We prove a commutative Gelfand--Naimark type theorem, by showing that the set $C_s(X)$ of continuous bounded (real or complex valued) functions with separable support on a locally separable metrizable space $X$ (provided with the supremum norm) is a Banach algebra, isometrically isomorphic to $C_0(Y)$, for some unique (up to homeomorphism) locally compact Hausdorff space $Y$. The space $Y$, which we explicitly construct as a subspace of the Stone--Čech compactification of $X$, is countably compact, and if $X$ is non-separable, is moreover non-normal; in addition $C_0(Y)=C_{00}(Y)$. When the underlying field of scalars is the complex numbers, the space $Y$ coincides with the spectrum of the ${C}^*$-algebra $C_s(X)$. Further, we find the dimension of the algebra $C_s(X)$.
The Banach-Mazur problem, which asks if every infinite-dimensional Banach space has an infinite-dimensional separable quotient space, has remained unsolved for 85 years, but has been answered in the affirmative for special cases such as reflexive Banach spaces. It is also known that every infinite-dimensional non-normable Fréchet space has an infinite-dimensional separable quotient space, namely $\mathbb{R}^ω$. It is proved in this paper that every infinite-dimensional Fréchet space (including every infinite-dimensional Banach space), indeed every locally convex space which has a subspace whic…
We study the class of compact spaces that appear as structure spaces of separable Banach lattices. In other words, we analyze what $C(K)$ spaces appear as principal ideals of separable Banach lattices. Among other things, it is shown that every such compactum $K$ admits a strictly positive regular Borel measure of countable type that is analytic, and in the nonmetrizable case these compacta are saturated with copies of $β\mathbb N$. Some natural questions about this class are left open.
Let X be a separable metric space and let βbe the strict topology on the space of bounded continuous functions on X, which has the space of τ-additive Borel measures as a continuous dual space. We prove a Banach-Dieudonneé type result for the space of bounded continuous functions equipped with β. As a consequence, this space is hypercomplete and a Pták space. Additionally, the closed graph, inverse mapping and open mapping theorems holds for linear maps between space of this type.
We prove that if $ C $ is a family of separable Banach spaces which is analytic with respect to the Effros-Borel structure and none member of $ C $ is isometrically universal for all separable Banach spaces, then there exists a separable Banach space with a monotone Schauder basis which is isometrically universal for $ C $ but still not for all separable Banach spaces. We also establish an analogous result for the class of strictly convex spaces.
In this note, we provide a starlike and normal tiling in any separable Banach space. That means, there are positive constants r and R (not depending on the separable Banach space) such that every tile of this tiling is starlike, contains a ball of radius r and is contained in a ball of radius R.
We introduce and study the notion of overcomplete set in a Banach space, that subsumes and extends the classical concept of overcomplete sequence in a (separable) Banach space. We give existence and non-existence results of overcomplete sets for a wide class of (non-separable) Banach spaces and we study to which extent properties of overcomplete sequences are retained by every overcomplete set.
We prove in this short report that for arbitrary weak converging sequence of sigma-finite Borelian measures in the separable Banach space there is a compact embedded separable subspace such that this measures not only are concentrated in this subspace but weak converge therein.
Paley-Wiener theorem for frames for Hilbert spaces, Banach frames, Schauder frames and atomic decompositions for Banach spaces are known. In this paper, we derive Paley-Wiener theorem for p-approximate Schauder frames for separable Banach spaces. We show that our results give Paley-Wiener theorem for frames for Hilbert spaces.
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 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.
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.
This paper considers explicit constructions of Auerbach bases in separable Banach spaces. Answering the question of A. Pełczy{ń}ski, we prove by construction the existence of Auerbach basis in arbitrary subspace of $c_0$ of finite codimension and in the space $C(K)$ for $K$ compact countable metric space.
We show that the classes of separable reflexive Banach spaces and of spaces with separable dual are strongly bounded. This gives a new proof of a recent result of E. Odell and Th. Schlumprecht, asserting that there exists a separable reflexive Banach space containing isomorphic copies of every separable uniformly convex Banach spaces.
We study a relation between three different formulations of theorems on separable determination - one using the concept of rich families, second via the concept of suitable models and third, a new one, suggested in this paper, using the notion of $ω$-monotone mappings. In particular, we show that in Banach spaces all those formulations are in a sense equivalent and we give a positive answer to two questions of O. Kalenda and the author. Our results enable us to obtain new statements concerning separable determination of $σ$-porosity (and of similar notions) in the language of rich families; th…
In this note we show that, if $\mcB$ is separable Banach space, then there is a biorthogonal system $\{x_n, x_n^*\}$ such that, the closed linear span of $\{x_n\},\bar{\left\langle {\{x_n\}}\right\rangle}=\mcB$ and $\left\| {x_n} \right\|\left\| {x_n^*} \right\| = 1$ for all $n$.
We prove the existence of a separable approximately ultra-homogeneous Banach lattice $\mathfrak{BL}$ that is isometrically universal for separable Banach lattices. This is done by showing that the class of Banach lattices has the Amalgamation Property, and thus finitely generated Banach lattices form a metric Fraïssé class. Some additional results about the structural properties of $\mathfrak{ BL}$ are also proven.
Concepts of g-fusion frame and gf-Riesz basis in a Hilbert to a Banach space is being presented. Some properties of g-fusion frame and gf-Riesz basis in Banach space have been developed. We discuss perturbation results of g-fusion frame in a Banach space. Finally, we construct g-p-fusion frames in Cartesian product of Banach spaces and tensor product of Banach spaces.
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 …
Using a variation of the Murphy-Varopoulos Theorem, we give a new proof of the following R.J.Loy Theorem: Let A be a separable Banach*-algebra with center Z such that ZA has at most countable codimension, then every positive linear functional on A is continuous.
Let $X$ be a Banach space with a separable dual $X^{*}$. Let $Y\subset X$ be a closed subspace, and $f:Y\to\mathbb{R}$ a $C^{1}$-smooth function. Then we show there is a $C^{1}$ extension of $f$ to $X$.
This paper deals with the extension of a classical theorem by R. Phelps on the Gâteaux differentiability of Lipschitz functions on separable Banach spaces to the non-separable case. The extension of the theorem is not possible for general non-separable Banach spaces, as shown by Larman. Therefore, we will work on some important non-separable spaces. We consider a norm of the non-separable space $\ell^{\infty}(\mathbb{R})$, showing that it complies with the aforementioned theorem thesis. We also consider other examples as $L^{\infty}(\mathbb{R})$ and $NBV[a,b]$, showing in all cases their sets …
A set $E$ in a Banach space $X$ is compactivorous if for every compact set $K$ in $X$ there is a nonempty, (relatively) open subset of $K$ which can be translated into $E$. In a separable Banach space, this is a sufficient condition which guarantees the Haar nonnegligibility of Borel subsets. We give some characterisations of this property in both separable and nonseparable Banach spaces and prove an extension of the main theorem to countable products of locally compact Polish groups.
In this paper, we study the descriptive complexity of some inevitable classes of Banach spaces. Precisely, as shown in [Go], every Banach space either contains a hereditarily indecomposable subspace or an unconditional basis, and, as shown in [FR], every Banach space either contains a minimal subspace or a continuously tight subspace. In these notes, we study the complexity of those inevitable classes as well as the complexity of containing a subspace in any of those classes.
Recent developments in Banach space theory provided unexpected examples of unital Banach algebras that are isomorphic to Calkin algebras of Banach spaces, however no example of a unital Banach algebra that cannot be realised as a~Calkin algebra has been found so far. This naturally led to the question of possible limitations of such assignments. In the present note we provide examples of unital Banach algebras meeting the necessary density condition for being the Calkin algebra of a separable Banach space that are not isomorphic to Calkin algebras of such spaces, nonetheless. The examples may …
We construct Zachary space in $\R^\infty$ and find that this is a Banach space of functions of bounded mean oscillation with order $p, 1\leq p \leq \infty$ containing the function of bounded mean oscillation $BMO[\R_I^\infty]$ as a dense continuous embedding. As an application of $\R_I^\infty$ we construction $\mcB,$ where $\mcB $ is separable Banach space and finally we construct $\mcZ^p[\mcB]$.
Let X be a separable real Banach space having a k-times continuously Fréchet differentiable (i.e. C^k-smooth) norm where k=1,...,\infty. We show that any equivalent norm on X can be approximated uniformly on bounded sets by C^k-smooth norms.
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 $Γ$.
It is proved that the relation of isomorphism between separable Banach spaces is a complete analytic equivalence relation, i.e., that any analytic equivalence relation Borel reduces to it. Thus, separable Banach spaces up to isomorphism provide complete invariants for a great number of mathematical structures up to their corresponding notion of isomorphism. The same is shown to hold for (1) complete separable metric spaces up to uniform homeomorphism, (2) separable Banach spaces up to Lipschitz isomorphism, and (3) up to (complemented) biembeddability, (4) Polish groups up to topological isomo…
The Separable Quotient Problem and the Lotz Property in Banach Spaces
Universidade Estadual da Paraíba（帕拉伊巴州立大学）；Instituto de Matemática Interdisciplinar (IMI)（交叉数学研究所）；Universidad Complutense de Madrid（马德里康普顿斯大学）；Universidade Federal da Paraíba（帕拉伊巴联邦大学）；Centro de Ensino Superior do Seridó Universidade Federal do Rio Grande do Norte（塞里多高等教育中心北里奥格兰德联邦大学）
We construct a continuum of mutually non-isomorphic separable Banach spaces which are complemented in each other. Consequently, the Schroeder-Bernstein Index of any of these spaces is $2^{\aleph_0}$. Our construction is based on a Banach space introduced by W. T. Gowers and B. Maurey in 1997. We also use classical descriptive set theory methods, as in some work of V. Ferenczi and C. Rosendal, to improve some results of P. G. Casazza and of N. J. Kalton on the Schroeder-Bernstein Property for spaces with an unconditional finite-dimensional Schauder decomposition.
While the classic separable quotient problem remains open, we survey general results related to this problem and examine the existence of a particular infinitedimensional separable quotient in some Banach spaces of vector-valued functions, linear operators and vector measures. Most of the results presented are consequence of known facts, some of them relative to the presence of complemented copies of the classic sequence spaces c_0 and l_p, for 1 <= p <= \infty. Also recent results of Argyros - Dodos - Kanellopoulos, and Sliwa are provided. This makes our presentation supplementary to a previo…
We prove in this article that every Borelian measure, for example, the distribution of a random variable, in separable Banach space has a support which is compact embedded Banach subspace; and prove that if the norm of the random variable belongs to some exponential Orlicz space, then the new subspace can be choose such that the norm of this variable in the new space also belongs to other exponential Orlicz space.
Famous Naimark-Han-Larson dilation theorem for frames in Hilbert spaces states that every frame for a separable Hilbert space $\mathcal{H}$ is image of a Riesz basis under an orthogonal projection from a separable Hilbert space $\mathcal{H}_1$ which contains $\mathcal{H}$ isometrically. In this paper, we derive dilation result for p-approximate Schauder frames for separable Banach spaces. Our result contains Naimark-Han-Larson dilation theorem as a particular case.
We construct a compact linearly ordered space $K$ of weight aleph one, such that the space $C(K)$ is not isomorphic to a Banach space with a projectional resolution of the identity, while on the other hand, $K$ is a continuous image of a Valdivia compact and every separable subspace of $C(K)$ is contained in a 1-complemented separable subspace. This answers two questions due to O. Kalenda and V. Montesinos.
It is shown that every separable reflexive Banach space is a quotient of a reflexive Hereditarily Indecomposable space, which yields that every separable reflexive Banach is isomorphic to a subspace of a reflexive Indecomposable space. Furthermore, every separable reflexive Banach space is a quotient of a reflexive complementably $\ell_p$ saturated space with $1<p<\infty$ and of a $c_0$ saturated space.
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 …
The purpose of this paper is to construct a new class of separable Banach spaces $\K^p[\mathbb{B}], \; 1\leq p \leq \infty$. Each of these spaces contain the $ \mcL^p[\mathbb{B}] $ spaces, as well as the space $\mfM[\R^\iy]$, of finitely additive measures as dense continuous compact embeddings. These spaces are of interest because they also contain the Henstock-Kurzweil integrable functions on $\mathbb{B}$. Finally, we offer a interesting approach to the Fourier transform on $\K^p[\mathbb{B}].$
If a separable Banach space contains an isometric copy of every separable reflexive Fréchet smooth Banach space, then it contains an isometric copy of every separable Banach space. The same conclusion holds if we consider separable Banach spaces with Fréchet smooth dual space. This improves a result of G. Godefroy and N. J. Kalton.
This note has two objectives. The first objective is show that, even if a separable Banach space does not have a Schauder basis (S-basis), there always exists Hilbert spaces $\mcH_1$ and $\mcH_2$, such that $\mcH_1$ is a continuous dense embedding in $\mcB$ and $\mcB$ is a continuous dense embedding in $\mcH_2$. This is the best possible improvement of a theorem due to Mazur (see \cite{BA} and also \cite{PE1}). The second objective is show how $\mcH_2$ allows us to provide a positive answer to the Marcinkiewicz-basis (M-basis) problem.
We study existence and partial regularity relative to the weighted Steiner problem in Banach spaces. We show $C^1$ regularity almost everywhere for almost minimizing sets in uniformly rotund Banach spaces whose modulus of uniform convexity verifies a Dini growth condition.
We show that there exists a separable reflexive Banach space into which every separable uniformly convex Banach space isomorphically embeds. This solves a problem of J. Bourgain. We also give intrinsic characterizations of separable reflexive Banach spaces which embed into a reflexive space with a block $q$-Hilbertian and/or a block $p$-Besselian finite dimensional decomposition.
We characterize those classes $\ccc$ of separable Banach spaces admitting a separable universal space $Y$ (that is, a space $Y$ containing, up to isomorphism, all members of $\ccc$) which is not universal for all separable Banach spaces. The characterization is a byproduct of the fact, proved in the paper, that the class $\mathrm{NU}$ of non-universal separable Banach spaces is strongly bounded. This settles in the affirmative the main conjecture form \cite{AD}. Our approach is based, among others, on a construction of $\llll_\infty$-spaces, due to J. Bourgain and G. Pisier. As a consequence w…
We compare two types of universal operators constructed relatively recently by Cabello Sánchez, and the authors. The first operator $Ω$ acts on the Gurarii space, while the second one $P_S$ has values in a fixed separable Banach space $S$. We show that if $S$ is the Gurarii space, then both operators are isometric. We also prove that, for a fixed space $S$, the operator $P_S$ is isometrically unique. Finally, we show that $Ω$ is generic in the sense of a natural infinite game.
We present a new proof of Zippin's Embedding Theorem, that every separable reflexive Banach space embeds into one with shrinking and boundedly complete basis, and every Banach space with a separable dual embeds into one with a shrinking basis. This new proof leads to improved versions of other embedding results.
Haar null sets were introduced by J.P.R. Christensen in 1972 to extend the notion of sets with zero Haar measure to nonlocally compact Polish groups. In 2013, U.B. Darij defined a categorical version of Haar null sets, which he named Haar meagre sets. The present paper aims to show that, whenever $C$ is a closed, convex subset of a separable Banach space, $C$ is Haar null if and only if $C$ is Haar meagre. We then use this fact to improve a theorem of E. Matoušková and to solve a conjecture proposed by Esterle, Matheron and Moreau. Finally, we apply the main theorem to find a characterisation …
We show that any separable stable Banach space can be represented as a group of isometries on a separable reflexive Banach space, which extends a result of S. Guerre and M. Levy. As a consequence, we can then represent homeomorphically its space of types.
We introduce and study the notion of space of almost universal complemented disposition (a.u.c.d.) as a generalization of Kadec space. We show that every Banach space with separable dual is isometrically contained as a $1$-complemented subspace of a separable a.u.c.d. space and that all a.u.c.d. spaces with $1$-FDD are isometric and contain isometric $1$-complemented copies of every separable Banach space with $1$-FDD. We then study spaces of universal complemented disposition (u.c.d.) and provide different constructions for such spaces. We also consider spaces of u.c.d. with respect to separa…
We consider the problem of determining the complexity of the uniform homeomorphism relation between separable Banach spaces in the Borel reducibility hierarchy of analytic equivalence relations. We prove that the complete $K_σ$ equivalence relation is Borel reducible to the uniform homeomorphism relation, and we also determine the possible complexities of the relation when restricted to some small classes of Banach spaces. Moreover, we determine the exact complexity of the local equivalence relation between Banach spaces, namely that it is bireducible with $K_σ$. Finally, we construct a class …
We use a method involving elementary submodels and a partial converse of Foran lemma to prove separable reduction theorems concerning Suslin sigma-P-porous sets where "P" can be from a rather wide class of porosity-like relations in complete metric spaces. In particular, we separably reduce the notion of Suslin cone small set in Asplund spaces. As an application we prove a theorem stating that a continuous approximately convex function on an Asplund space is Frechet differentiable up to a cone small set.
We characterize those classes $\mathcal{C}$ of separable Banach spaces for which there exists a separable Banach space $Y$ not containing $\ell_1$ and such that every space in the class $\mathcal{C}$ is a quotient of $Y$.
Employing a construction of Tsirelson-like spaces due to Argyros and Deliyanni, we show that the class of all Banach spaces which are isomorphic to a subspace of $c_{0}$ is a complete analytic set with respect to the Effros Borel structure of separable Banach spaces. Moreover, the classes of all separable spaces with the Schur property and of all separable spaces with the Dunford-Pettis property are $\mathbfΠ^{1}_{2}$-complete.
