arXiv subjects
Publications and source records attributed to Thomas Schlumprecht.
We show that if a sequence of expander graphs equi-coarsely embeds into a Banach space, then this Banach space fails Kasparov and Yu's Property (H). Consequently, no Banach space with Property (H) can be coarsely universal for all countable groups. We provide a Lean verification of our results.
For two countable ordinals $α$ and $β$, a basis of a Banach space $X$ is said to be $(α, β)$-quasi-greedy if it is 1) quasi-greedy, 2) $\mathcal{S}_α$-unconditional but not $\mathcal{S}_{α+1}$-unconditional, and 3) $\mathcal{S}_β$-democratic but not $\mathcal{S}_{β+1}$-democratic. If $α$ or $β$ is replaced with $\infty$, then the basis is required to be unconditonal or democratic, respectively. Previous work constructed a $(0,0)$-quasi-greedy basis, an $(α, \infty)$-quasi-greedy basis, and an $(\infty, α)$-quasi-greedy basis. In this paper, we construct $(α, β)$-quasi-greedy bases for $β\le α+1$ (except the already solved case $α= β= 0$).
We consider the problem of whether there is a sequence of homeomorphisms $(F_k)_k$ between the unit spheres of the $k$-dimensional Banach spaces $\ell_\infty^k$ and $\ell_1^k$ which is also equi-uniformly continuous. We prove that this cannot be the case if the sequence $(F_k)_k$ either (1) does not increase support sizes (which is a property strictly weaker than support preservation) or (2) is step preserving (which is a property strictly weaker than being equivariant with respect to permutations of the canonical basis). We also provide quantitative estimates relating the moduli of uniform continuity of the maps to the dimension of the spaces. This gives partial answers to a question of W. B. Johnson and it is related to the problem of whether $c_0$ has Kasparov and Yu's Property (H). Our results also apply to more general spaces other than $\ell_1$ such as spaces with unconditional bases which are not equivalent to the standard $c_0$ basis. Finally, we derive an asymptotic concentration inequality that must be satisfied by step preserving equi-uniformly continuous maps defined on the positive parts of these unit spheres.
We prove that for every countable ordinal $ξ$, the Tsirelson's space $T_ξ$ of order $ξ$, is naturally, i.e., via the identity, $3$-isomorphc to its modified version. For the first step, we prove that the Schreier family $\mathcal{S}_ξ$ is the same as its modified version $ \mathcal{S}^M_ξ$, thus answering a question by Argyros and Tolias. As an application, we show that the algebra of linear bounded operators on $T_ξ$ has $2^{\mathfrak c}$ closed ideals.
Let $(h_I)$ denote the standard Haar system on $[0,1]$, indexed by $I\in \mathcal D$, the set of dyadic intervals and $h_I\otimes h_J$ denote the tensor product $(s,t)\mapsto h_I(s) h_J(t)$, $I,J\in \mathcal D$. We consider a class of two-parameter function spaces which are completions of the linear span $\mathcal{V}(δ^2)$ of $h_I\otimes h_J$, $I,J\in \mathcal D$. This class contains all the spaces of the form $X(Y)$, where $X$ and $Y$ are either the Lebesgue spaces $L_p[0,1]$ or the Hardy spaces $H_p[0,1]$, $1\le p<\infty$. We say that $D\colon X(Y)\to X(Y)$ is a Haar multiplier if $D(h_I\otimes h_J) = d_{I,J} h_I\otimes h_J$, where $d_{I,J}\in \mathbb{R}$, and ask which more elementary operators factor through $D$. A decisive role plays the {\em Capon projection} $\mathcal{C}\colon \mathcal{V}(δ^2)\to \mathcal{V}(δ^2)$ given by $\mathcal{C} h_I\otimes h_J = h_I\otimes h_J$ if $|I|\leq |J|$, and $\mathcal{C} h_I\otimes h_J = 0$ if $|I| > |J|$, as our main result highlights: Given any bounded Haar multiplier $D\colon X(Y)\to X(Y)$, there exist $λ,μ\in \mathbb{R}$ such that \begin{equation*} \text{$λ\mathcal{C} + μ(\mathrm{Id}-\mathcal{C})$ approximately $1$-projectionally factors through $D$,} \end{equation*} i.e., for all $η>0$, there exist bounded operators $A,B$ so that $AB$ is the identity operator $\mathrm{Id}$, $\|A\|\cdot\|B\|=1$ and $\|λ\mathcal{C} + μ(\mathrm{Id}-\mathcal{C}) - ADB\|<η$. Additionally, if $\mathcal{C}$ is unbounded on $X(Y)$, then $λ= μ$ and then $\mathrm{Id}$ either factors through $D$ or $\mathrm{Id}-D$.
These notes present a basic survey on Transportation cost spaces (aka Lipschitzfree spaces, Wasserstein spaces) and their bi-Lipschitz and linear embeddings into $L_1$ spaces. To make these notes as self-contained as possible, we added the proofs of several relevant results from computational graph theory in the appendix.
By discretizing an argument of Kislyakov, Naor and Schechtman proved that the 1-Wasserstein metric over the planar grid $\{0,1,\dots n\}^2$ has $L_1$-distortion bounded below by a constant multiple of $\sqrt{\log n}$. We provide a new "dimensionality" interpretation of Kislyakov's argument, showing that, if $\{G_n\}_{n=1}^\infty$ is a sequence of graphs whose isoperimetric dimension and Lipschitz-spectral dimension equal a common number $δ\in [2,\infty)$, then the 1-Wasserstein metric over $G_n$ has $L_1$-distortion bounded below by a constant multiple of $(\log |G_n|)^{\frac{1}δ}$. We proceed to compute these dimensions for $\oslash$-powers of certain graphs. In particular, we get that the sequence of diamond graphs $\{\mathsf{D}_n\}_{n=1}^\infty$ has isoperimetric dimension and Lipschitz-spectral dimension equal to 2, obtaining as a corollary that the 1-Wasserstein metric over $\mathsf{D}_n$ has $L_1$-distortion bounded below by a constant multiple of $\sqrt{\log| \mathsf{D}_n|}$. This answers a question of Dilworth, Kutzarova, and Ostrovskii and exhibits only the third sequence of $L_1$-embeddable graphs whose sequence of 1-Wasserstein metrics is not $L_1$-embeddable.
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.
We formulate general conditions which imply that $L(X,Y)$, the space of operators from a Banach space $X$ to a Banach space $Y$, has $2^{\mathfrak c}$ closed ideals where $\mathfrak c$ is the cardinality of the continuum. These results are applied to classical sequence spaces and Tsirelson type spaces. In particular, we prove that the cardinality of the set of closed ideals in $L(\ell_p\oplus\ell_q)$ is exactly $2^{\mathfrak c}$ for all $1<p<q<\infty$, which in turn gives an alternate proof of the recent result of Johnson and Schechtman that $L(L_p)$ also has $2^{\mathfrak c}$ closed ideals for $1<p\neq 2<\infty$.
Theorem A and Theorem B of [1] state that for $1<p<\infty$ the lattice of closed ideals of $\mathcal{L}(\ell_p,c_0)$, $\mathcal{L}(\ell_p,\ell_\infty)$ and of $\mathcal{L}(\ell_1,\ell_p)$ are at least of cardinality $2^ω$. Here we show that the cardinality of the lattice of closed ideals of $\mathcal{L}(\ell_p,c_0)$, $\mathcal{L}(\ell_p,\ell_\infty)$ and of $\mathcal{L}(\ell_1,\ell_p)$, is at least $2^{2^ω}$, and thus equal to it.
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.
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}.
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.
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.
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.
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.
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 every infinite dimensional Banach space. In particular, we disprove the conjecture that the separable infinite dimensional Hilbert space coarsely embeds into every infinite dimensional Banach space. The rigidity result follows from a new concentration inequality for Lipschitz maps on the infinite Hamming graphs and taking values in $T^*$, and from the embeddability of the infinite Hamming graphs into Banach spaces that admit spreading models not isomorphic to $c_0$. Also, a purely metric characterization of finite dimensionality is obtained.
We generalize and prove a result which was first shown by Zippin, and was explicitly formulated by Benyamini.