Some consequences of von Neumann algebra uniqueness

TL;DR

This paper explores the implications of von Neumann algebra uniqueness theorems, providing new results on algebraic equivalence of representations, a proof relating injectivity and extreme amenability, and connections to the Connes embedding problem.

Contribution

It offers novel insights into von Neumann algebra theory, including a characterization of algebraic equivalence of type III representations and a new proof linking injectivity with extreme amenability.

Findings

Algebraic equivalence of type III representations characterized by automorphisms.

New proof of injectivity and extreme amenability equivalence.

Connes embedding problem linked to topological group properties.

Abstract

In this note, we derive some consequences of the von Neumann algebra uniqueness theorems developed in a previous paper (see arXiv:1207.6741v1). In particular, 1) we solvein a paper of Futamura, Kataoka, and Kishimoto, by proving that if A is a separable simple nuclear C*-algebra and for \pi_1 and \pi_2 are type III representations of A on a separable Hilbert space, then for \pi_1 and \pi_2 being algebraically equivalent, it is necessary and sufficient that there is an automorphism \alpha of A such that \pi_1 composed with \alpha, and \pi_2 are quasi-equivalent. 2) we give a new (short) proof of the equivalence of injectivity and extreme amenability (of the corresponding unitary group) for countably decomposable properly infinite von Neumann algebras. 3) using ideas of Pestov, we show that the Connes embedding problem is equivalent to many topological groups having the Kirchberg property.