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arXiv · 1709.06034

Decomposibility and norm convergence properties in finite von Neumann algebras

Ken Dykema·Joseph Noles·Dmitriy Zanin
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Abstract

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 |T^n|^{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 II_1-factor.

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Ken Dykema, Joseph Noles, Dmitriy Zanin. 2017-09-18. Decomposibility and norm convergence properties in finite von Neumann algebras. https://arxiv.org/abs/1709.06034

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