Uniqueness for embeddings of nuclear $C^*$-algebras into type II$_{1}$ factors
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
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2026
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| _version_ | 1866909040815112192 |
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| author | Hua, Shanshan White, Stuart |
| author_facet | Hua, Shanshan White, Stuart |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_08779 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Uniqueness for embeddings of nuclear $C^*$-algebras into type II$_{1}$ factors Hua, Shanshan White, Stuart Operator Algebras 46L35, 46L05 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. |
| title | Uniqueness for embeddings of nuclear $C^*$-algebras into type II$_{1}$ factors |
| topic | Operator Algebras 46L35, 46L05 |
| url | https://arxiv.org/abs/2601.08779 |