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

TL;DR

This paper proves uniqueness theorems for nuclear maps from certain $C^*$-algebras into ultraproducts of finite von Neumann factors, with implications for quasidiagonal approximations and maps into II$_1$ factors.

Contribution

It introduces a $KK$-uniqueness theorem tailored to the setting of nuclear maps into ultraproducts of finite von Neumann factors, extending classification results.

Findings

Uniqueness up to unitary conjugacy for maps into ultraproducts of finite von Neumann factors.

Approximate unitary equivalence of maps into II$_1$ factors based on trace agreement.

Application to quasidiagonal approximations of $C^*$-algebras.

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 $(\mathcal M,\tau_{\calM})$ 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.