Javier H. Guachalla
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.
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We extend the well known criteria of reflexivity of Banach lattices due to Lozanovsky and Lotz to the class of finitely generated Banach $C(K)$- modules. Namely we prove that a finitely generated Banach $C(K)$-module is reflexive if and only if it does not contain any subspace isomorphic to either $l^1$ or $c_0$.
In this paper we investigate some reflexivity-type properties of separable measurable Banach bundles over a $σ$-finite measure space. Our two main results are the following: - The fibers of a bundle are uniformly convex (with a common modulus of convexity) if and only if the space of its $L^p$-sections is uniformly convex for every $p\in(1,\infty)$. - The fibers of a bundle are reflexive if and only if the space of its $L^p$-sections is reflexive. These results generalise the well-known corresponding ones for Lebesgue-Bochner spaces.
Non-reflexivity of the Banach space $ΛBV^{(p)}$
研究巴拿赫空间\(ΛBV^{(p)}\)的非自反性，基于前人对有界变差函数空间非自反性的推广成果进行改进。
We obtain a strengthening of the principle of local reflexivity in a general form. The added strength makes local reflexivity operators respect given subspaces. Applications are given to bounded approximation properties of pairs, consisting of a Banach space and its subspace.
We prove that a Banach space X is not super-reflexive if and only if the hyperbolic infinite tree embeds metrically into X. We improve one implication of J.Bourgain's result who gave a metrical characterization of super-reflexivity in Banach spaces in terms of uniforms embeddings of the finite trees. A characterization of the linear type for Banach spaces is given using the embedding of the infinite tree equipped with a suitable metric.
Given an arbitrary $p$-Banach ideal $(0 < p \leq 1)$, we ask for geometrical properties of this ideal which are sufficient (and necessary) to allow a transfer of the principle of local reflexivity to this operator class.
We investigate possible quantifications of R. C. James' classical work on bases and reflexivity of Banach spaces. By introducing new quantities measuring how far a basic sequence is from being shrinking and/or boundedly complete, we prove quantitative versions of James' famous characterisations of reflexivity in terms of bases. Furthermore, we establish quantitative versions of James' characterisations of reflexivity of Banach spaces with unconditional bases.
In this paper, we prove the equivalence of reflexive Banach spaces and those Banach spaces which satisfy the following form of Bernstein's Lethargy Theorem. Let $X$ be an arbitrary infinite-dimensional Banach space, and let the real-valued sequence $\{d_n\}_{n\ge1}$ decrease to $0$. Suppose that $\{Y_n\}_{n\ge1}$ is a system of strictly nested subspaces of $X$ such that $\overline Y_n \subset Y_{n+1}$ for all $n\ge1$ and for each $n\ge1$, there exists $y_n\in Y_{n+1}\backslash Y_n$ such that the distance $ρ(y_n,Y_n)$ from $y_n$ to the subspace $Y_n$ satisfies $$ ρ(y_n,Y_n)=\|y_n\|. $$ Then, th…
Characterizations of reflexive Banach spaces
本文综述自反 Banach 空间基于收敛性、弱紧性、仿射集和双正交系统的刻画，并讨论含 Schauder 基空间中与 $\ell^1$、$c_0$ 同构及非自反性的关系。
We prove that if the group of isometries of C(X,E) is algebraically reflexive, then the group of n-isometries is also algebraically reflexive. Here, X is a compact Hausdorff space and E is a Banach space. As a corollary to this, we establish the algebraic reflexivity of the set of generalized bi-circular projections on C(X,E).
We define two metrics $d_{1,α}$ and $d_{\infty,α}$ on each Schreier family $\mathcal{S}_α$, $α<ω_1$, with which we prove the following metric characterization of reflexivity of a Banach space $X$: $X$ is reflexive if and only if there is an $α<ω_1$, so that there is no mapping $Φ:\mathcal{S}_α\to X$ for which $$ cd_{\infty,α}(A,B)\le \|Φ(A)-Φ(B)\|\le C d_{1,α}(A,B) \text{ for all $A,B\in\mathcal{S}_α$.}$$ Secondly, we prove for separable and reflexive Banach spaces $X$, and certain countable ordinals $α$ that $\max(\text{ Sz}(X),\text{ Sz}(X^*))\le α$ if and only if $({\mathcal S}_α, d_{1,α})$…
We investigate the reflexivity of the isometry group and the automorphism group of some important metric linear spaces and algebras. The paper consists of the following sections: 1. Preliminaries. 2. Sequence spaces. 3. Spaces of measurable functions. 4. Hardy spaces. 5. Banach algebras of holomorphic functions. 6. Frechet algebras of holomorphic functions. 7. Spaces of continuous functions.
We discuss the reflexivity of hyperexpansions and their Cauchy dual operators. In particular, we show that any cyclic completely hyperexpansive operator is reflexive. We also establish the reflexivity of the Cauchy dual of an arbitrary $2$-hyperexpansive operator. As a consequence, we deduce the reflexivity of the so-called Bergman-type operator, that is, a left-invertible operator $T$ satisfying the inequality $TT^* + (T^*T)^{-1} \leqslant 2 I_{\mathcal H}.$
We study random unconditional convergence for a basis in a Banach space. The connections between this notion and classical unconditionality are explored. In particular, we analyze duality relations, reflexivity, uniqueness of these bases and existence of unconditional subsequences.
In this paper we prove that if $E$ and $F$ are reflexive Banach spaces and $G$ is a closed linear subspace of the space $\mathcal{P}_{w}(^{n}E;F)$ of all $n$-homogeneous polynomials from $E$ to $F$ which are weakly continuous on bounded sets, then $G$ is either reflexive or non-isomorphic to a dual space. This result generalizes \cite[Theorem 2]{FEDER} and gives the solution to a problem posed by Feder \cite[Problem 1]{FED}.
We characterize the metric spaces whose free space has the bounded approximation property through a Lipschitz analogue of the local reflexivity principle. We show that there exist compact metric spaces whose free spaces fail the approximation property.
We investigate for a bounded semigroup of linear operators $S$ on a Banach space $E$ and a vector $x \in E$, when relative compactness of $S(I-T)x$ for every $T \in S$ implies relative compactness of the orbit $Sx$. In particular, we derive characterizations of separable Banach spaces not containing $\mathrm{c}_0$ and of reflexivity of Banach spaces with a Schauder basis in terms of such compactness results.
The property of being shift invariant and being reflexive or transitive in the case of the space of (asymmetric) truncated Toeplitz operators, and the space of (asymmetric) dual truncated operators is investigated. Most of the results obtained are new even for the symmetric case. A characterization of asymmetric dual truncated Toeplitz operators is also given.
Reflexivity of linear n-normed space with respect to b-linear functional
本文延续既有研究，利用 Hahn-Banach 定理探讨线性 n-赋范空间上有界 b-线性泛函，定义相应自反性，并引入向量序列的强、弱收敛概念及其性质。
Reflexivity of Grothendieck Spaces and Strongly Normal Sequences
该研究刻画了格罗滕迪克空间自反性与正规序列子序列行为的等价关系，给出正规序列含强正规子序列的商准则，并构造了相关实例。
This paper studies Schauder frames in Banach spaces, a concept which is a natural generalization of frames in Hilbert spaces and Schauder bases in Banach spaces. The associated minimal and maximal spaces are introduced, as are shrinking and boundedly complete Schauder frames. Our main results extend the classical duality theorems on bases to the situation of Schauder frames. In particular, we will generalize James' results on shrinking and boundedly complete bases to frames. Secondly we will extend his characterization of the reflexivity of spaces with unconditional bases to spaces with uncond…
The framework of the paper is that of the full Fock space ${\Cal F}^2({\Cal H}_n)$ and the Banach algebra $F^\infty$ which can be viewed as non-commutative analogues of the Hardy spaces $H^2$ and $H^\infty$ respectively. An inner-outer factorization for any element in ${\Cal F}^2({\Cal H}_n)$ as well as characterization of invertible elements in $F^\infty$ are obtained. We also give a complete characterization of invariant subspaces for the left creation operators $S_1,\cdots, S_n$ of ${\Cal F}^2({\Cal H}_n)$. This enables us to show that every weakly (strongly) closed unital subalgebra of $\{…
It is shown that a separable Banach space $X$ can be given an equivalent norm $|\!|\!|\cdot |\!|\!|$ with the following properties:\quad If $(x_n)\subseteq X$ is relatively weakly compact and $\lim_{m\to\infty} \lim_{n\to\infty}\break |\!|\!| x_m + x_n |\!|\!| = 2\lim_{m\to\infty} |\!|\!| x_m|\!|\!|$ then $(x_n)$ converges in norm. This yields a characterization of reflexivity once proposed by V.D.~Milman. In addition it is shown that some spreading model of a sequence in $(X, |\!|\!|\cdot |\!|\!|)$ is 1-equivalent to the unit vector basis of $\ell_1$ (respectively, $c_0$) implies that $X$ con…
In this paper, we show that under the condition $ 1<p_-, q_-, p_+, q_+<\infty$, the space $\ell^{q(\cdot)} (L^{p(\cdot)})$ is reflexive. In this way we give an answer to open problem posed by Hästö in 2017 about the reflexivity of the variable mixed Lebesgue-sequence spaces $\ell^{q(\cdot)} (L^{p(\cdot)})$. What is important here is that the dual space of $\ell^{q(\cdot)} (L^{p(\cdot)})$ is specified. As its direct corollary, we show that the corresponding Besov space $B^{s(\cdot)}_{p(\cdot)q(\cdot)}$ is reflexive.
We study associate and double associate spaces of two-weighted Sobolev spaces of the first order on real half-line and we show that unlike the notion of duality the associativity is divided into two cases which we call "strong" and "weak" ones with the division of the second associativity into four cases. On the way we prove that the Sobolev space of compactly supported functions possess weak associated reflexivity and the double weak-strong associate space is vacuous. The case of power weights was recently characterized by reduction to Cesàro or Copson type spaces [18].
Birkhoff-Orthogonal Takahashi-von Neumann-Jordan Type Constants in Banach Spaces
本文引入巴拿赫空间的伯克霍夫正交高桥-冯·诺依曼-乔丹型几何常数CtB(X)，刻画一致非平方空间，给出超自反性与不动点性质的充分条件，获其在拉东平面的精确值。
We show that the duals of Banach algebras of scalar-valued bounded holomorphic functions on the open unit ball $B_E$ of a Banach space $E$ lack weak$^*$-strongly exposed points. Consequently, we obtain that some Banach algebras of holomorphic functions on an arbitrary Banach space have the Daugavet property which extends the observation of P. Wojtaszczyk. Moreover, we present a new denseness result by proving that the set of norm-attaining vector-valued holomorphic functions on the open unit ball of a dual Banach space is dense provided that its predual space has the metric $π$-property. Besid…
It is well known that in the calculus of variations and in optimization there exist many formulations of the fundamental propositions on the attainment of the infima of sequentially weakly lower semicontinuous coercive functions on reflexive Banach spaces. By either some constructive skills or the regularization skill by inf--convolutions we show in this paper that all these formulations together with their important variants are equivalent to each other and equivalent to the reflexivity of the underlying space. Motivated by this research, we also give a characterization for a normed space to …
In this paper we study the reflexivity of a unital strongly closed algebra of operators with complemented invariant subspace lattice on a Banach space. We prove that if such an algebra contains a complete Boolean algebra of projections of finite uniform multiplicity and with the direct sum property, then it is reflexive, i.e. it contains every operator that leaves invariant every closed subspace in the invariant subspace lattice of the algebra. In particular, such algebras coincide with their bicommutant.
We study the Bishop-Phelps-Bollobás property for numerical radius (in short, BPBp-nu) and find sufficient conditions for Banach spaces ensuring the BPBp-nu. Among other results, we show that $L_1(μ)$-spaces have this property for every measure $μ$. On the other hand, we show that every infinite-dimensional separable Banach space can be renormed to fail the BPBp-nu. In particular, this shows that the Radon-Nikodým property (even reflexivity) is not enough to get BPBp-nu.
By using the Principle of Local Reflexivity (PLR), we prove that for every two Banach spaces $E$ and $X$ there exists a suitable ultrafilter $\mathcal{U}$ such that $ \mathcal{F}(E,X)^*,$ the dual space of the finite rank operators, can be isomorphically identified with certain quotient of the ultrapower space $(E\widehat{\otimes} X^*)_\mathcal{U}$, of the projective tensor product space $E\widehat{\otimes} X^*.$ This generalizes the identity $\mathcal B(E,X^{**})\cong\mathcal B(E,X)^{**}$, where $E$ is finite-dimensional. We then serve our main result to improve some results on the reflexivit…
On T-orthogonality in Banach spaces
该研究引入Banach空间中基于有界线性算子T的T-正交性概念，探究其与空间几何性质的关联，刻画保T-正交的算子并借此刻画Hilbert空间。
G-strong subdifferentiability and applications to norm attaining subspaces
本文在群不变映射框架下研究自反性与强次可微性，证明对偶范数G-强次可微蕴含Banach空间G-自反，并推广多个经典泛函分析定理，最终给出范数可达泛函集合中存在闭Banach子空间的充分条件。
In this paper the weak fixed point property ($w$-FPP) and the fixed point property (FPP) in Variable Lebesgue Spaces are studied. Given $(Ω,Σ,μ)$ a $σ$-finite measure and $p(\cdot)$ a variable exponent function, the $w$-FPP is completely characterized for the variable Lebesgue space $L^{p(\cdot)}(Ω)$ in terms of the exponent function $p(\cdot)$ and the absence of an isometric copy of $L_1[0,1]$. In particular, every reflexive $L^{p(\cdot)}(Ω)$ has the FPP and our results bring to light the existence of some nonreflexive variable Lebesgue spaces satisfying the $w$-FPP, in sharp contrast with th…
We establish some general principles and find some counter-examples concerning the Pontryagin reflexivity of precompact groups and P-groups. We prove in particular that: (1) A precompact Abelian group G of bounded order is reflexive iff the dual group $\hat{G}$ has no infinite compact subsets and every compact subset of G is contained in a compact subgroup of G. (2) Any extension of a reflexive P-group by another reflexive P-group is again reflexive. We show on the other hand that an extension of a compact group by a reflexive $ω$-bounded group (even dual to a reflexive P-group) can fail to be…
The operator space analogue of the {\em strong form} of the principle of local reflexivity is shown to hold for any von Neumann algebra predual, and thus for any $C^{*}$-algebraic dual. This is in striking contrast to the situation for $C^{*}$-algebras, since, for example, $K(H)$ does not have that property. The proof uses the Kaplansky density theorem together with a careful analysis of two notions of integrality for mappings of operator spaces.
A study on $\mr{F}$-simultaneous approximative $τ$-compactness property in Banach spaces
本文研究了Banach空间中F-同时近似τ-紧性性质，探讨了其与反射性、Fréchet光滑性和Kadec-Klee性质的关系。
We find a class of metric structures which do not admit bilipschitz embeddings into Banach spaces with the Radon-Nikodým property. Our proof relies on Chatterji's (1968) martingale characterization of the RNP and does not use the Cheeger's (1999) metric differentiation theory. The class includes the infinite diamond and both Laakso (2000) spaces. We also show that for each of these structures there is a non-RNP Banach space which does not admit its bilipschitz embedding. We prove that a dual Banach space does not have the RNP if and only if it admits a bilipschitz embedding of the infinite dia…
The duality of uniform approximation property for Banach spaces is well known. In this note, we establish, under the assumption of local reflexivity, the duality of uniform approximation property in the category of operator spaces.
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…
A Banach space $X$ has \textit{property $(K)$}, whenever every weak* null sequence in the dual space admits a convex block subsequence $(f_{n})_{n=1}^\infty$ so that $\langle f_{n},x_{n}\rangle\to 0$ as $n\to \infty$ for every weakly null sequence $(x_{n})_{n=1}^\infty$ in $X$; $X$ has \textit{property $(μ^{s})$} if every weak$^{*}$ null sequence in $X^{*}$ admits a subsequence so that all of its subsequences are Cesàro convergent to $0$ with respect to the Mackey topology. Both property $(μ^{s})$ and reflexivity (or even the Grothendieck property) imply property $(K)$. In the present paper we…
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
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.
New Generalized Dual Banach Frames
本文系统研究了Banach空间中的g-对偶Banach框架，建立了其存在条件，探讨了与其他广义对偶概念的关系，并证明了其在有界扰动下的稳定性。
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.
A Banach space contains asymptotically isometric copies of $\ell_1$ if and only if its dual space contains an isometric copy of $L_1$.
Mordukhovich derivatives of the normalized duality mapping in Banach spaces
本文研究了Banach空间中规范化对偶映射的Mordukhovich导数的性质，考察了一致凸且一致光滑的Banach空间lp以及一般Banach空间L1和C[0,1]三种情形。
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.
Necessary and sufficient conditions for Banach space to be(isometrically isomorphic to) a dual space will be given.
Sun dual theory for bi-continuous semigroups
本文在局部凸拓扑的温和假设下，为双连续半群发展了sun共轭理论，并讨论了sun自反性与Favard空间，扩展了van Neerven的经典结果。
A non RNP Banach space E is constructed such that $E^{*}$ is separable and RNP is equivalent to PCP on the subsets of E.
We use Birkhoff-James' orthogonality in Banach spaces to provide new conditions for the converse of the classical Riesz's representation theorem.
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…
The preduals of Banach space valued Bourgain-Morrey spaces
本文引入Banach空间值Bourgain-Morrey空间，证明其预对偶为块空间，并研究其完备性、对偶、反射性及Hardy-Littlewood极大算子的有界性。
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 …
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