Kadison-Singer problem from a banach algebra perspective

Abstract

In 1959, Kadison and Singer asked whether every pure state of the diagonal subspace D(ℓ2) of B(ℓ2) has a unique pure state extension to B(ℓ2). This problem has remained open until 2013; in 2013 it has been solved by a team of computer scientists. In my Master thesis, which is largely based on a paper by Akemann, Tanbay and Ulger, ¨ I have tried to learn this problem and the approach considered in this paper. We identify D(ℓ2) with C(βN). For t in βN, δt is the Dirac measure at t considered as a functional on C(βN). We denote by [δt ] the set of the states of B(ℓ2) that extend δt . Our main aim is to understand how large the set [δt ] is. Using the fact that the von Neumann algebra B(ℓ2) has the Pelczýnski’s property (V ), it is proven that either the set [δt ] lies in a finite dimension subspace of B(ℓ2) ∗ or, in its weak-star topology, it contains a homeomorphic copy of βN. We study this result under the so far directly unproven knowledge that [δt ] is a singleton.