  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 $M$.
