Natural Isomorphism from a Linear Map's Image to Complement of Nullspace

Some equivalence relations which are Borel reducible to isomorphism between separable Banach spaces

We improve the known results about the complexity of the relation of isomorphism between separable Banach spaces up to Borel reducibility, and we achieve this using the classical spaces $c_0$, $\ell_p$ and $L_p$, $1 \leq p <2$. More precisely, we show that the relation $E_{K_σ}$ is Borel reducible to isomorphism and complemented biembeddability between subspaces of $c_0$ or $\ell_p, 1 \leq p <2$. We show that the relation $E_{K_σ} \otimes =^+$ is Borel reducible to isomorphism, complemented biembeddability, and Lipschitz equivalence between subspaces of $L_p, 1 \leq p <2$.

Stabilizing isomorphisms from $\ell_p(\ell_2)$ into $L_p[0,1]$

Let $1<p\not=2<\infty$, $ε>0$ and let $T:\ell_p(\ell_2)\overset{into}{\rightarrow}L_p[0,1]$ be an isomorphism. Then there is a subspace $Y\subset \ell_p(\ell_2)$ $(1+ε)$-isomorphic to $\ell_p(\ell_2)$ such that: $T_{|Y}$ is an $(1+ε)$-isomorphism and $T(Y)$ is $K_p$-complemented in $L_p[0,1]$, with $K_p$ depending only on $p$. Moreover, $K_p\le (1+ε)γ_p$ if $p>2$ and $K_p\le (1+ε)γ_{p/(p-1)}$ if $1<p<2$, where $γ_r$ is the $L_r$ norm of a standard Gaussian variable.

The complexity of classifying separable Banach spaces up to isomorphism

It is proved that the relation of isomorphism between separable Banach spaces is a complete analytic equivalence relation, i.e., that any analytic equivalence relation Borel reduces to it. Thus, separable Banach spaces up to isomorphism provide complete invariants for a great number of mathematical structures up to their corresponding notion of isomorphism. The same is shown to hold for (1) complete separable metric spaces up to uniform homeomorphism, (2) separable Banach spaces up to Lipschitz isomorphism, and (3) up to (complemented) biembeddability, (4) Polish groups up to topological isomo…

0<p≤1时向量值$H_p$空间之间的同构与无条件结构的唯一性

Isomorphisms between vector-valued $H_p$-spaces for $0<p\le 1$ and uniqueness of unconditional structure

本文通过识别Banach值Hardy空间并发展新格估计技术，证明$0<p<1$时$H_p(\mathbb{D},\ell_2)$具有唯一原子格结构。

III型与半有限非交换$L_p$空间的Banach非同构性

Banach non-isomorphism of type III and semifinite noncommutative $L_p$-spaces

针对具有可分预对偶的非零III型von Neumann代数M与半有限von Neumann代数N，证明M_*不同构于N_*的子空间，1<p<∞且p≠2时L_p(M)不同构于L_p(N)的补子空间，其同构类可区分两类代数。