Classifying $^*$-homomorphisms I: Unital simple nuclear $C^*$-algebras

José R. Carrión; James Gabe; Christopher Schafhauser; Aaron Tikuisis; Stuart White

Classifying $^$-homomorphisms I: Unital simple nuclear $C^$-algebras

Operator Algebras 2023-12-25 v3 K-Theory and Homology

Authors: José R. Carrión , James Gabe , Christopher Schafhauser , Aaron Tikuisis , Stuart White

Abstract

We classify the unital embeddings of a unital separable nuclear $C^*$ -algebra satisfying the universal coefficient theorem into a unital simple separable nuclear $C^*$ -algebra that tensorially absorbs the Jiang--Su algebra. This gives a new and essentially self-contained proof of the stably finite case of the unital classification theorem: unital simple separable nuclear $C^*$ -algebras that absorb the Jiang--Su algebra tensorially and satisfy the universal coefficient theorem are classified by Elliott's invariant of $K$ -theory and traces.

Keywords

cuntz semigroup c*-algebras graph algebras

Cite

@article{arxiv.2307.06480,
  title  = {Classifying $^*$-homomorphisms I: Unital simple nuclear $C^*$-algebras},
  author = {José R. Carrión and James Gabe and Christopher Schafhauser and Aaron Tikuisis and Stuart White},
  journal= {arXiv preprint arXiv:2307.06480},
  year   = {2023}
}

Comments

132 pages; added some remarks on the classification of C*-dynamics at the end of section 9 and corrected some typos