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Residual finiteness

Definition

A C*-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
Every type I C*-algebra with a faithful tracial state is RFD

Quantum group results

Stability results

Known examples

$\hat S_N$ for any $N$ (see free quantum permutation groups) [BCF18]
$\hat H^{s+}_N$ for $N\ge 4$ and $1\le s\le\infty$ [BCF18]

Known non-examples

Relation with other properties

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

$\Gamma$ has property (F) [BBCW17]
$L^\infty(G)$ has the Connes embedding property [BBCW17]

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.

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

Table of Contents

Residual finiteness

Definition

C*-algebraic results