Residual finiteness

Residual finiteness

Definition

A $*$ -algebra $A$ is called residually finite-dimensional (RFD for short) if there exists a set $\{\pi_i\}_{i\in I}$ of representations $\pi_i\colon A\to M_{n_i}(\C)$ such that $\bigoplus_{i\in I}\pi_i\colon A\to \prod_{i\in I}M_{n_i}(\C)$ is faithful.

Equivalently, $A$ is RFD if finite-dimensional representations separate its points. That is, for every $a\in A$ , there is a finite-dimensional representation $\pi$ such that $\pi(a)\neq 0$ .

A discrete quantum group $\Gamma$ is called residually finite if $\C\Gamma$ is RFD.

C*-algebraic results

Any abelian C*-algebra is RFD
Any abelian $*$ -algebra that embeds into some C*-algebra is RFD
Every type I C*-algebra with a faithful tracial state is RFD

Quantum group results

Stability results

If $\hat G$ and $\hat H$ are finitely generated and residually finite, then $\widehat{\langle G,H\rangle}$ is residually finite. [BCF18]

Known examples

$\hat U_N^+$ and $\hat O_N^+$ for $N\neq 3$ [Chi15]
$\hat S_N$ 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)

Known non-examples

Relation with other properties

If $\Gamma=\hat G$ is residually finite, then

$\Gamma$ has property (F) [BBCW17]
$\Gamma$ has the Connes embedding property [BBCW17]
$\Gamma$ is unimodular [Sol05]

Quantum group $\Gamma=\hat G$ is residually finite if

$\Gamma$ has property (T) and property (F) [BBCW17]

References

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

[Chi15] Alexandru Chirvasitu, 2015. Residually finite quantum group algebras. Journal of Functional Analysis, 268(11), pp.3508–3533.

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

[Sol05] Piotr M. Sołtan, 2005. Quantum Bohr compactification. Illinois Journal of Mathematics, 49(4), Duke University Press, pp.1245–1270.

Table of Contents