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Summary of results on approximation properties for discrete QGs

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Mutual relationships

Stability results

	Direct	Free	Subgroups	Quotients	Dual top. gen.
	Products		Subgroups	Quotients	Dual top. gen.
Kirchberg (F)
Resid. finite					yes
Kazhdan (T)				yes
Haagerup		yes	yes
Amenable		no	yes

Concrete examples

	$\hat U_N^+$	$\hat O_N^+$	$\hat H_N^+$	$\hat H_N^{s+}$	$\hat S_N^+$	$\hat O_N^*$	$\hat U_F^+$	$\hat O_F^+$
	free CMQG duals (assuming $N$ high enough)					half-lib. duals	non-unimod.		Groups
Kirchberg (F)	yes ¹⁾	yes ¹⁾	yes²⁾	yes²⁾	yes
Resid. finite	yes ¹⁾	yes ¹⁾	yes²⁾	yes²⁾	yes		no
Kazhdan (T)	no	no	no	no	no	no	no
Haagerup	yes	yes	yes	yes	yes	yes	yes	yes
Amenable	no	$N=2$			$N=4$	yes	no	$N=2$

In addition, any finite quantum group satisfies all the listed approximation properties. Any Abelian discrete quantum group (dual of a compact group) satisfies all the listed approximation properties except for property (T) (Abelian discrete QG has (T) iff it is finite).

¹⁾ Proven for $N\neq 3$

²⁾ Proven for $N\ge 4$

Table of Contents

Summary of results on approximation properties for discrete QGs

Mutual relationships

Stability results

Concrete examples

Further reading

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Table of Contents

Summary of results on approximation properties for discrete QGs

Mutual relationships

Stability results

Concrete examples

Further reading

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