  Using various finite dimensional approximation properties, four convex
subsets of the tracial space of a unital C*-algebra are defined. Applications
of these tracial invariants include: (1) An analogue of Szego's limit theorem
for arbitrary self adjoint operators. (2) A McDuff factor embeds into the
ultrapower of the hyperfinite II_1 factor if and only if it contains a weakly
dense operator system which is injective. (3) There exists a simple,
quasidiagonal, real rank zero C*-algebra with non-hyperfinite II_1 factor
representations and which is not tracially AF. This answers negatively
questions of Sorin Popa and, respectively, Huaxin Lin.
