====== Summary of results on approximation properties for discrete QGs ======

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

===== Mutual relationships =====

Residually finite ⇒ property (F)

Amenable ⇒ Haagerup

Property (T) and (F) ⇒ Residually finite

Property (T) and Haagerup ⇔ finite

===== Stability results =====

|                                     ^                 Products       ^^^^ Subgroups ^ Quotients ^ Dual top. gen. ^
| :::                                 ^ Direct ^ Free ^ Wreath ^ Free wr. ^ :::       ^ :::       ^ :::            ^
^[[kirchberg_property|Kirchberg (F)]] |        |      |        |          |           |           |                |
^[[residual_finiteness|Resid. finite]]|        |      |        |          |           |           | yes            |
^[[kazhdan_property|Kazhdan (T)]]     |        |      |        |          |           | yes       |                |
^[[haagerup_property|Haagerup]]       |        | yes  |        |          | yes       |           |                |
^[[amenability|Amenable]]             | yes    | no   |        |          | yes       |           |                |


===== Concrete examples =====

|                                     ^              free CMQG duals (assuming $N$ high enough)                ^^^^^    half-lib. duals         ^^      non-unimod.              ^^  Groups  ^
| :::                                 ^ $\hat U_N^+$  ^ $\hat O_N^+$  ^ $\hat H_N^+$ ^ $\hat H_N^{s+}$^$\hat S_N^+$^ $\hat O_N^*$ ^ $\hat H_N^*$ ^ $\hat U_F^+$   ^ $\hat O_F^+$  ^          ^
^[[kirchberg_property|Kirchberg (F)]] | yes [(t:FO)]  | yes [(t:FO)]  | yes[(t:FH)]  | yes[(t:FH)]  | yes          |              |              |                |               |
^[[residual_finiteness|Resid. finite]]| yes [(t:FO)]  | yes [(t:FO)]  | yes[(t:FH)]  | yes[(t:FH)]  | yes          |              |              |         no                    ||
^[[kazhdan_property|Kazhdan (T)]]     | no            | no            | no           | no           | no           | no           |              |         no                    ||
^[[haagerup_property|Haagerup]]       | yes           | yes           | yes          | yes          | yes          | yes          |              | yes            | yes           |          |
^[[amenability|Amenable]]             | no            | $N=2$         |              |              | $N=4$        | yes          |              | no             | $N=2$         |          |

[(t:FO>Proven for $N\neq 3$)]
[(t:FH>Proven for $N\ge 4$)]

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).

~~REFNOTES t ~~

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


  * Michael Brannan. //[[https://arxiv.org/abs/1605.01770|Approximation properties for locally compact quantum groups]]//, 2016.
  * Nathanial P. Brown and Narutaka Ozawa, //C*-algebras and Finite-Dimensional Approximations//, [[https://bookstore.ams.org/gsm-88|American Mathematical Society]], 2008.