Sorin Popa (University of California, Los Angeles)

 

A famous problem going back to Murray and von Neumann (1936-1943), asks whether the $\text{II}_1$ factors $L(\Bbb F_n)$, obtained as the centralizers in ${\cal B}(\ell^2 \Gamma)$ of the left regular representation of the free groups on $n$ generators, are non-isomorphic for different $n$. While this is still open, its "group measure space" version, asking whether the crossed product $\text{II}_1$ factors $L^\infty(X)\rtimes \Bbb F_n$, arising from free ergodic probability measure preserving actions $\Bbb F_n\curvearrowright X$, are non-isomorphic for $n= 2, 3, ...$, independently of the actions, has recently been settled by Stefaan Vaes and myself. I will comment on this result, as well as on some related problems.