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\(\newcommand{\Cat}{{\rm Cat}} \) \(\newcommand{\A}{\mathcal A} \) \(\newcommand{\freestar}{ \framebox[7pt]{$\star$} }\)

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.