A Beurling-Blecher-Labuschagne Theorem for Noncommutative Hardy Spaces Associated with Semifinite von Neumann Algebras

Abstract

In 2008, Blecher and Labuschagne extended Beurling's classical theorem to H ∞invariant subspaces of L p (M, τ ) for a finite von Neumann algebra M with a finite, faithful, normal tracial state τ when 1 ≤ p ≤ ∞. In this paper, using Arveson's non-commutative Hardy space H ∞ in relation to a von Neumann algebra M with a semifinite, faithful, normal tracial weight τ , we prove a Beurling-Blecher-Labuschagne theorem for H ∞ -invariant spaces of L p (M, τ ) when 0 < p ≤ ∞. The proof of the main result relies on proofs of density theorems for L p (M, τ ) and semifinite versions of several other known theorems from the finite case. Using the main result, we are able to completely characterize all H ∞ -invariant subspaces of L p (M⋊ α Z, τ ), where M ⋊ α Z is a crossed product of a semifinite von Neumann algebra M by the integer group Z and H ∞ is a non-selfadjoint crossed product of M by Z + . As an example, we characterize all H ∞

A Beurling-Chen-Hadwin-Shen Theorem for Noncommutative Hardy Spaces Associated with Semifinite von Neumann Algebras with Unitarily Invariant Norms

“…(iv) K = Y ⊕ row (⊕ row λ∈Λ H α u λ ). Similar to Sager's result in [35] for L p spaces, we prove a Beurling-Chen-Hadwin-Shen theorem for the crossed product of a von Neumann algebra M by a trace-preserving action β with a unitarily invariant, locally • 1 -dominating, mutually continuous with respect to the trace τ .…”

“…In [35], L. Sager extends the work of Blecher and Labuschagne in [6] from a finite von Neumann algebra to von Neumann algebras M with a semifinite, normal, faithful tracial weight τ . Suppose 0 < p ≤ ∞, and M is a von Neumann algebra with a semifinite, faithful, normal tracial weight τ .…”

“…However, many of the methods used by Chen, Hadwin and Shen no longer apply when M is a semifinite von Neumann algebra. We use a similar method to extend their theorem as in Sager's work on L p (M, τ ) spaces (see [35]). We therefore prove a series of density lemmas for the L α (M, τ ) spaces.…”

“…Sager proved in [35] that, given a von Neumann algebra M with a semifinite, faithful, normal tracial state τ , and a trace-preserving *-automorphism β of M, consider the crossed product of M by the action β, M⋊ β Z, and the extended semifinite, faithful, normal tracial state τ . Let H ∞ be the weak *-closed non-self-adjoint subalgebra M ⋊ β Z + of M ⋊ β Z.…”

A Beurling-Blecher-Labuschagne theorem for Haagerup noncommutative $L^p$ spaces

“…In [6], Blecher and Labuschagne extend the classical Beurling's theorem to describe closed A-invariant subspaces in noncommutative space L p (M) with 1 ≤ p ≤ ∞. Sager [20] extends the work of Blecher and Labuschagne from a finite von Neumann algebra to semifinite von Neumann algebras, proved a Beurling-Blecher-Labuschagne theorem for A-invariant spaces of L p (M) when 0 < p ≤ ∞. The Beurling theorem has been generalized to the setting of unitarily invariant norms on finite and semifinite von Neumann algebras (see [3], [9], [21]).…”