Injective von Neumann Algebras

Abstract

We have finally come to the first highlight of the entire book. Both sections are devoted to the proof of the equivalence of the injectivity and the approximately finite dimensionality of a von Neumann algebra, Corollary 1.8 and Theorem 1.9. The implication: the AFD ⇒ the injectivity for a von Neumann algebra is an easy consequence of averaging over the unitary groups of generating finite dimensional subalgebras. Thus, the hard part is the reverse implication: the injectivity ⇒ the AFD. Thanks to the preparation in the last chapter, Chapter XV, the proof of this implication for a properly infinite von Neumann algebra M is relatively easy as the injectivity implies that the identity map of M is approximable by a sequence of completely positive maps of finite ranks and we know the form of each approximating CP maps of finite rank. The hard part is the case of type II₁. It requires a considerable patience of the reader. As it is such an important result in the theory of operator algebras, we will present second approach for the type II₁ case in §2.