Uniqueness for embeddings of nuclear $C^*$-algebras into type II$_{1}$ factors

Abstract

Let $A$ be a separable, unital and exact $C^*$-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 $(\mathcal M,τ_{\calM})$ is a II$_1$ factor, a pair $ϕ,ψ:A\to\mathcal M$ of unital, injective and nuclear maps are norm approximately unitarily equivalent if and only if $τ_{\mathcal M}\circϕ=τ_{\mathcal M}\circψ$. 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.