DETERMINANTS AND TRACES FOR THE COMMUTING OPERATORS ON A FINITE VECTOR SPACE

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.