Set theory and C* algebras

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5. Pure states

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Problem 5.1.
[Kadison-Singer] Does every pure state of the atomic masa in $\mathcal{B}\left( H\right)$ extend uniquely to a pure state of $\mathcal{B}\left( H\right)$ ?
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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?
Its negation is known to follow from the Continuum Hypothesis (Akemann-Weaver).
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Problem 5.3.
Consider the following statement: every pure state on $\mathcal{B}\left( H\right)$ is diagonalizable. Is it consistent with with ZFC?
Its negation is known to follow from the Continuum Hypothesis (Akemann-Weaver) or even from Martin’s Axiom (Farah-Weaver).

Cite this as: AimPL: Set theory and C* algebras, available at http://aimpl.org/settheorycstar.