====== Kirchberg factorization property ======

===== Definition =====

Let $G$ be a compact quantum group. The discrete dual $\hat G$ is said to have the **Kirchberg factorization property** or **property (F)** if the Haar state $h$ on $C(G)$ is amenable. [(ref:BW16)]

===== Results =====

The following quantum groups have (F)

  * $\hat U_N^+$, $\hat O_N^+$ for $N\neq 3$ [(ref:BW16)]
  * $\hat S_N$ for any $N$ [(ref:BCF18)]
  * $\hat H^{s+}_N$ for $N\ge 4$ and $1\le s\le\infty$ [(ref:BCF18)]
  * Any Abelian discrete quantum group (i.e. dual of a compact group)

===== Relation with other properties =====

If $\Gamma=\hat G$ has (F), then

    * $\Gamma$ has the Connes embedding property [(ref:BBCW17)]

Discrete quantum group $\Gamma=\hat G$ has (F) if

  * $\Gamma$ is [[residual_finiteness|residually finite]] [(ref:BBCW17)]

===== References =====
[(ref:BW16>>
author    : Angshuman Bhattacharya, Shuzhou Wang
title     : Kirchberg's factorization property for discrete quantum groups
journal   : Bulletin of the London Mathematical Society
volume    : 48
number    : 5
pages     : 866–876
year      : 2016
doi {10.1112/blms/bdw048},
url       : https://doi.org/10.1112/blms/bdw048
)]

[(ref:BBCW17>>
title     : Kirchberg factorization and residual finiteness for discrete quantum groups
year      : 2017
author    : Angshuman Bhattacharya, Michael Brannan, Alexandru Chirvasitu, Shuzhou Wang
url       : https://arxiv.org/abs/1712.08682
)]

[(ref:BCF18>>
title     : Topological generation and matrix models for quantum reflection groups
year      : 2018
author    : Michael Brannan, Alexandru Chirvasitu, Amaury Freslon
url       : https://arxiv.org/abs/1808.08611
)]

[(ref:Chi15>>
title     : Residually finite quantum group algebras
journal   : Journal of Functional Analysis
volume    : 268
number    : 11
pages     : 3508–3533
year      : 2015
doi       : https://doi.org/10.1016/j.jfa.2015.01.013
url       : http://www.sciencedirect.com/science/article/pii/S0022123615000373
author    : Alexandru Chirvasitu
)]

[(ref:Sol05>>
author    : Piotr M. Sołtan
doi       : 10.1215/ijm/1258138137
journal   : Illinois Journal of Mathematics
number    : 4
pages     : 1245–1270
publisher : Duke University Press
title     : Quantum Bohr compactification
url       : https://doi.org/10.1215/ijm/1258138137
volume    : 49
year      : 2005
)]

~~REFNOTES ref ~~