  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 \pi_1 and \pi_2 are type III
representations of A on a separable Hilbert space, then for \pi_1 and \pi_2
being algebraically equivalent, it is necessary and sufficient that there is an
automorphism \alpha of A such that \pi_1 composed with \alpha, and \pi_2 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.
