Free noncommutative probability theory, random matrices and von Neumann algebras of free groups

Dan Voiculescu (University of California, Berkeley, USA)

In noncommutative probability theory, replacing tensor products by free products in the definition of independence yields a highly noncommutative theory ([29]). Large random matrices provide an asymptotic model for this free probability theory ([26]). Other models are given by convolution operators on free groups and by creation and annihilation operators on the full Fock space of Botlzmann statistics ([22]). Free independence also occurs in random permutations ([11], [12],[3]) and in asymptotics of group representations ([4]).

Operations on free random variables define nonlinar convolution operations on their distributions ([22--24]). This free harmonic analysis has classified the freely infinitely divisible and stable laws ([23], [10], [1], [2]). The semicircle law becomes the analogue of the Gaussian distribution and the complex Burger equation plays the role of the one-dimensional heat equation ([22],[23]). From the point of view of the combinatorics of cummulants, passing from the usual to the free context corresponds to replacing the lattice of all partitions by the lattice of noncrossing partitions ([20], see also [13]).

Free probability theory and especially its random matrix model have led to important progress in the study of the von Neumann algebras of free groups ([25],[6--8],[15--17]). This includes results on self-similarity (von Neumann fundamental group [25,[15]) and examples of subfactors of free group factors for all admissible indices $<4$ ([16]). New isomorphisms among free product von Neumann algebras have been obtained ([25],[6],[8]) and von Neumann algebras of free groups on fractional numbers of generators have been introduced ([7],[17]).

Further, free random variables played a role in the solution of the problem of quasitraces for exact $C^*$-algebras ([9]).

Usual entropy is inadequate for free random variables ([21]). There is another quantity, the free entropy of a system of noncommutative random variables, which is adapted to the free context ([27]).

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