A note on proper asymptotic uniqueness for semifinite factors

Abstract

Let $\mathcal{A}$ be a separable nuclear C*-algebra, and let $\mathcal{M}$ be a semifinite von Neumann factor with separable predual. Let $ϕ, ψ: \mathcal{A} \rightarrow \mathcal{M}$ be essential trivial extensions with $ϕ(a) - ψ(a) \in \mathcal{K}_{\mathcal{M}}$ for all $a \in \mathcal{A}$ such that either both $ϕ$ and $ψ$ (and hence $\mathcal{A}$) are unital or both $ϕ$ and $ψ$ have large complement. Then $ϕ$ and $ψ$ are properly asymptotically unitarily equivalent if and only if $[ϕ, ψ]_{CS} = 0$ in $KK(\mathcal{A}, \mathcal{C}(S \mathcal{K}_{\mathcal{M}}))$.