Consequences of the Marcus/Spielman/Stivastava solution to the Kadison-Singer Problem

Abstract

It is known that the famous, intractible 1959 Kadison-Singer problem in $C^{*}$-algebras is equivalent to fundamental unsolved problems in a dozen areas of research in pure mathematics, applied mathematics and Engineering. The recent surprising solution to this problem by Marcus, Spielman and Srivastava was a significant achievement and a significant advance for all these areas of research. We will look at many of the known equivalent forms of the Kadison-Singer Problem and see what are the best new theorems available in each area of research as a consequence of the work of Marcus, Spielman and Srivastave. In the cases where {\it constants} are important for the theorem, we will give the best constants available in terms of a {\it generic constant} taken from \cite{MSS}. Thus, if better constants eventually become available, it will be simple to adapt these new constants to the theorems.