Uniqueness for embeddings of nuclear C*-algebras into type II_1 factors

Journal article

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 norm-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_kn )_n of matrix algebras, we obtain a uniqueness result for quasidiagonal approximations of A. Secondly, when (M, τM) is a II₁ factor, a pair ϕ, ψ : A → M of unital, injective and nuclear maps are norm approximately unitarily equivalent if and only if τM ◦ ϕ = τM ◦ ψ.

The main strategy is to use Schafhauser’s classification of lifts along the trace-kernel extension from [72]. Since our codomains may lack the tensorial absorption properties needed in [72], 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 ([56]).

Authors: Hua, S; White, S

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Elsevier
• Journal: Advances in Mathematics
• Volume: 499
• Article number: 111032
• Publication date: 2026-05-21
• Acceptance date: 2026-05-14
• DOI: 10.1016/j.aim.2026.111032
• EISSN: 1090-2082
• ISSN: 0001-8708
• Language: English
• Keywords: C*-algebras; classification; II1 factors; K1-injectivity; KK-uniqueness