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Journal of Operator Theory

Volume 59, Issue 1, Winter 2008  pp. 29-51.

A Beurling theorem for noncommutative $L^p$

Authors David P. Blecher (1) and Louis E. Labuschagne (2)
Author institution: (1) Department of Mathematics, University of Houston, Houston, TX 77204-3008, USA
(2) Department of Mathematical Sciences, P.O. Box 392, 0003 UNISA, South Africa

Summary:  We extend Beurling's invariant subspace theorem, by characterizing subspaces $K$ of the noncommutative $L^p$ spaces which are invariant with respect to Arveson's maximal subdiagonal algebras, sometimes known as noncommutative $H^\infty$. We show that a certain subspace, and a certain quotient, of $K$ are $L^p({\mathcal D})$-modules in the recent sense of Junge and Sherman, and therefore have a nice decomposition into cyclic submodules. This is used, together with earlier results of Nakazi and Watatani, to give our Beurling theorem. We also give general inner-outer factorization formulae for elements in the noncommutative $L^p$.

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