Given an n-tuple {b_1, ..., b_n} of self-adjoint operators in a finite vonNeumann algebra M and a faithful, normal tracial state tau on M, we define amap Psi from M to R^{n+1} by Psi(a) = (tau(a), tau(b_1a), ..., tau(b_na)). Theimage of the positive part of the unit ball under Psi is called the spectralscale of {b_1, .., b_n} relative to tau and is denoted by B. In a previouspaper with Nik Weaver we showed that the geometry of B reflects spectral datafor real linear combinations of the operators {b_1, .., b_n}. For example, weshowed that an exposed face in B is determined by a certain pair of spectralprojections of a real linear combination of {b_1, .., b_n}. In the presentpaper we extend this study to faces that are not exposed. We completelydescribe the structure of arbitrary faces of B in terms of {b_1, .., b_n} andtau. We also study faces of convex, compact sets that are exposed by more thanone hyperplane of support. Although many of the conclusions of this studyinvolve too much notation to fit nicely in an abstract, there are two resultsthat 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 iscountable, then N is abelian. Corollary 5.6: B has a finite number of extremepoints if and only if N is abelian and finite dimensional.
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