The Kadison–Singer and Paulsen problems in finite frame theory

Abstract

We now know that some of the basic open problems in frame theory are equivalent to fundamental open problems in a dozen areas of research in both pure and applied mathematics, engineering, and others. These problems include the 1959 Kadison–Singer problem in C ∗-algebras, the paving conjecture in operator theory, the Bourgain–Tzafriri conjecture in Banach space theory, the Feichtinger conjecture and the Rϵ-conjecture in frame theory, and many more. In this chapter we will show these equivalences among others. We will also consider a slight weakening of the Kadison–Singer problem called the Sundberg problem. Then we will look at the recent advances on another deep problem in frame theory called the Paulsen problem. In particular, we will see that this problem is also equivalent to a fundamental open problem in operator theory. Namely, if a projection on a finite dimensional Hilbert space has a nearly constant diagonal, how close is it to a constant diagonal projection?.