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Quantized Commutant Lifting and Noncommutative Poisson Transforms
by
Gelu Popescu
University of Texas at San Antonio
A noncommutative multi-dimensional analogue of Parrott's generalization of the Sz.-Nagy-Foias commutant lifting theorem is obtained. This yields interpolation theorems (e.g. Sarason, Nevanlinna-Pick, Caratheodory) for the WOT-closed algebra generated by the spatial tensor product of the noncommutative analytic Toeplitz algebra Fn and an arbitrary von Neumann algebra M. In particular, we obtain interpolation theorems for bounded analytic functions from the open unit ball of Cn into a von Neumann algebra.
Quantized noncommutative Poisson transforms are introduced and used to extend the noncommutative von Neumann inequality, and to provide a quantized Fn-functional calculus for contractive sequences of operators on Hilbert spaces. Commutative versions of all these results are also considered.
Date received: April 8, 1999
Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cacw-19.