Proceedings of the Edinburgh Mathematical Society (Series 2)

Research Article

!COMMUTANT LIFTING, TENSOR ALGEBRAS, AND FUNCTIONAL CALCULUS

Gelu Popescua1

a1 Division of Mathematics and Statistics, The University of Texas at San Antonio, San Antonio, TX 78249, USA (gpopescu@math.utsa.edu)

Abstract

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

(Received October 20 1998)

Keywords

  • commutant lifting;
  • tensor algebras;
  • functional calculus;
  • interpolation