A Beurling theorem for noncommutative L^p

TL;DR

This paper generalizes Beurling's theorem to noncommutative L^p spaces, providing invariant subspace characterizations, inner-outer factorizations, and new insights into noncommutative H^p theory.

Contribution

It extends classical invariant subspace results to noncommutative L^p spaces, introducing decomposition and factorization techniques for noncommutative H^p spaces.

Findings

Characterization of invariant subspaces under noncommutative H^

Inner-outer factorization formulas for noncommutative L^p elements

Decomposition of subspaces into cyclic modules

Abstract

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.