An extension of the Beurling-Chen-Hadwin-Shen theorem for noncommutative Hardy spaces associated with finite von Neumann algebras

Abstract

In 2015, Yanni Chen, Don Hadwin and Junhao Shen proved a noncommutative version of Beurling’s theorem for a continuous unitarily invariant norm α on a tracial von Neumann algebra (M, τ) such that α is one dominating with respect to τ . The role of H∞ is played by a maximal subdiagonal algebra A. In the talk, we first will show that if α is a continuous normalized unitarily invariant norm on (M, τ), then there exists a faithful normal tracial state ρ on M and a constant c > 0 such that α is a c times one norm-dominating norm on (M,ρ). Moreover, ρ(x) = τ(xg), where x ∈M , g is positive in L1(Z, τ), where Z is the center of M . Here c and ρ are not unique. However, if there is a c and ρ so that the Fuglede-Kadison determinant of g is positive, then Beurling-Chen-Hadwin-Shen theorem holds for L(α)(M, τ). The key ingredients in the proof of our result include a factorization theorem and a density theorem for for L(α)(M,ρ). Talk time: 2016-07-19 03:00 PM— 2016-07-19 03:20 PM Talk location: Cupples I Room 207