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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K. Dykema: This work was supported by a grant from the Simons Foundation/SFARI (524187, K.D.). J. Noles: Portions of this work are included in the thesis of J. Noles for partial fulfillment of the requirements to obtain a Ph.D. degree at Texas A&M University. D. Zanin: Research supported by ARC.
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Dykema, K., Noles, J. & Zanin, D. Decomposability and Norm Convergence Properties in Finite von Neumann Algebras. Integr. Equ. Oper. Theory 90, 54 (2018). https://doi.org/10.1007/s00020-018-2480-4
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DOI: https://doi.org/10.1007/s00020-018-2480-4