary results with applictions to computations of the bounded Hochschild cohomology groups H^1(A,A^*) with emphasis on A=A(X), the approximable operators on a Banach space X, are given.
A theorem of Delorme states that every unitary representation of a connected Lie group with nontrivial reduced first cohomology has a finite-dimensional subrepresentation. More recently Shalom showed that such a property is inherited by cocompact lattices and stable under coarse equivalence among amenable countable discrete groups. We give a new geometric proof of Delorme's theorem which extends to a larger class of groups, including solvable $p$-adic algebraic groups, and finitely generated solvable groups with finite Prüfer rank. Moreover all our results apply to isometric representations in…
Let $M_{C}=\left(\begin{array}{cc}A&C\\0&B\\\end{array} \right)$ is a 2-by-2 upper triangular operator matrix acting on the Banach space $X\oplus Y$ or Hilbert space $H\oplus K$. For the most import spectra such as spectrum, essential spectrum and Weyl spectrum, the characterizations of the perturbations of $M_{C}$ on Banach space have been presented. However, the characterization for the Browder spectrum on the Banach space is still unknown. The goal of this paper is to present some necessary and sufficient conditions for $M_C$ to be Browder for some $C\in B(Y,X)$ by using an alternative appr…
Let $A$ be a Banach algebra and $M$ be a Banach right $A$-module. A linear map $δ: M\to M$ is called a generalized derivation if there exists a derivation $d : A \to A$ such that $$δ(xa)=δ(x)a + x d(a) \quad (a \in A, x \in M).$$ In this paper, we associate a triangular Banach algebra ${\mathcal T}$ to Banach $A$-module $M$ and investigate the relation between generalized derivations on $M$ and derivations on ${\mathcal T}$. In particular, we prove that the so-called generalized first cohomology group of $M$ is isomorphic to the first cohomology group of ${\mathcal T}$.
In this paper, we study the notion of Connes amenability for a class of $I\times{I}$-upper triangular matrix algebra $UP(I,\mathcal{A})$, where $\mathcal{A}$ is a dual Banach algebra with a non-zero $wk^\ast$-continuous character and $I$ is a totally ordered set. For this purpose, we characterize the $ϕ$-Connes amenability of a dual Banach algebra $\mathcal{A}$ through the existence of a specified net in $\mathcal{A}\hat{\otimes}\mathcal{A}$, where $ϕ$ is a non-zero $wk^\ast$-continuous character. Using this, we show that $UP(I,\mathcal{A})$ is Connes amenable if and only if $I$ is singleton a…
In this paper, we establish the Pexiderized stability of coboundaries and cocycles and use them to investigate the Hyers--Ulam stability of some functional equations. We prove that for each Banach algebra $A$, Banach $A$-bimodule $X$ and positive integer $n, H^n(A,X)=0$ if and only if the $n$-th cohomology group approximately vanishes.
Let $X$ and $Y$ be Banach spaces, $A\in B(X)$, $B\in B(Y)$, $C\in B(Y, X)$, $M_{C}=({cc}A&C 0&B)$ be the operator matrix acting on the Banach space $X\oplus Y$. In this paper, we give out 20 kind spectra structure of $M_C$, decide 18 kind spectra filling-in-hole properties of $M_C$, and present 10 examples to show that some conclusions about the spectra structure or filling-in-hole properties of $M_C$ are not true.
Let A and U be Banach algebras such that U is also a Banach A- bimodule with compatible algebra operations, module actions and norm. By defining an approprite action, we turn l1-direct product A item 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 litem U and every semidirect products of Banach algebras can be represented as this form. In this paper we consider the Banach algebra A litem U as mentioned and study the derivations on it. In fact we consider the automatic continui…
We initiate a study of cohomological aspects of weakly almost periodic group representations on Banach spaces, in particular, isometric representations on reflexive Banach spaces. Using the Ryll-Nardzewski fixed point Theorem, we prove a vanishing result for the restriction map (with respect to a subgroup) in the reduced cohomology of weakly periodic representations. Combined with the Alaoglu-Birkhoff decomposition theorem, this generalizes and complements theorems on continuous group cohomology by several authors.
In this article we study cohomology of a group with coefficients in representations on Banach spaces and its stability under deformations. We show that small, metric deformations of the representation preserve vanishing of cohomology. As applications we obtain deformation theorems for fixed point properties on Banach spaces. In particular, our results yield fixed point theorems for affine actions in which the linear part is not uniformly bounded. Our proofs are effective and allow for quantitative estimates.
Let A be a Banach algebra and I be a closed ideal of A. We say that A is amenable relative to I, if A/I is an amenable Banach algebra. We study the relative amenability of Banach algebras and investigate the relative amenability of triangular Banach algebras and Banach algebras associated to locally compact groups. We generalize some of the previous known results by applying the concept of relative amenability of Banach algebras, especially, we present a generalization of Johnson's theorem in the concept of relative amenability.
Let B be any Lp space for p in (1,infty) or any Banach space isomorphic to a Hilbert space, and k be a nonnegative integer. We show that if n is at least 4, then the universal lattice Gamma =SL_n (Z[x1,...,xk]) has property (F_B) in the sense of Bader--Furman--Gelander--Monod. Namely, any affine isometric action of Gamma on B has a global fixed point. The property of having (F_B) for all B above is known to be strictly stronger than Kazhdan's property (T). We also define the following generalization of property (F_B)$ for a group: the boundedness property of all affine quasi-actions on B. We n…
The main objective of this paper is to determine the simplicial and cyclic cohomology groups of the Cuntz semigroup algebra $\ell^1(\Cuntz)$. In order to do so, we first establish some general results which can be used when studying simplicial and cyclic cohomology of Banach algebras in general. We then turn our attention to $\ell^1(\Cuntz)$, showing that the cyclic cohomology groups of degree $n$ vanish when $n$ is odd and are one-dimensional when $n$ is even ($n\ge 2$). Using the Connes-Tzygan exact sequence, these results are used to show that the simplicial cohomology groups of degree $n$ …
In this paper, we study the notion of approximately bi at Banach algebras for second dual Banach algebras and semigroup algebras. We show that for a locally compact group G, if S(G)?? is approximately bi at, then G is amenable group. Also we give some conditions which the second dual of a Triangular Banach algebra is never approximately bi at. For a uniformly locally finite semigroup S, we show that `1(S) is approximately bi at if and only if `1(S) is bi at.
Let $\mathcal{L}$ be a completely distributive commutative subspace lattice or a subspace lattice with two atoms, we use a unified approach to study the derivations, homomorphisms on $\mathrm{Alg} \mathcal{L}$. We verify that the multiplier algebra of $\mathrm{Alg} \mathcal{L}\cap \mathcal{K}(\mathcal{H})$ is isomorphic to $\mathrm{Alg} \mathcal{L}$ and $\mathrm{Alg} \mathcal{L}$ is zero product determined. For $T$ in $M_{n}(\mathbb{C})$, $n\geq 2$, we show that $\mathcal{A}_{T}$ is zero product determined if and only if every local derivation from $\mathcal{A}_{T}$ into any Banach $\mathcal{A…
We study the relation between module and Hochschild cohomology groups of Banach algebras with a compatible module structure. More precisely, we show that for every commutative Banach $ \mathcal{A} $-$ \mathfrak{A}$-bimodule $ X $ and every $ k \in \mathbb{N}$, the seminormed spaces $ \mathcal{H}^{k}_{\mathfrak{A}} (\mathcal{A},X^*)$ and $ \mathcal{H}^k (\frac{\mathcal{A}}{J}, X^*) $ are isomorphic, where $ J $ is the closed ideal of $ \mathcal{A} $ generated by the elements of the form $ a (α\cdot b)-(a\cdot α)b$ with $ a,b \in \mathcal{A} $ and $α\in \mathfrak{A}. $ As an example, we calculat…
In this paper, we apply the theory of algebraic cohomology to study the amenability of Thompson's group $\mathcal{F}$. We introduce the notion of unique factorization semigroup which contains Thompson's semigroup $\mathcal{S}$ and the free semigroup $\mathcal{F}_n$ on $n$ generators ($\geq2$). Let $\mathfrak{B}(\mathcal{S})$ and $\mathfrak{B}(\mathcal{F}_n)$ be the Banach algebras generated by the left regular representations of $\mathcal{S}$ and $\mathcal{F}_n$, respectively. It is proved that all derivations on $\mathfrak{B}(\mathcal{S})$ and $\mathfrak{B}(\mathcal{F}_n)$ are automatically c…
We extend a p-adic spectral theorem of M. M. Vishik to a certain class of p-adic Banach algebras. This class includes inductive limits of finite-dimensional p-adic Banach algebras of the form B(X), where X is a finite-dimensional p-adic Banach space. In particular, we present a p-adic spectral theorem for p-adic UHF and p-adic triangular UHF (TUHF) Banach algebras.
We investigate the higher-dimensional amenability of tensor products $\A \ptp \B$ of Banach algebras $\A$ and $\B$. We prove that the weak bidimension $db_w$ of the tensor product $\A \ptp \B$ of Banach algebras $\A$ and $\B$ with bounded approximate identities satisfies \[ db_w \A \ptp \B = db_w \A + db_w \B. \] We show that it cannot be extended to arbitrary Banach algebras. For example, for a biflat Banach algebra $\A$ which has a left or right, but not two-sided, bounded approximate identity, we have $db_w \A \ptp \A \le 1$ and $db_w \A + db_w \A =2.$ We describe explicitly the continuous …
Let $\cal A$ be a Banach algebra. We study those closed ideals $I$ of $\cal A$ for which the first cohomology group of $\cal A$ with coefficients in $I^*$ is trivial; i.e. $H^1(\cal A,I^*)=\{0\}$. We investigate such closed ideals when $\cal A$ is weakly amenable or biflat. Also we give some hereditary properties of ideal amenability.
Let $A$ be a unital commutative Banach algebra with maximal ideal space $X.$ We determine the rational H-type of the group $GL_n (A)$ of invertible n by n matrices with coefficients in A, in terms of the rational cohomology of $X.$ We also address an old problem of J. L. Taylor. Let $Lc_n (A)$ denote the space of "last columns" of $GL_n (A).$ For $n > 1 + s/2,$ we construct a natural isomorphism from the rational Cech cohomology group $H^s (X; Q)$ to the rational homotopy group $π_{2 n - 1 - s} (Lc_n (A)) \otimes Q,$ which shows that the rational cohomology groups of $X$ are determined by a to…
In this paper, we introduce a notion of approximate Connes-biprojectivity for dual Banach algebras. We study the relation between approximate Connes-biprojectivity, Johnson pseudo-Connes amenability and $φ$-Connes amenability. We propose a criterion to show that some certain dual triangular Banach algebras are not approximately Connes-biprojective. Next we show that for a locally compact group $G$, the Banach algebra $M(G)$ is approximately Connes-biprojective if and only if $G$ is amenable. Finally for an infinite commutative compact group $G$ we show that the Banach algebra $L^2(G)$ with con…
Let $X$ be a left introverted subspace of dual of a Banach algebra. We study $Z_t(X^*),$ the topological center of Banach algebra $X^*$. We fined the topological center of $(X\cA)^*$, when $\cA$ has a bounded right approximate identity and $\cA\subseteq X^*.$ So we introduce a new notation of amenability for a dual Banach algebra $\cal A$. A dual Banach algebra $\cal A$ is weakly Connes-amenable if the first $weak^*-$continuous cohomology group of $\cal A$ with coefficients in $\cal A$ is zero; i.e., $H^1_{w^*}(\cal A, \cal A)=\{o\}$. We study the weak Connes-amenability of some dual Banach al…
In this short note, we give some new results on continuous bounded cohomology groups of topological semigroups with values in complex field. We show that the second continuous bounded cohomology group of a compact metrizable semigroup, is a Banach space. Also, we study cohomology groups of amenable topological semigroups, and we show that cohomology groups of rank greater than one of a compact left or right amenable semigroup, are trivial. Also, we give some examples and applications about topological lattices.
In this paper, we study the structure of Lipschitz algebras under the notions of approximate biflatness and Johnson pseudo-contractibility. We show that for a compact metric space $X,$ the Lipschitz algebras $Lip_α(X)$ and $\ell ip_α(X)$ are approximately biflat if and only if $X$ is finite, provided that $0<α<1$. We give an enough and sufficient condition that a vector-valued Lipschitz algebras is Johnson pseudo-contractible for each $α>0.$ We also show that some triangular Banach algebras are not approximately biflat.
The first part of this paper is devoted to the theory of Poisson-Lie groups in the Banach setting. Our starting point is the straightforward adaptation of the notion of Manin triples to the Banach context. The difference with the finite-dimensional case lies in the fact that a duality pairing between two non-reflexive Banach spaces is necessary weak (as opposed to a strong pairing where one Banach space can be identified with the dual space of the other). The notion of generalized Banach Poisson manifolds introduced in this paper is compatible with weak duality pairings between the tangent spa…
Let $A$ and $B$ be Banach algebras and let $T$ be an algebra homomorphism from $B$ into $A$. The Cartesian product space $A\times B$ by $T$- Lau product and $\ell^{1}$- norm becomes a Banach algebra $A\times_{T}B$. We investigate the notions such as injectivity, projectivity and flatness for the Banach algebra $A\times_{T}B$. We also characterize Hochschild cohomology for the Banach algebra $A\times_{T}B$.
We introduce notions of compactness and weak compactness for multilinear maps from a product of normed spaces to a normed space, and prove some general results about these notions. We then consider linear maps $T:A\to B$ between Banach algebras that are "close to multiplicative" in the following senses: the failure of multiplicativity, defined by $S_T(a,b)=T(a)T(b)-T(ab)$, is compact [respectively weakly compact]. We call such maps cf-homomorphisms [respectively wcf-homomorphisms]. We also introduce a number of other, related definitions. We state and prove some general theorems about these ma…
Let ${\mathcal A}$ and ${\frak A}$ be Banach algebras such that ${\mathcal A}$ is a Banach ${\frak A}$-bimodule with compatible actions. We define the product ${\cal A}\rtimes{\frak A}$, which is a strongly splitting Banach algebra extension of ${\frak A}$ by $\cal A$. After characterization of the multiplier algebra, topological centre, (maximal) ideals and spectrum of ${\cal A}\rtimes{\frak A}$, we restrict our investigation to the study of semisimplicity, regularity, Arens regularity of ${\cal A}\rtimes{\frak A}$ in relation to that of the algebras $\cal A$, $\frak A$ and the action of $\fr…
In this paper, we present a new characterization of g-Drazin inverse in a Banach algebra. We prove that an element a is a Banach algebra has g-Drazin inverse if and only if there exists $x\in A$ such that $ax=xa, a-a^2x\in A^{qnil}$. we obtain the sufficient and necessary conditions for the existence of the g-Drain inverse for certain $2 \times 2$ anti-triangular matrices over a Banach algebra. These extend the results of Koliha (Glasgow Math. J., 38(1996), 367-381), Nicholson (Comm. Algebra,27(1999), 3583-3592 and Zou et al. (Studia Scient. Math. Hungar., 54(2017,489-508).
We show a structural property of cohomology with coefficients in an isometric representation on a uniformly convex Banach space: if the cohomology group $H^1(G,π)$ is reduced, then, up to an isomorphism, it is a closed complemented, subspace of the space of cocycles and its complement is the subspace of coboundaries.
Let $G$ be a finitely generated, infinite group, let $p>1$, and let $L^p(G)$ denote the Banach space $\{\sum_{x\in G} a_xx \mid \sum_{x\in G} |a_x |^p < \infty \}$. In this paper we will study the first cohomology group of $G$ with coefficients in $L^p(G)$, and the first reduced $L^p$-cohomology space of $G$. Most of our results will be for a class of groups that contains all finitely generated, infinite nilpotent groups.
Garland's method with Banach coefficients
本文建立适用于所有自反 Banach 空间的 Garland 方法，并据此获得多类 Banach 系数上同调消失准则，用于改进随机群不动点与共形维数结果及 p-Schatten 稳定性判据。
In a Banach algebra, we introduce a new type of generalized inverse called g$π$-Hirano inverse. Firstly, several existence criteria and the equivalent definition of this inverse are investigated. Then, we discuss the relationship between the g$π$-Hirano invertibility of $a$, $b$ and that of the sum $a+b$ under some weaker conditions. Finally, as applications to the previous additive results, some equivalent characterizations for the g$π$-Hirano invertibility of the anti-triangular matrix over Banach algebras are obtained.In particular, some results in this paper are different from the correspo…
In this paper, for a Banach algebra A, we introduced the new notions of approximately left $ϕ$-biprojective and approximately left character biprojective, where $ϕ$ is a non-zero multiplicative linear functional on A. We show that for SIN group G, Segal algebra S(G) is approximately left $ϕ_1$- biprojective if and only if G is amenable, where $ϕ_1$ is the augmentation character on S(G). Also we showed that the measure algebra M(G) is approximately left character biprojective if and only if G is discrete and amenable. For a Clifford semigroup S, we show that `1(S) is approximately left characte…
Some remarks on acyclicity in bounded cohomology
本文证明群同态的核有界无圈当且仅当该同态对所有对偶赋范模诱导有界上同调同构，并推广到道路连通空间间的映射，核心是证明有界无圈群对所有对偶赋范平凡模具有平凡有界上同调。
Żuk proved that if a finitely generated group admits a Cayley graph such that the Laplacian on the links of this Cayley graph has a spectral gap $> \frac{1}{2}$, then the group has property (T), or equivalently, every affine isometric action of the group on a Hilbert space has a fixed point. We prove that the same holds for affine isometric actions of the group on a uniformly curved Banach space (for example an $L^p$-space with $1 < p < \infty$ or an interpolation space between a Hilbert space and an arbitrary Banach space) as soon as the Laplacian on the links has a two-sided spectral gap $>1…
We introduce two notions of amenability for a Banach algebra $\cal A$. Let $I$ be a closed two-sided ideal in $\cal A$, we say $\cal A$ is $I$-weakly amenable if the first cohomology group of $\cal A$ with coefficients in the dual space $I^*$ is zero; i.e., $H^1({\cal A},I^*)=\{0\}$, and, $\cal A$ is ideally amenable if $\cal A$ is $I$-weakly amenable for every closed two-sided ideal $I$ in $\cal A$. We relate these concepts to weak amenability of Banach algebras. We also show that ideal amenability is different from amenability and weak amenability. We study the $I$-weak amenability of a Bana…
We introduce two notions of amenability for a Banach algebra $\cal A$. Let $n\in \Bbb N$ and let $I$ be a closed two-sided ideal in $\cal A$, $\cal A$ is $n-I-$weakly amenable if the first cohomology group of $\cal A$ with coefficients in the n-th dual space $I^{(n)}$ is zero; i.e., $H^1({\cal A},I^{(n)})=\{0\}$. Further, $\cal A$ is n-ideally amenable if $\cal A$ is $n-I-$weakly amenable for every closed two-sided ideal $I$ in $\cal A$. We find some relationships of $n-I-$ weak amenability and $m-J-$ weak amenability for some different m and n or for different closed ideals $I$ and $J$ of $\c…
Measurable bounded cohomology of $t$-discrete measured groupoids via resolutions
本文通过同调代数方法定义了 $t$-离散测度群胚的有界上同调，扩展了经典理论，并证明可利用顺从空间或强边界进行计算。
Taking the l^1-completion and the topological dual of the singular chain complex gives rise to l^1-homology and bounded cohomology respectively. In contrast to l^1-homology, major structural properties of bounded cohomology are well understood by the work of Gromov and Ivanov. Based on an observation by Matsumoto and Morita, we derive a mechanism linking isomorphisms on the level of homology of Banach chain complexes to isomorphisms on the level of cohomology of the dual Banach cochain complexes and vice versa. Therefore, certain results on bounded cohomology can be transferred to l^1-homology…
The aim of this paper is to extend the framework of the spectral method for proving property (T) to the class of reflexive Banach spaces and present a condition implying that every affine isometric action of a given group $G$ on a reflexive Banach space $X$ has a fixed point. This last property is a strong version of Kazhdan's property (T) and is equivalent to the fact that $H^1(G,π)=0$ for every isometric representation $π$ of $G$ on $X$. The condition is expressed in terms of $p$-Poincaré constants and we provide examples of groups, which satisfy such conditions and for which $H^1(G,π)$ vani…
In this paper, we study some cohomlogical properties of Banach algebras. For a Banach algebra $A$ and a Banach $A$-bimodule $B$, we investigate the vanishing of the first Hochschild cohomology groups $H^1(A^n,B^m)$ and $H_{w^*}^1(A^n,B^m)$, where $0\leq m,n\leq 3$. For amenable Banach algebra $A$, we show that there are Banach $A$-bimodules $C$, $D$ and elements $\mathfrak{a}, \mathfrak{b}\in A^{**}$ such that $$Z^1(A,C^*)=\{R_{D^{\prime\prime}(\mathfrak{a})}:~D\in Z^1(A,C^*)\}=\{L_{D^{\prime\prime}(\mathfrak{b})}:~D\in Z^1(A,D^*)\}.$$ where, for every $b\in B$, $L_{b}(a)=ba$ and $R_{b}(a)=a b…
Building on the recent determination of the simplicial cohomology groups of the convolution algebra ${\ell}^1({\mathbb Z}_+^k)$ [Gourdeau, Lykova, White, 2005] we investigate what can be said for cohomology of this algebra with more general symmetric coefficients. Our approach leads us to a discussion of Harrison homology and cohomology in the context of Banach algebras, and a development of some of its basic features. As an application of our techniques we reprove some known results on second-degree cohomology.
Classically, Tannaka-Krein duality allows us to reconstruct a (co)algebra from its category of representation. In this paper we present an approach that allows us to generalise this theory to the setting of Banach spaces. This leads to several interesting applications in the directions of analytic quantum groups, bounded cohomology and galois cohomology. A large portion of this paper is dedicated to such examples.
We study bounded actions of groups and semigroups $G$ on exact sequences of Banach spaces from the point of view of quasilinear maps, characterize the actions on the twisted sum space by commutator estimates and introduce the associated notions of $G$-centralizer and $G$-equivariant map. We will show that when (A) $G$ is an amenable group and (U) the target space is complemented in its bidual by a $G$-equivariant projection, then uniformly bounded compatible families of operators generate bounded actions on the twisted sum space; that compatible quasilinear maps are linear perturbations of $G$…
We study the cohomology of isometric group actions on (super) reflexive Banach spaces with a focus on the relation between finite conjugacy classes and split exactness of cochain complexes. In particular, we show that, if a uniformly convex Banach module has no almost invariant vectors under the FC-centre of the acting group, then the associated cochain complex is split exact. Other similar rigidity results are established that are related to prior work of Bader - Furman - Gelander - Monod, Bader - Rosendal - Sauer and Nowak.
We prove that for every group $G$ and any two sets $I,J$, the Brandt semigroup algebras $\ell(B(I,G))$ and $\ell(B(J,G))$ are Morita equivalent with respect to the Morita theory of self-induced Banach algebras introduced by Gronbaek. As applications, we show that if $G$ is an amenable group, then for a wide class of Banach $\ell(B(I,G))$-bimodules $E$, and every $n>0$, the bounded Hochschild cohomology groups $H^n(\ell(B(I,G)),E^*)$ are trivial, and also, the notion of approximate amenability is not Morita invariant.
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