• Bulletin of the Korean Mathematical Society
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• v.47 no.6
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• pp.1205-1211
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• 2010

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.

• Korean Journal of Mathematics
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• v.16 no.1
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• pp.115-122
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• 2008

In the present article, we give a set of axioms for determinants and traces of the l-tuples of commuting operators on a fixed finite dimensional vector space over a field when $l{\geq}2$. We describe them with or without a coherence assumption especially when k is the field of real numbers. Under the coherence assumption, it turns out that there are only a trivial determinant and trace over arbitrary field k. This leads us to formulate a more appropriate definition of the determinants. In this case, the set of determinants can be described in terms of the Milnor's K-theory. As for the traces, it is not clear to us how to correctly formulate a definition except for certain cases.