Uniqueness for embeddings of nuclear $C^{\ast }$-algebras into type II${}_1$ factors

Let $A$ be a separable, unital and exact $C^{\ast }$-algebra satisfying the universal coefficient theorem. We prove uniqueness theorems up to unitary conjugacy for unital, full and nuclear maps from $A$ into ultraproducts of finite von Neumann factors: any two such maps agreeing on traces and total $K$-theory are unitarily equivalent. There are two consequences. Firstly if one takes the factors to be a sequence $(M_{k_n}{)}_n$ of matrix algebras, we obtain a uniqueness result for quasidiagonal approximations of $A$. Secondly, when $(\mathcal{M},{\tau }_{\text{calM}})$ is a II${}_1$ factor, a pair $\phi ,\psi :A\to \mathcal{M}$ of unital, injective and nuclear maps are norm approximately unitarily equivalent if and only if ${\tau }_{\mathcal{M}}\circ \phi ={\tau }_{\mathcal{M}}\circ \psi$. The main strategy is to use Schafhauser’s classification of lifts along the trace–kernel extension. Since our codomains may lack the tensorial absorption properties needed in this work, the main new ingredient is a suitable $KK$-uniqueness theorem tailored to our situation. This is inspired by $KK$-uniqueness theorems of Loreaux, Ng and Sutradhar.