====== 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. [(ref:BCF18)]
==== Known examples ====

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

==== Known non-examples ====


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

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

  * $\Gamma$ has [[kirchberg_property|property (F)]] [(ref:BBCW17)]
  * $\Gamma$ has the Connes embedding property [(ref:BBCW17)]
  * $\Gamma$ is [[unimodularity|unimodular]] [(ref:Sol05)]

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

  * $\Gamma$ has [[kazhdan_property|property (T)]] and [[kirchberg_property|property (F)]] [(ref:BBCW17)]


===== Further reading =====

  * Nathanial P. Brown and Narutaka Ozawa, //C*-algebras and Finite-Dimensional Approximations//, [[https://bookstore.ams.org/gsm-88|American Mathematical Society]], 2008.
  * Amaury Freslon, //Applications of Noncrossing Partitions to Quantum Groups//, lecture notes, 2019. [[https://www.math.u-psud.fr/~freslon/Documents/Talks/Copenhaguen_2019.pdf|Available on-line]].

===== References =====


[(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 ~~