Mathematical Sciences: Models and Linearization for Freeness in Operator Algebras

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9622798 Alexandru Nica The problems addressed by the project are in free probability theory - a line of research in operator algebras which parallels aspects of classical probability, in a context where tensor products are replaced by free products, and independent random variables are replaced by free random variables. The proposed research reflects the interplay between two main directions in the development of the theory, namely that: while (on one hand) studying the parallelism between freeness and independence is of intrinsic interest, we have (on the other hand) that free probability is a powerful tool for attacking hard problems on free products of operator algebras. The main object of study, called the R-transform, is a free non-commutative analog for the logarithm of the Fourier transform of a (multivariable) probability distribution. In recent work, the multivariable R-transfopm has emerged as an elegant and at the same time far-reaching instrument in the study of the freeness phenomenon; we present problems that naturally come in the continuation of this promising direction of research. Problems related to a remarkable interpolation, depending on a parameter q in 0,1 , between the R-transform and its commutative counterpart; and problems related to an asymptotic model for freeness, based on random permutations, are also discussed. The notion of random variable, and the concept of independence for random variables are basic elements in the language of probability theory. Random variables can be added and multiplied together, and built in their definition is the fact that the product of two random variables f and g does not depend on the order in which they are considered (always fg = gf). In the field of free probability, the latter assumption is dropped, and the concept of independence is replaced by the one of freeness - a condition originally inspired from the study of certain algebras associated to free groups. While free pr obability was started in the 80's as part of an effort to approach some long-standing problems in operator algebras, it is at present clear that we have to do with a fertile mathematical theory, connected to other branches of mathematics and science; recent contributions in the field were made, e.g., by quantum physicists and algebraic combinatorists. The present project is concerned with topics from free harmonic analysis - the `free analog' for the Fourier analysis methods classically used in the study of random variables.