Eli Shamovich, Pure Mathematics, University of Waterloo
"Free Function Theory on the Noncommutative Ball"
Free or noncommutative functions (nc-functions for short) were introduced by Taylor in 1972, in order to facilitate functional calculus for tuples of noncommuting operators. Since then the theory of nc-functions took on a life of its own with applications to free probability, control theory, real algebraic geometry and more. In fact, bounded nc-functions arose in operator algebras, implicitly at first, in the works of Popescu and Davidson and Pitts. The theory of noncommutative reproducing kernels was recently developed by Ball, Marx and Vinnikov and it “interpolates” between the classical theory of reproducing kernels and the theory of completely positive maps. In this talk I will introduce nc-functions and nc reproducing kernel Hilbert spaces of nc-functions. Then I will proceed to discuss applications to operator algebras and in particular, the isomorphism problem for multiplier algebras on sub varieties of the noncommutative ball. This talk is based on joint work with Guy Salomon and Orr Shalit.