Operator-Valued Free Probability Theory

ShlyakhtenkoShlyakhtenko将研究与沃库列斯库的自由概率理论有关的一类算子代数。这项研究的主要重点放在研究算子值自由随机变量及其生成的算子代数。对evon Neumann代数的分析与自由Araki-Wood因子的分类目标密切相关，后者是ITPFI III型因子的自由概率类似物。这种分析对于理解合并自由积代数的子因子和自同构也是很重要的。本研究的目标之一是发展基于自由熵的技术来处理算子值随机变量。自由概率理论是一种与基本概率理论高度非对易的平行理论。矩阵值随机变量(如随机矩阵)自然符合非对易概率框架；大型随机矩阵的渐近行为由矩阵值自由随机变量很好地模拟。在数学上的应用包括算子代数理论、子因子理论、遍历理论以及与某些物理模型有关的随机矩阵理论。

AbstractShlyakhtenkoShlyakhtenko will study classes of operator algebras arising in connectionwith Voiculescu's free probability theory. The main emphasis of thisresearch is placed on the study of operator-valued free randomvariables and operator algebras that they generate. Analysis of thesevon Neumann algebras is intimately tied with the goal ofclassification of free Araki-Woods factors, which are free probabilityanalogs of ITPFI type III factors. Such analysis is also importantfor understanding of subfactors and automorphisms of amalgamated freeproduct algebras. One of the goals of the present research is todevelop free entropy-based techniques for dealing with operator-valuedrandom variables.Free probability theory is a highly non-commutative parallel to basicprobability theory. Matrix-valued random variables (such as randommatrices) naturally fit in the non-commutative probability framework;the asymptotic behavior of large random matrices is well-modeled bymatrix-valued free random variables. Applications are in mathematicsto the theory of operator algebras, subfactors, ergodic theory, aswell as the theory of random matrices, which have connections withcertain physical models.