DOI QR코드

DOI QR Code

TRIVIALITY OF A TRACE ON THE SPACE OF COMMUTING TRACE-CLASS SELF-ADJOINT OPERATORS

  • Myung, Sung (DEPARTMENT OF MATHEMATICS EDUCATION INHA UNIVERSITY)
  • Received : 2009.04.13
  • Published : 2010.11.30

Abstract

In the present article, we investigate the possibility of a real-valued map on the space of tuples of commuting trace-class self-adjoint operators, which behaves like the usual trace map on the space of trace-class linear operators. It turns out that such maps are related with continuous group homomorphisms from the Milnor's K-group of the real numbers into the additive group of real numbers. Using this connection, it is shown that any such trace map must be trivial, but it is proposed that the target group of a nontrivial trace should be a linearized version of Milnor's K-theory as with the case of universal determinant for commuting tuples of matrices rather than just the field of constants.

Keywords

traces;commuting operators;Milnor's K-theory

Acknowledgement

Supported by : INHA University

References

  1. J. A. Erdos, On the trace of a trace class operator, Bull. London Math. Soc. 6 (1974), 47-50. https://doi.org/10.1112/blms/6.1.47
  2. R. V. Kadison and J. R. Ringrose, Fundamentals of the theory of operator algebras. Vol. I, Elementary theory. Reprint of the 1983 original. Graduate Studies in Mathematics, 15. American Mathematical Society, Providence, RI, 1997.
  3. R. V. Kadison and J. R. Ringrose, Fundamentals of the theory of operator algebras. Vol. II, Advanced theory. Corrected reprint of the 1986 original. Graduate Studies in Mathematics, 16. American Mathematical Society, Providence, RI, 1997. pp. i-xxii and 399-1074.
  4. J. Milnor, Algebraic K-theory and quadratic forms, Invent. Math. 9 (1969/1970), 318-344. https://doi.org/10.1007/BF01425486
  5. J. Milnor, Introduction to Algebraic K-theory, Annals of Mathematics Studies, No. 72. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1971.
  6. S. Myung, On multilinearity and skew-symmetry of certain symbols in motivic cohomology of fields, Math. Res. Lett. 16 (2009), no. 2, 303-322. https://doi.org/10.4310/MRL.2009.v16.n2.a8
  7. S. Myung, Transfer maps and nonexistence of joint determinant, Linear Algebra Appl. 431 (2009), no. 9, 1633-1651. https://doi.org/10.1016/j.laa.2009.05.036
  8. Yu. P. Nesterenko and A. A. Suslin, Homology of the general linear group over a local ring, and Milnor's K-theory, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 1, 121-146; translation in Math. USSR-Izv. 34 (1990), no. 1, 121-145.
  9. M. Walker, Motivic complexes and the K-theory of automorphisms, Thesis, University of Illinois, 1996.