Non-amenable Factors
Von Neumann algebras of type I are always amenable, but for the other types there are an uncountable number of different non-amenable factors, which seem very hard to classify, or even distinguish from each other. Nevertheless Voiculescu has shown that the class of non-amenable factors coming from the group-measure space construction is disjoint from the class coming from group von Neumann algebras of free groups. Later Narutaka Ozawa proved that group von Neumann algebras of hyperbolic groups yield prime type II1 factors, i.e. ones that cannot be factored as tensor products of type II1 factors, a result first proved by Leeming Ge for free group factors using Voiculescu's free entropy. Popa's work on fundamental groups of non-amenable factors represents another significant advance. The theory of factors "beyond the hyperfinite" is rapidly expanding at present, with many new and surprising results; it has close links with rigidity phenomena in geometric group theory and ergodic theory.
Read more about this topic: Von Neumann Algebra
Famous quotes containing the word factors:
“Girls tend to attribute their failures to factors such as lack of ability, while boys tend to attribute failure to specific factors, including teachers’ attitudes. Moreover, girls avoid situations in which failure is likely, whereas boys approach such situations as a challenge, indicating that failure differentially affects self-esteem.”
—Michael Lewis (late–20th-century)