Speaker: Mingchu Gao
Affiliation: Louisiana College
Title: Prime von Neumann Algebras.
Time and Place: Wednesday, October 3, 2:00-2:55pm, Milner 317.
Abstract: This is joint work with Marius Junge.
We proved that the free product von Neumann algebra of separable injective von Neumann algebras is prime if the free product algebra is diffuse. Moreover, it is proved that a non-injective von Neumann sub-algebra of the free product algebra is prime if there is a normal conditional expectation from the free product algebra onto the sub-algebra. Our result provides many examples of primes von Neumann algebras, which include free group factors, free Araki-Woods factors and more. To prove the result, we generalize Ozawa's solid (finite) von Neumann algebras to general von Neumann algebras. We combine the generalized solid von Neumann algebra technique with Shlyakhtenko's matrix model technique for free Araki-Wood factors to get a proof of the result. We also introduce the concept of Ozawa property (OP) relating to the (AO) condition. We prove that Ozawa property implies the (AO) condition, and Ozawa property in the standard representation is stable under free products. Finally, we get another approach to the main result by applying Ozawa property and techniques in graph C*-algebras.