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Commutants mod normed ideals

  • Dan-Virgil VoiculescuEmail author
Chapter

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.

Notes

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. 1.Department of MathematicsUniversity of California at BerkeleyBerkeleyUSA

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