We study representations of Banach algebras on reflexive Banach spaces. Algebras which admit such representations which are bounded below seem to be a good generalisation of Arens regular Banach algebras; this class includes dual Banach algebras as defined by Runde, but also all group algebras, and all discrete (weakly cancellative) semigroup algebras. Such algebras also behave in a similar way to C$^*$- and W$^*$-algebras; we show that interpolation space techniques can be used in the place of GNS type arguments. We define a notion of injectivity for dual Banach algebras, and show that this is equivalent to Connes-amenability. We conclude by looking at the problem of defining a well-behaved tensor product for dual Banach algebras.
We characterize projective and injective Banach modules in approximate terms, generalizing thereby a characterization of contractible Banach algebras given by F. Ghahramani and R. J. Loy. As a corollary, we show that each uniformly approximately amenable Banach algebra is amenable. Some applications to homological dimensions of Banach modules and algebras are also given.
In this paper, we study the notion of $ϕ$-injectivity in the special case that $ϕ=0$. For an arbitrary locally compact group $G$, we characterize the 0-injectivity of $L^{1}(G)$ as a left $L^{1}(G)$ module. Also, we show that $L^{1}(G)^{**}$ and $L^{p}(G)$ for $1<p<\infty$ are 0-injective Banach $L^{1}(G)$ modules.
In this book we study metric and topological versions of projectivity injectivity and flatness of Banach modules over Banach algebras. These two non-standard versions of Banach homology theories are studied in parallel under unified approach.
In this paper, we investigate the notion $ϕ$-injectivity for Banach $A$-modules, where $ϕ$ is a character on $A.$ We obtain some hereditary properties of $ϕ$-injectivity for certain classes of Banach modules related to closed ideals. These results allow us to study $ϕ$-injectivity of certain Banach $A$-modules in commutative case, specially $\ell^{1}$-semilattice algebras. As an application, we give an example of a non-injective Banach module which is $ϕ$-injective for each character $ϕ.$
We first unify all notions of partial injectivity appearing in the literature ---(universal) separable injectivity, (universal) $\aleph$-injectivity --- in the notion of $(α, β)$-injectivity ($(α, β)_λ$-injectivity if the parameter $λ$ has to be specified). Then, extend the notion of space of universal disposition to space of universal $(α, β)$-disposition. Finally, we characterize the $1$-complemented subspaces of spaces of universal $(α, β)$-disposition as precisely the spaces $(α, β)_1$-injective.
We study distinguished objects in the category $\mathcal{BL}$ of Banach lattices and lattice homomorphisms. The free Banach lattice construction introduced by de Pagter and Wickstead generates push-outs, and combining this with an old result of Kellerer on marginal measures, the amalgamation property of Banach lattices is established. This will be the key tool to prove that $L_1([0,1]^{\mathfrak{c}})$ is separably $\mathcal{BL}$-injective, as well as to give more abstract examples of Banach lattices of universal disposition for separable sublattices. Finally, an analysis of the ideals on $C(Δ,…
Let $A$ be a dual Banach algebra with predual $A_\ast$ and consider the following assertions: (A) $A$ is Connes-amenable; (B) $A$ has a normal, virtual diagonal; (C) $A_\ast$ is an injective $A$-bimodule. For general $A$, all that is known is that (B) implies (A) whereas, for von Neumann algebras, (A), (B), and (C) are equivalent. We show that (C) always implies (B) whereas the converse is false for $A = M(G)$ where $G$ is an infinite, locally compact group. Furthermore, we present partial solutions towards a characterization of (A) and (B) for $B(G)$ in terms of $G$: For amenable, discrete $G…
We find large classes of injective and projective $p$-multinormed spaces. In fact, these classes are universal, in the sense that every $p$-multinormed space embeds into (is a quotient of) an injective (resp. projective) $p$-multinormed space. As a consequence, we show that any $p$-multinormed space has a canonical representation as a subspace of a quotient of a Banach lattice.
In this paper we deal with two weaker forms of injectivity which turn out to have a rich structure behind: separable injectivity and universal separable injectivity. We show several structural and stability properties of these classes of Banach spaces. We provide natural examples of (universally) separably injective spaces, including $\mathcal L_\infty$ ultraproducts built over countably incomplete ultrafilters, in spite of the fact that these ultraproducts are never injective. We obtain two fundamental characterizations of universally separably injective spaces: a) A Banach space $E$ is unive…
In a locally $λ$-presentable category, with $λ$ a regular cardinal, classes of objects that are injective with respect to a family of morphisms whose domains and codomains are $λ$-presentable, are known to be characterized by their closure under products, $λ$-directed colimits and $λ$-pure subobjects. Replacing the strict commutativity of diagrams by "commutativity up to $\varepsilon$", this paper provides an "approximate version" of this characterization for categories enriched over metric spaces. It entails a detailed discussion of the needed $\varepsilon$-generalizations of the notion of $λ…
Properties of categories enriched over the category of metric spaces are investigated and applied to a study of constructions known from that category and the category of Banach spaces. For every class of morphisms satisfying a mild smallness condition we prove the corresponding approximate-injectivity class is weakly reflective, and we study the properties of the reflection morphisms. As an application we present a new categorical proof of the essential uniqueness of the Gurarii space.
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 $A\times_{T}B$.
Building on our previous work, we study the non-relative homology of quantum group convolution algebras. Our main result establishes the equivalence of amenability of a locally compact quantum group $\mathbb{G}$ and 1-injectivity of $L^{\infty}(\widehat{\mathbb{G}})$ as an operator $L^1(\widehat{\mathbb{G}})$-module. In particular, a locally compact group $G$ is amenable if and only if its group von Neumann algebra $VN(G)$ is 1-injective as an operator module over the Fourier algebra $A(G)$. As an application, we provide a decomposability result for completely bounded $L^1(\widehat{\mathbb{G}}…
We introduce an equivariant version of the weak expectation property (WEP) at the level of operator modules over completely contractive Banach algebras $A$. We prove a number of general results---for example, a characterization of the $A$-WEP in terms of an appropriate $A$-injective envelope, and also a characterization of those $A$ for which $A$-WEP implies WEP. In the case of $A=L^1(G)$, we recover the $G$-WEP for $G$-$C^*$-algebras in recent work of Buss--Echterhoff--Willett. When $A=A(G)$, we obtain a dual notion for operator modules over the Fourier algebra. These dual notions are related…
In this paper, we prove that every completely contractive dual Banach algebra is completely isometric to a $w^\ast$-closed subalgebra the operator space of completely bounded linear operators on some reflexive operator space.
A Banach space $X$ is elastic if there is a constant $K$ so that whenever a Banach space $Y$ embeds into $X$, then there is an embedding of $Y$ into $X$ with constant $K$. We prove that $C[0,1]$ embeds into separable infinite dimensional elastic Banach spaces, and therefore they are universal for all separable Banach spaces. This confirms a conjecture of Johnson and Odell. The proof uses incremental embeddings into $X$ of $C(K)$ spaces for countable compact $K$ of increasing complexity. To achieve this we develop a generalization of Bourgain's basis index that applies to unconditional sums of …
In this paper, among other things, we study those conditions under which the weighted semigroup algebra $\ell^1(S,ω)$ is a dual Banach algebra with respect to predual $c_0(S)$. Some useful examples, illustrating the results, are also included.
We begin the study of characterizations of recently defined approximate Schauder frame (ASF) and its duals for separable Banach spaces. We show that, under some conditions, both ASF and its dual frames can be characterized for Banach spaces. We also give an operator-theoretic characterization for similarity of ASFs. Our results encode the results of Holub, Li, Balan, Han, and Larson. We also address orthogonality of ASFs.
We define a Banach algebra A to be dual if $A = (A_\ast)^\ast$ for a closed submodule $A_\ast$ of $A^\ast$. The class of dual Banach algebras includes all $W^\ast$-algebras, but also all algebras M(G) for locally compact groups G, all algebras L(E) for reflexive Banach spaces E, as well as all biduals of Arens regular Banach algebras. The general impression is that amenable, dual Banach algebras are rather the exception than the rule. We confirm this impression. We first show that under certain conditions an amenable dual Banach algebra is already super-amenable and thus finite-dimensional. We…
We answer, by counterexample, several open questions concerning algebras of operators on a Hilbert space. The answers add further weight to the thesis that, for many purposes, such algebras ought to be studied in the framework of operator spaces, as opposed to that of Banach spaces and Banach algebras. In particular, the `nonselfadjoint analogue' of a W*-algebra resides naturally in the category of dual operator spaces, as opposed to dual Banach spaces. We also show that an automatic w*-continuity result in the preceding paper of the authors is sharp.
Let $G$ be a locally compact group, and let $WAP(G)$ denote the space of weakly almost periodic functions on $G$. We show that, if $G$ is a $[SIN]$-group, but not compact, then the dual Banach algebra $WAP(G)^\ast$ does not have a normal, virtual diagonal. Consequently, whenever $G$ is an amenable, non-compact $[SIN]$-group, $WAP(G)^\ast$ is an example of a Connes-amenable, dual Banach algebra without a normal,virtual diagonal.
This paper contains a study of the separable form $J_s(\cdot)$ of the classical Jung constant. We first establish, following Davis \cite{davis}, that a Banach space $X$ is $1$-separably injective if and only if $J_s(X)=1$. This characterization is then used for the understanding of new $1$-separably injective spaces. The last section establishes the inequality $\frac{1}{2}K(Y)J_s(X)\leq e(Y,X)$ connecting the separable Jung constant, Kottman's constant and the extension constant for Lipschitz maps, which is then used to obtain a simple proof of the equality $K(X,c_0)=e(X,c_0)$ of Kalton and a …
In the first part of the paper, we present a short survey of the theory of multipliers, or double centralisers, of Banach algebras and completely contractive Banach algebras. Our approach is very algebraic: this is a deliberate attempt to separate essentially algebraic arguments from topological arguments. We concentrate upon the problem of how to extend module actions, and homomorphisms, from algebras to multiplier algebras. We then consider the special cases when we have a bounded approximate identity, and when our algebra is self-induced. In the second part of the paper, we mainly concentra…
To every Banach space V we associate a compact right topological affine semigroup E(V). We show that a separable Banach space V is Asplund if and only if E(V) is metrizable, and it is Rosenthal (i.e. it does not contain an isomorphic copy of $l_1$) if and only if E(V) is a Rosenthal compactum. We study representations of compact right topological semigroups in E(V). In particular, representations of tame and HNS-semigroups arise naturally as enveloping semigroups of tame and HNS (hereditarily non-sensitive) dynamical systems, respectively. As an application we obtain a generalization of a theo…
In this paper we present a method to obtain Banach spaces of universal and almost-universal disposition with respect to a given class $\mathfrak M$ of normed spaces. The method produces, among other, the Gurari\uı space $\mathcal G$ (the only separable Banach space of almost-universal disposition with respect to the class $\mathfrak F$ of finite dimensional spaces), or the Kubis space $\mathcal K$ (under {\sf CH}, the only Banach space with the density character the continuum which is of universal disposition with respect to the class $\mathfrak S$ of separable spaces). We moreover show that $…
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 paper we define the module extension dual Banach algebras and we use this Banach algebras to finding the relationship between $weak^*-$continuous homomorphisms of dual Banach algebras and Connes-amenability. So we study the $weak^*-$continuous derivations on module extension dual Banach algebras.
A dual Banach algebra is a Banach algebra which is a dual space, with the multiplication being separately weak$^*$-continuous. We show that given a unital dual Banach algebra $\mc A$, we can find a reflexive Banach space $E$, and an isometric, weak$^*$-weak$^*$-continuous homomorphism $π:\mc A\to\mc B(E)$ such that $π(\mc A)$ equals its own bicommutant.
We shall develop a notion of amenability for dual Banach algebras, namely weak Connes amenability, which will play the role that weak amenability does for usual Banach algebras
Given a separable Banach space $E$, we construct an extremely non-complex Banach space (i.e. a space satisfying that $\|Id + T^2\|=1+\|T^2\|$ for every bounded linear operator $T$ on it) whose dual contains $E^*$ as an $L$-summand. We also study surjective isometries on extremely non-complex Banach spaces and construct an example of a real Banach space whose group of surjective isometries reduces to $\pm Id$, but the group of surjective isometries of its dual contains the group of isometries of a separable infinite-dimensional Hilbert space as a subgroup.
We extend the well-known criterion of Lotz for the dual Radon-Nikodym property (RNP) of Banach lattices to finitely generated Banach $C(K)$-modules and Banach $C(K)$-modules of finite multiplicity. Namely, we prove that if $X$ is a Banach space from one of these classes then its Banach dual $X^\star$ has the RNP iff $X$ does not contain a closed subspace isomorphic to $\ell^1$.
Let $ ( X , d ) $ be a metric space and $ A $ be a commutative Banach algebra such that $ Δ( A ) $ be nonempty. In this paper, we provide necessary and sufficient conditions, under which $ C_{BSE} ( Δ(A) ) $ is a Dual Banach algebra. Moreover, we provide some conditions for that $ Lip_α ( X , A ) $ is a Dual Banach algebra.
In this paper, we introduce a new notion of biprojectivity, called Connes-biprojective, for dual Banach algebras. We study the relation between this new notion to Connes-amenability and we show that, for a given dual Banach algebra $ \mathcal{A} $, it is Connes-amenable if and only if $ \mathcal{A} $ is Connes-biprojective and has a bounded approximate identity. Also, for an Arens regular Banach algebra $ \mathcal{A} $, we show that if $ \mathcal{A} $ is biprojective, then the dual Banach algebra $ \mathcal{A} ^{**} $ is Connes-biprojective.
In this paper we present new characterizations of Banach spaces whose duals are isomorphic to $l_{1}(Γ),$ extending results of Stegall, Lewis-Stegall and Cilia-D'Anna-Gutiérrez.
Spaces of homogeneous polynomials on a Banach space are frequently equipped with quasinorms instead of norms. In this paper we develop a technique to replace the original quasi-norm by a norm in a dual preserving way, in the sense that the dual of the space with the new norm coincides with the dual of the space with the original quasi-norm. Applications to problems on the existence and approximation of solutions of convolution equations and on hypercyclic convolution operators on spaces of entire functions are provided.
Given a Banach space. We show that its three times dual space can be written as a direct sum. Then being one of the sumands null is a necessary and sufficient condition for the dual space to be reflexive. We end with an application of this result to the known relationship between the reflexivity of a Banach space and that of its dual space.
In this paper, we introduce a new notion of biprojectivity, called $WAP$-biprojectivity for $F(\mathcal{A})$, the enveloping dual Banach algebra associated to a Banach algebra $\mathcal{A}$. We find some relations between Connes biprojectivity, Connes amenability and this new notion. We show that, for a given dual Banach algebra $\mathcal{A}$, if $F(\mathcal{A})$ is Connes amenable, then $\mathcal{A}$ is Connes amenable. For an infinite commutative compact group $G$, we show that the convolution Banach algebra $F(L^2(G))$ is not $WAP$-biprojective. Finally, we provide some examples of the enve…
We investigate topologically left Noetherian Banach algebras. We show that if $G$ is a compact group, then $L^{\, 1}(G)$ is topologically left Noetherian if and only if $G$ is metrisable. We prove that, given a Banach space $E$ such that $E'$ has BAP, the algebra of compact operators $\mathcal{K}(E)$ is topologically left Noetherian if and only if $E'$ is separable; it is topologically right Noetherian if and only if $E$ is separable. We then give some examples of dual Banach algebras which are topologically left Noetherian in the weak*-topology. Finally we give a unified approach to classifyi…
A topological space is said to be sequential if every sequentially closed subspace is closed. We consider Banach spaces with weak*-sequential dual ball. In particular, we show that if $X$ is a Banach space with weak*-sequentially compact dual ball and $Y \subset X$ is a subspace such that $Y$ and $X/Y$ have weak*-sequential dual ball, then $X$ has weak*-sequential dual ball. As an application we obtain that the Johnson-Lindenstrauss space $JL_2$ and $C(K)$ for $K$ scattered compact space of countable height are examples of Banach spaces with weak*-sequential dual ball, answering in this way a …
We prove, under certain conditions, the existence of zeros for a weakly continuous operator on a paracompact topological space into the dual of a Banach space.
Let $A$ be a commutative semisimple Arens regular unital Banach algebra. The correlation between the BSE-property of the Banach algebra $A$ and its second duals are assessed. It is found and approved that if $A$ is a BSE-algebra, then so is $A^{**}$. The opposite correlation will hold in certain conditions. The correlation of the BSE-norm property of the Banach algebra and its second dual are assessed and examined. It is revealed that, if $A$ is a commutative Arens regular unital Banach algebra where $A^{**}$ is semisimple, then, $A$ and $A^{**}$ are BSE-norm algebra.
A Banach space $X$ is said to have the $\mathsf{SVM}$ (stability of vector measures) property if there exists a constant $v<\infty$ such that for any algebra of sets $\mathcal F$, and any function $ν\colon\mathcal F\to X$ satisfying $$\|ν(A\cup B)-ν(A)-ν(B)\|\leq 1\quad{for disjoint}A,B\in\mathcal F,$$there is a vector measure $μ\colon\mathcal F\to X$ with $\|ν(A)-μ(A)\|\leq v$ for all $A\in\mathcal F$. If this condition is valid when restricted to set algebras $\mathcal F$ of cardinality less than some fixed cardinal number $κ$, then we say that $X$ has the $κ$-$\mathsf{SVM}$ property. The le…
We show that any non-zero Banach space with a separable dual contains a totally disconnected, closed and bounded subset S of Hausdorff dimension 1 such that every Lipschitz function on the space is Fréchet differentiable somewhere in S.
We present a unified approach to the processes of inversion and duality for quasilinear and $1$-quasilinear maps; in particular, for centralizers and differentials generated by interpolation methods.
In this paper we study the Connes amenability of the second dual of Arens regular Banach algebras. Of course we provide a partial answer to the question posed by Volker Runde. Also we fined the necessary and sufficient conditions for the second dual of an Arens regular module extension Banach algebra to be Connes amenable when the module is reflexive.
The main result: the dual of separable Banach space $X$ contains a total subspace which is not norming over any infinite dimensional subspace of $X$ if and only if $X$ has a nonquasireflexive quotient space with the strictly singular quotient mapping.
See title. (A Banach space is said to be L-embedded if it is complemented in its bidual such that the norm between the two complementary subspaces is additive.)
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.
Let \(X\) be a compact metric space and \(E\) be a Banach space. \(\lip (X, E)\) denotes the Banach space of all \(E\)-valued little Lipschitz functions on \(X\). We show that \(\lip (X, E)^{**}\) is isometrically isomorphic to Banach space of \(E^{**}\)-valued Lipschitz functions \(\Lip(X, E^{**})\) under several conditions. Moreover, we describe the isometric isomorphism from \(\lip (X, E)^{**}\) to \(\Lip (X, E^{**})\).
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…
