Analysis Seminar: Von Neumann inequalities reduce to finite dimensions

Abstract: Let p be a polynomial in n variables.  We are interested in estimating the norm of p applied to n commuting operators on Hilbert space of norm 1 or less.  We say that the von Neumann inequalities hold when this norm is no greater than the sup norm of p over the unit polydisk in n space. This condition is valid for all polynomials in 1 or 2 variables plus many polynomials in 3 or more variables.  In general, it is unknown for which p it holds.  We shall show that estimating such norms can always be reduced to finite dimensional Hilbert space. The proof combines elementary ideas from C* algebras and harmonic analysis.

Host: John McCarthy