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 $\lambda\in(0,1]$?



Factor states on C*-algebras
By the main result of [this paper of Odile Maréchal](https://mathscinet.ams.org/mathscinet-getitem?mr=430797), which was generalizing [James Glimm's famous paper](https://mathscinet.ams.org/mathscinet-getitem?mr=124756), 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 $\lambda \in (0,1]$.


