Decomposibility and norm convergence properties in finite von Neumann algebras

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We study Schur-type upper triangular forms for elements, T, of von Neumann algebras equipped with faithful, normal, tracial states. These were introduced in a paper of Dykema, Sukochev and Zanin; they are based on Haagerup-Schultz projections. We investigate when the s.o.t.-quasinilpotent part of this decomposition of T is actually quasinilpotent. We prove implications involving decomposability and strong decomposability of T. We show this is related to norm convergence properties of the sequence |Tⁿ|¹/ⁿ which, by a result of Haagerup and Schultz, is known to converge in strong operator topology. We introduce a Borel decomposability, which is a property appropriate for elements of finite von Neumann algebras, and show that the circular operator is Borel decomposable. We also prove the existence of a thin-spectrum s.o.t.-quasinilpotent operator in the hyperfinite II₁-factor.