2012/08/01 by Thierry Giordano, Giordano, Thierry, Ping W. Ng +1 · 1 citation
Mathematics · #46L05 #46L10 #FOS: Mathematics #Operator Algebras (math.OA) #math.OA #msc:46L05 #msc:46L10
paper · pdf · doi:10.48550/arxiv.1208.0120
arxiv created 2012/08/01 · arxiv updated 2012/08/02
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 π1 and π2 are type III representations of A on a separable Hilbert space, then for π1 and π2 being algebraically equivalent, it is necessary and sufficient that there is an automorphism αof A such that π1 composed with α, and π2 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.