Regarding the hyperfinite $II\_{1}$ factor $R$ as $C^{\*}$-algebra, is it known whether any two irreducible representations of $R$ are unitarily equivalent? If it is known that there exists a pair of irreducible representations that are not unitarily equivalent, please provide a reference.


(Note: I am not requiring that the representations be normal!!)



Does the hyperfinite II_1 factor admit two irreducible representations that are not unitarily equivalent?
There are $2^c$ mutually non-equivalent irreducible representations. Since $\ell\_\infty(N)$ has $2^c$ many pure states (there are $2^c$ many ultrafilters on $N$), any $C^\*$-algebra containing $\ell\_\infty(N)$ has at least as much pure states. Since $R$ has $c$ many unitary elements, there are $2^c$ many mutually non-equivalent pure states. This is a very old observation, but I don't know a specific reference. (It was noted in our paper too. <http://xxx.lanl.gov/abs/math.OA/0110152>) 


