  Given an n-tuple {b_1, ..., b_n} of self-adjoint operators in a finite von
Neumann algebra M and a faithful, normal tracial state tau on M, we define a
map Psi from M to R^{n+1} by Psi(a) = (tau(a), tau(b_1a), ..., tau(b_na)). The
image of the positive part of the unit ball under Psi is called the spectral
scale of {b_1, .., b_n} relative to tau and is denoted by B. In a previous
paper with Nik Weaver we showed that the geometry of B reflects spectral data
for real linear combinations of the operators {b_1, .., b_n}. For example, we
showed that an exposed face in B is determined by a certain pair of spectral
projections of a real linear combination of {b_1, .., b_n}. In the present
paper we extend this study to faces that are not exposed. We completely
describe the structure of arbitrary faces of B in terms of {b_1, .., b_n} and
tau. We also study faces of convex, compact sets that are exposed by more than
one hyperplane of support. Although many of the conclusions of this study
involve too much notation to fit nicely in an abstract, there are two results
that give their flavor very well. Let N be the algebra generated by {b_1, ...,
b_n} and the identity. Theorem 6.1: If the set of extreme points of B is
countable, then N is abelian. Corollary 5.6: B has a finite number of extreme
points if and only if N is abelian and finite dimensional.
