  It is shown in this paper that two positive elements of a C*-algebra agree on
all lower semicontinuous traces if and only if they are equivalent in the sense
of Cuntz and Pedersen. A similar result is also obtained in the more general
case where the two elements are comparable by their values on the lower
semicontinuous traces. This result is used to give a characterization of the
functions on the cone of lower semicontinuous traces of a stable C*-algebra
that arise from positive elements of the algebra.
