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KK-rigidity of simple nuclear C*-algebras

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arxiv 2408.02745 v2 pith:ZZLO3UCB submitted 2024-08-05 math.OA

classification math.OA
keywords algebrasnuclearsimpleunitalequivalentisomorphicmathcalonly
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abstract

It is shown that if $A$ and $B$ are unital separable simple nuclear $\mathcal Z$-stable C$^*$-algebras and there is a unital embedding $A \rightarrow B$ which is invertible on $KK$-theory and traces, then $A \cong B$. In particular, two unital separable simple nuclear $\mathcal Z$-stable C$^*$-algebras which either have real rank zero or unique trace are isomorphic if and only if they are homotopy equivalent. It is further shown that two finite strongly self-absorbing C$^*$-algebras are isomorphic if and only if they are $KK$-equivalent in a unit-preserving way.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. A unified approach for classifying simple nuclear $C^\ast$-algebras

    math.OA 2024-12 conditional novelty 8.0 of 10

    A new proof of the Kirchberg–Phillips theorem that unifies the stably finite and purely infinite classification frameworks and recovers Kirchberg's Geneva theorems as corollaries.

  2. A homotopy rigidity theorem for $\mathcal{Z}_0$-stable $\mathrm{C}^\ast$-algebras

    math.OA 2025-05 conditional novelty 7.0 of 10

    Trace-preserving homotopy equivalence implies isomorphism for simple, separable, nuclear Z0-stable C*-algebras, without assuming the UCT.

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