Skip to main content
Log in

Quantum Calculus and Ideals in the Algebra of Compact Operators

  • Published:
Proceedings of the Steklov Institute of Mathematics Aims and scope Submit manuscript

Abstract

One of the goals of noncommutative geometry is to translate the basic notions of analysis into the language of Banach algebras. This translation is based on the quantization procedure. The arising operator calculus is called, following Connes, the quantum calculus. In this paper we give several assertions from this calculus concerning the interpretation of Schatten ideals of compact operators in a Hilbert space in terms of function theory. The main focus is on the case of Hilbert-Schmidt operators.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. L. V. Ahlfors, Conformal Invariants: Topics in Geometric Function Theory (McGraw-Hill, New York, 1973).

    MATH  Google Scholar 

  2. A. Connes, Noncommutative Geometry (Academic, San Diego, CA, 1994).

    MATH  Google Scholar 

  3. V. V. Peller, Hankel Operators and Their Applications (Springer, New York, 2003).

    Book  Google Scholar 

Download references

Acknowledgments

I am grateful to G. A. Karapetyan and A. N. Karapetyants, my colleagues from the Russian-Armenian RFBR grant 18-51-05009, with whom I discussed the idea of this paper and its results.

Funding

The work was supported in part by the Russian Foundation for Basic Research, project nos. 16-01-00117, 16-52-12012, and 18-51-05009.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to A. G. Sergeev.

Additional information

This article was submitted by the author simultaneously in Russian and English

Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2019, Vol. 306, pp. 227–234.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sergeev, A.G. Quantum Calculus and Ideals in the Algebra of Compact Operators. Proc. Steklov Inst. Math. 306, 212–219 (2019). https://doi.org/10.1134/S0081543819050183

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0081543819050183

Navigation