### Abstract

The relative commutant A ′ ∩A U A′∩AU of a strongly self-absorbing algebra A is indistinguishable from its ultrapower A U AU. This applies both to the case when A is the hyperfinite II 1 1 factor and to the case when it is a strongly self-absorbing C ∗ C∗-algebra. In the latter case, we prove analogous results for ℓ ∞ (A)/c 0 (A) ℓ∞(A)/c0(A) and reduced powers corresponding to other filters on N N. Examples of algebras with approximately inner flip and approximately inner half-flip are provided, showing the optimality of our results. We also prove that strongly self-absorbing algebras are smoothly classifiable, unlike the algebras with approximately inner half-flip.

Original language | English |
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Pages (from-to) | 363–387 |

Number of pages | 25 |

Journal | Selecta Mathematica |

Volume | 23 |

Issue number | 1 |

Early online date | 29 Apr 2016 |

DOIs | |

Publication status | Published - 1 Jan 2017 |

### Keywords

- central sequence algebra
- relative commutant
- approximately inner half-flip
- continuous model theory
- Strongly self-absorbing C ∗

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## Cite this

Farah, I., Hart, B., Rordam, M., & Tikuisis, A. (2017). Relative commutants of strongly self-absorbing C∗-algebras.

*Selecta Mathematica*,*23*(1), 363–387. https://doi.org/10.1007/s00029-016-0237-y