Mathematical Sciences: Problems in the Theory of Operator Algebras

小行星9500308 自由独立性、基于自由乘积而不是张量积的非交换独立性以及相应的自由概率理论将被用来研究自由群的II型因子。这将特别涉及自由熵，自由调和分析和渐近随机矩阵模型。考虑的其他问题，将在超有限的背景下， 将是关于近似方法的动力学熵（作为一个非交换替代麦克米兰斯定理）和作用的Macaev理想之间的连接熵和扰动的运营商。 算子是矩阵的无限维推广。有一个自然的乘法和加法运算符，因此，一些运算符的集合可以被认为是数学结构称为算子代数。 这个项目涉及这些非交换算子代数和概念之间的接口从一个新兴领域的非交换概率论。我们将进一步研究算子代数与动力系统中出现的不变量之间的联系。 ***

9500308 Voiculescu Free independence , the noncommutative independence based on free products instead of tensor products and the corresponding free probability theory will be used to study the type II factors of free groups . This will especially involve free entropy, free harmonic analysis and asymptotic random matrix models. Other problems considered , will be in the hyperfinite context and will be about the approximation approach to dynamical entropy (as a non-commutative substitute for Mc Millans theorem ) and the role of the Macaev ideal in the connection between entropy and perturbations of operators . Operators are infinite dimensional generalizations of matrices. There is a natural multiplication and addition of operators and, consequently, some collections of operators can be considered as mathematical structures called operator algebras. This project involves the interface between these non-commutative operator algebras and concepts from an emerging field of non-commutative probability theory. Further connections between operator algebras and invariants that arise in dynamical systems will be investigated. ***