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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.
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참고문헌 (9)
참고문헌 신청
J. A. Erdos / 1974 / On the trace of a trace class operator / Bull. London Math. Soc. 6 : 47 ~ 50
J. Milnor / 1970 / Algebraic K-theory and quadratic forms / Invent. Math. 9 : 318 ~ 344
J. Milnor / 1971 / Introduction to Algebraic K-theory, Annals of Mathematics Studies, No. 72 / Princeton University Press
M. Walker / 1996 / Motivic complexes and the K-theory of automorphisms / University of Illinois
R. V. Kadison / 1997 / Fundamentals of the theory of operator algebras. Vol. Ⅰ, Elementary theory / American Mathematical Society
J. Milnor / 1970 / Algebraic K-theory and quadratic forms / Invent. Math. 9 : 318 ~ 344
J. Milnor / 1971 / Introduction to Algebraic K-theory, Annals of Mathematics Studies, No. 72 / Princeton University Press
M. Walker / 1996 / Motivic complexes and the K-theory of automorphisms / University of Illinois
R. V. Kadison / 1997 / Fundamentals of the theory of operator algebras. Vol. Ⅰ, Elementary theory / American Mathematical Society
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