5. Pure states

Problem 5.1.
[KadisonSinger] Does every pure state of the atomic masa in $\mathcal{B}\left( H\right) $ extend uniquely to a pure state of $\mathcal{B}\left( H\right) $? 
Problem 5.2.
Consider the following statement: for every pure state $\phi$ of $\mathcal{B}\left( H\right) $ there is a masa $\mathcal{A}$ such that $\phi\restriction \mathcal{A}$ is multiplicative (i.e., pure). Is it consistent with ZFC? 
Problem 5.3.
Consider the following statement: every pure state on $\mathcal{B}\left( H\right) $ is diagonalizable. Is it consistent with with ZFC?
Cite this as: AimPL: Set theory and C* algebras, available at http://aimpl.org/settheorycstar.