Joint Traces and determinants for commuting operators

초록

Determinants and Traces play important roles in the theory of linear operators. We first give a set of axioms for determinants of the l-tuples of commuting operators on a vector space over a field when l>1. The set of determinants then can be described in terms of the Milnor's K-theory. As for the traces, it is not clear how to correctly formulate a definition except for the tuples of commuting trace-class self-adjoint operators on a Hilbert space. We also relate them to 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.