arXiv subjects

Pavlos Motakis

Publications and source records attributed to Pavlos Motakis.

At least 37 records · Page 2Linked to original sources

arXiv2021

On uniqueness and plentitude of subsymmetric sequences

We explore the diversity of subsymmetric basic sequences in spaces with a subsymmetric basis. We prove that the subsymmetrization $Su(T^*)$ of Tsirelson's original Banach space provides the first known example of a space with a unique subsymmetric basic sequence that is additionally non-symmetric. Contrastingly, we provide a criterion for a space with a subsymmetric basis to contain a continuum of nonequivalent subsymmetric basic sequences and apply it to $Su(T^*)^*$. Finally, we provide a criterion for a subsymmetric sequence to be equivalent to the unit vector basis of some $\ell_p$ or $c_0$.

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arXiv2021

The space $L_1(L_p)$ is primary for $1<p<\infty$

The classical Banach space $L_1(L_p)$ consists of measurable scalar functions $f$ on the unit square for which $$\|f\| = \int_0^1\Big(\int_0^1 |f(x,y)|^p dy\Big)^{1/p}dx < \infty.$$ We show that $L_1(L_p)$ $(1 < p < \infty)$ is primary, meaning that, whenever $L_1(L_p) = E\oplus F$ then either $E$ or $F$ is isomorphic to $L_1(L_p)$. More generally we show that $L_1(X)$ is primary, for a large class of rearrangement invariant Banach function spaces.

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arXiv2020

Stochastic approximation of lamplighter metrics

We observe that embeddings into random metrics can be fruitfully used to study the $L_1$-embeddability of lamplighter graphs or groups, and more generally lamplighter metric spaces. Once this connection has been established, several new upper bound estimates on the $L_1$-distortion of lamplighter metrics follow from known related estimates about stochastic embeddings into dominating tree-metrics. For instance, every lamplighter metric on a $n$-point metric space embeds bi-Lipschitzly into $L_1$ with distortion $O(\log n)$. In particular, for every finite group $G$ the lamplighter group $H = \mathbb{Z}_2\wr G$ bi-Lipschitzly embeds into $L_1$ with distortion $O(\log\log|H|)$. In the case where the ground space in the lamplighter construction is a graph with some topological restrictions, better distortion estimates can be achieved. Finally, we discuss how a coarse embedding into $L_1$ of the lamplighter group over the $d$-dimensional infinite lattice $\mathbb{Z}^d$ can be constructed from bi-Lipschitz embeddings of the lamplighter graphs over finite $d$-dimensional grids, and we include a remark on Lipschitz free spaces over finite metric spaces.

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arXiv2020

On the bi-Lipschitz geometry of lamplighter graphs

In this article we start a systematic study of the bi-Lipschitz geometry of lamplighter graphs. We prove that lamplighter graphs over trees bi-Lipschitzly embed into Hamming cubes with distortion at most~$6$. It follows that lamplighter graphs over countable trees bi-Lipschitzly embed into $\ell_1$. We study the metric behaviour of the operation of taking the lamplighter graph over the vertex-coalescence of two graphs. Based on this analysis, we provide metric characterizations of superreflexivity in terms of lamplighter graphs over star graphs or rose graphs. Finally, we show that the presence of a clique in a graph implies the presence of a Hamming cube in the lamplighter graph over it. An application is a characterization in terms of a sequence of graphs with uniformly bounded degree of the notion of trivial Bourgain-Milman-Wolfson type for arbitrary metric spaces, similar to Ostrovskii's characterization previously obtained in \cite{ostrovskii:11}.

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arXiv2020

A new coarsely rigid class of Banach spaces

We prove that the class of reflexive asymptotic-$c_0$ Banach spaces is coarsely rigid, meaning that if a Banach space $X$ coarsely embeds into a reflexive asymptotic-$c_0$ space $Y$, then $X$ is also reflexive and asymptotic-$c_0$. In order to achieve this result we provide a purely metric characterization of this class of Banach spaces. This metric characterization takes the form of a concentration inequality for Lipschitz maps on the Hamming graphs, which is rigid under coarse embeddings. Using an example of a quasi-reflexive asymptotic-$c_0$ space, we show that this concentration inequality is not equivalent to the non equi-coarse embeddability of the Hamming graphs.

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arXiv2020

The geometry of Hamming-type metrics and their embeddings into Banach spaces

Within the class of reflexive Banach spaces, we prove a metric characterization of the class of asymptotic-$c_0$ spaces in terms of a bi-Lipschitz invariant which involves metrics that generalize the Hamming metric on $k$-subsets of $\mathbb{N}$. We apply this characterization to show that the class of separable, reflexive, and asymptotic-$c_0$ Banach spaces is non-Borel co-analytic. Finally, we introduce a relaxation of the asymptotic-$c_0$ property, called the asymptotic-subsequential-$c_0$ property, which is a partial obstruction to the equi-coarse embeddability of the sequence of Hamming graphs. We present examples of spaces that are asymptotic-subsequential-$c_0$. In particular $T^*(T^*)$ is asymptotic-subsequential-$c_0$ where $T^*$ is Tsirelson's original space.

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arXiv2020

Coarse and Lipschitz universality

In this paper we provide several \emph{metric universality} results. We exhibit for certain classes $\cC$ of metric spaces, families of metric spaces $(M_i, d_i)_{i\in I}$ which have the property that a metric space $(X,d_X)$ in $\cC$ is coarsely, resp. Lipschitzly, universal for all spaces in $\cC$ if the collection of spaces $(M_i,d_i)_{i\in I}$ equi-coarsely, respectively equi-Lipschitzly, embeds into $(X,d_X)$. Such families are built as certain Schreier-type metric subsets of $\co$. We deduce a metric analog to Bourgain's theorem, which generalized Szlenk's theorem, and prove that a space which is coarsely universal for all separable reflexive asymptotic-$c_0$ Banach spaces is coarsely universal for all separable metric spaces. One of our coarse universality results is valid under Martin's Axiom and the negation of the Continuum Hypothesis. We discuss the strength of the universality statements that can be obtained without these additional set theoretic assumptions. In the second part of the paper, we study universality properties of Kalton's interlacing graphs. In particular, we prove that every finite metric space embeds almost isometrically in some interlacing graph of large enough diameter.

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arXiv2020

Non-asymptotic $\ell_1$ spaces with unique $\ell_1$ asymptotic model

A recent result of Freeman, Odell, Sari, and Zheng states that whenever a separable Banach space not containing $\ell_1$ has the property that all asymptotic models generated by weakly null sequences are equivalent to the unit vector basis of $c_0$ then the space is Asymptotic $c_0$. We show that if we replace $c_0$ with $\ell_1$ then this result is no longer true. Moreover, a stronger result of B. Maurey - H. P. Rosenthal type is presented, namely, there exists a reflexive Banach space with an unconditional basis admitting $\ell_1$ as a unique asymptotic model whereas any subsequence of the basis generates a non-Asymptotic $\ell_1$ subspace.

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arXiv2019

Continuous factorization of the identity matrix

We investigate conditions under which the identity matrix $I_n$ can be continuously factorized through a continuous $N\times N$ matrix function $A$ with domain in $\mathbb{R}$. We study the relationship of the dimension $N$, the diagonal entries of $A$, and the norm of $A$ to the dimension $n$ and the norms of the matrices that witness the factorization of $I_n$ through $A$.

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arXiv2019

On the complete separation of asymptotic structures in Banach spaces

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 constraints that uses sequences of functionals with increasing weights.

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arXiv2019

Asymptotically symmetric spaces with hereditarily non-unique spreading models

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.

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arXiv2018

Strategically reproducible bases and the factorization property

We introduce the concept of strategically reproducible bases in Banach spaces and show that operators which have large diagonal with respect to strategically reproducible bases are factors of the identity. We give several examples of classical Banach spaces in which the Haar system is strategically reproducible: multi-parameter Lebesgue spaces, mixed-norm Hardy spaces and most significantly the space $L^1$. Moreover, we show the strategical reproducibility is inherited by unconditional sums.

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arXiv2017

Algebras of diagonal operators of the form scalar-plus-compact are Calkin algebras

For every Banach space $X$ with a Schauder basis consider the Banach algebra $\mathbb{R} I\oplus\mathcal{K}_\mathrm{diag}(X)$ of all diagonal operators that are of the form $λI + K$. We prove that $\mathbb{R} I\oplus\mathcal{K}_\mathrm{diag}(X)$ is a Calkin algbra i.e., there exists a Banach space $\mathcal{Y}_X$ so that the Calkin algebra of $\mathcal{Y}_X$ is isomorphic as a Banach algebra to $\mathbb{R} I\oplus\mathcal{K}_\mathrm{diag}(X)$. Among other applications of this theorem we obtain that certain hereditarily indecomposable spaces and the James spaces $J_p$ and their duals endowed with natural multiplications are Calkin algebras, that all non-reflexive Banach spaces with unconditional bases are isomorphic as Banach spaces to Calkin algebras, and that sums of reflexive spaces with unconditional bases with certain James-Tsirelson type spaces are isomorphic as Banach spaces to Calkin algebras.

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arXiv2016

A study of conditional spreading sequences

It is shown that every conditional spreading sequence can be decomposed into two well behaved parts, one being unconditional and the other being convex block homogeneous, i.e. equivalent to its convex block sequences. This decomposition is then used to prove several results concerning the structure of spaces with conditional spreading bases as well as results in the theory of conditional spreading models. Among other things, it is shown that the space $C(ω^ω)$ is universal for all spreading models, i.e., it admits all spreading sequences, both conditional and unconditional, as spreading models. Moreover, every conditional spreading sequence is generated as a spreading model by a sequence in a space that is quasi-reflexive of order one.

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arXiv2016

The scalar-plus-compact property in spaces without reflexive subspaces

A hereditarily indecomposable Banach space $\mathfrak{X}_{\mathfrak{nr}}$ is constructed that is the first known example of a $\mathscr{L}_\infty$-space not containing $c_0$, $\ell_1$, or reflexive subspaces and answers a question posed by J. Bourgain. Moreover, the space $\mathfrak{X}_{\mathfrak{nr}}$ satisfies the "scalar-plus-compact" property and it is the first known space without reflexive subspaces having this property. It is constructed using the Bourgain-Delbaen method in combination with a recent version of saturation under constraints in a mixed-Tsirelson setting. As a result, the space $\mathfrak{X}_{\mathfrak{nr}}$ has a shrinking finite dimensional decomposition and does not contain a boundedly complete sequence.

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arXiv2016

A metric interpretation of reflexivity for Banach spaces

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 $({\mathcal S}_α, d_{1,α})$ does not bi-Lipschitzly embed into $X$. Here $\text{Sz}(Y)$ denotes the Szlenk index of a Banach space $Y$.

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arXiv2015

On the structure of separable $\mathcal{L}_\infty$-spaces

Based on a construction method introduced by J. Bourgain and F. Delbaen, we give a general definition of a Bourgain-Delbaen space and prove that every infinite dimensional separable $\mathcal{L}_\infty$-space is isomorphic to such a space. Furthermore, we provide an example of a $\mathcal{L}_\infty$ and asymptotic $c_0$ space not containing $c_0$.

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arXiv2015

A dual method of constructing hereditarily indecomposable Banach spaces

A new method of defining hereditarily indecomposable Banach spaces is presented. This method provides a unified approach for constructing reflexive HI spaces and also HI spaces with no reflexive subspace. All the spaces presented here satisfy the property that the composition of any two strictly singular operators is a compact one. This yields the first known example of a Banach space with no reflexive subspace such that every operator has a non-trivial closed invariant subspace.

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