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 superstable Banach space.
We prove that, for $1 < p \neq q < \infty$, there does not exist any coarse Lipschitz embedding between the two James spaces $J_p$ and $J_q$, and that, for $1 < p < q < \infty$ and $1 < r < \infty$ such that $r \notin \{p,q\}$, $J_r$ does not coarse Lipschitz embed into $J_p \oplus J_q$.
In these notes, we study the relation between uniform and coarse embeddings between Banach spaces. In order to understand this relation better, we also look at the problem of when a coarse embedding can be assumed to be topological. Among other results, we show that if a Banach space $X$ uniformly embeds into a minimal Banach space $Y$, then $X$ simultaneously coarsely and uniformly embeds into $Y$, and if a Banach space $X$ coarsely embeds into a minimal Banach space $Y$, then $X$ simultaneously coarsely and homeomorphically embeds into $Y$ by a map with uniformly continuous inverse.
We study how well a quasi-Banach space can be coarsely embedded into a Hilbert space. Given any quasi-Banach space X which coarsely embeds into a Hilbert space, we compute its Hilbert space compression exponent. We also show that the Hilbert space compression exponent of X is equal to the supremum of the amounts of snowflakings of X which admit a bi-Lipschitz embedding into a Hilbert space.
Let $λ$ be a large enough cardinal number (assuming GCH it suffices to let $λ=\aleph_ω$). If $X$ is a Banach space with $\text{dens}(X)\geλ$, which admits a coarse (or uniform) embedding into any $c_0(Γ)$, then $X$ fails to have nontrivial cotype, i.e. $X$ contains $\ell_\infty^n$ $C$-uniformly for every $C>1$. In the special case when $X$ has a symmetric basis, we may even conclude that it is linearly isomorphic with $c_0(\text{dens}X)$.
We give an almost complete description of the coarse and uniform embeddability between Orlicz sequence spaces. We show that the embeddability between two Orlicz sequence spaces is in most cases determined only by the values of their upper Matuszewska-Orlicz indices. On the other hand, we present examples which show that sometimes the embeddability is not determined by the values of these indices.
In this paper, we study the coarse embedding into Banach space. We proved that under certain conditions, the property of embedding into Banach space can be preserved under taking the union the metric spaces. For a group $G$ strongly relative hyperbolic to a subgroup $H$, we proved that if $H$ admits a coarse embedding into a uniformly convex Banach space, so is $B(n)=\{g\in G|\abs{g}_{S\cup\mathscr{H}}\leq n\}$.
In this paper, we study nonlinear embeddings between Banach spaces. More specifically, the goal of this paper is to study weaker versions of coarse and uniform embeddability, and to provide suggestive evidences that those weaker embeddings may be stronger than one would think. We do such by proving that many known results for coarse and uniform embeddings remain valid for those weaker notions of embeddings.
We construct the first example of a coarsely non-amenable (= without Guoliang Yu's property A) metric space with bounded geometry which coarsely embeds into a Hilbert space.
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 generalize the notions of asymptotic dimension and coarse embeddings from metric spaces to quantum metric spaces in the sense of Kuperberg and Weaver. We show that quantum asymptotic dimension behaves well with respect to metric quotients and direct sums, and is preserved under quantum coarse embeddings. Moreover, we prove that a quantum metric space that equi-coarsely contains a sequence of reflexive quantum expanders must have infinite asymptotic dimension. This is done by proving a quantum version of a vertex-isoperimetric inequality for expanders, based upon a previously known edge-isop…
In this paper we prove that $L_{p}$ does not admit an equivariant coarse embedding into $\ell_p$ i.e there is no proper, affine, isometric action of $L_{p}$, viewed as a group under addition with the standard metric $|| . ||_p$, on $\ell_p$. This is done by showing that representations of $L_{p}$ into $ Isom(\ell_p)$ has to be trivial, which allows us to reduce the question to bi-Lipschitz setting.
We study embeddings of uniform Roe algebras which have "large range" in their codomain and the relation of those with coarse quotients between metric spaces. Among other results, we show that if $Y$ has property A and there is an embedding $Φ:\mathrm{C}^*_u(X)\to \mathrm{C}^*_u(Y)$ with "large range" and so that $Φ(\ell_\infty(X))$ is a Cartan subalgebra of $\mathrm{C}^*_u(Y)$, then there is a bijective coarse quotient $X\to Y$. This shows that the large scale geometry of $Y$ is, in some sense, controlled by the one of $X$. For instance, if $X$ has finite asymptotic dimension, so does $Y$.
We construct a finitely generated group which is an extension of two finitely generated groups coarsely embeddable into Hilbert space but which itself does not coarsely embed into Hilbert space. Our construction also provides a new infinite monster group: the first example of a finitely generated group that does not coarsely embed into Hilbert space and yet does not contain a weakly embedded expander.
We investigate how coarse embeddability of box spaces into Hilbert space behaves under group extensions. In particular, we prove a result which implies that a semidirect product of a finitely generated free group by a finitely generated residually finite amenable group has a box space which coarsely embeds into Hilbert space. This provides a new class of examples of metric spaces with bounded geometry which coarsely embed into Hilbert space but do not have property A, generalising the example of Arzhantseva, Guentner and Spakula.
The nonlinear geometry of operator spaces has recently started to be investigated. Many notions of nonlinear embeddability have been introduced so far, but, as noticed before by other authors, it was not clear whether they could be considered ``correct notions''. The main goal of these notes is to provide the missing evidence to support that \emph{almost complete coarse embeddability} is ``a correct notion''. This is done by proving results about the complete isomorphic theory of $\ell_1$-sums of certain operators spaces. Several results on the complete isomorphic theory of $c_0$-sums of opera…
The main purpose of the paper is to prove the following results: Let $A$ be a locally finite metric space whose finite subsets admit uniformly bilipschitz embeddings into a Banach space $X$. Then $A$ admits a bilipschitz embedding into $X$. Let $A$ be a locally finite metric space whose finite subsets admit uniformly coarse embeddings into a Banach space $X$. Then $A$ admits a coarse embedding into $X$. These results generalize previously known results of the same type due to Brown-Guentner (2005), Baudier (2007), Baudier-Lancien (2008), and the author (2006, 2009). One of the main steps in th…
In this paper, we prove the coarse geometric $\ell^p$-Novikov Conjecture for metric spaces with bounded geometry which admit a coarse embedding into a simply connected complete Riemannian manifold of nonpositive sectional curvature.
If $X$ is an almost transitive Banach space with amenable isometry group (for example, if $X=L^p([0,1])$ with $1\leqslant p<\infty$) and $X$ admits a uniformly continuous map $X\oversetϕ\longrightarrow E$ into a Banach space $E$ satisfying $$\inf_{\|x-y\|=r} \| ϕ(x)-ϕ(y)\|>0 $$ for some $r>0$, then $X$ admits a simultaneously uniform and coarse embedding into a Banach space $V$ that is finitely representable in $L^2(E)$.
The main result is a submetric characterization of the class of Banach spaces admitting an equivalent norm with Rolewicz's property ($β$). As applications we prove that up to renorming, property ($β$) is stable under coarse Lipschitz embeddings and coarse quotients.
We study uniform and coarse embeddings between Banach spaces and topological groups. A particular focus is put on equivariant embeddings, i.e., continuous cocycles associated to continuous affine isometric actions of topological groups on separable Banach spaces with varying geometry.
In this paper fundamental nonlinear geometries of Lebesgue sequence spaces are studied in their quantitative aspects. Applications of this work are a positive solution to the strong embeddability problem from $\ell_q$ into $\ell_p$ ($0<p<q\le 1$) and new insights on the coarse embedabbility problem from $L_q$ into $\ell_q$, $q>2$. Relevant to geometric group theory purposes, the exact $\ell_p$-compressions of $\ell_2$ are computed. Finally coarse deformation of metric spaces with property A and locally compact amenable groups is investigated.
We introduce two notions of coarse embeddability between operator spaces: almost complete coarse embeddability of bounded subsets and spherically-complete coarse embeddability. We provide examples showing that these notions are strictly weaker than complete isomorphic embeddability - in fact, they do not even imply isomorphic embeddability. On the other hand, we show that, despite their nonlinearity, the existence of such embeddings provides restrictions on the linear operator space structures of the spaces. Examples of nonlinear equivalences between operator spaces are also provided.
In this paper we show that a countable, residually amenable group admits a proper isometric affine action on some uniformly convex Banach space if and only if one (or equivalently, all) of its box families admits a fibred cofinitely-coarse embedding into some uniformly convex Banach space.
We study the nonlinear embeddability of Banach spaces and the equi-embeddability of the family of Kalton's interlaced graphs $([\mathbb N]^k,d_{\mathbb K})_k$ into dual spaces. Notably, we define and study a modification of Kalton's property $\mathcal Q$ that we call property $\mathcal{Q}_p$ (with $p \in (1,+\infty]$). We show that if $([\mathbb N]^k,d_{\mathbb K})_k$ equi-coarse Lipschitzly embeds into $X^*$, then the Szlenk index of $X$ is greater than $ω$, and that this is optimal, i.e., there exists a separable dual space $Y^*$ that contains $([\mathbb N]^k,d_{\mathbb K})_k$ equi-Lipschitz…
We show that there exists a strong uniform embedding from any proper metric space into any Banach space without cotype. Then we prove a result concerning the Lipschitz embedding of locally finite subsets of $\mathcal{L}_{p}$-spaces. We use this locally finite result to construct a coarse bi-Lipschitz embedding for proper subsets of any $\mathcal{L}_p$-space into any Banach space $X$ containing the $\ell_p^n$'s. Finally using an argument of G. Schechtman we prove that for general proper metric spaces and for Banach spaces without cotype a converse statement holds.
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…
We generalize the construction of Arzhantseva, Guentner and Spakula of a box space of the free group which admits a coarse embedding into Hilbert space. We show that for a finitely generated free group, the box space corresponding to the derived $m$-series (for any integer $m\geq 2$) coarsely embeds into Hilbert space. This gives new examples of metric spaces with bounded geometry which coarsely embed into Hilbert space but do not have Yu's property A.
$K$-theory of ghostly ideals for $\ell^p$-coarsely embeddable spaces
Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS)（上海数学与交叉学科研究院）；Research Institute of Intelligent Complex Systems, Fudan University（复旦大学智能复杂系统研究院）；Centre for Mathematical Sciences, Lund University（隆德大学数学科学中心）；Research Center for Operator Algebras, School of Mathematical Sciences, East China Normal University（华东师范大学数学科学学院算子代数研究中心）
