Speaker: Kenley Jung
Affiliation: UCLA
Title: Strongly 1-Bounded von Neumann algebras
Time and Place: Monday, 3/20, 3:00-3:55pm, Milner 216.
Abstract: Using Voiculescu's microstates we introduce a notion of strongly 1- boundedness for a von Neumann algebra. Given a finite set F of selfadjoint elements in a tracial von Neumann algebra, we say that F is 1-bounded if its free 1-packing entropy is finite. A von Neumann algebra is strongly 1-bounded if it has a 1-bounded generating set such that there exists an element in the set with finite free entropy. The free entropy dimension of any generating set for a strongly 1-bounded von Neumann algebra is either 1 or -infinity. It follows that such von Neumann algebras are not isomorphic to the free product von Neumann algebras or the property T-perturbation algebras examined by Nate Brown. Examples of strongly 1-bounded von Neumann algebras include those which are not prime, those with property Gamma, the von Neumann algebras of SL_n(Z), n>2 and those with a Cartan subalgebra. That these examples are strongly 1-bounded follow from results by Voiculescu, and Ge and Shen. We show that an amalgamated free product of strongly 1-bounded von Neumann algebras over a common diffuse algebra or the normalizer of a strongly 1-bounded von Neumann algebra is again strongly 1-bounded. In particular, the free group factors cannot be isomorphic to an amalgamated free product of strongly 1-bounded von Neumann algebras over a diffuse von Neumann subalgebra. The proof uses geometric measure theory and Besicovitch's classification of 1-sets.