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Simultaneous Upper Triangular Forms for Commuting Operators in a Finite von Neumann Algebra

Part of: General theory of linear operators Individual linear operators as elements of algebraic systems

Published online by Cambridge University Press:  15 May 2019

Ian Charlesworth ,
Ken Dykema ,
Fedor Sukochev  and
Dmitriy Zanin
Ian Charlesworth
Affiliation:
Department of Mathematics, UC–Berkeley, Berkeley, CA94720-3840, USA Email: ilc@math.berkeley.edu
Ken Dykema
Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX77843-3368, USA Email: kdykema@math.tamu.edu
Fedor Sukochev
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Kensington, NSW, Australia Email: f.sukochev@unsw.edu.aud.zanin@unsw.edu.au
Dmitriy Zanin
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Kensington, NSW, Australia Email: f.sukochev@unsw.edu.aud.zanin@unsw.edu.au
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Abstract

The joint Brown measure and joint Haagerup–Schultz projections for tuples of commuting operators in a von Neumann algebra equipped with a faithful tracial state are investigated, and several natural properties are proved for these. It is shown that the support of the joint Brown measure is contained in the Taylor joint spectrum of the tuple, and also in the ostensibly smaller left Harte spectrum. A simultaneous upper triangularization result for finite commuting tuples is proved, and the joint Brown measure and joint Haagerup–Schultz projections are shown to behave well under the Arens multivariate holomorphic functional calculus of such a commuting tuple.

Keywords

finite von Neumann algebrajoint spectral distribution measureinvariant projectionholomorphic functional calculus

MSC classification

Primary: 47C15: Operators in $C^*$- or von Neumann algebras
Secondary: 47A60: Functional calculus
Type
Article
Information
Canadian Journal of Mathematics , Volume 72 , Issue 5 , October 2020 , pp. 1188 - 1245
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Copyright
© Canadian Mathematical Society 2019

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Footnotes

Author I. C. was supported by a grant from the NSF (DMS-1803557). Author K. D. was supported by a grant from the Simons Foundation/SFARI (524187, K.D.) and by a grant from the NSF (DMS-1800335). Author F. S. was supported by the ARC.

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