Studies in Free Probability and Operator Algebras

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The main themes of the project are the analytic side of free probability and almost-normal operators modulo the Hilbert-Schmidt class. This will involve on the one hand further study of the "free Riemann sphere," a highly noncommutative analogue of the Riemann sphere, and of corresponding highly noncommutative functions. On the other hand, the almost-normal operator questions are related to the study of certain Banach algebras of operators on Hilbert space and their K-theory. The operator theory part of the project also has connections to noncommutative probability, since the operator algebras considered appear naturally in noncommutative potential theory and the analogue of Fisher information in free probability involves, in the simplest cases, almost-normal operators. Free probability is a probability framework adapted to deal with random variables that exhibit the highest degree of noncommutativity. The theory has important connections with operator algebra theory, with random matrix theory, and with certain topics in combinatorics. Studies in free probability often benefit from these connections and, in return, may be of interest in these other fields. In particular, users of random matrices in certain models of multiuser wireless communications and in certain physics models have found the free probability approach to random matrices useful.