A non-commutative Beurling's theorem with respect to unitarily invariant norms

2015/05/15 by Chen, Yanni, Hadwin, Don, Shen, Junhao
#FOS: Mathematics #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.1505.03952

Abstract

In 1967, Arveson invented a non-commutative generalization of classical H^∞, known as finite maximal subdiagonal subalgebras, for a finite von Neumann algebra \mathcal M with a faithful normal tracial state τ. In 2008, Blecher and Labuschagne proved a version of Beurling's theorem on H^∞-right invariant subspaces in a non-commutative L^p(\mathcal M,τ) space for 1≤ p≤ ∞. In the present paper, we define and study a class of norms N_c(\mathcal M, τ) on M, called normalized, unitarily invariant, \Vert ⋅ \Vert₁-dominating, continuous norms, which properly contains the class \ \Vert ⋅ \Vert_p:1≤ p< ∞ \. For α∈ N_c(\mathcal M, τ), we define a non-commutative L^α(M,τ) space and a non-commutative H^α space. Then we obtain a version of the Blecher-Labuschagne-Beurling invariant subspace theorem on H^∞-right invariant subspaces in a non-commutative L^α(M,τ) space. Key ingredients in the proof of our main result include a characterization theorem of H^α and a density theorem for L^α(\mathcal M,τ).

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