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