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

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 $\Vert x\Vert ={\sup }_n\Vert x_n\Vert$. 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 $M$.