The positive Schur property on positive projective tensor products and spaces of regular multilinear operators

Complete boundedness of multiple operator integrals

In this paper, we characterize the multiple operator integrals mappings which are bounded on the Haagerup tensor product of spaces of compact operators. We show that such maps are automatically completely bounded and prove that this is equivalent to a certain factorization property of the symbol associated to the operator integral mapping. This generalizes a result by Juschenko-Todorov-Turowska on the boundedness of continuous multilinear Schur multipliers.

On projective Banach lattices of the form C(K) and FBL[E]

We show that if a Banach lattice is projective, then every bounded sequence that can be mapped by a homomorphism onto the basis of $c_0$ must contain an $\ell_1$-subsequence. As a consequence, if Banach lattices $\ell_p$ or $FBL[E]$ are projective, then $p=1$ or $E$ has the Schur property, respectively. On the other hand, we show that $C(K)$ is projective whenever $K$ is an absolute neighbourhood retract, answering a question by de Pagter and Wickstead.

$λ$-tensor product of operator spaces

We propose a theory of $λ$-tensor product of operator spaces which extends the theory of Blecher-Paulsen and Effros-Ruan for the operator space projective tensor product \cite{blecp}, \cite{effros}, \cite{eff} and that of Rajpal-Kumar-Itoh for the Schur tensor product \cite{vandee4} of operator spaces.

Equivalence after extension and Schur coupling for Fredholm operators on Banach spaces

Schur coupling (SC) and equivalence after extension (EAE) are important relations for bounded operators on Banach spaces. It has been known for 30 years that the former implies the latter, but only recently Ter Horst, Messerschmidt, Ran and Roelands disproved the converse by constructing a pair of Fredholm operators which are EAE, but not SC. Motivated by this result, we investigate when EAE and SC coincide for Fred\-holm operators. Fredholm operators which are EAE have the same Fredholm index. Surprisingly, we find that for each integer $k$ and every pair of Banach spaces $(\mathcal{X},\mathc…

巴拿赫格代数的Fremlin张量积的张量积

The tensor product of the Fremlin tensor product of Banach lattice algebras

该论文研究巴拿赫格代数的Fremlin投射张量积的Arens正则性，引入L-弱几乎周期泛函并刻画其性质，证明其Arens正则性的蕴含关系、反例及充分条件，还得出巴拿赫f-代数的Fremlin投射张量积的Arens正则性结论。

The Dunford-Pettis property on tensor products

We show that, in some cases, the projective and the injective tensor products of two Banach spaces do not have the Dunford-Pettis property (DPP). As a consequence, we obtain that $(c_0\hat{\otimes}_πc_0)^{**}$ fails the DPP. Since $(c_0\hat{\otimes}_πc_0)^{*}$ does enjoy it, this provides a new space with the DPP whose dual fails to have it. We also prove that, if $E$ and $F$ are ${\mathscr L}_1$-spaces, then $E\hat{\otimes}_εF$ has the DPP if and only if both $E$ and $F$ have the Schur property. Other results and examples are given.

关于 Schatten p-类算子的射影张量积的 Arens 正则性

On arens regularity of projective tensor product of Schatten p-class operators

本文利用 Ülger 的双正则性条件，通过具体例子证明了 Schatten p-类算子射影张量积的 Arens 非正则性，并展示了该条件在判定 Arens 正则性中的有效性与便利性。

(E,F)-multipliers and applications

For two given symmetric sequence spaces $E$ and $F$ we study the $(E,F)$-multiplier space, that is the space all of matrices $M$ for which the Schur product $M\ast A$ maps $E$ into $F$ boundedly whenever $A$ does. We obtain several results asserting continuous embedding of $(E,F)$-multiplier space into the classical $(p,q)$-multiplier space (that is when $E=l_p$, $F=l_q$). Furthermore, we present many examples of symmetric sequence spaces $E$ and $F$ whose projective and injective tensor products are not isomorphic to any subspace of a Banach space with an unconditional basis, extending classi…

格罗滕迪克功能积分的弗比尼定理

A Fubini Theorem for Grothendieck Functional Integrals

本文提出格罗滕迪克功能积分的概念，证明其具有希尔伯特空间表示定理，并通过抽象弗比尼定理证明积分顺序可交换，同时扩展到多重张量积环境。

Geometry of $C^*$-algebras, the bidual of their projective tensor product, and completely bounded module maps

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…

On the strong subdifferentiability of the homogeneous polynomials and (symmetric) tensor products

In this paper, we study the (uniform) strong subdifferentiability of the norms of the Banach spaces $\mathcal{P}(^N X, Y^*)$, $X \hat{\otimes}_π\cdots \hat{\otimes}_πX$ and $\hat{\otimes}_{π_s,N} X$. Among other results, we characterize when the norms of the spaces $\mathcal{P}(^N \ell_p, \ell_{q}), \mathcal{P}(^N l_{M_1}, l_{M_2})$, and $\mathcal{P}(^N d(w,p), l_{M_2})$ are strongly subdifferentiable. Analogous results for multilinear mappings are also obtained. Since strong subdifferentiability of a dual space implies reflexivity, we improve some known results on the reflexivity of spaces of…