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!COMMUTANT LIFTING, TENSOR ALGEBRAS, AND FUNCTIONAL CALCULUS

Published online by Cambridge University Press:  20 January 2009

Gelu Popescu
Affiliation:
Division of Mathematics and Statistics, The University of Texas at San Antonio, San Antonio, TX 78249, USA (gpopescu@math.utsa.edu)
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Abstract

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A non-commutative multivariable analogue of Parrott’s generalization of the Sz.-Nagy–Foia\c{s} commutant lifting theorem is obtained. This yields Tomita-type commutant results and interpolation theorems (e.g. Sarason, Nevanlinna–Pick, Carathéodory) for $F_n^\infty\,\bar{\otimes}\,\M$, the weakly-closed algebra generated by the spatial tensor product of the non-commutative analytic Toeplitz algebra $F_n^\infty$ and an arbitrary von Neumann algebra $\M$. In particular, we obtain interpolation theorems for bounded analytic functions from the open unit ball of $\mathbb{C}^n$ into a von Neumann algebra.

A variant of the non-commutative Poisson transform is used to extend the von Neumann inequality to tensor algebras, and to provide a generalization of the functional calculus for contractive sequences of operators on Hilbert spaces. Commutative versions of these results are also considered.

AMS 2000 Mathematics subject classification: Primary 47L25; 47A57; 47A60. Secondary 30E05

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 2001