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

Abstract

Given an n-tuple {b 1 ,...,b n } of self-adjoint operators in a finite von Neumann algebra M and a faithful, normal tra-cial state r on M, we define a map Ψ from M to R n by Ψ(a) = (τ(a),τ(b 1 a),...,τ(bna)). The image of the positive part of the unit ball under Ψ is called the spectral scale of {b 1 ,...,b n } relative to r 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 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 R n . Using this new method, we completely describe the structure of arbitrary faces of B in terms of {b 1 ,...,b n } 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 {b 1 ,...,b n }. 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 = {b 1 ,...,b n } is abelian. Corollary 5.6: B has a finite number of extreme points if and only if N is abelian and has finite dimension.