Kirchberg factorization property

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. [BW16]

Results

The following quantum groups have (F)

$\hat U_N^+$ , $\hat O_N^+$ for $N\neq 3$ [BW16]
$\hat S_N$ for any $N$ [BCF18]
$\hat H^{s+}_N$ for $N\ge 4$ and $1\le s\le\infty$ [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 [BBCW17]

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

$\Gamma$ is residually finite [BBCW17]

References

[BW16] Angshuman Bhattacharya, Shuzhou Wang, 2016. Kirchberg's factorization property for discrete quantum groups. Bulletin of the London Mathematical Society, 48(5), pp.866–876.

[BCF18] Michael Brannan, Alexandru Chirvasitu, Amaury Freslon, 2018. Topological generation and matrix models for quantum reflection groups.

[BBCW17] Angshuman Bhattacharya, Michael Brannan, Alexandru Chirvasitu, Shuzhou Wang, 2017. Kirchberg factorization and residual finiteness for discrete quantum groups.

Table of Contents

Kirchberg factorization property

Definition

Results

Relation with other properties

References