Decomposibility and norm convergence properties in finite von Neumann algebras

2017/09/18 by Ken Dykema, Dykema, Ken, Joseph Noles +3
Mathematics · #Advanced Operator Algebra Research #Random Matrices and Applications #Algebraic structures and combinatorial models

paper · pdf · doi:10.48550/arxiv.1709.06034

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 |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.

Related

• Upper triangular Toeplitz matrices and real parts of quasinilpotent operators
• A decomposition theorem in II₁-factors
• Holomorphic functional calculus on upper triangular forms in finite von Neumann algebras
• An upper triangular decomposition theorem for some unbounded operators affiliated to II₁-factors
• Upper Triangular Forms and Spectral Orderings in a II₁-factor