  In the present article, we investigate a 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.
