Constructive Proof of the Carpenter’s Theorem

We give a constructive proof of Carpenter’s Theorem due to Kadison [14, 15]. Unlike the original proof our approach also yields the real case of this theorem. 1. Kadison’s theorem In [14] and [15] Kadison gave a complete characterization of the diagonals of orthogonal projections on a Hilbert space H. Theorem 1.1 (Kadison). Let {di}i∈I be a sequence in [0, 1]. Define a = ∑ di<1/2 di and b = ∑ di≥1/2 (1− di). There exists a projection P with diagonal {di} if and only if one of the following holds (i) a, b <∞ and a− b ∈ Z, (ii) a =∞ or b =∞. The goal of this paper is to give a constructive proof of the sufficiency direction of Kadison’s theorem. Kadison [14, 15] referred to the necessity part of Theorem 1.1 as the Pythagorean Theorem and the sufficiency as Carpenter’s Theorem. Arveson [3] gave a necessary condition on the diagonals of a certain class of normal operators with finite spectrum. When specialized to the case of two point spectrum Arveson’s theorem yields the Pythagorean Theorem, i.e., the necessity of (i) or (ii) in Theorem 1.1. Whereas Kadison’s original proof is a beautiful direct argument, Arveson’s proof uses the Fredholm Index Theory. In contrast, until very recently there were no proofs of Carpenter’s Theorem other than the original one by Kadison, although its extension for II1 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 Carpenter’s Theorem which has two main advantages over the original. First, the original proof does not yield the real case, which ours does. Second, our proof is constructive in the sense that it gives a concrete algorithmic process for finding the desired projection. This is distinct from Kadison’s original proof, which is mostly existential. The paper is organized as follows. In Section 2 we state preliminary results such as finite rank Horn’s theorem. These results are then used in Section 3 to show the sufficiency of (i) in Theorem 1.1. The key role in the proof is played by a lemma from [8] which enables Date: September 12, 2013. 2000 Mathematics Subject Classification. Primary: 42C15, 47B15, Secondary: 46C05.