A Beurling theorem for noncommutative L^p

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.

16 pages

References in corpus (1)

• Logmodularity and isometries of operator algebras

Cited by in corpus (12)

• Riesz and Szegö type factorizations for noncommutative Hardy spaces
• Von Neumann algebraic H^p theory
• A Beurling-Chen-Hadwin-Shen Theorem for Noncommutative Hardy Spaces Associated with Semifinite von Neumann Algebras with Unitarily Invariant Norms
• Applications of the Fuglede-Kadison determinant: Szegö's theorem and outers for noncommutative $H^p$
• Complex interpolation of noncommutative Hardy spaces associated semifinite von Neumann algebra
• Noncommutative $H^p$ spaces associated with type 1 subdiagonal algebras
• Subdiagonal algebras with the Beurling type invariant subspaces
• Noncommutative function theory and unique extensions
• Maximality and finiteness of type 1 subdiagonal algebras
• A Beurling-Blecher-Labuschagne theorem for Haagerup noncommutative $L^p$ spaces
• Outers for noncommutative H^p revisited
• Helson and subdiagonal operator algebras