Abstract: An old conjecture of Higman from the 1930s states that non-isomoprih torsion free groups $\Gamma$ have non-isomorphic (complex) algebras $\Bbb C\Gamma$. The group algebra $\Bbb C\Gamma$ acts by left convolution on the Hilbert space $\ell^2 \Gamma$ and its closure in the week operator topology gives rise to the group von Neumann algebra $L(\Gamma) \subset \mathcal B(\ell^2\Gamma)$. But the isomorphism class of these much bigger algebras may dramatically "forget" the initial data $\Gamma$. For instance, all wreath product groups $\Gamma = H \wr \Lambda$, with $H, \Lambda$ infinite amenable (e.g. $\Gamma_n=\Bbb Z \wr \Bbb Z^n$, $n=1, 2,É$) give rise to isomorphic $L(\Gamma)$'s (Connes 1976). A famous problem of Murray-von Neumann (1943) asks whether the von Neumann algebras $L(\Bbb F_n)$ (II$_1$ factors), arising from the free groups $\Bbb F_n$, $2\leq n \leq \infty$, are non-isomorphic for distinct $n$'s. While this so called free group factor problem is still open, its "group measure space" version, showing that the von Neumann algebras arising from free ergodic actions of $\Bbb F_n$ on probability measure spaces are non-isomoprphic for different $n$, independently of the actions, has recently been settled by Stefaan Vaes and myself. This shows in particular that $L(\Bbb Z \wr \Bbb F_n)$, $n=1, 2, É,$ are non-isomorphic. I will comment on this result and on the related free group factor problem.
A reception will follow the Feb 2 lecture in 239 Altgeld Hall from 5-6 p.m.