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.