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residual_finiteness

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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. [BCF17]

Known examples

Known non-examples

Relation with other properties

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

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

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

Further reading

  • Nathanial P. Brown and Narutaka Ozawa, C*-algebras and Finite-Dimensional Approximations, American Mathematical Society, 2008.
  • Amaury Freslon, Applications of Noncrossing Partitions to Quantum Groups, lecture notes, 2019. Available on-line.

References

[Chi15] Alexandru Chirvasitu, 2015. Residually finite quantum group algebras. Journal of Functional Analysis, 268(11), pp.3508–3533.
[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.
[Sol05] Piotr M. Sołtan, 2005. Quantum Bohr compactification. Illinois Journal of Mathematics, 49(4), Duke University Press, pp.1245–1270.
residual_finiteness.1567766515.txt.gz · Last modified: 2021/11/23 11:56 (external edit)

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