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 $\varphi ,\psi :\mathcal{A}\to \mathcal{M}$ be essential trivial extensions with $\varphi (a)-\psi (a)\in {\mathcal{K}}_{\mathcal{M}}$ for all $a\in \mathcal{A}$ such that either both $\varphi$ and $\psi$ (and hence $\mathcal{A}$) are unital or both $\varphi$ and $\psi$ have large complement. Then $\varphi$ and $\psi$ are properly asymptotically unitarily equivalent if and only if $[\varphi ,\psi {]}_{CS}=0$ in $KK(\mathcal{A},\mathcal{C}(S{\mathcal{K}}_{\mathcal{M}}))$.

22 pages