Factorization of completely bounded bilinear operators and injectivity

与双线性形式和 Schur 乘子相关的唯一矩阵分解

Unique Matrix Factorizations associated to Bilinear Forms and Schur Multipliers

本文利用算子空间理论研究复 m×n 矩阵的 Grothendieck 型范数最优分解，证明四类分解中三类唯一、一类在特定条件下唯一。

Bilinear forms, Schur multipliers, complete boundedness and duality

Grothendieck's inequalities for operators and bilinear forms imply some factorization results for complex m x n matrices. Based on the theory of operator spaces and completely bounded mappings we present norm optimal versions of these results and two norm optimal factorization results related to the Schur product. We show that the spaces of respectively bilinear forms and Schur multipliers are conjugate duals to each other with respect to their completely bounded norms.

Bilinear operator multipliers into the trace class

Given Hilbert spaces $H_1,H_2,H_3$, we consider bilinear maps defined on the cartesian product $S^2(H_2,H_3)\times S^2(H_1,H_2)$ of spaces of Hilbert-Schmidt operators and valued in either the space $B(H_1,H_3)$ of bounded operators, or in the space $S^1(H_1,H_3)$ of trace class operators. We introduce modular properties of such maps with respect to the commutants of von Neumann algebras $M_i\subset B(H_i)$, $i=1,2,3$, as well as an appropriate notion of complete boundedness for such maps. We characterize completely bounded module maps $u\colon S^2(H_2,H_3)\times S^2(H_1,H_2)\to B(H_1,H_3)$ by…

Norm optimal factorizations of scalar and block matrices

For an $m \times n$ complex matrix $X$ of rank $r$ with Schur multiplier $S_X$ we show that there exist an $ r \times m $ complex matrix $L$ and an $ r\times n $ complex matrix $R$ such that $X = L^*R$ and $\|S_X\|\, =\, \|\mathrm{diag} (L^*L) \|^{\frac{1}{2}} \| \mathrm{diag} (R^*R) \| ^{\frac{1}{2}},$ and the norm condition is optimal. Let the completely bounded norm of the bilinear form $B_X$ induced by $X$ on $(\mathbb{C}^m, \|.\|_\infty) \times (\mathbb{C}^n, \|.\|_\infty)$ be denoted $\|B_X\|_{cb},$ then $X$ has a factorization $ X = Δ(η)^* C Δ(ξ)$ with $η$ in $\mathbb{C}^m,$ $ξ$ in $\ma…

C*-代数上多重线性映射的Jordan范数与Grothendieck不等式

Jordan norms for multilinear maps on C*-algebras and Grothendieck's inequalities

该研究提出C*-代数上多重线性映射完全有界范数的推广形式Jordan范数，通过映射分解定义，并证明C*-代数对上的有界双线性型均为Jordan有界，且其Jordan范数不超过原范数的两倍。

Grothendieck's Theorem for Operator Spaces

We prove several versions of Grothendieck's Theorem for completely bounded linear maps $T\colon E \to F^*$, when E and F are operator spaces. We prove that if E,F are $C^*$-algebras, of which at least one is exact, then every completely bounded $T\colon E \to F^*$ can be factorized through the direct sum of the row and column Hilbert operator spaces. Equivalently T can be decomposed as $T=T_r+T_c$ where $T_r$ (resp. $T_c$) factors completely boundedly through a row (resp. column) Hilbert operator space. This settles positively (at least partially) some earlier conjectures of Effros-Ruan and Bl…