A Study of Bi-Free Probability

算子代数是研究向量空间之间的连续线性映射（称为算子）的数学领域。 算子代数的一个重要子领域是Voiculescu在20世纪80年代引入的非交换概率理论，称为自由概率。 通过研究自由独立性，这是一个不需要交换的随机变量的独立性的概念，在算子代数中取得了一些进展。 此外，自由概率与其他领域有许多应用和关系，如组合数学、大群表示论、数学物理、量子信息论、随机矩阵理论和无线通信。最近引入了自由概率的一种推广，称为双自由概率。 双自由独立性的概念将自由独立性的概念扩展到非交换随机变量对。 这种对双变量系统的扩展允许更广泛的行为被观察和建模。 例如，研究者证明了单变量系统的所有可能的独立性概念都可以通过双自由独立性来研究。 因此，双自由概率可以看作是一种普适的非交换概率论。 因此，这种更大的普遍性允许双自由概率研究自由概率未触及的问题。本提案的目的是继续研究者对双自由概率的研究。 这个提议的主要目标是开发一个熵的双自由模拟，它将量化一个算子集合与双自由独立的接近程度。 自由熵的概念在算子代数中有许多重要的应用和意义，双自由熵的概念将推广这些结果。 除了主要目标外，研究者还将考虑几个相关主题。 这些主题的例子包括对随机变量对的分布特性的更深入的理解，由运算符对的集合生成的代数是否同构，在运算符代数中出现双自由独立族的例子，以及是否有双自由概率扩展到任意数量的变量系统。这一提议的主要意义在于，自由概率极大地影响了算子代数和其他数学领域，对双自由概率的研究也会产生同样的影响。 这个项目的主要好处是自由概率理论的扩展，从而允许改进的应用程序。 最终的结果将是更深入地了解双自由概率论，它对其他数学理论的影响，以及该理论可以提供的应用。

Operator Algebras is the area of mathematics where collections of continuous linear maps (known as operators) between vector spaces are studied. One important subarea of Operator Algebras is a non-commutative probability theory introduced by Voiculescu in the 1980s known as free probability. By studying free independence, which is a notion of independence for random variables that need not commute, several advancements in Operator Algebras were made. Additionally free probability has many applications and relations to other areas, such as combinatorics, representation theory of large groups, mathematical physics, quantum information theory, random matrix theory, and wireless communications.******Recently a generalization of free probability known as bi-free probability was introduced. The notion of bi-free independence extends the notion of free independence to pairs of non-commutative random variables. This extension to two-variable systems allows for a wider variety of behaviours to be observed and modelled. For example, the investigator demonstrated that all possible notions of independence for one-variable systems can be studied via bi-free independence. Thus bi-free probability may be viewed as a universal non-commutative probability theory. Consequently, this greater generality allows for bi-free probability to investigate problems untouched by free probability.******The purpose of this proposal is to continue the investigator's study of bi-free probability. The main goal of this proposal is to develop a bi-free analogue of entropy, which would quantify how close to being bi-freely independent a collection of operators are. The notion of free entropy has had many important applications and implications in Operator Algebras and a notion of bi-free entropy will extend these results. In addition to the main goal, there are several related topics the investigator will consider. Examples of such topics include a deeper understanding of the distributional properties of pairs of random variables, whether the algebras generated by collections of pairs of operators are isomorphic, where examples of bi-free independent families occur in Operator Algebras, and whether there is an extension of bi-free probability to systems of an arbitrary number of variables.******The main significance of this proposal is that free probability substantially influenced Operator Algebras and other areas of mathematics, and a study of bi-free probability will have the same impact. The main benefit of this project is an extension of free probability theory thereby allowing improved applications. The ultimate outcome will be a deeper understanding of bi-free probability theory, its implications to other mathematical theories, and the applications this theory can provide.