Proceedings of the Edinburgh Mathematical Society (Series 2)

Research Article


Gelu Popescua1

a1 Division of Mathematics and Statistics, The University of Texas at San Antonio, San Antonio, TX 78249, USA (


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)


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