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Given an n-tuple
{b1,…,bn} of self-adjoint operators in a finite von Neumann algebra M and a
faithful, normal tracial state τ on M, we define a map Ψ from M to ℝn by
Ψ(a) = (τ(a),τ(b1a),…,τ(bna)). The image of the positive part of the unit
ball under Ψ is called the spectral scale of {b1,…,bn} relative to τ 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
{b1,…,bn}. 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 the bi’s. In
the present paper we extend this study to faces that are not exposed. In
order to do this we need to introduce a recursive method for describing
faces of compact convex sets in ℝn. Using this new method, we completely
describe the structure of arbitrary faces of B in terms of {b1,…,bn} and τ. We
also study faces of convex, compact sets that are exposed by more than one
hyperplane of support (we call these sharp faces). When such faces appear on B,
they signal the existence of commutativity among linear combinations of
the operators {b1,…,bn}. 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. Theorem 6.1: If the set of extreme points
of B is countable, then N = {b1,…,bn}′′ is abelian. Corollary 5.6: B has a
finite number of extreme points if and only if N is abelian and has finite
dimension.
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