Actions of Compact Quantum Groups II Examples, spectral components - PowerPoint PPT Presentation

Actions of Compact Quantum Groups II Examples, spectral components and Podle´ s algebra Kenny De Commer (VUB, Brussels, Belgium)

Examples Coinvariants Isotypical components Algebraic actions Outline Examples of compact quantum group actions The C ∗ -algebra of coinvariants for an action Isotypical components Algebraic actions

Examples Coinvariants Isotypical components Algebraic actions Recall: actions on locally compact quantum spaces Definition Right action X � G : ◮ Compact quantum group G , ◮ C ∗ -algebra C 0 ( X ) , ◮ non-degenerate ∗ -homomorphism, right coaction α : C 0 ( X ) → C 0 ( X ) ⊗ C ( G ) s.t. ◮ coaction property, ( α ⊗ id G ) ◦ α = (id X ⊗ ∆) ◦ α, ◮ density: [ α ( C 0 ( X ))(1 X ⊗ C ( G ))] = C 0 ( X ) ⊗ C ( G ) .

Examples Coinvariants Isotypical components Algebraic actions Actions from representations Example Let G CQG, π G -representation. Then G � B ( H π ) , Ad π : B ( H π ) → B ( H π ) ⊗ C ( G ) , ξη ∗ �→ δ π ( ξ ) δ π ( η ) ∗ = U π ( ξη ∗ ⊗ 1) U ∗ π . For G classical Hausdorff group, π representation: (Ad π ) g ( x ) = π ( g ) xπ ( g ) ∗ , x ∈ B ( H π ) .

Examples Coinvariants Isotypical components Algebraic actions Universal C ∗ -envelopes Definition Let O ( X ) ∗ -algebra. Then O ( X ) admits universal C ∗ -envelope if � x � u = sup {� λ ( x ) � | λ non-degenerate ∗ -representation O ( X ) → B ( H λ ) } < ∞ . Then � �� � � C 0 ( X u ) ∼ Im O ( X ) → B ( H λ ) . = λ Remark: ◮ O ( X ) → C 0 ( X u ) not necessarily injective... ◮ C 0 ( X u ) could be zero! Examples ◮ Examples: C u ( G ) , O n , C ( S N − 1 ) , . . . + ◮ Non-example: C [ x ] , x ∗ = x .

Examples Coinvariants Isotypical components Algebraic actions C ∗ -algebraic actions from algebraic actions Lemma Let O ( X ) ∗ -algebra with Hopf ∗ -algebraic coaction α : O ( X ) → O ( X ) ⊗ alg O ( G ) . Assume O ( X ) admits a universal C ∗ -envelope. Then α extends to coaction α u : C 0 ( X u ) → C 0 ( X u ) ⊗ C ( G u ) . Hopf ∗ -algebraic coaction: ◮ ( α ⊗ id) ◦ α = (id ⊗ ∆) ◦ α , ◮ (id X ⊗ ǫ ) α = id X .

Examples Coinvariants Isotypical components Algebraic actions Proof ◮ Existence α u as ∗ -homomorphism: clear by universality. ◮ α u coaction: ◮ Coaction property: clear by continuity. ◮ Density: ◮ Write α ( a ) = a (0) ⊗ a (1) , (id ⊗ ∆) α ( a ) = a (0) ⊗ a (1) ⊗ a (2) , . . . ◮ Then α ( a (0) )(1 ⊗ S ( a (1) )) = a (0) ⊗ a (1) S ( a (2) ) = a (0) ⊗ ǫ ( a (1) )1 = a ⊗ 1 . ◮ Hence α ( O ( X ))(1 X ⊗ O ( G )) = O ( X ) ⊗ alg O ( G ) . ◮ Hence [ α u ( C 0 ( X u ))(1 ⊗ C ( G u ))] = C 0 ( X u ) ⊗ C ( G u ) .

Examples Coinvariants Isotypical components Algebraic actions Actions from representations II Definition Let H finite dimensional Hilbert space. Cuntz C ∗ -algebra O ( H ) , ◮ H ⊆ O ( H ) linearly, ◮ ξ ∗ η = � ξ, η � , ◮ � i ξ i ξ ∗ i = 1 for { ξ i } o.n. basis. Example O n = O ( C n ) . Example Let G CQG, π G -representation. Then action G � O ( H π ) , α π : O ( H π ) → O ( H π ) ⊗ C ( G ) , ξ �→ δ π ( ξ ) .

Examples Coinvariants Isotypical components Algebraic actions Liberated and free Definition (Wang) Universal C ∗ -algebra C ( O + N ) , C ∗ ( u ij | 1 ≤ i, j ≤ N, u ∗ ij = u ij and U = ( u ij ) i,j unitary ) is CQG by � ∆( u ij ) = u ik ⊗ u kj . k Example S N − 1 � O + N by + � α ( V i ) = V j ⊗ u ji . j

Examples Coinvariants Isotypical components Algebraic actions Half-classical revisited Definition (Wang) Universal C ∗ -algebra C (Sym + n ) = C ( O + ( n )) / < u ij − u 2 ij > is CQG by � ∆( u ij ) = u ik ⊗ u kj . k Example Let X n = { 1 , 2 , . . . , n } . Then X n � Sym + n , � α : C ( X n ) → C ( X n ) ⊗ C (Sym + n ) , δ i �→ δ j ⊗ u ji . j

Examples Coinvariants Isotypical components Algebraic actions Actions of discrete group duals Example (C ∗ -algebraic bundles and Γ -graded C ∗ -algebras) ◮ Γ discrete group. ◮ Banach spaces A g with ‘associative’ contractive multiplication A g × A h → A gh . ◮ ∗ : A g → A g − 1 antilinear, ‘involutive’, isometric. ◮ � b ∗ b � = � b � 2 for b ∈ A g , ◮ b ∗ b ≥ 0 in (the C ∗ -algebra) A e for b ∈ A g . Then � Γ � A = universal C ∗ -envelope ⊕ g A g , α : A → A ⊗ C ∗ (Γ) , a g �→ a g ⊗ λ g .

Examples Coinvariants Isotypical components Algebraic actions The C ∗ -algebra of coinvariants Definition α � G . Space of orbits: Y = X / G , given by C ∗ -algebra Let X C 0 ( Y ) = { a ∈ C 0 ( X ) | α ( a ) = a ⊗ 1 G } . Examples α ◮ G � C 0 ( X ) C 0 ( Y ) = C 0 ( X ) G = { a ∈ C 0 ( X ) | α g ( a ) = a for all g ∈ G } . α ◮ X � G : C 0 ( X ) G = { G -constant continuous functions on X } ∼ { continuous functions on X/G } . =

Examples Coinvariants Isotypical components Algebraic actions Intertwiners and fixed point subalgebras Definition (Space of intertwiners) Let π 1 and π 2 G -representations. Then Mor( π 1 , π 2 ) = { T : H 1 → H 2 | δ 2 ◦ T = ( T ⊗ id) ◦ δ 1 } ⊆ B ( H 1 , H 2 ) . Lemma ◮ T ∈ Mor( π 1 , π 2 ) , T ′ ∈ Mor( π 2 , π 3 ) ⇒ T ′ ◦ T ∈ Mor( π 1 , π 3 ) . ◮ T ∈ Mor( π 1 , π 2 ) ⇒ T ∗ ∈ Mor( π 2 , π 1 ) . Example B ( H π ) α π = Mor( π, π ) , α π ( ξη ∗ ) = δ π ( ξ ) δ π ( η ) ∗ = U π ( ξη ∗ ⊗ 1) U ∗ π .

Examples Coinvariants Isotypical components Algebraic actions Commuting actions Definition β α Let X � G and H � X . Commutation of α and β : ( β ⊗ id G ) α = (id G ⊗ α ) β. Example β α Assume X � G and H � X commute. Then H � X / G , β | C ( X / G ) : C ( X / G ) → C ( H ) ⊗ C ( X / G ) .

Examples Coinvariants Isotypical components Algebraic actions Invariant functionals on CQG Theorem (Woronowicz) G compact quantum group. ∃ ! state ϕ on C ( G ) , Haar state, s.t. (id ⊗ ϕ )∆( f ) = ( ϕ ⊗ id)∆( f ) = ϕ ( f )1 G , ∀ f ∈ C ( G ) . For G compact Hausdorff group, µ Haar (probability) measure , � ϕ ( f ) = f ( g )d µ ( g ) . G Lemma ϕ faithful on O ( G ) : ϕ ( h ∗ h ) = 0 ∀ h ∈ O ( G ) , ⇒ h = 0 . In fact: if h ∈ O ( G ) positive in C ( G ) and ϕ ( h ) = 0 , then h = 0 .

Examples Coinvariants Isotypical components Algebraic actions Conditional expectation Lemma (Integration over fibers) α � G , Y = X / G . Conditional expectation onto C 0 ( Y ) X E Y : C 0 ( X ) → C 0 ( X ) , a �→ (id ⊗ ϕ ) α ( a ) , ◮ range C 0 ( Y ) , ◮ idempotent, ◮ completely positive, ◮ bimodular: E Y ( bac ) = bE Y ( a ) c, a ∈ C 0 ( X ) , b, c ∈ C 0 ( Y ) , ◮ non-degenerate: [ C 0 ( X ) C 0 ( Y )] = C 0 ( X ) . Non-degeneracy: ‘Every point of X is in an orbit (point of Y )’.

Examples Coinvariants Isotypical components Algebraic actions Examples Examples α ◮ G � C 0 ( X ) � E Y ( a ) = α g ( a )d µ ( g ) . G α ◮ X � G : integration over orbits, � E Y ( f )( xG ) = f ( xg )d µ ( g ) . G Example S 1 � O n : � S 1 z N − M ( V i 1 . . . V i N V ∗ E Y ( V i 1 . . . V i N V ∗ j 1 . . . V ∗ j 1 . . . V ∗ j M ) = j M )d z δ M,N V i 1 . . . V i N V ∗ j 1 . . . V ∗ = j N .

Examples Coinvariants Isotypical components Algebraic actions Proof (of properties E Y ) ◮ Range ⊆ C 0 ( Y ) : � � α ( E Y ( a )) = α (id ⊗ ϕ ) α ( a ) = (id ⊗ id ⊗ ϕ )(( α ⊗ id) α ( a )) = (id ⊗ id ⊗ ϕ )((id ⊗ ∆) α ( a )) = (id ⊗ ϕ )( α ( a )) ⊗ 1 G = E Y ( a ) ⊗ 1 G . ◮ Trivially, E Y ( b ) = b for b ∈ C 0 ( Y ) . ◮ Trivially, E Y completely positive (state + ∗ -homs c.p.). ◮ Trivially, E Y C 0 ( Y ) -bimodular. ◮ Non-degenerate: u α bounded approximate unit C 0 ( X ) , ∀ b ∈ C 0 ( X ) , E Y ( u α ) b = (id ⊗ ϕ )( α ( u α )( b ⊗ 1)) → b, since b ⊗ 1 ∈ [ α ( C 0 ( X ))(1 ⊗ C ( G ))] .

Examples Coinvariants Isotypical components Algebraic actions Results on G -representations Definition ◮ π indecomposable: π ≇ π 1 ⊕ π 2 . ◮ π irreducible: T ∈ Mor( π ′ , π ) , T ∗ T = id ⇒ TT ∗ = id or 0 . Proposition Let G compact quantum group. ◮ G -representation indecomposable ⇔ irreducible. ◮ G -representation ∼ = direct sum irreducibles.

Examples Coinvariants Isotypical components Algebraic actions Isotypical components Definition α X � G , π G -representation. Intertwiner space Mor( π, α ) : Mor( π, α ) = { T : H π → C 0 ( X ) | α ( Tξ ) = ( T ⊗ id) δ π ( ξ ) } . π irreducible: π -isotypical component (or π -spectral subspace) C 0 ( X ) π = { Tξ | ξ ∈ H π , T ∈ Mor( π, α ) } ⊆ C 0 ( X ) . Note: Each C 0 ( X ) π is C 0 ( Y ) -bimodule.

Examples Coinvariants Isotypical components Algebraic actions The Podle´ s subalgebra Theorem (Podle´ s) α � G . Then ∗ -algebra (unital if X compact) Let X O G ( X ) = linear span { C 0 ( X ) π | π irreducible } ⊆ C 0 ( X ) . Definition O G ( X ) Podle´ s subalgebra of C 0 ( X ) .