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Commutants mod normed ideals

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Advances in Noncommutative Geometry

Abstract

To Alain Connes’ non-commutative geometry the normed ideals of compact operators are purveyors of infinitesimals. A numerical invariant, the modulus of quasicentral approximation, plays a key role in perturbations from these ideals. New structure is provided by commutants mod normed ideals of n-tuples of operators and by their Calkin algebras. I review the modulus of quasicentral approximation, the relation to invariance of absolutely continuous spectra, to dynamical entropy and the hybrid generalization. I then discuss commutants mod normed ideals, their Banach space duality properties, K-theory aspects, the case of the Macaev ideal. Sample open problems are included.

Dedicated to Alain Connes on the occasion of his 70th birthday.

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Acknowledgements

This research was supported in part by NSF Grant DMS-1665534. Part of this paper was written while visiting IPAM in Spring 2018 for the Quantitative Linear Algebra program, which was supported by a NSF grant.

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Authors and Affiliations

  1. Department of Mathematics, University of California at Berkeley, Berkeley, CA, USA

    Dan-Virgil Voiculescu

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  1. Dan-Virgil Voiculescu
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Correspondence to Dan-Virgil Voiculescu .

Editor information

Editors and Affiliations

  1. Department of Physics Institute for Mathematics, Astrophysics and Particle Physics, Nijmegen, The Netherlands

    Ali Chamseddine

  2. Department of Mathematics, Johns Hopkins University, Baltimore, MD, USA

    Caterina Consani

  3. Department of Mathematics, Pennsylvania State University, University Park, PA, USA

    Nigel Higson

  4. Department of Mathematics, The University of Western Ontario, London, ON, Canada

    Masoud Khalkhali

  5. Department of Mathematics, Ohio State University, Columbus, OH, USA

    Henri Moscovici

  6. Department of Mathematics, Texas A&M University, College Station, TX, USA

    Guoliang Yu

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Voiculescu, DV. (2019). Commutants mod normed ideals. In: Chamseddine, A., Consani, C., Higson, N., Khalkhali, M., Moscovici, H., Yu, G. (eds) Advances in Noncommutative Geometry. Springer, Cham. https://doi.org/10.1007/978-3-030-29597-4_10

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