C*-algebras with the weak expectation property and a multivariable analogue of Ando's theorem on the numerical radius
C*-algebras with the weak expectation property and a multivariable analogue of Ando's theorem on the numerical radius
复制标题
具有弱期望性质的C*-代数和安藤数值半径定理的多变量模拟
DOI：
--
复制
发表时间：
2011
期刊：
影响因子：
--
通讯作者：
V. Paulsen
中科院分区：
文献类型：
--
作者：
D. Farenick;A. Kavruk;V. Paulsen
T.Ando的一个经典定理将数值半径至多为1的算子刻画为允许某个正2x2算子矩阵补全的算子。在这篇文章中，我们考虑Ando定理的变体，其中算子(和矩阵补全)被限制在给定的C*-代数上。通过考虑n×n个矩阵的完备化，将Ando定理推广到多变量情形。证明了这些Ando定理的推广形式成立的C*-代数正是具有弱期望性质(WEP)的C*-代数。我们还证明了B(H)的C*-子代数A有WEP当且仅当当B(H)上的某个3×3(算子)矩阵完成问题可以在B(H)上的矩阵中求解时，它也可以在A上的矩阵中求解。这最后一个结果给出了WEP的一个刻划，它是空间的，但与C*-代数的特殊表示无关。这给出了内射von Neumann代数的一个新刻画。我们还给出了关于3x3矩阵补的Connes嵌入问题的一个新的等价形式。
A classic theorem of T. Ando characterises operators that have numerical radius at most one as operators that admit a certain positive 2x2 operator matrix completion. In this paper we consider variants of Ando's theorem, in which the operators (and matrix completions) are constrained to a given C*-algebra. By considering nxn matrix completions, an extension of Ando's theorem to a multivariable setting is made. We show that the C*-algebras in which these extended formulations of Ando's theorem hold true are precisely the C*-algebras with the weak expectation property (WEP). We also show that a C*-subalgebra A of B(H) has WEP if and only if whenever a certain 3x3 (operator) matrix completion problem can be solved in matrices over B(H), it can also be solved in matrices over A. This last result gives a characterisation of WEP that is spatial and yet is independent of the particular representation of the C*-algebra. This leads to a new characterisation of injective von Neumann algebras. We also give a new equivalent formulation of the Connes Embedding Problem as a problem concerning 3x3 matrix completions.
影响因子：
1.7
作者：
A. Kavruk;V. Paulsen;I. Todorov;M. Tomforde
通讯作者：
A. Kavruk;V. Paulsen;I. Todorov;M. Tomforde