Michael Jury, University of Florida
"A Tour of Noncommutative Function Theory"
A holomorphic function of one or several complex variables is in a natural way a generalization of a polynomial in these variables. By the phrase “noncommutative function” we refer to a class of functions which in an analogous way generalize the polynomials in noncommuting variables X1 X2, … ,Xd (say, p(X1, X2) = X1X2X1+X1X2-X2X1). Such a polynomial can be evaluated at d-tuples of n x n matrices, for any size n. (One then thinks of the domain of pas graded set, with “levels” corresponding to each size n.) I will introduce the notions of a “noncommutative set” and “noncommutative functions” on these sets, describe some of their basic properties, and give examples. We will then tour some of the recent results in this area (which has seen considerable growth in the past few years), with a particular emphasis on the interplay between the commutative and noncommutative theories. Generally, noncommutative functions exhibit a great deal more rigidity than their commutative counterparts, so that one may try to prove results about ordinary holomorphic functions by first passing to a noncommutative “lift,” applying results from noncommutative function theory, and then pushing the result back down.
MC 5501