From injectivity to approximation properties for von Neumann algebras

Abstract: A von Neumann algebra M is called injective if there is a projection P:B(H) -> M with ||P||= 1. This is the analogue for von Neumann algebras of amenability for discrete groups, and it notoriously fails when M = M(F) is the von Neumann algebra of a non-commutative free group F. We will introduce the class of ''seemingly injective'' von Neumann algebras. This includes M(F). We show that M is seemingly injective iff it has the (matricial) weak* positive metric approximation property (AP in short). This is parallel to Connes's characterization of injectivity by the weak* completely positive AP. We show that M(F) is isomorphic to B(H) as Banach spaces when F is countable. Lastly we discuss several open questions that might be related to Kazhdan's property (T) for groups.

operator algebras

Audience: researchers in the topic

Conference on operator algebras and related topics in Istanbul, 2021