Majorisation and the Carpenter’s Theorem

Constructive Proof of the Carpenter's Theorem

“…In contrast, until very recently there were no proofs of the carpenter's theorem other than the original one by Kadison, although its extension for II 1 factors was studied by Argerami and Massey [2]. A notable exception is a recent paper by Argerami [1] about which we became aware only after completing this work. In this paper we give an alternative proof of the carpenter's theorem which has two main advantages over the original.…”

“…Among several equivalent ways of introducing the spectral function of an SI space, the most relevant definition uses a range function. The spectral function of an SI space V is a measurable mappingσ V : R d → [0,1] given by(5.1) σ V (ξ + k) = P J (ξ)e k 2 = P J (ξ)e k , e k for ξ ∈T d , k ∈ Z d ,where {e k } k∈Z d denotes the standard basis of 2 (Z d ) andT d = [−1/2, 1/2) d . In other words, {σ V (ξ + k)} k∈Z dis a diagonal of a projection P J (ξ).Note that σ V (ξ) is well defined for a.e.…”