University of California, Berkeley
Election Year: 2006
Primary Section: 11, Mathematics
Membership Type: Member
Studying operator algebra problems, I was led to introduce free probability theory. This is a highly noncommutative probability theory where the random variables , like the quantum observables, are operators on Hilbert space, but where independence is defined in a new way so that the freely independent random variables are, in general, quite far from commuting . The theory runs parallel to a surprisingly large part of basic probability theory. For instance, there is a free central limit theorem, where the Gauss law is replaced by the semicircle law. The semicircle law had appeared earlier in Wigner's work on random matrices and it provided the first clue to a connection with random matrix theory. I later showed that the large N limit of random matrices provides an aymptotic model of free probability theory. This led to applications of free probability to random matrices and in a converse direction to many applications of random matrices to the von Neumann algebra problems which had been my starting point. Also, several applications to operator algebras appeared when I found that I could develop information theoretic concepts in free probability, similar to those around Shannon's differential entropy. From time to time, I return from the probabilistic exploration of the highest degree of noncommutativity to pure operator theory methods.