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In this note, we derive some consequences of the von Neumann algebrauniqueness theorems developed in a previous paper (see arXiv:1207.6741v1). Inparticular,  1) we solvein a paper of Futamura, Kataoka, and Kishimoto, by proving that ifA is a separable simple nuclear C*-algebra and for \pi_1 and \pi_2 are type IIIrepresentations of A on a separable Hilbert space, then for \pi_1 and \pi_2being algebraically equivalent, it is necessary and sufficient that there is anautomorphism \alpha of A such that \pi_1 composed with \alpha, and \pi_2 arequasi-equivalent.  2) we give a new (short) proof of the equivalence of injectivity and extremeamenability (of the corresponding unitary group) for countably decomposableproperly infinite von Neumann algebras.  3) using ideas of Pestov, we show that the Connes embedding problem isequivalent to many topological groups having the Kirchberg property.

Some consequences of von Neumann algebra uniqueness