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In Davidson and Szarek's article "Local Operator Theory, Random Matrices and Banach Spaces" in the Handbook of the Geometry of Banach Spaces, the authors discuss the (now solved) Kadison-Singer problem of unique extensions of pure states from atomic masas to $\mathcal{B}(H)$. After describing the form of pure states on atomic masas (really, point evaluations on $\beta\mathbb{N}$), they say:

This suggests the open question raised in [99, §5] whether every pure state on $\mathcal{L}(\ell_2)$ is an extension of a pure state on some discrete masa, or indeed a given masa. However, it has been shown in [6], cf. [4, §7.3], that the two questions are in fact equivalent.

  • [4] is Anderson, J., Extensions, Restrictions, and Representations of States on C*-Algebras, Trans. AMS, 1979
  • [6] is Anderson, J., Extreme Points in Sets of Positive Linear Maps on $\mathcal{B}(H)$, J. Funct. Anal., 1979
  • [99] is Kadison, R., Singer, I., Extensions of Pure States, Amer. J. Math., 1959

I assume that the "two questions" being referred to are (1) whether a given pure state restricts to a pure state (i.e., a homomorphism to $\mathbb{C}$) on some atomic masa, and (2) whether the same is true on a given masa. There is also the related question (1'), usually credited to Kadison and Singer, of whether the same is true on some masa.

My question is, are these statements in fact equivalent?

Skimming through [6], I didn't find a direct answer to my question, though I may have missed something. Corollary 7.3 in [4] contains the relevant fact that if $h$ is a non-zero homomorphism to $\mathbb{C}$ on a non-atomic masa, then it extends to a pure state on $\mathcal{B}(H)$ which itself restricts to a homomorphism on a particular atomic masa. However, the extension of $h$ need not be unique.

Relevant is the paper by Akemann and Weaver (PNAS, 2008), producing a pure state which is not pure on any masa, in the presence of the continuum hypothesis. That paper, and others I've looked at with refinements of their construction, do not seem to mention the equivalence of these two problems, hence my surprise.

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I just ran across this question and apologize for not noticing it earlier. I think the statement is incorrect --- I'm not aware of such an equivalence. As far as I know it is open whether there are models of ZFC in which every pure state is pure on some masa, but some pure state is not pure on any atomic masa.

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