
THE
FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES

JanuaryJune
2014
Thematic Program on Abstract Harmonic Analysis, Banach
and Operator Algebras
Coxeter
Lecture Series
May 27,28,29,
2014 at 3:30
p.m.
Fields Institute, Room 230
Sorin
Popa
University of California, Los Angeles
On $\text{II}_1$ factors arising from free groups acting on
spaces




On $\text{II}_1$ factors arising from free groups acting on
spaces
A famous problem going back to Murray and von Neumann (19361943),
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 nonisomorphic
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 nonisomoprphic
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.

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