A Geometric Spectral Theory for n-tuples of Self-Adjoint Operators in Finite von Neumann Algebras

Abstract

Suppose b1, …, bn are self-adjoint elements in a finite von Neumann algebra M with trace τ and define a map Ψ from M to complex (n+1)-space by the formula Ψ(x)=(τ(x), τ(b1x), …, τ(bnx)). Next let B denote the image of the positive unit ball of M under the map Ψ. B is called the spectral scale of τ, b1, …, bn. It is clearly compact and convex. The main theme of this work is that the geometry of the spectral scale B reflects spectral data for the bi's. For example, in the finite dimensional case the operators commute if and only if the spectral scale is a polytope. Thus, one can “see” that the operators commute from the shape of spectral scale. In the case of a single operator, where the scale lies in the plane, the slopes of the boundary fill out the spectrum of the operator, corners correspond to gaps in the spectrum, and flat sports indicate eigenvalues. Analogous results hold when there is more than one operator. In the commutative setting, the spectral scale “determines” the (n+1)-tuple (τ, b1, …, bn). However, an example is given that shows this is not generally true in the noncommutative case. Finally, a matricial version of the spectral scale is shown to be sufficient to completely determine the (n+1)-tuple (τ, b1, …, bn).