A result about Maximal injectivity in von Neumann algebras

A result about Maximal injectivity in von
Neumann algebras
Mohan Ravichandran, Sabanci University
Abstract
Thursday, May 26, 4:20PM, SU Colloquium
The most studied von Neumann algebras are those that have trivial center
and have no minimal projections: the class of II1 factors. Given a discrete
group G, all of whose conjugacy classes save the trivial one are innite, the
double commutant of the natural representation of the group ring CG in
B(`2(G)), denoted LG is a II1 factor. A von Neumann algebra is said to
be amenable if it is a direct limit of matrix algebras and an early result of
von Neumann says that there is a unique amenable II1 factor - The so called
hypernite II1 factor, R. The group algebra LG is amenable i G is.
II1 factors were the setting for a celebrated set of questions asked by
Richard Kadison in 1967. One of them was the following - Is a maximal
amenable subalgebra of a II1 factor necessarily R? I will start o by ex-
plaining why this seemed reasonable. The question was unexpectedly solved
in the negative in 1982 by Sorin Popa, who showed for instance that there
exist abelian maximal amenable von Neumann algebras. I will talk about a
result in this vein: Let F2 =< a; b > be the free group on two generators,
LF2 the group von Neumann algebra and A  LF2 the abelian von Neumann
subalgebra consisting of all radial elements in LF2. Then, A is maximal in-
jective in LF2. I will conclude with some extensions of this result.
The rst half of the talk will introduce some useful notions in operator
algebras and will be of benet to any graduate student interested in analysis.