A Beurling theorem for noncommutative L^p

Details A Beurling theorem for noncommutative L^p A Beurling theorem for noncommutative L^p

2005-10-17

arxiv.org/abs/math/0510358v1

Mathematics

Operator Algebras

16 pages

Scientific paper

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 $H^\infty$. It is significant 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. We also give general inner-outer factorization formulae for elements in the noncommutative $L^p$. These facts generalize the classical ones, and should be useful in the future development of noncommutative $H^p$ theory. In addition, these results characterize maximal subdiagonal algebras.

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