A Constructive Proof of Gleason's Theorem

Abstract

Gleason's theorem states that any totally additive measure on the closed subspaces, or projections, of a Hilbert space of dimension greater than two is given by a positive operator of trace class. In this paper we give a constructive proof of that theorem. A measure μ on the projections of a real or complex Hilbert space assigns to each projection P a nonnegative real number μ(P) such that if σ = ∑Pi, where the Pi are mutually orthogonal, then μ(σ) =∑μ(Pi). Such a measure is determined by its values on the one-dimensional projections. Let W be the measure of the identity projection, and Px the projection onto the 1-dimensional space spanned by the unit vector x. Then the measure μ is determined by the real-valued function f(x) = μ(Px) on the unit sphere, a function which has the property that [see pdf for formula] for each orthonormal basis E. Gleason calls such a function f a frame function of weight W. If T is a positive operator of trace class, then f(x) = ‹Tx,x› is a frame function. Gleason's theorem is that every frame function arises in this way. The original reference for Gleason's theorem is [4], which can also be found in Hooker [6]. Cooke, Keane and Moran [3] gave a proof that is elementary in the sense that it does not appeal to the theory of representations of the orthogonal group, which the original proof does. However, some of the reasoning in [3] seems hopelessly nonconstructive, so we follow the general outline of [4] until we come to the end of the 3-dimensional real case, at which point we modify some arguments in [3] rather than attempt a constructive development of the necessary representation theory. Any Hermitian form B on a finite-dimensional inner product space gives rise to a frame function f(x) = B(x; x) whose weight is equal to the trace of the matrix of B. The essence of Gleason's theorem is the following converse. -- from Introduction