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

TL;DR

The paper discusses the implications of the Marcus/Spielman/Stivastava solution to the longstanding Kadison-Singer problem, highlighting new theorems and optimal constants across various mathematical and engineering fields.

Contribution

It analyzes the impact of the solution on equivalent forms of the problem and provides the best known constants for related theorems in multiple research areas.

Findings

New theorems derived from the solution across different fields.

Optimal constants for related theorems are identified.

Simplified adaptation of constants if improved values are found.

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.