Gilles Pisier
Let F(X) denote the norm closure of the space of all finite rank operators on a Banach space X. We show that there are Banach spaces X for which the product map $a\otimes b\to ab$ does not define a surjective map from the projective tensor product $F(X) \hat\otimes F(X)$ onto F(X).
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Let $E,F$ be exact operators (For example subspaces of the $C^*$-algebra $K(H)$ of all the compact operators on an infinite dimensional Hilbert space $H$). We study a class of bounded linear maps $u\colon E\to F^*$ which we call tracially bounded. In particular, we prove that every completely bounded (in short $c.b.$) map $u\colon E\to F^*$ factors boundedly through a Hilbert space. This is used to show that the set $OS_n$ of all $n$-dimensional operator spaces equipped with the $c.b.$ version of the Banach Mazur distance is not separable if $n>2$. As an application we show that there is more …
A characterisation of the Daugavet property in spaces of vector-valued Lipschitz functions
本文研究了向量值 Lipschitz 函数空间的结构，证明了长度、非一致离散或无界度量空间下相关空间的 Daugavet 性质与八面体性质，并刻画了使算子空间恒为八面体的 Banach 空间，解决了多个公开问题。
The set of elementary tensors is weakly closed in projective tensor products
本文证明 Banach 空间投影张量积中的初等张量集合为弱闭集，并据此解决文献问题，确认弱零序列对应初等张量序列若弱收敛则必弱零。
It is shown that if $C_1$ and $C_2$ are maximal abelian self-adjoint subalgebras (masas) of C*-algebras $A_1$ and $A_2$, respectively, then the completion $C_1\otimes C_2$ of the algebraic tensor product $C_1\odot C_2$ of $C_1$ and $C_2$ in any C*-tensor product $A_1\otimes_β A_2$ is maximal abelian provided that $C_1$ has the extension property of Kadison and Singer and $C_2$ contains an approximate identity for $A_2$. An example is given to show that $C_1\otimes C_2$ can fail to be a masa in $A_1\otimes_β A_2$ with $A_1$ and $A_2$ unital if neither $C_1$ nor $C_2$ has the extension property.…
Weak operator Daugavet property and weakly open sets in tensor product spaces
本文研究了张量积空间中的直径二性质与Daugavet性质，基于弱算子Daugavet性质（WODP）推导出一系列新的充分条件，改进了现有文献结果并推进了相关公开问题的解决。
In this paper, we introduce a concept of norm-attainment in the projective symmetric tensor product $\widehat{\otimes}_{π,s,N} X$ of a Banach space $X$, which turns out to be naturally related to the classical norm-attainment of $N$-homogeneous polynomials on $X$. Due to this relation, we can prove that there exist symmetric tensors that do not attain their norms, which allows us to study the problem of when the set of norm-attaining elements in $\widehat{\otimes}_{π,s,N} X$ is dense. We show that the set of all norm-attaining symmetric tensors is dense in $\widehat{\otimes}_{π,s,N} X$ for a l…
We examine some structural properties of (injective and projective) tensor products of $\ell_p$-spaces (projections, complemented subspaces, reflexivity, isomorphisms, etc.). We combine these results with combinatorial arguments to address the question of primarity for these spaces and their duals. Our main results are: \medbreak \item{(1)} If $1<p<\infty$, then $B(\ell_p)\approx B(L_p)$ ($B(X)$ consists of the bounded linear operators on $X$). \medbreak \item{(2)} If ${1\over p_i}+{1\over p_j}\leq1$ for every $i\neq j$, or if all of the $p_i$'s are equal, then $\ell_{p_1}\hat{\otimes}\cdots \…
One of the most fundamental questions in quantum information theory is PPT-entanglement of quantum states, which is an NP-hard problem in general. In this paper, however, we prove that all PPT $(\overlineπ_A\otimes π_B)$-invariant quantum states are separable if and only if all extremal unital positive $(π_B,π_A)$-covariant maps are decomposable where $π_A,π_B$ are unitary representations of a compact group and $π_A$ is irreducible. Moreover, an extremal unital positive $(π_B,π_A)$-covariant map $\mathcal{L}$ is decomposable if and only if $\mathcal{L}$ is completely positive or completely cop…
A New Crossnorm That Preserves Unconditional Bases in Banach Spaces
针对Banach空间中保持无条件基的张量范数存在性这一公开问题，构造了一种新的合理交叉范数，其虽非一致但能使对应张量积完备化空间具有无条件Schauder基。
Stability results of octahedrality in tensor product spaces
该研究构造反例证明巴拿赫空间的八面体范数与强直径二性质不会通过射影/内射张量积自动从单因子继承，解决了领域内的公开问题。
Optimal integral representations in projective tensor products: countability and topology
该研究探讨射影张量积最优积分表示的可数性与拓扑问题，通过构造仿射嵌入证明存在仅具不可数最优积分表示的张量，并明确不同拓扑下范数可达张量类的等价性及相关空间性质。
Given two Banach spaces $X$ and $Y$, we analyze when the projective tensor product $X\widehat{\otimes}_πY$ has Corson's property (C) or is weakly Lindelöf determined (WLD), subspace of a weakly compactly generated (WCG) space or subspace of a Hilbert generated space. For instance, we show that: (i) $X\widehat{\otimes}_πY$ is WLD if and only if both $X$ and $Y$ are WLD and all operators from $X$ to $Y^*$ and from $Y$ to $X^*$ have separable range; (ii) $X\widehat{\otimes}_πY$ is subspace of a WCG space if the same holds for both $X$ and $Y$ under the assumption that every operator from $X$ to $…
Let $\mathcal{A}$ be a $C^*$-algebra, and consider the Banach algebra $\mathcal{A} \otimes_γ\mathcal{A}$, where $\otimes_γ$ denotes the projective Banach space tensor product; if $\mathcal{A}$ is commutative, this is the Varopoulos algebra $V_\mathcal{A}$. It has been an open problem for more than 35 years to determine precisely when $\mathcal{A} \otimes_γ\mathcal{A}$ is Arens regular. Even the situation for commutative $\mathcal{A}$, in particular the case $\mathcal{A} = \ell_\infty$, has remained unsolved. We solve this classical question for arbitrary $C^*$-algebras by using von Neumann alg…
Let H^2_m be the Drury-Arveson (DA) module which is the reproducing kernel Hilbert space with the kernel function (z, w) \in B^m \times B^m \raro (1 - <z,w>)^{-1}. We investigate for which multipliers θ: \mathbb{B}^m \raro \cll(\cle, \cle_*) the quotient module \clh_θ is similar to H^2_m \otimes \clf for some Hilbert space \clf, where M_θ is the corresponding multiplication operator in \cll(H^2_m \otimes \cle, H^2_m \otimes \cle_*) for Hilbert spaces \cle and \cle_* and \clh_θ is the quotient module (H^2_m \otimes \cle_*)/ {clos} [M_θ(H^2_m \otimes \cle)]. We show that a necessary condition is…
The tensorization problem for Sobolev spaces asks for a characterization of how the Sobolev space on a product metric measure space $X\times Y$ can be determined from its factors. We show that two natural descriptions of the Sobolev space from the literature coincide, $W^{1,2}(X\times Y)=J^{1,2}(X,Y)$, thus settling the tensorization problem for Sobolev spaces in the case $p=2$, when $X$ and $Y$ are infinitesimally quasi-Hilbertian, i.e. the Sobolev space $W^{1,2}$ admits an equivalent renorming by a Dirichlet form. This class includes in particular metric measure spaces $X,Y$ of finite Hausdo…
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