The Kadison-Singer Problem

Specialeforsvar ved Christine Broby Nielsen

Titel: The Kadison-Singer problem

Abstract:  This thesis will concern the problem raised by Kadison and Singer in 1959 in their article "Extensions of pure states" [1], in which it was asked whether pure states on a maximal abelian sub-$C^*$-algebra (MASA) of $B(H)$, the bounded operators of Hilbert space $H$, can be extended uniquely to a pure state on $B(H)$. They proved that there is no unique extension if the MASA is continuous, such as $L^\infty(]0, 1[) \subset B(H)$ where $H = L^2(]0, 1[)$. However, they were not able to prove nor disprove this in the case of the discrete MASA, $\ell^\infty\subset B(H)$ where $H = \ell^2$. Hence this was up for discussion. I continue to follow the problem, and the manner in which the problem changed from the original phrasing. Joel Anderson [2] managed to relate the problem to a notion of compressibility, where he gives an equivalent formulation. From thereon, that subsequent formulation is then found to be analogous to looking at weakly paveable elements in [3] by Akemann, Anderson and Tanbay. This notion of weak paveability caused Nik Weaver [4] to be able to generalise to a combinatorial version, which was the basis to the manner in which Marcus, Spielman and Srivastava [9] were able to solve the problem in 2013.

Vejleder: Mikael Rørdam
Censor:   Wojciech Szymanski, Syddansk Universitet