Linearly rigid metric spaces and the embedding problem

等距刚性和Orlicz序列空间的Fraïssé性质

Isometric rigidity and Fraïssé properties of Orlicz sequence spaces

论文研究了Orlicz序列空间的近等距刚性性质，证明了某些空间guarded Fraïssé但非ω-范畴，且不含ℓ₂的拷贝，回答了Cúth-de Rancourt-Doucha的问题。

Linear versus lattice embeddings between Banach lattices

A well-known classical result states that $c_0$ is linearly embeddable in a Banach lattice if and only if it is lattice embeddable. Improving results of H.P.~Lotz, H.P.~Rosenthal and N.~Ghoussoub, we prove that $C[0,1]$ shares this property with $c_0$. Furthermore, we show that any infinite-dimensional sublattice of $C[0,1]$ is either lattice isomorphic to $c_0$ or contains a sublattice isomorphic to $C[0,1]$. As a consequence, it is proved that for a separable Banach lattice $X$ the following conditions are equivalent: (1) $X$ is linearly embeddable in a Banach lattice if and only if it is la…

通过确定性降维映射将稀疏向量从 $\ell_2$ 线性嵌入到 $\ell_1$

Linearly Embedding Sparse Vectors from $\ell_2$ to $\ell_1$ via Deterministic Dimension-Reducing Maps

本文研究从 $\ell_2$ 到 $\ell_1$ 的稀疏向量限制等距嵌入的确定性构造，在 $m \asymp s^4$ 范围内提出实数情形的准确定性拉斯维加斯构造和复数情形的显式戈洛姆尺构造，并将低秩矩阵扩展留作开放问题。

On the uniform Roe algebra as a Banach algebra and embeddings of $\ell_p$ uniform Roe algebras

We work on $\ell_p$ uniform Roe algebras associated to metric spaces, and on their mutual embedding. We generalize results of I. Farah and the authors to mutual embeddings of uniform Roe algebras of operators on $\ell_p$ spaces. Simultaneously, we obtain rigidity results for the classic uniform Roe $\mathrm{C}^*$-algebras which depend only on their Banach algebra structure.

On uniform and coarse rigidity of $L^p([0,1])$

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 $L^2(E)$.

Best constants for Lipschitz embeddings of metric spaces into c_0

We answer a question of Aharoni by showing that every separable metric space can be Lipschitz 2-embedded into $c_0$ and this result is sharp; this improves earlier estimates of Aharoni, Assouad and Pelant. We use our methods to examine the best constant for Lipschitz embeddings of the classical $\ell_p-$spaces into $c_0$ and give other applications. We prove that if a Banach space embeds almost isometrically into $c_0$, then it embeds linearly almost isometrically into $c_0$. We also study Lipschitz embeddings into $c_0^+$.