Some consequences of von Neumann algebra uniqueness

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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.