Noncommutative Probability Aspects of Operator Algebra Theory

文章摘要：丹·沃库列斯库将研究与自由概率有关的算子代数。他将强调自由熵在算子代数中的应用，以及自由熵方法的发展和统一所带来的算子代数和随机矩阵理论中的问题。在物理学的某些模型中，也有与主场的联系。丹·沃库列斯库还打算研究差商和希尔伯特变换的非交换推广，与自由概率中出现的一类余代数的对偶有关。自由概率理论是与基本概率理论高度非对易平行的理论，它模拟大型随机矩阵的渐近行为，随着其大小的增加。一个主要的焦点将是自由熵，在此背景下扮演香农信息论熵角色的量。在数学中的应用包括冯·诺依曼代数和随机矩阵，以及物理学中的一些模型。

AbstractVoiculescuDan Voiculescu will study operator algebras in connection with free probability. He will emphasize applications of free entropy to operator algebras as well as the problems in operator algebras and random matrix theory posed by the development and unification of approaches to free entropy. There are also connections to masterfields in certain models in physics. Dan Voiculescu also intends to study noncommutative generalizations of the difference - quotient and of the Hilbert transform, in connection with duality for a certain class of coalgebras arising in free probability. Free probability theory is a highly noncommutative parallel to basic probability theory , modelling the asymptotic behavior of large random matrices as their size increases. One main focus will be on free entropy, the quantity which plays in this context the role of Shannon's information-theoretic entropy. Applications are in mathematics to von Neumann algebras and to random matrices, as well as to some models in physics .