Grants and Contributions:
Grant or Award spanning more than one fiscal year. (2017-2018 to 2022-2023)
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.