Kadison's Pythagorean Theorem and essential codimension

TL;DR

This paper interprets Kadison's Pythagorean theorem's integrality condition using the concept of essential codimension and index of projection pairs, linking it to previous results on diagonals of projections and selfadjoint operators.

Contribution

It provides a new interpretation of the integrality condition in Kadison's theorem via essential codimension and index theory, unifying different proofs and extending understanding.

Findings

The integrality condition relates to the essential codimension of projection pairs.

The same techniques clarify the integer in diagonals of selfadjoint operators with finite spectrum.

Connections are established between Kadison's theorem and index theory of Fredholm pairs.

Abstract

Kadison's Pythagorean theorem (2002) provides a characterization of the diagonals of projections with a subtle integrality condition. Arveson (2007), Kaftal, Ng, Zhang (2009), and Argerami (2015) all provide different proofs of that integrality condition. In this paper we interpret the integrality condition in terms of the essential codimension of a pair of projections introduced by Brown, Douglas and Fillmore (1973), or, equivalently of the index of a Fredholm pair of projections introduced by Avron, Seiler, and Simon (1994). The same techniques explain the integer occurring in the characterization of diagonals of selfadjoint operators with finite spectrum by Bownik and Jasper (2015).