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.