This is an old revision of the document!

Summary of results on approximation properties for discrete QGs

More information including citations of original sources is available in the individual articles.

	$\hat U_N^+$	$\hat O_N^+$	$\hat H_N^+$	$\hat H_N^{s+}$	$\hat S_N^+$
	free CMQG duals (assuming $N$ high enough)
Kirchberg (F)	yes ¹⁾	yes ¹⁾	yes²⁾	yes²⁾	yes
Resid. finite	yes ¹⁾	yes ¹⁾	yes²⁾	yes²⁾	yes
Kazhdan (T)	no	no			no
Haagerup	yes	yes	yes	yes	yes
Amenable		$N=2$

¹⁾ Proven for $N\neq 3$

²⁾ Proven for $N\ge 4$

Further reading