### Abstract

K

K-theory. Finally, we use the equivariant Cuntz semigroup to classify locally representable actions on direct limits of one-dimensional NCCW-complexes, generalizing work of Handelman and Rossmann.

Original language | English |
---|---|

Pages (from-to) | 189-241 |

Number of pages | 53 |

Journal | Proceedings of the London Mathematical Society |

Volume | 114 |

Issue number | 2 |

Early online date | 11 Jan 2017 |

DOIs | |

Publication status | Published - Feb 2017 |

### Fingerprint

### Keywords

- 46L55 (primary)
- 46L35
- 46L80 (secondary)

### Cite this

*Proceedings of the London Mathematical Society*,

*114*(2), 189-241. https://doi.org/10.1112/plms.12001

**The equivariant Cuntz semigroup.** / Gardella, Eusebio ; Santiago, Luis.

Research output: Contribution to journal › Article

*Proceedings of the London Mathematical Society*, vol. 114, no. 2, pp. 189-241. https://doi.org/10.1112/plms.12001

}

TY - JOUR

T1 - The equivariant Cuntz semigroup

AU - Gardella, Eusebio

AU - Santiago, Luis

N1 - Funded by SFB 878 Groups, Geometry and Actions Humboldt Foundation

PY - 2017/2

Y1 - 2017/2

N2 - We introduce an equivariant version of the Cuntz semigroup that takes an action of a compact group into account. The equivariant Cuntz semigroup is naturally a semimodule over the representation semiring of the given group. Moreover, this semimodule satisfies a number of additional structural properties. We show that the equivariant Cuntz semigroup, as a functor, is continuous and stable. Moreover, cocycle conjugate actions have isomorphic associated equivariant Cuntz semigroups. One of our main results is an analog of Julg's theorem: the equivariant Cuntz semigroup is canonically isomorphic to the Cuntz semigroup of the crossed product. We compute the induced semimodule structure on the crossed product, which in the abelian case is given by the dual action. As an application of our results, we show that freeness of a compact Lie group action on a compact Hausdorff space can be characterized in terms of a canonically defined map into the equivariant Cuntz semigroup, extending results of Atiyah and Segal for equivariant KK-theory. Finally, we use the equivariant Cuntz semigroup to classify locally representable actions on direct limits of one-dimensional NCCW-complexes, generalizing work of Handelman and Rossmann.

AB - We introduce an equivariant version of the Cuntz semigroup that takes an action of a compact group into account. The equivariant Cuntz semigroup is naturally a semimodule over the representation semiring of the given group. Moreover, this semimodule satisfies a number of additional structural properties. We show that the equivariant Cuntz semigroup, as a functor, is continuous and stable. Moreover, cocycle conjugate actions have isomorphic associated equivariant Cuntz semigroups. One of our main results is an analog of Julg's theorem: the equivariant Cuntz semigroup is canonically isomorphic to the Cuntz semigroup of the crossed product. We compute the induced semimodule structure on the crossed product, which in the abelian case is given by the dual action. As an application of our results, we show that freeness of a compact Lie group action on a compact Hausdorff space can be characterized in terms of a canonically defined map into the equivariant Cuntz semigroup, extending results of Atiyah and Segal for equivariant KK-theory. Finally, we use the equivariant Cuntz semigroup to classify locally representable actions on direct limits of one-dimensional NCCW-complexes, generalizing work of Handelman and Rossmann.

KW - 46L55 (primary)

KW - 46L35

KW - 46L80 (secondary)

U2 - 10.1112/plms.12001

DO - 10.1112/plms.12001

M3 - Article

VL - 114

SP - 189

EP - 241

JO - Proceedings of the London Mathematical Society

JF - Proceedings of the London Mathematical Society

SN - 0024-6115

IS - 2

ER -