EXTREME POINTS RELATED TO MATRIX ALGEBRAS

Let A denote the set {$a{\in}M_n{\mid}a{\geq}0$, $tr(a)=1$}, $St(M_n)$ the set of all states on $M_n$, and $PS(M_n)$ the set of all pure states on $M_n$. We show that there are one-to-one correspondences between A and $St(M_n)$, and between the set of all extreme points of A and $PS(M_n)$. We find a necessary and sufficient condition for a state on $M_{n1}{\oplus}{\cdots}{\oplus}M_{nk}$ to be extended to a pure state on $M_{n1}+{\cdots}+_{nk}$.