A geometric spectral theory for n-tuples of self-adjoint operators in a finite von Neumann algebra: II
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.