approximation_properties_summary
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approximation_properties_summary [2019/10/04 08:41] d.gromada created |
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| More information including citations of original sources is available in the individual articles. | More information including citations of original sources is available in the individual articles. | ||
| - | | ^ free CMQG duals (assuming $N$ high enough) ^^^^^ | + | ===== Mutual relationships ===== |
| - | | ::: ^ $\hat U_N^+$ ^ $\hat O_N^+$ ^ $\hat H_N^+$ ^ $\hat H_N^{s+}$^$\hat S_N^+$^ | + | |
| - | ^Kirchberg (F) | yes [(t:FO)] | yes [(t:FO)] | yes[(t:FH)] | yes[(t:FH)] | yes | | + | Residually finite ⇒ property (F) |
| - | ^Resid. finite | yes [(t:FO)] | yes [(t:FO)] | yes[(t:FH)] | yes[(t:FH)] | yes | | + | |
| - | ^Kazhdan (T) | no | no | | | no | | + | Amenable ⇒ Haagerup |
| - | ^Haagerup | yes | yes | yes | yes | yes | | + | |
| - | ^Amenable | | $N=2$ | | | | | + | 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:FO>Proven for $N\neq 3$)] | ||
| [(t:FH>Proven for $N\ge 4$)] | [(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 ~~ | ~~REFNOTES t ~~ | ||
approximation_properties_summary.1570178508.txt.gz · Last modified: 2021/11/23 11:56 (external edit)
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