Hostname: page-component-8448b6f56d-42gr6 Total loading time: 0 Render date: 2024-04-24T19:39:14.226Z Has data issue: false hasContentIssue false
  • English
  • Français

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

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

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
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
© Canadian Mathematical Society 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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.

References

Albrecht, E., On joint spectra. Studia Math. 64(1979), 263–271. https://doi.org/10.4064/sm-64-3-263-271.CrossRefGoogle Scholar
Arens, R., The analytic-functional calculus in commutative topological algebras. Pacific J. Math. 11(1961), 405–429.CrossRefGoogle Scholar
Bourbaki, N., Éléments de mathématique. Fasc. XXXII. Théories spectrales. Actualités Scientifiques et Industrielles, 1332, Hermann, Paris, 1967.Google Scholar
Brown, L. G., Lidskii’s theorem in the type II case. In: Geometric methods in operator algebras (Kyoto, 1983). Pitman Res. Notes Math. Ser., 123, Longman Sci. Tech., Harlow, 1986, pp. 1–35.Google Scholar
Dixmier, J., Von Neumann algebras. In: North-Holland Mathematical Library, 27. North-Holland Publishing, Amsterdam, 1981.Google Scholar
Dykema, K., Noles, J., Sukochev, F., and Zanin, D., On reduction theory and Brown measure for closed unbounded operators. J. Funct. Anal. 371(2016), 3403–3422. https://doi.org/10.1016/j.jfa.2016.09.015CrossRefGoogle Scholar
Dykema, K., Sukochev, F., and Zanin, D., A decomposition theorem in II1-factors. J. Reine Angew. Math. 708(2015), 97–114. https://doi.org/10.1515/crelle-2013-0084.Google Scholar
Dykema, K., Sukochev, F., and Zanin, D., Holomorphic functional calculus on upper triangular forms in finite von Neumann algebras. Illinois J. Math. 59(2015), 819–824.CrossRefGoogle Scholar
Dykema, K., Sukochev, F., and Zanin, D., An upper triangular decomposition theorem for some unbounded operators affiliated to II1-factors. Israel J. Math. 222(2017), 645–709. https://doi.org/10.1007/s11856-017-1603-yCrossRefGoogle Scholar
Fack, T. and Kosaki, H., Generalized s-numbers of 𝜏-measurable operators. Pacific J. Math. 123(1986), 269–300.CrossRefGoogle Scholar
Folland, G. B., Real analysis. Second ed., John Wiley & Sons, New York, 1999.Google Scholar
Haagerup, U. and Schultz, H., Brown measures of unbounded operators affiliated with a finite von Neumann algebra. Math. Scand. 100(2007), 2, 209–263. https://doi.org/10.7146/math.scand.a-15023CrossRefGoogle Scholar
Haagerup, U. and Schultz, H., Invariant subspaces for operators in a general II1-factor. Publ. Math. Inst. Hautes Études Sci. 109(2009), 19–111. https://doi.org/10.1007/s10240-009-0018-7CrossRefGoogle Scholar
Harte, R. E., Spectral mapping theorems. Proc. Roy. Irish Acad. Sect. A 72(1972), 89–107.Google Scholar
Müller, V., On the Taylor functional calculus. Studia Math. 150(2002), 79–97. https://doi.org/10.4064/sm150-1-6CrossRefGoogle Scholar
Putinar, M., Uniqueness of Taylor’s functional calculus. Proc. Am. Math. Soc. 89(1983), 647–650. https://doi.org/10.2307/2044599Google Scholar
Raeburn, I. and Sinclair, A. M., The C ∗-algebra generated by two projections. Math. Scand. 65(1989), 278–290. https://doi.org/10.7146/math.scand.a-12283.CrossRefGoogle Scholar
Schultz, H., Brown measures of sets of commuting operators in a type II1 factor. J. Funct. Anal. 236(2006), 457–489. https://doi.org/10.1016/j.jfa.2006.03.003.CrossRefGoogle Scholar
Taylor, J. L., The analytic-functional calculus for several commuting operators. Acta Math. 125(1970), 1–38. https://doi.org/10.1007/BF02392329.CrossRefGoogle Scholar
Taylor, J. L., A joint spectrum for several commuting operators. J. Funct. Anal. 6(1970), 172–191. https://doi.org/10.1016/0022-1236(70)90055-8CrossRefGoogle Scholar
Vasilescu, F.-H., A characterization of the joint spectrum in Hilbert spaces. Rev. Roumaine Math. Pures Appl. 22(1977), 1003–1009.Google Scholar
Vasilescu, F.-H., A Martinelli type formula for the analytic functional calculus. Rev. Roumaine Math. Pures Appl. 23(1978), 1587–1605.Google Scholar
Vasilescu, F.-H., Analytic functional calculus and Martinelli’s formula. In: Romanian-Finnish Seminar on complex analysis. Lecture Notes in Math., 743, Springer, Berlin, 1979, pp. 693–701.CrossRefGoogle Scholar
Waelbrock, L., Le calcule symbolique dans les alg ‘ebres commutatives. J. Math. Pures Appl. 33(1954), 147–186.Google Scholar