Why the set of pure state ‎is ‎weak* ‎compact?

Question

Let ‎$‎‎A$ ‎be a‎ ‎C*-algebra‎.

‎ ‎$‎S(A)‎$ ‎is ‎the ‎set ‎of ‎state ‎on ‎‎$‎‎A$ and $‎‎PS(A)$ ‎is ‎the ‎set ‎of ‎pure ‎state ‎on ‎‎$‎‎A$.

‎ ‎ I ‎know ‎that ‎if ‎‎$‎‎A$ ‎is ‎unital ‎then ‎‎$‎‎S(A)$ ‎is ‎weak* ‎compat.‎

‎I know that extreme points of $S(A)$ is $PS(A)$

I ‎want ‎to ‎prove ‎‎$‎‎PS(A)$ ‎is ‎weak* ‎compact.so I‎ ‎should ‎show ‎that ‎‎$‎‎PS(A)$ ‎is ‎weak* ‎closed.(when $A$ is unital)

‎ Q: ‎‎My question is:" Why $‎PS(A)‎$ ‎is ‎weak* ‎compact?‎" " Is the extreme point of compact set compact? "

Answer 1

Let $\{\phi\}$ be a net of pure states on $A$ and assume that it is $w^*$-convergent to $\phi$. Clearly $\phi$ is a positive linear functional on $A$.

To prove $\phi$ is a pure state, it is enough to show that support of $\phi$ is a minimal projection in $A^{**}$. Assume $q$ is a non-trivial projection majorized by the support of $\phi$. We have that $q\phi_i q$ is $w^*$-convergent to $q\phi q$ then there is a subnet $\{\phi_j\}$ such that $q\phi_jq=\phi_j$. It implies that $\phi=q\phi q$ which means that $q\geq$supp($\phi$). Hence support of $\phi$ is minimal.

Answer 2

The set of pure states of a unital C$^*$-algebra is weak$^*$-compact when $A$ is abelian, but not in general.

This has been known since at least Glimm's thesis, where he proved that the set of pure states of a UHF algebra is not closed.