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Which C$^*$-algebras admit factor states for which the von Neumann algebra it generates in the corresponding GNS representation is a type III$_1$ factor? For example, do all purely infinite algebras admit such a state?

Even more generally: what about a type III$_\lambda$ factor for $\lambda\in(0,1]$?

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    $\begingroup$ This is a good question, why the downvote? $\endgroup$
    – Nik Weaver
    May 1, 2021 at 1:41

1 Answer 1

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By the main result of this paper of Odile Maréchal, which was generalizing James Glimm's famous paper, the following extreme dichotomy holds. If $A$ is any separable C*-algebra, precisely one of the following statements holds.

  1. $A$ is of type I: for every representation $\pi$ of $A$, we have that $\pi(A)''$ is a von Neumann algebra of type I.
  2. For every infinite injective factor $M$, there exists a representation $\pi$ of $A$ such that $\pi(A)'' \cong M$.

In particular, every separable C*-algebra that is not of type I, admits states $\omega_\lambda$ such that $\pi_{\omega_\lambda}(A)''$ is the unique injective factor of type III$_\lambda$, with $\lambda \in (0,1]$.

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  • $\begingroup$ Very helpful! Thanks Stefaan! $\endgroup$
    – Isaac
    May 1, 2021 at 17:12

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