Kadison–singer from Mathematical Physics: an Introduction

We give an informal overview of the Kadison–Singer extension problem with emphasis on its initial connections to Dirac’s formulation of quantum mechanics. Let H be an infinite dimensional separable Hilbert space, and B (H) the algebra of all bounded operators in H. In the language of operator algebras, the Kadison–Singer problem asks whether or not for a given MASA D in B (H), every pure state on D has a unique extension to a pure state on B (H). In other words, are these pure-state extensions unique? It was shown recently by Pete Casazza and co-workers that this problem is closely connected to central open problems in other parts of mathematics (harmonic analysis, combinatorics (via Anderson pavings), Banach space theory, frame theory), and applications (signal processing, internet coding, coding theory, and more).