The main purpose of this paper is to study some geometric and topological properties on $ c_0$ sum of the finite dimensional Banach lattice $\ell_2^n$, its dual and its bidual. Among other results, we show that the Banach lattices $c_0(\ell_2^n)$ has the strong Gelfand-Philips property, but does not have the positive Grothendieck property. We also prove that the closed unit ball of $l_{\infty}(\ell_2^n)$ is an almost limited set.
By extending the notion of Arens regularity of bilinear mappings, we say that a bounded trilinear map on Banach spaces id Aron--Berner regular when all its six Aron-Berner extensions to the bidual spaces coincide. We give some results on the Aron-Berner regularity of certain trilinear maps. We then focus on the bidual, $E^{**},$ of a Jordan banach triple system $(E,π)$, and investigate those conditions under which $E^{**}$ is itself a Jordan Banach triple system under each of the Aron-Berner extensions of the triple product $π.$ We also compare these six triple products with those arising from…
If $E$ is a Banach space, any element $x^{**}$ in its bidual $E^{**}$ is an affine function on the dual unit ball $B_{E^*}$ that might possess variety of descriptive properties with respect to the weak* topology. We prove several results showing that descriptive properties of $x^{**}$ are quite often determined by the behaviour of $x^{**}$ on the set of extreme points of $B_{E^*}$, generalizing thus results of J. Saint Raymond and F. Jellett. We also prove several results on relation between Baire classes and intrinsic Baire classes of $L_1$-preduals which were introduced by S.A. Argyros, G. G…
We prove that every separable Banach space containing $\ell_1$ can be equivalently renormed so that its bidual space is octahedral, which answers, in the separable case, a question by Godefroy in 1989. As a direct consequence, we obtain that every dual Banach space, with a separable predual, failing to be strongly regular (that is, without convex combinations of slices with diameter arbitrarily small for some closed, convex and bounded subset) can be equivalently renormed with a dual norm to satisfy the strong diameter two property (that is, such that every convex combination of slices in its …
We prove that all Arens extensions of finite rank Riesz multimorphisms taking values in Archimedean Riesz spaces coincide and are Riesz multimorphisms. Partial results for arbitrary Riesz multimorphisms are obtained. We also prove that, for a class of Banach lattices $F$, which includes $F = c_0, \ell_p, c_0(\ell_p), \ell_p(c_0), \ell_p(\ell_s), 1 < p,s < \infty$, among many others, all Aron-Berner extensions of $F$-valued Riesz multimorphisms between Banach lattices are Riesz multimorphisms.
Under certain hypotheses on the Banach space $X$, we prove that the set of analytic functions in $\mathcal{A}_u(X)$ (the algebra of all holomorphic and uniformly continuous functions in the ball of $X$) whose Aron-Berner extensions attain their norms, is dense in $\mathcal{A}_u(X)$. The result holds also for functions with values in a dual space or in a Banach space with the so-called property $(β)$. For this, we establish first a Lindenstrauss type theorem for continuous polynomials. We also present some counterexamples for the Bishop-Phelps theorem in the analytic and polynomial cases where …
We prove that the bidual of a Beurling algebra on $\mathbb{Z}$, considered as a Banach algebra with the first Arens product, can never be semisimple. We then show that ${\rm rad\,}(\ell^{\, 1}(\oplus_{i=1}^\infty \mathbb{Z})")$ contains nilpotent elements of every index. Each of these results settles a question of Dales and Lau. Finally we show that there exists a weight $ω$ on $\mathbb{Z}$ such that the bidual of $\ell^{\, 1}(\mathbb{Z}, ω)$ contains a radical element which is not nilpotent.
We show that a representation of a Banach algebra $A$ on a Banach space $X$ can be extended to a canonical representation of $A^{**}$ on $X$ if and only if certain orbit maps $A\to X$ are weakly compact. When this is the case, we show that the essential space of the representation is complemented if $A$ has a bounded left approximate identity. This provides a tool to disregard the difference between degenerate and nondegenerate representations. Our results have interesting consequences both in C*-algebras and in abstract harmonic analysis. For example, a C*-algebra $A$ has an isometric represe…
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.)
We study smoothness and strict convexity of (the bidual) of Banach spaces in the presence of diameter 2 properties. We prove that the strong diameter 2 property prevents the bidual from being strictly convex and being smooth, and we initiate the investigation whether the same is true for the (local) diameter 2 property. We also give characterizations of the following property for a Banach space $X$: "For every slice $S$ of $B_X$ and every norm-one element $x$ in $S$, there is a point $y\in S$ in distance as close to 2 as we want." Spaces with this property are shown to have non-smooth bidual.
In this paper we study different aspects of the representation of weak*-compact convex sets of the bidual $X^{**}$ of a separable Banach space $X$ via a nested sequence of closed convex bounded sets of $X$.
We generalize an important class of Banach spaces, namely the $M$-embedded Banach spaces, to the non-commutative setting of operator spaces. The one-sided $M$-embedded operator spaces are the operator spaces which are one-sided $M$-ideals in their second dual. We show that several properties from the classical setting, like the stability under taking subspaces and quotients, unique extension property, Radon Nikod$\acute {\rm{y}}$m Property and many more, are retained in the non-commutative setting. We also discuss the dual setting of one-sided $L$-embedded operator spaces.
In this note the following version of Phillips' lemma is proved. The L-projection of an L-embedded space - that is of a Banach space which is complemented in its bidual such that the norm between the two complementary subspaces is additive - is weak-weakly sequentially continuous.
In this note the following is proved. Separable L-embedded spaces - that is separable Banach spaces which are complemented in their biduals such that the norm between the two complementary subspaces is additive - have property (X) which, by a result of Godefroy and Talagrand, entails uniqueness of the space as a predual.
We continue with the study of octahedral norms in the context of spaces of Lipschitz functions and in their duals. First, we prove that the norm of $\mathcal F(M)^{**}$ is octahedral as soon as $M$ is unbounded or is not uniformly discrete. Further, we prove that a concrete sequence of uniformly discrete and bounded metric spaces $(K_m)$ satisfies that the norm of $\mathcal F(K_m)^{**}$ is octahedral for every $m$. Finally, we prove that if $X$ is an arbitrary Banach space and the norm of $\operatorname{Lip}_0(M)$ is octahedral, then the norm of $\operatorname{Lip}_0(M,X^\ast)$ is octahedral. …
For Banach spaces X and Y, we establish a natural bijection between preduals of Y and preduals of L(X,Y) that respect the right L(X)-module structure. If X is reflexive, it follows that there is a unique predual making L(X) into a dual Banach algebra. This removes the condition that X have the approximation property in a result of Daws. We further establish a natural bijection between projections that complement Y in its bidual and L(X)-linear projections that complement L(X,Y) in its bidual. It follows that Y is complemented in its bidual if and only if L(X,Y) is (either as a module or as a B…
We show that finite dimensional Banach spaces fail to be uniformly non locally almost square. Moreover, we construct an equivalent almost square bidual norm on $\ell_\infty.$ As a consequence we get that every dual Banach space containing $c_0$ has an equivalent almost square dual norm. Finally we characterize separable real almost square spaces in terms of their position in their fourth duals.
In this supplement to [GJ1], [GJ3], we give an intrinsic characterization of (bounded, linear) operators on Banach lattices which factor through Banach lattices not containing a copy of $c_0$ which complements the characterization of [GJ1], [GJ3] that an operator admits such a factorization if and only if it can be written as the product of two operators neither of which preserves a copy of $c_0$. The intrinsic characterization is that the restriction of the second adjoint of the operator to the ideal generated by the lattice in its bidual does not preserve a copy of $c_0$. This property of an…
Let $\mathcal{A}$ be a $C^*$-algebra, and consider the Banach algebra $\mathcal{A} \otimes_γ\mathcal{A}$, where $\otimes_γ$ denotes the projective Banach space tensor product; if $\mathcal{A}$ is commutative, this is the Varopoulos algebra $V_\mathcal{A}$. It has been an open problem for more than 35 years to determine precisely when $\mathcal{A} \otimes_γ\mathcal{A}$ is Arens regular. Even the situation for commutative $\mathcal{A}$, in particular the case $\mathcal{A} = \ell_\infty$, has remained unsolved. We solve this classical question for arbitrary $C^*$-algebras by using von Neumann alg…
Nonreflexive Banach spaces that are complemented in their bidual by an L-projection - like preduals of von Neumann algebras or the Hardy space $H^1$ - contain, roughly speaking, many copies of $l^1$ which are very close to isometric copies. Such $l^1$-copies are known to fail the fixed point property. Similar dual results hold for $c_0$.
On the $L$-embeddability of Lipschitz-free spaces in their biduals
IMPA（巴西数学纯应用与计算研究所）；Department of Mathematics and Statistics, University of North Carolina at Charlotte（北卡罗来纳大学夏洛特分校数学与统计系）；Université Marie et Louis Pasteur, CNRS, Laboratoire de mathématiques de Besançon (UMR 6623)（玛丽和路易·巴斯德大学，法国国家科学研究中心，贝桑松数学实验室）；Department of Mathematics and Statistics, York University（约克大学数学与统计系）；Faculty of Information Technology, Czech Technical University in Prague（布拉格捷克理工大学信息技术学院）；Texas A&M University（德州农工大学）
Let $X$ be a Banach space. Then $X$ is complemented in the bidual $X^{**}$ if and only if there exists an invariant mean $\ell_\infty(G, X)\to X$ with respect to a free Abelian group $G$ of rank equal to the cardinality of $X^{**}$, and this happens if and only if there exists an invariant mean with respect to the additive group of $X^{**}$. This improves upon previous results due to Bustos Domecq =and the second-named author, where certain idempotent semigroups of cardinality equal to the cardinality of $X^{**}$ were considered, and answers a question of J.M.F. Castillo (private communication…
Let $X$ be a Banach space and $Γ\subseteq X^*$ a total linear subspace. We study the concept of $Γ$-integrability for $X$-valued functions $f$ defined on a complete probability space, i.e. an analogue of Pettis integrability by dealing only with the compositions $\langle x^*,f \rangle$ for $x^*\in Γ$. We show that $Γ$-integrability and Pettis integrability are equivalent whenever $X$ has Plichko's property ($\mathcal{D}'$) (meaning that every $w^*$-sequentially closed subspace of $X^*$ is $w^*$-closed). This property is enjoyed by many Banach spaces including all spaces with $w^*$-angelic dual…
We study the spectrum $M_b(U)$ of the algebra of bounded type holomorphic functions on a complete Reinhardt domain in a symmetrically regular Banach space $E$ as an analytic manifold over the bidual of the space. In the case that $U$ is the unit ball of $\ell_p$, $1<p<\infty$, we prove that each connected component of $M_b(B_{\ell_p})$ naturally identifies with a ball of a certain radius. We also provide estimates for this radius and in many natural cases we have the precise value. As a consequence, we obtain that for connected components different from that of evaluations, these radii are str…
Several new characterizations of Banach spaces containing a subspace isomorphic to $\ell^1$, are obtained. These are applied to the question of when $\ell^1$ embeds in the injective tensor product of two Banach spaces.
A well known argument of James yields that if a Banach space $X$ contains $\ell_1^n$'s uniformly then $X$ contains $\ell_1^n$'s almost isometrically. In the first half of the paper we extend this idea to the ordinal $\ell_1$-indices of Bourgain. In the second half we use our results to calculate the $\ell_1$-index of certain Banach spaces. Furthermore we show that the $\ell_1$-index of a separable Banach space not containing $\ell_1$ must be of the form $ω^α$ for some countable ordinal $α$.
A Hereditarily Indecomposable asymptotic $\ell_2$ Banach space is constructed. The existence of such a space answers a question of B. Maurey and verifies a conjecture of W.T. Gowers.
The class of countably intersected families of sets is defined. For any such family we define a Banach space not containing $\ell^{1}(\NN )$. Thus we obtain counterexamples to certain questions related to the heredity problem for W.C.G. Banach spaces. Among them we give a subspace of a W.C.G. Banach space not containing $\ell^{1}(\NN )$ and not being itself a W.C.G. space.
We introduce and investigate a class of non-separable tree-like Banach spaces. As a consequence, we prove that we can not achieve a satisfactory extension of Rosenthal's $\ell_1$-theorem to spaces of the type $\ell_1(κ)$, for $κ$ an uncountable cardinal.
Nonatomic bounded sequences in $\ell_1$, that is, those giving rise to nonatomic submeasures on $\mathbb N$ are introduced and shown to form a closed subspace nonat$(\ell_1)$ of $\ell_\infty(\ell_1)$, and some spaces of relevant operators in $\ell_1$ are considered.
We prove that, when $2<p<\infty$, in the free Banach lattice generated by $\ell_p$ (respectively by $c_0$), the absolute values of the canonical basis form an $\ell_r$-sequence, where $\frac{1}{r} = \frac{1}{2} + \frac{1}{p}$ (respectively an $\ell_2$-sequence). In particular, in any Banach lattice, the absolute values of any $\ell_p$ sequence always have an upper $\ell_r$-estimate. Quite surprisingly, this implies that the free Banach lattices generated by the nonseparable $\ell_p(Γ)$ for $2<p<\infty$, as well as $c_0(Γ)$, are weakly compactly generated whereas this is not the case for $1\leq…
We present condition on higher order asymptotic behaviour of basic sequences in a Banach space ensuring the existence of bounded non-compact strictly singular operator on a subspace. We apply it in asymptotic $\ell_p$ spaces, $1\leq p<\infty$, in particular in convexified mixed Tsirelson spaces and related asymptotic $\ell_p$ HI spaces.
In this note we study Banach spaces of traces of real polynomials on $\mathbb R^n$ to compact subsets equipped with supremum norms from the point of view of Geometric Functional Analysis.
Two examples of asymptotic $\ell_{1}$ Banach spaces are given. The first, $X_{u}$, has an unconditional basis and is arbitrarily distortable. The second, $X$, does not contain any unconditional basic sequence. Both are spaces of the type of Tsirelson. We thus answer a question raised by W.T.Gowers.
We give sufficient conditions on a Banach space $X$ which ensure that $\ell_{\infty}$ embeds in $\mathcal{L}(X)$, the space of all operators on $X$. We say that a basic sequence $(e_n)$ is quasisubsymmetric if for any two increasing sequences $(k_n)$ and $(\ell_n)$ of positive integers with $k_n \leq \ell_n$ for all $n$, we have that $(e_{k_n})$ dominates $(e_{\ell_n})$. We prove that if a Banach space $X$ has a seminormalized quasisubsymmetric basis then $\ell_{\infty}$ embeds in $\mathcal{L}(X)$.
Extending recent results by Cascales, Kadets, Orihuela and Wingler (2016), Kadets and Zavarzina (2017), and Zavarzina (2017) we demonstrate that for every Banach space $X$ and every collection $Z_i, i\in I$ of strictly convex Banach spaces every non-expansive bijection from the unit ball of $X$ to the unit ball of sum of $Z_i$ by $\ell_1$ is an isometry.
In this paper we first show that if $X$ is a Banach space and $α$ is a left invariant crossnorm on $\ell_\infty\otimes X$, then there is a Banach lattice $L$ and an isometric embedding $J$ of $X$ into $L$, so that $I\otimes J$ becomes an isometry of $\ell_\infty\otimes_αX$ onto $\ell_\infty\otimes_m J(X)$. Here $I$ denotes the identity operator on $\ell_\infty$ and $\ell_\infty\otimes_m J(X)$ the canonical lattice tensor product. This result is originally due to G. Pisier (unpublished), but our proof is different. We then use this to characterize the Gordon-Lewis property $\GL$ in terms of emb…
We study the Bishop-Phelps-Bollobás property for operators from $\ell_\infty ^4 $ to a Banach space. For this reason we introduce an appropiate geometric property, namely the AHSp-$\ell_\infty ^4$. We prove that spaces $Y$satisfying AHSp-$\ell_\infty ^4$ are precisely those spaces $Y$ such that $(\ell_\infty^4,Y)$ has the Bishop-Phelps-Bollobás property. We also provide classes of Banach spaces satisfying this condition. For instance, finite-dimensional spaces, uniformly convex spaces, $C_0(L)$ and $L_1 (μ)$ satisfy AHSp-$\ell_\infty ^4 $.
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 construct a class of minimal trees and use these trees to establish a number of coloring theorems on general trees. Among the applications of these trees and coloring theorems are quantification of the Bourgain $\ell_p$ and $c_0$ indices, dualization of the Bourgain $c_0$ index, establishing sharp positive and negative results for constant reduction, and estimating the Bourgain $\ell_p$ index of an arbitrary Banach space $X$ in terms of a subspace $Y$ and the quotient $X/Y$.
Using upper $\ell_p$-estimates for normalized weakly null sequence images, we describe a new family of operator ideals $\mathcal{WD}_{\ell_p}^{(\infty,ξ)}$ with parameters $1\leq p\leq\infty$ and $1\leqξ\leqω_1$. These classes contain the completely continuous operators, and are distinct for all choices $1\leq p\leq\infty$ and, when $p\neq 1$, for all choices $ξ\neqω_1$. For the case $ξ=1$, there exists an ideal norm $\|\cdot\|_{(p,1)}$ on the class $\mathcal{WD}_{\ell_p}^{(\infty,1)}$ under which it forms a Banach ideal.
We classify the closed ideals of bounded operators acting on the Banach spaces $\left(\bigoplus_{n \in \mathbb{N}} \ell_2^n\right)_{c_0} \oplus c_0(Γ)$ and $\left(\bigoplus_{n \in \mathbb{N}} \ell_2^n\right)_{\ell_1} \oplus \ell_1(Γ)$ for every uncountable cardinal $Γ$.
We characterize $k-$smoothness of an element on the unit sphere of a finite-dimensional polyhedral Banach space. Then we study $k-$smoothness of an operator $T \in \mathbb{L}(\ell_{\infty}^n,\mathbb{Y}),$ where $\mathbb{Y}$ is a two-dimensional Banach space with the additional condition that $T$ attains norm at each extreme point of $B_{\ell_{\infty}^{n}}.$ We also characterize $k-$smoothness of an operator defined between $\ell_{\infty}^3$ and $\ell_{1}^3.$
We introduce a notion of p-orthogonality in a general Banach space $1 \le p \le \infty$. We use this concept to characterize $\ell_p$-spaces among Banach spaces and also among complete order smooth p-normed spaces. We further introduce a notion of $p$-orthogonal decomposition in order smooth p-normed spaces. We prove that if the $\infty$-orthogonal decomposition holds in an order smooth $\infty$-normed space, then the 1-orthogonal decomposition holds in the dual space. We also give an example to show that the above said decomposition may not be unique.
We prove that a non ergodic Banach space must be near Hilbert. In particular, $\ell_p$ ($2<p<\infty$) is ergodic. This reinforces the conjecture that $\ell_2$ is the only non ergodic Banach space. As an application of our criterion for ergodicity, we prove that there is no separable Banach space which is complementably universal for the class of all subspaces of $\ell_p$, for $1\leq p <2$. This solves a question left open by W. B. Johnson and A. Szankowski in 1976.
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…
We show that the dual of every infinite-dimensional Lipschitz-free Banach space contains an isometric copy of $\ell_\infty$ and that it is often the case that a Lipschitz-free Banach space contains a $1$-complemented subspace isometric to $\ell_1$. Even though we do not know whether the latter is true for every infinite-dimensional Lipschitz-free Banach space, we show that the space is never rotund. Further, in the last section we survey the relations between "isometric embedding of~$\ell_\infty$ into the dual" and "containing as good copy of~$\ell_1$ as possible" in a general Banach space.
