  For A a separable unital C*-algebra and M a separable McDuff II_1-factor, we
show that the space Hom_w(A,M) of weak approximate unitary equivalence classes
of unital *-homomorphisms A \rightarrow M may be considered as a closed,
bounded, convex subset of a separable Banach space -- a variation on N. Brown's
convex structure Hom(N,R^U). When A is nuclear, Hom_w(A,M) is affinely
homeomorphic to the trace space of A, but in general Hom_w(A,M) and the trace
space of A do not share the same data (several examples are provided). We
characterize extreme points of Hom_w(A,M) in the case where either A or M is
amenable; and we give two different conditions -- one necessary and the other
sufficient -- for extremality in general. The universality of C*(F_\infty) is
reflected in the fact that for any unital separable A, Hom_w(A,M) may be
embedded as a face in Hom_w(C*(F_\infty),M). We also extend Brown's
construction to apply more generally to Hom(A,M^U).
