Le probl\`eme de Kadison-Singer (The Kadison-Singer problem)

TL;DR

This paper discusses the resolution of the Kadison-Singer problem, which questions the unique extension of pure states in operator algebras, culminating in a 2013 proof involving polynomial zero estimates.

Contribution

It summarizes the historical development and the final proof of the Kadison-Singer problem, linking it to polynomial zero estimates and random matrix theory.

Findings

Kadison-Singer problem was affirmatively solved in 2013

The solution involves bounds on zeros of expected characteristic polynomials

The problem relates to extensions of pure states in operator algebras

Abstract

In 1959, R.V. Kadison and I.M. Singer asked whether each pure state of the algebra of bounded diagonal operators on $\ell^2$, admits a unique state extension to $B(\ell^2)$. The positive answer was given in June 2013 by A. Marcus, D. Spielman and N. Srivastava, who took advantage of a series of translations of the original question, due to C. Akemann, J. Anderson, N. Weaver,... Ultimately, the problem boils down to an estimate of the largest zero of the expected characteristic polynomial of the sum of independent random variables taking values in rank 1 positive matrices in the algebra of n-by-n matrices.