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

Shanshan Hua, Stuart White

TL;DR

This work establishes uniqueness up to unitary conjugacy for unital, full, and nuclear maps from a separable, exact $C^*$-algebra $A$ with the UCT into ultraproducts of finite von Neumann factors, and specializes to II$_1$-factors. The authors develop a KK- and KL- uniqueness framework tailored to trace–kernel extensions, using a new $K_1$-injectivity result for Paschke duals under relatively purely large extensions, and then apply Schafhauser–style lifting to obtain unitary equivalence from trace and total $K$-theory data. The approach allows passing to ultraproducts and separable subextensions, yielding a novel uniqueness theorem for quasidiagonal approximations and a norm-approximate unitary equivalence result for embeddings into II$_1$ factors. Central to the method are the trace–kernel extension, a relatively purely large ideal in the multiplier algebra, and a $KL_{ ext{nuc}}$-based obstruction vanishing grounded in the universal multicoefficient theorem. The results significantly advance embeddings into II$_1$- and ultraproduct factors and provide robust tools for quasidiagonality and operator-algebra classification in this setting.

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 $τ_{\calM}\circφ=τ_{\calM}\circψ$. 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.

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

TL;DR

This work establishes uniqueness up to unitary conjugacy for unital, full, and nuclear maps from a separable, exact -algebra with the UCT into ultraproducts of finite von Neumann factors, and specializes to II-factors. The authors develop a KK- and KL- uniqueness framework tailored to trace–kernel extensions, using a new -injectivity result for Paschke duals under relatively purely large extensions, and then apply Schafhauser–style lifting to obtain unitary equivalence from trace and total -theory data. The approach allows passing to ultraproducts and separable subextensions, yielding a novel uniqueness theorem for quasidiagonal approximations and a norm-approximate unitary equivalence result for embeddings into II factors. Central to the method are the trace–kernel extension, a relatively purely large ideal in the multiplier algebra, and a -based obstruction vanishing grounded in the universal multicoefficient theorem. The results significantly advance embeddings into II- and ultraproduct factors and provide robust tools for quasidiagonality and operator-algebra classification in this setting.

Abstract

Let be a separable, unital and exact -algebra satisfying the universal coefficient theorem. We prove uniqueness theorems up to unitary conjugacy for unital, full and nuclear maps from into ultraproducts of finite von Neumann factors: any two such maps agreeing on traces and total -theory are unitarily equivalent. There are two consequences. Firstly if one takes the factors to be a sequence of matrix algebras, we obtain a uniqueness result for quasidiagonal approximations of . Secondly, when is a II factor, a pair of unital, injective and nuclear maps are norm approximately unitarily equivalent if and only if . 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 -uniqueness theorem tailored to our situation. This is inspired by -uniqueness theorems of Loreaux, Ng and Sutradhar.
Paper Structure (16 sections, 50 theorems, 86 equations)

This paper contains 16 sections, 50 theorems, 86 equations.

Table of Contents

  1. Preliminaries
  2. Multiplier algebras of $C^*$-algebras with real rank zero
  3. Cuntz equivalence
  4. $KK$-theory
  5. Total $K$-theory and the universal coefficient theorem
  6. Absorption and uniqueness theorems for $KK$-theory
  7. Sufficient condition for $K_{1}$-injectivity
  8. Codomain hypotheses, the trace--kernel extension and separabilisation arguments
  9. A relatively purely large ideal in $\mathcal{M}(J)$
  10. $K_{1}$-injectivity of relative commutants
  11. Outline of the proof of Lemma \ref{['main']}
  12. Proof of Lemma \ref{['main']}
  13. Uniqueness theorems
  14. Main uniqueness theorem
  15. Uniqueness of embeddings into ultraproducts of finite factors
  16. ...and 1 more sections

Key Result

Theorem 1

Let $A$ be a $C^*$-algebra and let $\mathcal{H}$ be a Hilbert space. Let $\phi,\psi\colon A\to\mathcal{B}(\mathcal{H})$ be non-degenerate $^*$-homomorphisms. Then $\phi$ and $\psi$ are approximately unitarily equivalent if and only if $\phi(a)$ and $\psi(a)$ have the same rank in $\mathcal{B}(\mathc

Theorems & Definitions (78)

  • Theorem : Hadwin's formulation of Voiculescu's theorem
  • Theorem A
  • Theorem : Ding--Hadwin, Hadwin--Li--Liu; Uniqueness for maps from ASH-algebras to II$_1$ factors
  • Theorem : A consequence of Gabe's treatment of Kirchberg's work
  • Theorem : Li--Shen--Shi; Uniqueness for full maps into II$_{\infty}$-factors
  • Theorem : Hadwin--Li--Liu; Uniqueness for maps into from ASH-algebras to II$_{\infty}$-factors
  • Theorem : A folklore consequence of Connes' theorem
  • Theorem : Ciuperca, Giordano, Ng and Niu
  • Theorem B
  • Theorem : Schafhauser's uniqueness of embeddings into $\mathcal{Q}_\omega$
  • ...and 68 more