Actions of Compact Quantum Groups I Definition Kenny De Commer - PowerPoint PPT Presentation

Actions of Compact Quantum Groups I Definition Kenny De Commer (VUB, Brussels, Belgium)

CQG Compact actions Non-compact actions Course material Material that will be treated: ◮ Actions and coactions of compact quantum groups. ◮ Actions on C ∗ -algebras and Hilbert modules. ◮ Crossed products. ◮ Free actions, ergodic actions, and their interrelationship.

CQG Compact actions Non-compact actions Outline Lecture I Compact quantum groups Actions of compact quantum groups on compact quantum spaces Actions on non-compact quantum spaces

CQG Compact actions Non-compact actions Compact quantum groups Definition (Woronowicz) Compact quantum group (CQG) G : ◮ unital C ∗ -algebra C ( G ) , ◮ unital ∗ -homomorphism, comultiplication ∆ : C ( G ) → C ( G ) ⊗ C ( G ) s.t. ◮ coassociativity: (∆ ⊗ id) ◦ ∆ = (id ⊗ ∆) ◦ ∆ , ◮ cancellation: [( C ( G ) ⊗ 1 G )∆( C ( G ))] = [∆( C ( G ))(1 G ⊗ C ( G ))] = C ( G ) ⊗ C ( G ) . Here: [ S ] = closed linear span of S (in some Banach space) .

CQG Compact actions Non-compact actions Classical CQG Lemma X, Y compact Hausdorff: C ( X ) ⊗ C ( Y ) ∼ = C ( X × Y ) , ( a ⊗ b )( x, y ) = a ( x ) b ( y ) . Example G compact Hausdorff group ⇒ CQG ( C ( G ) , ∆) , ∆ : C ( G ) → C ( G ) ⊗ C ( G ) , f �→ (∆( f ) : ( g, h ) �→ f ( gh )) . Conversely: CQG G with C ( G ) commutative ⇓ G = Spec( C ( G )) compact Hausdorff group .

CQG Compact actions Non-compact actions C ( G ) -corepresentations Definition Unitary C ( G ) -corepresentation: ◮ finite dimensional Hilbert space H , ◮ U ∈ B ( H ) ⊗ C ( G ) s.t. ◮ U unitary, ◮ (id ⊗ ∆)( U ) = U 12 U 13 , where U 12 = U ⊗ 1 etc. U ∈ B ( H ) ⊗ C ( G ) � δ : H → H ⊗ C ( G ) , ξ �→ U ( ξ ⊗ 1 G ) s.t. . . . ?

CQG Compact actions Non-compact actions G -representations Definition G compact quantum group. (Continuous finite dimensional unitary left) G -representation π : ◮ finite dimensional Hilbert space H π , ◮ linear map δ π : H π → H π ⊗ C ( G ) s.t. ◮ right comodule: (id ⊗ ∆) ◦ δ π = ( δ π ⊗ id) ◦ δ π , ◮ isometric: δ π ( ξ ) ∗ δ π ( η ) = � ξ, η � 1 C ( G ) , ◮ density: [ δ π ( H )(1 ⊗ C ( G ))] = H ⊗ C ( G ) . ◮ Here H π ∼ = B ( C , H π ) , so ( ξ ⊗ a ) ∗ ( η ⊗ b ) = ξ ∗ η ⊗ a ∗ b ∼ = � ξ, η � a ∗ b. ◮ Density condition automatically satisfied. ◮ C ( G ) -corepresentations ↔ G -representations.

CQG Compact actions Non-compact actions Classical representations Example Let G compact Hausdorff group. Then G -representations as compact quantum group � G -representations as compact group by δ π : H π → H π ⊗ C ( G ) ∼ = C ( G, H π ) � π : G × H π → H π , ( g, ξ ) �→ π ( g ) ξ = δ π ( ξ )( g ) .

CQG Compact actions Non-compact actions The canonical Hopf ∗ -algebra Theorem (Woronowicz) Let O ( G ) = { ( ξ ∗ ⊗ id) δ π ( η ) | π G -representation, ξ, η ∈ H π } . Then ◮ ( O ( G ) , ∆) Hopf ∗ -algebra, ( O ( G ) , ∆ , ǫ, S ) , ◮ O ( G ) dense in C ( G ) , ◮ ( O ( G ) , ∆) unique dense Hopf ∗ -algebra, ◮ δ π : H → H ⊗ O ( G ) is O ( G ) -comodule: ◮ (id ⊗ ∆) ◦ δ π = ( δ π ⊗ id) ◦ δ π , ◮ (id H ⊗ ǫ ) δ π = id H .

CQG Compact actions Non-compact actions Notation (Sweedler-Heynemann notation) h ∈ O ( G ) : ∆( h ) = h (1) ⊗ h (2) , (∆ ⊗ ι )∆( h ) = ∆ (2) ( h ) = h (1) ⊗ h (2) ⊗ h (3) , ... Example Let h ∈ O ( G ) . Then ∆( h (1) )(1 ⊗ S ( h (2) )) = h (1) ⊗ h (2) S ( h (3) ) = h (1) ⊗ ǫ ( h (2) )1 h ⊗ 1 . = Hence (Linear span) ∆( O ( G ))(1 ⊗ O ( G )) = O ( G ) ⊗ alg O ( G ) .

CQG Compact actions Non-compact actions Universal C ∗ -algebra Lemma G CQG. ◮ Universal C ∗ -envelope C ( G u ) of O ( G ) exists. ◮ CQG G u by ∆ u : C ( G u ) → C ( G u ) ⊗ C ( G u ) . Definition G u universal CQG (associated to G ).

CQG Compact actions Non-compact actions Right actions of compact quantum groups on C ∗ -algebras Definition (Podle´ s) Right action X � G : ◮ Compact quantum group G , ◮ C ∗ -algebra C ( X ) (with X ‘compact quantum space’), ◮ Unital ∗ -homomorphism, right coaction α : C ( X ) → C ( X ) ⊗ C ( G ) s.t. ◮ coaction property: ( α ⊗ id G ) ◦ α = (id X ⊗ ∆) ◦ α, ◮ density (Podle´ s condition): [ α ( C ( X ))(1 X ⊗ C ( G ))] = C ( X ) ⊗ C ( G ) .

CQG Compact actions Non-compact actions Right translations Example ∆ Let G compact quantum group. Then G � G by ∆ : C ( G ) → C ( G ) ⊗ C ( G ) .

CQG Compact actions Non-compact actions Half-classical case Lemma (All C ( G ) commutative) ◮ G compact Hausdorff group, ◮ C ∗ -algebra C ( X ) , α ◮ G � C ( X ) continuous action: ◮ ( g, a ) �→ α g ( a ) continuous, ◮ each α g ∗ -automorphism, ◮ α gh = α g ◦ α h , ◮ α e = id X , for e ∈ G identity element. ⇒ X � G , α : C ( X ) → C ( X ) ⊗ C ( G ) ∼ = C ( G, C ( X )) , a �→ ( α ( a ) : g �→ α g ( a )) .

CQG Compact actions Non-compact actions Proof, Part I ◮ Forgetting group structure: ◮ Using partitions of unity on G : ∼ = ◮ C ( X ) ⊗ C ( G ) → C ( G, C ( X )) by a ⊗ f �→ ( g �→ f ( g ) a ) . ◮ C ( X ) ⊗ C ( G ) ⊗ C ( G ) ∼ = C ( G × G, C ( X )) , etc. α ◮ continuous G � C ( X ) by unital ∗ -endomorphisms ⇔ α : C ( X ) → C ( G, C ( X )) unital ∗ -homomorphism. ◮ ((id ⊗ ∆) α )( a )( g, h ) = (( α ⊗ id) α )( a )( g, h ) ⇔ α gh ( a ) = α g ( α h ( a )) . Conclusion: one-to-one correspondence between ◮ α with coaction property, and ◮ actions of a group on a C ∗ -algebra by endomorphisms. To do : Density ⇔ α e = id C ( X ) for e unit G .

CQG Compact actions Non-compact actions Proof, Part II ◮ ∗ -homomorphism α : C ( X ) ⊗ C ( G ) → C ( X ) ⊗ C ( G ) , a ⊗ f �→ α ( a )(1 ⊗ f ) . � ◮ Density ⇔ � α surjective. ◮ On level of C ( G, C ( X )) ∼ = C ( X ) ⊗ C ( G ) : ∀ F ∈ C ( G, C ( X )) , α ( F )( g ) = α g ( F ( g )) . � α has inverse � ◮ Assume α e = id C ( X ) . Then � β , � β ( F )( g ) = α g − 1 ( F ( g )) . Hence range � α dense. ◮ If α e � = id C ( X ) ⇒ α e non-trivial idempotent ∗ -endomorphism. ◮ Put C ( X e ) = α e ( C ( X )) � = C ( X ) . ◮ ∀ g ∈ G : α g ( C ( X )) = α e ( α g ( C ( X ))) ⊆ C ( X e ) . ◮ ⇒ If a / ∈ C ( X e ) , then g �→ a not in range � α .

CQG Compact actions Non-compact actions Classical Example (All C ( G ) and C ( X ) commutative) G compact Hausdorff group, X compact Hausdorff space, X � G continuous ⇒ G � C ( X ) , α g ( f )( x ) = f ( x · g ) . Example Consider sphere � S N − 1 = { z = ( z 1 , . . . , z N ) ∈ R N | z 2 i = 1 } . i Then S N − 1 � O ( N ) by ( z, g ) �→ zg.

CQG Compact actions Non-compact actions Example: Half-classical I Example Cuntz algebras, � O n = C ∗ ( V 1 , . . . , V n | V ∗ V i V ∗ i V j = δ ij , i = 1) . i Then U ( n ) � O n by � α u ( V i ) = u ji V j . j In particular, S 1 � O n by α z ( V i ) = zV i .

CQG Compact actions Non-compact actions Example: Half-classical II Example (Banica) Free spheres, � C ( S N − 1 ) = < V 1 , . . . , V N | V i = V ∗ V 2 i , i = 1 } . + i Then O ( N ) � C ( S N − 1 ) by + � α g ( V i ) = g ji V j . j

CQG Compact actions Non-compact actions Left actions of compact quantum groups on C ∗ -algebras Definition (Podle´ s) Left action G � H : ◮ Compact quantum group G , ◮ C ∗ -algebra C ( X ) , ◮ Unital ∗ -homomorphism, left coaction α : C ( X ) → C ( G ) ⊗ C ( X ) s.t. ◮ coaction property: (id G ⊗ α ) ◦ α = (∆ ⊗ id X ) ◦ α, ◮ density: [( C ( G ) ⊗ 1 X ) α ( C ( X ))] = C ( X ) ⊗ C ( G ) .

CQG Compact actions Non-compact actions From left to right Definition Let G CQG. Then G op CQG by C ( G op ) = C ( G ) , ∆ G op = ∆ op G = ς ◦ ∆ , where ς : C ( G ) ⊗ C ( G ) → C ( G ) ⊗ C ( G ) , g ⊗ h �→ h ⊗ g. Lemma α op α � G op . ↔ G � X X

CQG Compact actions Non-compact actions Non-unital C ∗ -algebras Definition (Multiplier C ∗ -algebras) C 0 ( X ) non-unital C ∗ -algebra (‘locally compact quantum space’) . Multiplier C ∗ -algebra M ( C 0 ( X )) = C b ( X ) : ◮ Concrete: For C 0 ( X ) ⊆ B ( H ) with [ C 0 ( X ) H ] = H : C b ( X ) = { T ∈ B ( H ) | ∀ a ∈ C 0 ( X ) , Ta, aT ∈ C 0 ( X ) } . ◮ Abstract: C b ( X ) collection maps T : C 0 ( X ) → C 0 ( X ) s.t. ∃ T ∗ , ∀ a, b ∈ C 0 ( X ) , a ∗ ( Tb ) = ( T ∗ b ) ∗ a. If T ∈ C b ( X ) : T ( ab ) = T ( a ) b , and C 0 ( X ) ⊆ C b ( X ) . Example If X locally compact Hausdorff space, M ( C 0 ( X )) = C b ( X ) .

CQG Compact actions Non-compact actions Morphisms between locally compact quantum spaces Definition ∗ -homomorphism π : C 0 ( Y ) → M ( C 0 ( X )) non-degenerate: [ π ( C 0 ( Y )) C 0 ( X )] = C 0 ( X ) . Example Let X, Y locally compact Hausdorff spaces. ◮ Non-degenerate maps C 0 ( Y ) → C b ( X ) ⇔ continuous maps X → Y . ◮ Non-degenerate maps C 0 ( Y ) → C 0 ( X ) ⇔ continuous proper maps X → Y . ◮ Degenerate map C 0 ( Y ) → C b ( X ) : points of X to infinity. Lemma π : C 0 ( Y ) → C b ( X ) non-degenerate ⇒ ∃ ! π : C b ( Y ) → C b ( X ) .