The Kadison-Singer problem for the direct sum of matrix algebras

TL;DR

This paper investigates the extension of pure states from a diagonal subalgebra to the entire algebra of matrices, providing new results on the uniqueness of such extensions and the existence of non-multiplicative pure states under certain set-theoretic assumptions.

Contribution

It proves that a dense subset of singular pure states of the diagonal algebra extend uniquely, and shows the existence of non-multiplicative pure states assuming the Continuum hypothesis.

Findings

Normal pure states of D extend uniquely.

A weak* open, dense subset of singular pure states also extend uniquely.

Existence of pure states not multiplicative on any MASA under CH.

Abstract

Let $M_n$ denote the algebra of complex $n\times n $ matrices and write $M$ for the direct sum of the $M_n$. So a typical element of $M$ has the form \[x = x_1\oplus x_2 \... \oplus x_n \oplus \...,\] where $x_n \in M_n$ and $\|x\| = \sup_n\|x_n\|$. We set $D= \{\{x_n\} \in M: x_n$ is diagonal for all $N\}$. We conjecture (contra Kadison and Singer (1959)) that every pure state of $D$ extends uniquely to a pure state of $M$. This is known for the normal pure states of D, and we show that this is true for a (weak*) open, dense subset of all the singular pure states of $D$. We also show that (assuming the Continuum hypothesis) $M$ has pure states that are not multiplicative on any maximal abelian *-subalgebra of $M$.