Some consequences of von Neumann algebra uniqueness

In this note, we derive some consequences of the von Neumann algebra uniqueness theorems developed in the previous paper (Ciuperca et al. [2]). In particular:(1)We solve a question raised in Futamura et al. (2003) [6], by proving that if A is a separable simple nuclear C⁎-algebra and πi, i=1,2, are 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 α of A such that π1∘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 and Uspenskij (2006) [14], we show that the Connes embedding problem is equivalent to many topological groups having the Kirchberg property.