We study some questions concerning the structure of the set of spreading models of a separable infinite-dimensional Banach space $X$. In particular we give an example of a reflexive $X$ so that all spreading models of $X$ contain $\ell_1$ but none of them is isomorphic to $\ell_1$. We also prove that for any countable set $C$ of spreading models generated by weakly null sequences there is a spreading model generated by a weakly null sequence which dominates each element of $C$. In certain cases this ensures that $X$ admits, for each $α< ω_1$, a spreading model $(\tilde x_i^α)_i$ such that if $α< β$ then $(\tilde x_i^α)_i$ is dominated by (and not equivalent to) $(\tilde x_i^β)_i$. Some applications of these ideas are used to give sufficient conditions on a Banach space for the existence of a subspace and an operator defined on the subspace, which is not a compact perturbation of a multiple of the inclusion map.
The main result of this paper is a fixed point result relating the spreading model structure of Banach spaces and Schauder basis with not too large basis constant. As a striking consequence, we deduce that every super-reflexive space has the fixed point property thus solving a long-standing open question in metric fixed point theory.
We present a reflexive Banach space $\mathfrak{X}_{_{^\text{usm}}}$ which is Hereditarily Indecomposable and satisfies the following properties. In every subspace $Y$ of $\mathfrak{X}_{_{^\text{usm}}}$ there exists a weakly null normalized sequence $\{y_n\}_n$, such that every subsymmetric sequence $\{z_n\}_n$ is isomorphically generated as a spreading model of a subsequence of $\{y_n\}_n$. Also, in every block subspace $Y$ of $\mathfrak{X}_{_{^\text{usm}}}$ there exists a seminormalized block sequence $\{z_n\}$ and $T:\mathfrak{X}_{_{^\text{usm}}}\rightarrow\mathfrak{X}_{_{^\text{usm}}}$ an i…
It is shown that for every $k\in\mathbb{N}$ and every spreading sequence $\{e_n\}_{n\in\mathbb{N}}$ that generates a uniformly convex Banach space $E$, there exists a uniformly convex Banach space $X_{k+1}$ admitting $\{e_n\}_{n\in\mathbb{N}}$ as a $k+1$-iterated spreading model, but not as a $k$-iterated one.
We extend the classical Brunel-Sucheston definition of the spreading model by introducing the $\mathcal{F}$-sequences $(x_s)_{s\in\mathcal{F}}$ in a Banach space and the plegma families in $\mathcal{F}$ where $\mathcal{F}$ is a regular thin family. The new concept yields a transfinite increasing hierarchy of classes of spreading sequences. We explore the corresponding theory and we present examples establishing this hierarchy and illustrating the limitation of the theory.
We show that for every separable Banach space $X$, either $\spw(X)$ (the set of all spreading models of $X$ generated by weakly-null sequences in $X$, modulo equivalence) is countable, or $\spw(X)$ contains an antichain of the size of the continuum. This answers a question of S. J. Dilworth, E. Odell and B. Sari.
Extending the classical notion of the spreading model, the $k$-spreading models of a Banach space are introduced, for every $k\in\mathbb{N}$. The definition, which is based on the $k$-sequences and plegma families, reveals a new class of spreading sequences associated to a Banach space. Most of the results of the classical theory are stated and proved in the higher order setting. Moreover, new phenomena like the universality of the class of the 2-spreading models of $c_0$ and the composition property are established. As consequence, a problem concerning the structure of the $k$-iterated spread…
We introduce the higher order spreading models associated to a Banach space $X$. Their definition is based on $\ff$-sequences $(x_s)_{s\in\ff}$ with $\ff$ a regular thin family and the plegma families. We show that the higher order spreading models of a Banach space $X$ form an increasing transfinite hierarchy $(\mathcal{SM}_ξ(X))_{ξ<ω_1}$. Each $\mathcal{SM}_ξ(X)$ contains all spreading models generated by $\ff$-sequences $(x_s)_{s\in\ff}$ with order of $\ff$ equal to $ξ$. We also provide a study of the fundamental properties of the hierarchy.
Let $(e_i)_i$ denote the unit vector basis of $\ell_p$, $1\leq p< \infty$, or $c_0$. We construct a reflexive Banach space with an unconditional basis that admits $(e_i)_i$ as a uniformly unique spreading model while it has no subspace with a unique asymptotic model, and hence it has no asymptotic-$\ell_p$ or $c_0$ subspace. This solves a problem of E. Odell. We also construct a space with a unique $\ell_1$ spreading model and no subspace with a uniformly unique $\ell_1$ spreading model. These results are achieved with the utilization of a new version of the method of saturation under constrai…
Given a Banach space X, denote by SP_{w}(X) the set of equivalence classes of spreading models of X generated by normalized weakly null sequences in X. It is known that SP_{w}(X) is a semilattice, i.e., it is a partially ordered set in which every pair of elements has a least upper bound. We show that every countable semilattice that does not contain an infinite increasing sequence is order isomorphic to SP_{w}(X) for some separable Banach space X.
We investigate possible quantifications of the Banach-Saks property and the weak Banach-Saks property. We prove quantitative versions of relationships of the Banach-Saks property of a set with norm compactness and weak compactness. We further establish a quantitative version of the characterization of the weak Banach-Saks property of a set using uniform weak convergence and $\ell_1$-spreading models. We also study the case of the unit ball and in this case we prove a dichotomy which is an analogue of the James distortion theorem for $\ell_1$-spreading models.
A well known application of Ramsey's Theorem to Banach Space Theory is the notion of a spreading model (e'_i) of a normalized basic sequence (x_i) in a Banach space X. We show how to generalize the construction to define a new creature (e_i), which we call an asymptotic model of X. Every spreading model of X is an asymptotic model of X and in most settings, such as if X is reflexive, every normalized block basis of an asymptotic model is itself an asymptotic model. We also show how to use the Hindman-Milliken Theorem--a strengthened form of Ramsey's Theorem--to generate asymptotic models with …
In the first part of the paper we study the structure of Banach spaces with a conditional spreading basis. The geometry of such spaces exhibit a striking resemblance to the geometry of James' space. Further, we show that the averaging projections onto subspaces spanned by constant coefficient blocks with no gaps between supports are bounded. As a consequence, every Banach space with a spreading basis contains a complemented subspace with an unconditional basis. This gives an affirmative answer to a question of H. Rosenthal. The second part contains two results on Banach spaces $X$ whose asympt…
We study an ordinal rank on the class of Banach spaces with bases that quantifies the distortion of the norm of a given Banach space. The rank $AD(\cdot)$, introduced by P. Dodos, uses the transfinite Schreier familes and has the property that $AD(X) < ω_1$ if and only if $X$ is arbitrarily distortable. We prove several properties of this rank as well as some new results concerning higher order $\ell_1$ spreading models. We also compute this rank for for several Banach spaces. In particular, it is shown that class of Banach spaces $\mathfrak{X}^{ω^ξ}_{0,1}$ , which each admit $\ell_1$ and $c_0…
We consider problems concerning the partial order structure of the set of spreading models of Banach spaces. We construct examples of spaces showing that the possible structure of these sets include certain classes of finite semi-lattices and countable lattices, and all finite lattices.
We investigate the following property for Banach spaces. A Banach space $X$ satisfies the Uniform Approximation on Large Subspaces (UALS) if there exists $C>0$ with the following property: for any $A\in\mathcal{L}(X)$ and convex compact subset $W$ of $\mathcal{L}(X)$ for which there exists $\varepsilon>0$ such that for every $x\in X$ there exists $B\in W$ with $\|A(x)-B(x)\|\le\varepsilon\|x\|$, there exists a subspace $Y$ of $X$ of finite codimension and a $B\in W$ with $\|(A-B)|_Y\|_{\mathcal{L}(Y,X)}\leq C\varepsilon$. We prove that a class of separable Banach spaces including $\ell_p$, for…
We examine a variant of a Banach space $\mathfrak{X}_{0,1}$ defined by Argyros, Beanland, and the second named author that has the property that it admits precisely two spreading models in every infinite dimensional subspace. We prove that this space is asymptotically symmetric and thus it provides a negative answer to a problem of Junge, the first. named author, and Odell.
In this paper we show that every sequence (F_n) of finite dimensional subspaces of a real or complex Banach space with increasing dimensions can be ``refined'' to yield an F.D.D. (G_n), still having increasing dimensions, so that either every bounded sequence (x_n), with x_n in G_n for n in N, is weakly null, or every normalized sequence (x_n), with x_n in G_n for n in N, is equivalent to the unit vector basis of l_1. Crucial to the proof are two stabilization results concerning Lipschitz functions on finite dimensional normed spaces. These results also lead to other applications. We show, for…
It is shown that for each separable Banach space $X$ not admitting $\ell_1$ as a spreading model there is a space $Y$ having $X$ as a quotient and not admitting any $\ell_p$ for $1 \leq p < \infty$ or $c_0$ as a spreading model. We also include the solution to a question of W.B. Johnson and H.P. Rosenthal on the existence of a separable space not admitting as a quotient any space with separable dual.
It is proved that if a Banach space $X$ has a basis $(e_n)$ satisfying every spreading model of a normalized block basis of $(e_n)$ is 1-equivalent to the unit vector basis of $\ell_1$ (respectively, $c_0$) then $X$ contains $\ell_1$ (respectively, $c_0$). Furthermore Tsirelson's space $T$ is shown to have the property that every infinite dimensional subspace contains a sequence having spreading model 1-equivalent to the unit vector basis of $\ell_1$. An equivalent norm is constructed on $T$ so that $\|s_1+s_2\|<2$ whenever $(s_n)$ is a spreading model of a normalized basic sequence in $T$.
We generalize some results concerning the classical notion of a spreading model for the spreading models of order $ξ$. Among them, we prove that the set $SM_ξ^w(X)$ of the $ξ$-order spreading models of a Banach space $X$ generated by subordinated weakly null $\mathcal{F}$-sequences endowed with the pre-partial order of domination is a semi-lattice. Moreover, if $SM_ξ^w(X)$ contains an increasing sequence of length $ω$ then it contains an increasing sequence of length $ω_1$. Finally, if $SM_ξ^w(X)$ is uncountable, then it contains an antichain of size the continuum.
We present some new results on the symmetric Kottman's constant $K^s(X)$ of a Banach space $X$ and its relationship with the Kottman constant. We show that $K^s(X)>1$, for every infinite-dimensional Banach space, thereby solving a problem by J.M.F. Castillo and P.L. Papini. We also investigate such constant in the class of Banach spaces admitting $c_0$ spreading models, answering in particular one question from our previous joint paper with P. Hájek and T. Kania.
Krivine and Maurey proved in 1981 that every stable Banach space contains almost isometric copies of $\ell_p$, for some $p\in[1,\infty)$. In 1983, Raynaud showed that if a Banach space uniformly embeds into a superstable Banach space, then $X$ must contain an isomorphic copy of $\ell_p$, for some $p\in[1,\infty)$. In these notes, we show that if a Banach space coarsely embeds into a superstable Banach space, then $X$ has a spreading model isomorphic to $\ell_p$, for some $p\in[1,\infty)$. In particular, we obtain that there exist reflexive Banach spaces which do not coarsely embed into any sup…
We prove that every Banach space containing a subspace isomorphic to $\co$ fails the fixed point property. The proof is based on an amalgamation approach involving a suitable combination of known results and techniques, including James's distortion theorem, Ramsey's combinatorial theorem, Brunel-Sucheston spreading model techniques and Dowling, Lennard and Turett's fixed point methodology employed in their characterization of weak compactness in $\co$.
In these notes, we study nonlinear embeddings between Banach spaces which are also weakly sequentially continuous. In particular, our main result implies that if a Banach space $X$ coarsely (resp. uniformly) embeds into a Banach space $Y$ by a weakly sequentially continuous map, then every spreading model $(e_n)_n$ of a normalized weakly null sequence in $X$ satisfies \[\|e_1+\ldots+e_k\|_{\overlineδ_Y}\lesssim\|e_1+\ldots+e_k\|_S,\] where $\overlineδ_Y$ is the modulus of asymptotic uniform convexity of $Y$. Among other results, we obtain Banach spaces $X$ and $Y$ so that $X$ coarsely (resp. u…
We give examples of two Banach spaces. One Banach space has no spreading model which contains $\ell_p$ ($1\le p<\infty$) or $c_0$. The other space has an unconditional basis for which $\ell_p$ ($1\le p<\infty$) and $c_0$ are block finitely represented in all block bases.
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…
Using the hierarchy of weakly null sequences introduced by Argyros, Merkourakis, and Tsarpalias, we introduce two new families of operator classes. The first family simultaneously generalizes the completely continuous operators and the weak Banach-Saks operators. The second family generalizes the class $\mathfrak{DP}$. We study the distinctness of these classes, and prove that each class is an operator ideal. We also investigate the properties possessed by each class, such as injectivity, surjectivity, and identification of the dual class. We produce a number of examples, including the higher …
The main result of this article is a rigidity result pertaining to the spreading model structure for Banach spaces coarsely embeddable into Tsirelson's original space $T^*$. Every Banach space that is coarsely embeddable into $T^*$ must be reflexive and all its spreading models must be isomorphic to $c_0$. Several important consequences follow from our rigidity result. We obtain a coarse version of an influential theorem of Tsirelson: $T^*$ does not coarsely contain $c_0$ nor $\ell_p$ for $p\in[1,\infty)$. We show that there is no infinite dimensional Banach space that coarsely embeds into eve…
Let $E$ be a Banach space and, for any positive integer $n$, let ${\cal P}(^nE)$ denote the Banach space of continuous $n$-homogeneous polynomials on $E$. Davie and Gamelin showed that the natural extension mapping from ${\cal P}(^nE)$ to ${\cal P}(^nE^{\ast\ast})$ is an isometry into the latter space. Here, we investigate when there is a natural isomorphism between ${\cal P}(^nE)^{\ast\ast}$ and ${\cal P}(^nE^{\ast\ast})$. Among other things, we show that if $E$ satisfies: \break (a) no spreading model built on a normalised weakly null sequence has a lower $q$-estimate for any $q < \infty,$ (…
We define and study asymptotically symmetric Banach spaces (a.s.) and its variations: weakly a.s. (w.a.s.) and weakly normalized a.s. (w.n.a.s.). If X is a.s. then all spreading models of X are uniformly symmetric. We show that the converse fails. We also show that w.a.s. and w.n.a.s. are not equivalent properties and that Schlumprecht's space S fails to be w.n.a.s. We show that if X is separable and has the property that every normalized weakly null sequence in X has a subsequence equivalent to the unit vector basis of c_0 then X is w.a.s.. We obtain an analogous result if c_0 is replaced by …
A Banach space is {\it polynomially Schur} if sequential convergence against analytic polynomials implies norm convergence. Carne, Cole and Gamelin show that a space has this property and the Dunford-Pettis property if and only if it is Schur. Herein is defined a reasonable generalization of the Dunford--Pettis property using polynomials of a fixed homogeneity. It is shown, for example, that a Banach space will has the $P_N$ Dunford--Pettis property if and only if every weakly compact $N-$homogeneous polynomial (in the sense of Ryan) on the space is completely continuous. A certain geometric c…
The ``multiple of the inclusion plus compact problem'' which was posed by T.W. Gowers in 1996 and Th. Schlumprecht in 2003, asks whether for every infinite dimensional Banach space $X$ there exists a closed subspace $Y$ of $X$ and a bounded linear operator from $Y$ to $X$ which is not a compact perturbation of a multiple of the inclusion map from $Y$ to $X$. We give sufficient conditions on the spreading models of seminormalized basic sequences of a Banach space $X$ which guarantee that the ``multiple of the inclusion plus compact'' problem has an affirmative answer for $X$. Our results streng…
We consider a normalized basis in a Banach space with the following property: any normalized block sequence of the basis has a subsequence equivalent to the basis. We show that under uniformity or other natural assumptions, a basis with this property is equivalent to the unit vector basis of $c_0$ or $\ell_p$. We also address an analogous problem for spreading models.
Banach Spaces Generated by Finite-Valued Functions: Superreflexive Rigidity, Hankel Operators, and Universality
Konstantin Preslavsky University of Shumen（舒门康斯坦丁·普拉斯夫斯基大学）；University of Plovdiv Paisii Hilendarski（普罗夫迪夫帕伊西·希伦达斯基大学）；Technical University of Varna（瓦尔纳技术大学）
We introduce and study the Bourgain index of an operator between two Banach spaces. In particular, we study the Bourgain $\ell_p$ and $c_0$ indices of an operator. Several estimates for finite and infinite direct sums are established. We define classes determined by these indices and show that some of these classes form operator ideals. We characterize the ordinals which occur as the index of an operator and establish exactly when the defined classes are closed. We study associated indices for non-preservation of $\ell_p^ξ$ and $c_0^ξ$ spreading models and indices characterizing weak compactne…
The notion of $α$-large families of finite subsets of an infinite set is defined for every countable ordinal number $α$, extending the known notion of large families. The definition of the $α$-large families is based on the transfinite hierarchy of the Schreier families $\mathcal{S}_α, α<ω_1$. We prove the existence of such families on the cardinal number $2^{\aleph_0}$ and we study their properties. As an application, based on those families we construct a reflexive space $\mathfrak{X}_{2^{\aleph_0}}^α$, $α<ω_1$ with density the continuum, such that every bounded non norm convergent sequence …
For any closed subset $F$ of $[1,\infty]$ which is either finite or consists of the elements of an increasing sequence and its limit, a reflexive Banach space $X$ with a 1-unconditional basis is constructed so that in each block subspace $Y$ of $X$, $\ell_p$ is finitely block represented in $Y$ if and only if $p \in F$. In particular, this solves the question as to whether the stabilized Krivine set for a Banach space had to be connected. We also prove that for every infinite dimensional subspace $Y$ of $X$ there is a dense subset $G$ of $F$ such that the spreading models admitted by $Y$ are e…
For a large class of Banach spaces, a general construction of subspaces without local unconditional structure is presented. As an application it is shown that every Banach space of finite cotype contains either $l_2$ or a subspace without unconditional basis, which admits a Schauder basis. Some other interesting applications and corollaries follow.
In this note, we extend to the setting of quasi-reflexive spaces a classical result of N. Kalton and L. Randrianarivony on the coarse Lipschitz structure of reflexive and asymptotically uniformly smooth Banach spaces. As an application, we show for instance, that for $1\le q<p$, a $q$-asymptotically uniformly convex Banach space does not coarse Lipschitz embed into a $p$-asymptotically uniformly smooth quasi-reflexive Banach space. This extends a recent result of B.M. Braga.
We extend and generalize the result of Kalton and Swanson ($Z_2$ is a symplectic Banach space with no Lagrangian subspace) by showing that all higher order Rochgberg spaces $\mathfrak R^{(n)}$ are symplectic Banach spaces with no Lagrangian subspaces. The nontrivial symplectic structure on even spaces is the one induced by the natural duality; while the nontrivial symplectic structure on odd spaces requires perturbation with a complex structure. We will also study symplectic structures on general Banach spaces and, motivated by the unexpected appearance of complex structures, we introduce and …
This paper deals with the interplay of the geometry of the norm and the weak topology in Banach spaces. Both dual and intrinsic connections between weak forms of rotundity and smoothness ared discussed. Weakly exposed points, weakly locally uniformly rotund spaces, smoothness, duality and the interplay of all the above are studied.
Let $A$ be a commutative semisimple Banach algebra, $X$ be a locally compact Hausdorff topological space and $G$ be a locally compact topological group. In this paper, we investigate several properties of vector valued Banach algebras $C_0(X,A)$, $L^p(G,A)$, $l^p(X,A)$ and $l^{\infty}(X,A)$. We prove that these algebras are isomorphic with a $C^*-$algebra if and only if $A$ is so.
This note contains a corrected proof of the main result (which remains unchanged) from [K-T]. It was recently observed that an argument in a basic technical criterium has a gap.
In general Banach spaces, the metric projection map lacks the powerful properties it enjoys in Hilbert spaces. There are a few generalized projections that have been proposed in order to resolve many of the deficiencies of the metric projection. However, such notions are predominantly studied in Banach spaces with rich topological structures, such as uniformly convex Banach spaces. In this paper, we investigate two notions of generalized projection in general Banach spaces. Various examples are provided to demonstrate the proposed notions and the loss of structure in the generalized projection…
In this paper, we study the coarse Lipschitz geometry of Banach spaces with several asymptotic properties. Specifically, we look at asymptotically uniformly smoothness and convexity, and several distinct Banach-Saks-like properties. Among other results, we characterize the Banach spaces which are either coarsely or uniformly homeomorphic to $T^{p_1}\oplus \ldots \oplus T^{p_n}$, where each $T^{p_j}$ denotes the $p_j$-convexification of the Tsirelson space, for $p_1,\ldots,p_n\in (1,\ldots, \infty)$, and $2\not\in\{p_1,\ldots ,p_n\}$. We obtain applications to the coarse Lipschitz geometry of t…
M. Talagrand showed that, for the Cech-Stone compactification βω of the space of natural numbers, the norm and the weak topology generate different Borel structures in the Banach space C(βω). We prove that the Borel structures in C(βω) generated by the weak and the pointwise topology are also different. We also show that in C(ω*), where ω*=βω- ω, there is no countable family of pointwise Borel sets separating functions from C(ω*).
A well-known application of the Ramsey Theorem in the Banach Space Theory is the proof of the fact that every normalized basic sequence has a subsequence which generates a spreading model (the Brunel-Sucheston Theorem). Based on this application, as an intermediate step, we can talk about the notion of $(k,\varepsilon)-$oscillation stable sequence, which will be described and analyzed more generally in this article. Indeed, we introduce the notion $((\mathcal{B}_i)_{i=1}^k,\varepsilon)-$block oscillation stable sequence where $(\mathcal{B}_i)_{i=1}^k$ is a finite sequence of barriers and using…
