TOPOLOGICAL OBSTRUCTIONS C OLOURING S IMPLE C - ALGEBRAS S TUART W - PowerPoint PPT Presentation

C OLOURING S IMPLE C ˚ - ALGEBRAS Stuart White University of Glasgow and WWU Münster Abel Symposium 2015 BBSTWW := Bosa, Brown, Sato, Tikusis, W. , Winter C OLOURING S IMPLE C ˚ - ALGEBRAS S TUART W HITE (G LASGOW ) 1 / 7

C OLOURING VIA ORDER ZERO MAPS V IEW POINT Well behaved simple nuclear C ˚ -algebras exhibit “coloured” versions of properties of injective factors. infinite dim simple C ˚ -algebra A infinite dim factor M . D ˚ -hms M n Ñ M A could be projectionless ù U SE ORDER ZERO MAPS IN PLACE OF ˚ - HMS φ : A Ñ B is order zero if it is c.p. and preserves orthogonality. unital order zero maps are just ˚ -homomorphisms C OLOURING S IMPLE C ˚ - ALGEBRAS S TUART W HITE (G LASGOW ) 2 / 7

C OLOURING VIA ORDER ZERO MAPS V IEW POINT Well behaved simple nuclear C ˚ -algebras exhibit “coloured” versions of properties of injective factors. infinite dim simple C ˚ -algebra A infinite dim factor M . D ˚ -hms M n Ñ M A could be projectionless ù U SE ORDER ZERO MAPS IN PLACE OF ˚ - HMS φ : A Ñ B is order zero if it is c.p. and preserves orthogonality. unital order zero maps are just ˚ -homomorphisms C OLOURING : PROPERTY EXPRESSED WITH A UNITAL ˚ - HM Replace ˚ -hm by finitely many order zero maps, whose sum is unital (possibly replace exact statements by approximate statements). Number of summands ” number of colours C OLOURING S IMPLE C ˚ - ALGEBRAS S TUART W HITE (G LASGOW ) 2 / 7

C OLOURING VIA ORDER ZERO MAPS C OLOURING : PROPERTY EXPRESSED WITH A UNITAL ˚ - HM Replace ˚ -hm by finitely many order zero maps, whose sum is unital (possibly replace exact statements by approximate statements). Number of summands ” number of colours H YPERFINITENESS AND N UCLEAR D IMENSION VNA version One coloured version n -coloured version Hyperfinite AF C OLOURING S IMPLE C ˚ - ALGEBRAS S TUART W HITE (G LASGOW ) 2 / 7

� � C OLOURING VIA ORDER ZERO MAPS C OLOURING : PROPERTY EXPRESSED WITH A UNITAL ˚ - HM Replace ˚ -hm by finitely many order zero maps, whose sum is unital (possibly replace exact statements by approximate statements). Number of summands ” number of colours H YPERFINITENESS AND N UCLEAR D IMENSION VNA version One coloured version n -coloured version Hyperfinite AF Exist approximations id � M M ❆ ❆ ⑥ ❆ ⑥ ⑥ ❆ ⑥ ˚ hm ❆ ❆ ucp ❆ ⑥ ❆ ⑥ ⑥ F i C OLOURING S IMPLE C ˚ - ALGEBRAS S TUART W HITE (G LASGOW ) 2 / 7

� � � � C OLOURING VIA ORDER ZERO MAPS C OLOURING : PROPERTY EXPRESSED WITH A UNITAL ˚ - HM Replace ˚ -hm by finitely many order zero maps, whose sum is unital (possibly replace exact statements by approximate statements). Number of summands ” number of colours H YPERFINITENESS AND N UCLEAR D IMENSION VNA version One coloured version n -coloured version Hyperfinite AF Exist approximations Exist approximations id � M id � A M A ❆ ❄ ❆ ❄ ⑥ ⑧ ❆ ⑥ ❄ ⑧ ⑥ ❄ ❆ ⑥ ⑧ ˚ hm ❄ ˚ hm ❆ ❆ ❄ ucp ❆ ccp ❄ ⑥ ⑧ ❆ ❄ ⑥ ⑧ ⑧ ⑥ F i F i C OLOURING S IMPLE C ˚ - ALGEBRAS S TUART W HITE (G LASGOW ) 2 / 7

� � � � � � C OLOURING VIA ORDER ZERO MAPS C OLOURING : PROPERTY EXPRESSED WITH A UNITAL ˚ - HM Replace ˚ -hm by finitely many order zero maps, whose sum is unital (possibly replace exact statements by approximate statements). Number of summands ” number of colours H YPERFINITENESS AND N UCLEAR D IMENSION VNA version One coloured version n -coloured version Hyperfinite AF dim nuc p A q ď n ´ 1 Exist approximations Exist approximations Exist approximations id � M id � A id � A M A A ❆ ❄ ❄ ❆ ❄ ❄ ⑥ ⑧ ⑧ ❆ ⑥ ❄ ❄ ⑧ ⑧ ⑥ ⑧ ❄ ❄ ❆ ⑥ ⑧ ř n ´ 1 ˚ hm ❄ ˚ hm ❄ ❆ j “ 0 ccp ord 0 ❆ ❄ ❄ ucp ❆ ccp ❄ ccp ❄ ⑥ ⑧ ❆ ❄ ❄ ⑥ ⑧ ⑧ ⑧ ⑧ ⑥ F i F i F i C OLOURING S IMPLE C ˚ - ALGEBRAS S TUART W HITE (G LASGOW ) 2 / 7

� � � � � � C OLOURING VIA ORDER ZERO MAPS H YPERFINITENESS AND N UCLEAR D IMENSION VNA version One coloured version n -coloured version Hyperfinite AF dim nuc p A q ď n ´ 1 Exist approximations Exist approximations Exist approximations id id � A id � A � M M A A ❆ ❄ ❄ ❆ ❄ ❄ ⑥ ⑧ ⑧ ⑥ ❄ ❄ ⑧ ❆ ⑧ ⑥ ⑧ ❆ ❄ ❄ ⑥ ⑧ ř n ´ 1 ˚ hm ˚ hm ❆ ❄ ❄ j “ 0 ccp ord 0 ❆ ❄ ❄ ucp ❆ ccp ❄ ccp ❄ ⑥ ⑧ ❆ ❄ ❄ ⑥ ⑧ ⑧ ⑧ ⑧ ⑥ F i F i F i S OME EXAMPLES OF COLOURINGS Rohklin Theorems Rohklin Poperty Rohklin dim ú ú McDuff ( M – M b R ) A – A b UHF A – A b Z ú ú a.u. equivalence a.u. equivalence ? ú ú . . . . . . . . . ú ú C OLOURING S IMPLE C ˚ - ALGEBRAS S TUART W HITE (G LASGOW ) 2 / 7

C OLOURING VIA ORDER ZERO MAPS S OME EXAMPLES OF COLOURINGS hyperfinite AF dim nuc p A q ú ú Rohklin Theorems Rohklin Poperty Rohklin dim ú ú McDuff ( M – M b R ) A – A b UHF A – A b Z ú ú a.u. equivalence a.u. equivalence ? ú ú . . . . . . . . . ú ú Why? TOPOLOGICAL OBSTRUCTIONS C OLOURING S IMPLE C ˚ - ALGEBRAS S TUART W HITE (G LASGOW ) 2 / 7

C OLOURING VIA ORDER ZERO MAPS S OME EXAMPLES OF COLOURINGS hyperfinite AF dim nuc p A q ú ú Rohklin Theorems Rohklin Poperty Rohklin dim ú ú McDuff ( M – M b R ) A – A b UHF A – A b Z ú ú a.u. equivalence a.u. equivalence ? ú ú . . . . . . . . . ú ú Why? TOPOLOGICAL OBSTRUCTIONS C OLOURING S IMPLE C ˚ - ALGEBRAS S TUART W HITE (G LASGOW ) 2 / 7

C OLOURING VIA ORDER ZERO MAPS S OME EXAMPLES OF COLOURINGS hyperfinite AF dim nuc p A q ú ú Rohklin Theorems Rohklin Poperty Rohklin dim ú ú McDuff ( M – M b R ) A – A b UHF A – A b Z ú ú a.u. equivalence a.u. equivalence ? ú ú . . . . . . . . . ú ú Why? TOPOLOGICAL OBSTRUCTIONS C OLOURING S IMPLE C ˚ - ALGEBRAS S TUART W HITE (G LASGOW ) 2 / 7

C OLOURING VIA ORDER ZERO MAPS S OME EXAMPLES OF COLOURINGS hyperfinite AF dim nuc p A q ú ú Rohklin Theorems Rohklin Poperty Rohklin dim ú ú McDuff ( M – M b R ) A – A b UHF A – A b Z ú ú a.u. equivalence a.u. equivalence ? ú ú . . . . . . . . . ú ú Why? TOPOLOGICAL OBSTRUCTIONS C OLOURING S IMPLE C ˚ - ALGEBRAS S TUART W HITE (G LASGOW ) 2 / 7

C OLOURING VIA ORDER ZERO MAPS S OME EXAMPLES OF COLOURINGS hyperfinite AF dim nuc p A q ú ú Rohklin Theorems Rohklin Poperty Rohklin dim ú ú McDuff ( M – M b R ) A – A b UHF A – A b Z ú ú a.u. equivalence a.u. equivalence ? ú ú . . . . . . . . . ú ú Why? TOPOLOGICAL OBSTRUCTIONS C OLOURING S IMPLE C ˚ - ALGEBRAS S TUART W HITE (G LASGOW ) 2 / 7

C OLOURING VIA ORDER ZERO MAPS S OME EXAMPLES OF COLOURINGS hyperfinite AF dim nuc p A q ú ú Rohklin Theorems Rohklin Poperty Rohklin dim ú ú McDuff ( M – M b R ) A – A b UHF A – A b Z ú ú a.u. equivalence a.u. equivalence ? ú ú . . . . . . . . . ú ú Why? TOPOLOGICAL OBSTRUCTIONS C OLOURING S IMPLE C ˚ - ALGEBRAS S TUART W HITE (G LASGOW ) 2 / 7

C OLOURING VIA ORDER ZERO MAPS S OME EXAMPLES OF COLOURINGS hyperfinite AF dim nuc p A q ú ú Rohklin Theorems Rohklin Poperty Rohklin dim ú ú McDuff ( M – M b R ) A – A b UHF A – A b Z ú ú a.u. equivalence a.u. equivalence ? ú ú . . . . . . . . . ú ú Why? TOPOLOGICAL OBSTRUCTIONS C OLOURING S IMPLE C ˚ - ALGEBRAS S TUART W HITE (G LASGOW ) 2 / 7

C OLOURING VIA ORDER ZERO MAPS S OME EXAMPLES OF COLOURINGS hyperfinite AF dim nuc p A q ú ú Rohklin Theorems Rohklin Poperty Rohklin dim ú ú McDuff ( M – M b R ) A – A b UHF A – A b Z ú ú a.u. equivalence a.u. equivalence ? ú ú . . . . . . . . . ú ú Why? TOPOLOGICAL OBSTRUCTIONS C OLOURING S IMPLE C ˚ - ALGEBRAS S TUART W HITE (G LASGOW ) 2 / 7

C OLOURING VIA ORDER ZERO MAPS S OME EXAMPLES OF COLOURINGS hyperfinite AF dim nuc p A q ú ú Rohklin Theorems Rohklin Poperty Rohklin dim ú ú McDuff ( M – M b R ) A – A b UHF A – A b Z ú ú a.u. equivalence a.u. equivalence ? ú ú . . . . . . . . . ú ú Why? TOPOLOGICAL OBSTRUCTIONS C OLOURING S IMPLE C ˚ - ALGEBRAS S TUART W HITE (G LASGOW ) 2 / 7

C OLOURING VIA ORDER ZERO MAPS S OME EXAMPLES OF COLOURINGS hyperfinite AF dim nuc p A q ú ú Rohklin Theorems Rohklin Poperty Rohklin dim ú ú McDuff ( M – M b R ) A – A b UHF A – A b Z ú ú a.u. equivalence a.u. equivalence ? ú ú . . . . . . . . . ú ú Why? TOPOLOGICAL OBSTRUCTIONS C OLOURING S IMPLE C ˚ - ALGEBRAS S TUART W HITE (G LASGOW ) 2 / 7

C OLOURING VIA ORDER ZERO MAPS S OME EXAMPLES OF COLOURINGS hyperfinite AF dim nuc p A q ú ú Rohklin Theorems Rohklin Poperty Rohklin dim ú ú McDuff ( M – M b R ) A – A b UHF A – A b Z ú ú a.u. equivalence a.u. equivalence ? ú ú . . . . . . . . . ú ú Why? TOPOLOGICAL OBSTRUCTIONS C OLOURING S IMPLE C ˚ - ALGEBRAS S TUART W HITE (G LASGOW ) 2 / 7

C OLOURING VIA ORDER ZERO MAPS S OME EXAMPLES OF COLOURINGS hyperfinite AF dim nuc p A q ú ú Rohklin Theorems Rohklin Poperty Rohklin dim ú ú McDuff ( M – M b R ) A – A b UHF A – A b Z ú ú a.u. equivalence a.u. equivalence ? ú ú . . . . . . . . . ú ú Why? TOPOLOGICAL OBSTRUCTIONS C OLOURING S IMPLE C ˚ - ALGEBRAS S TUART W HITE (G LASGOW ) 2 / 7