residual_finiteness
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| ===== Definition ===== | ===== 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. | + | 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 $\pi(a)\neq 0$. | 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. | + | A discrete quantum group $\Gamma$ is called **residually finite** if $\C\Gamma$ is RFD. |
| ===== C*-algebraic results ===== | ===== C*-algebraic results ===== | ||
| * Any abelian C*-algebra is RFD | * 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 | * Every type I C*-algebra with a faithful tracial state is RFD | ||
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| ==== Stability 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 ==== | ==== Known examples ==== | ||
| - | * $\hat S_N$ for any $N$ (see [[free symmetric quantum group|free quantum permutation groups]]) [(ref:BCF18)] | + | * $\hat U_N^+$ 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)] | * $\hat H^{s+}_N$ 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 ==== | ==== Known non-examples ==== | ||
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| If $\Gamma=\hat G$ is residually finite, then | If $\Gamma=\hat G$ is residually finite, then | ||
| - | * $\Gamma$ has property (F) [(ref:BBCW17)] | + | * $\Gamma$ has [[kirchberg_property|property (F)]] [(ref:BBCW17)] |
| - | * $L^\infty(G)$ has the Connes embedding property [(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 | Quantum group $\Gamma=\hat G$ is residually finite if | ||
| - | * $\Gamma$ has property (T) and property (F) [(ref:BBCW17)] | + | * $\Gamma$ has [[kazhdan_property|property (T)]] and [[kirchberg_property|property (F)]] [(ref:BBCW17)] |
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| )] | )] | ||
| + | [(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 ~~ | ~~REFNOTES ref ~~ | ||
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