Actions of Compact Quantum Groups III Reduced and universal actions - PowerPoint PPT Presentation

Actions of Compact Quantum Groups III Reduced and universal actions Kenny De Commer (VUB, Brussels, Belgium)

Universal actions Hilbert modules Reduced actions Outline Universal actions Hilbert modules Reduced actions

Universal actions Hilbert modules Reduced actions Universal completions Proposition (H. Li) α � G . Then O G ( X ) admits universal C ∗ -completion C 0 ( X u ) . Let X

Universal actions Hilbert modules Reduced actions Proof ◮ Take T ∈ Mor( π, α ) . ◮ Let { e i } o.n. basis of H π , δ π ( e i ) = � j e j ⊗ u ji . ◮ Then T ( e j ) T ( e k ) ∗ ⊗ u ji u ∗ � � T ( e i ) T ( e i ) ∗ � � α = ki i i,j,k T ( e j ) T ( e j ) ∗ ⊗ 1 . � = j i T ( e i ) T ( e i ) ∗ ∈ C 0 ( X / G ) , so ◮ Hence x T = � λ ∗ -representation of O G ( X ) . � λ ( T ( e i )) � ≤ � x T � , ◮ O G ( X ) = span { Tξ | π, T ∈ Mor( π, α ) , ξ ∈ H π } , so � a � = sup { λ ( a ) | λ ∗ -representation } < ∞ . ∀ a ∈ O G ( X ) ,

Universal actions Hilbert modules Reduced actions Universal coaction Theorem (H. Li) α Let X � G . Then α alg extends to injective right coaction α u : C 0 ( X u ) → C 0 ( X u ) ⊗ C ( G u ) . Moreover ◮ C 0 ( Y u ) = C 0 ( Y ) , ◮ O G u ( X u ) = O G ( X ) . Remark: We will use the corresponding result for X = G .

Universal actions Hilbert modules Reduced actions Proof (Part I) (Cf. universal construction first lecture.) ◮ Existence α u : trivial. ◮ Coaction property: trivial. ◮ Density condition: via Hopf algebra theory (antipode) α ( O G ( X ))(1 ⊗ O ( G )) = O G ( X ) ⊗ alg O ( G ) . ◮ Injectivity: counit extends to C ( G u ) .

Universal actions Hilbert modules Reduced actions Proof (Part II) ◮ Let λ u : C 0 ( X u ) → C 0 ( X ) , λ u : C ( G u ) → C ( G ) . ◮ Then ( λ u ⊗ λ u ) ◦ α u = α ◦ λ u . ◮ Hence: λ u ( C 0 ( X u )) π = C 0 ( X ) π . ◮ To show: λ u injective on each C 0 ( X u ) π . ◮ If a n → b ∈ C 0 ( Y u ) with a n ∈ O G ( X ) , b n = E Y ( a n ) = (id ⊗ ϕ ) α ( a n ) → (id ⊗ ϕ ) α u ( b ) = b. But b n ∈ C 0 ( Y ) C ∗ -algebra, so b ∈ C 0 ( Y ) . ◮ Assume a ∈ C 0 ( X u ) π , λ u ( a ) = 0 . Then 0 = α ( λ u ( a ∗ a )) = ( λ u ⊗ λ u ) α u ( a ∗ a ) ∈ C ( X u ) ⊗ alg O ( G ) . ◮ Apply (id ⊗ ϕ ) , E Y u ( a ∗ a ) E Y u ( a ∗ a ) = 0 . � � λ u = 0 ⇒ ◮ But E Y u faithful on O G u ( X u ) , so a = 0 .

Universal actions Hilbert modules Reduced actions Right C ∗ -algebra valued inner products Definition Let C 0 ( X ) C ∗ -algebra. Let Γ( E ) (unital) right C 0 ( X ) -module. Right C 0 ( X ) -valued inner product on Γ( E ) : � · , · � : Γ( E ) × Γ( E ) → C 0 ( X ) , ( s, t ) → � s, t � s.t. ◮ � · , · � linear in second, anti-linear in first argument. ◮ � s, ta � = � s, t � a , ◮ � s, t � ∗ = � t, s � , ◮ � s, s � ≥ 0 , ◮ � s, s � = 0 ⇒ s = 0 .

Universal actions Hilbert modules Reduced actions Right pre-Hilbert modules Definition Right pre-Hilbert C 0 ( X ) -module: ◮ C ∗ -algebra C 0 ( X ) , ◮ right C 0 ( X ) -module Γ( E ) ◮ right C 0 ( X ) -valued inner product on Γ( E ) . Lemma If Γ( E ) right pre-Hilbert C 0 ( X ) -module, then norm � s � = �� s, s �� 1 / 2 , s ∈ Γ( E ) . Definition Γ( E ) right Hilbert C 0 ( X ) -module if Γ( E ) complete. � Γ( E ) Right pre-Hilbert ⇒ completion Γ( E ) Hilbert.

Universal actions Hilbert modules Reduced actions Hilbert bundles Example (Classical bundles) X compact Hausdorff space, E ։ π X locally trivial Hilbert bundle: ◮ E is locally compact Hausdorff space, ◮ each E x = π − 1 ( x ) is finite dimensional Hilbert space. ◮ the map E × X E = { ( e, f ) | π ( e ) = π ( f ) } → C , ( e, f ) �→ � e, f � is continuous. ◮ E is locally trivial: π − 1 ( U ) ∼ = U × C n , Then Γ( E ) = Γ( E ) = { continuous sections X → E } is Hilbert C ( X ) -module by ( sf )( x ) = s ( x ) f ( x ) , � s, t � ( x ) = � s ( x ) , t ( x ) � .

Universal actions Hilbert modules Reduced actions Example Example (Trivial Hilbert modules) Let C 0 ( X ) C ∗ -algebra, I set. Γ( E ) = l 2 ( I, C 0 ( X )) � a ∗ = { ( a i ) i ∈ I | i a i norm-convergent } i is Hilbert C 0 ( X ) -module by     a 1 a 1 a a 2 a 2 a  · a = � s, t � = s ∗ t.  ,   . . . . . .

Universal actions Hilbert modules Reduced actions Tensor product with Hilbert space Example (Tensor product with Hilbert space) Let ◮ C 0 ( X ) C ∗ -algebra, ◮ Γ( E ) right Hilbert C 0 ( X ) -module, ◮ H Hilbert space. Then right pre-Hilbert C 0 ( X ) -module Γ( E ) ⊗ alg H with inner product � s ⊗ ξ, t ⊗ η � = � ξ, η �� s, t � . ⇒ Completion Γ( E ) ⊗ H . When Γ( E ) = C 0 ( X ) and H = l 2 ( I ) , l 2 ( I, C 0 ( X )) ∼ = C 0 ( X ) ⊗ l 2 ( I ) .

Universal actions Hilbert modules Reduced actions Hilbert modules from conditional expectations Example Let C 0 ( X ) C ∗ -algebra. Let E Y faithful conditional expectation, E Y : C 0 ( X ) → C 0 ( Y ) ⊆ C 0 ( X ) . Then C 0 ( X ) pre-Hilbert C 0 ( Y ) -module by � a, b � Y = E Y ( a ∗ b ) . Remark: For E Y not faithful: first divide out submodule { a ∈ C 0 ( X ) | E Y ( a ∗ a ) = 0 } . Notation L 2 Y ( X ) : completed Hilbert C 0 ( Y ) -module of ( C 0 ( X ) , � · , · � Y )

Universal actions Hilbert modules Reduced actions Adjointable maps Definition Let Γ( E ) and Γ( F ) Hilbert C 0 ( X ) -modules. Linear map T : Γ( E ) → Γ( F ) adjointable if ∃ T ∗ : Γ( F ) → Γ( E ) s.t. � s, Tt � = � T ∗ s, t � , ∀ s, t. Then L (Γ( E ) , Γ( F )) = { T : Γ( E ) → Γ( F ) | T adjointable } .

Universal actions Hilbert modules Reduced actions Properties of adjointable maps Lemma ◮ Adjointable maps are bounded ( ⇐ Banach-Steinhaus). ◮ T adjointable ⇒ T module map, T ( ξa ) = T ( ξ ) a . ◮ L (Γ( E ) , Γ( F )) is a Banach space. ◮ L (Γ( E )) = L (Γ( E ) , Γ( E )) is C ∗ -algebra. ◮ U : Γ( E ) → Γ( F ) surjective linear isometry iff U ∈ L (Γ( E ) , Γ( F )) and unitary. Remark: U linear isometry � U ∈ L (Γ( E ) , Γ( F )) .

Universal actions Hilbert modules Reduced actions Left Hilbert modules Definition Left pre-Hilbert C 0 ( X ) -module: ◮ left C 0 ( X ) -module Γ( E ) , ◮ left C 0 ( X ) -valued inner product on Γ( E ) , ◮ � · , · � linear in first, anti-linear in second argument. ◮ � as, t � = a � s, t � , ◮ � s, t � ∗ = � t, s � , ◮ � s, s � ≥ 0 , ◮ � s, s � = 0 ⇒ s = 0 .

Universal actions Hilbert modules Reduced actions Examples Example For E Y conditional expectation, � x, y � Y = E Y ( xy ∗ ) . Example Let Γ( E ) right Hilbert C ( X ) -module. Then Γ( E ∗ ) = Γ( E ) ∗ = L (Γ( E ) , C ( X )) = { L ∗ ξ : η �→ � ξ, η � | ξ ∈ Γ( E ) } left Hilbert C 0 ( X ) -module by � L, M � = LM ∗ , ( aL )( s ) = a ( L ( s )) , where we use L ( C ( X )) ∼ = C ( X ) by T �→ T (1 X ) .

Universal actions Hilbert modules Reduced actions An equivariance property Lemma α Let X � G with Y = X / G . Then ( E Y ⊗ id) α ( a ) = E Y ( a ) ⊗ 1 , a ∈ C 0 ( X ) . Proof. We have ( E Y ⊗ id) α ( a ) = (id ⊗ ϕ ⊗ id)(( α ⊗ id) α ( a )) = (id ⊗ ( ϕ ⊗ id) ◦ ∆))( α ( a )) = (id ⊗ ϕ ) α ( a ) ⊗ 1 G = E Y ( a ) ⊗ 1 G .

Universal actions Hilbert modules Reduced actions The implementing unitary Lemma α � G with Y = X / G . Then Let X O G ( X ) ⊗ alg O ( G ) → O G ( X ) ⊗ alg O ( G ) , a ⊗ g �→ α ( a )(1 ⊗ g ) completes to a unitary map U α : L 2 Y ( X ) ⊗ L 2 ( G ) → L 2 Y ( X ) ⊗ L 2 ( G ) . Proof. ◮ Isometric: � α ( a )(1 ⊗ g ) , α ( b )(1 ⊗ h ) � = ( E Y ⊗ ϕ )((1 X ⊗ g ∗ ) α ( a ∗ b )(1 X ⊗ h )) ϕ ( g ∗ h ) E Y ( a ∗ b ) = = � a ⊗ g, b ⊗ h � . ◮ Surjective: range dense by algebraic surjectivity.

Universal actions Hilbert modules Reduced actions The reduced C ∗ -algebra Lemma The non-degenerate ∗ -homomorphisms π red : C 0 ( X ) → L ( L 2 π red : C ( G ) → B ( L 2 ( G )) Y ( X )) , by left multiplication satisfy ( π red ⊗ π red )( α ( a )) = U α ( π red ( a ) ⊗ 1) U ∗ α . Moreover, π red is injective on O G ( X ) . Proof. ◮ π red well-defined and non-degenerate: basic (positivity E Y ). ◮ U α implements α : check on O G ( X ) . ◮ E Y faithful on O G ( X ) , so π red injective on O G ( X ) .

Universal actions Hilbert modules Reduced actions The reduced coaction Theorem (H. Li) α Let X � G , C 0 ( X red ) = π red ( C 0 ( X )) , C ( G red ) = π red ( C ( G )) . Then α red : C 0 ( X red ) → C 0 ( X red ) ⊗ C ( G red ) ⊆ L ( L 2 Y ( X ) ⊗ L 2 ( G )) , a �→ U α ( π red ( a ) ⊗ 1) U ∗ α α red defines injective right coaction X red � G red . Moreover, O G red ( X red ) = O G ( X ) and C 0 ( Y red ) = C 0 ( Y ) .

Universal actions Hilbert modules Reduced actions Proof ◮ U α implements coaction α red : check on O G ( X ) . ◮ α red injective: obvious. ◮ For λ red : C 0 ( X ) → C 0 ( X red ) , ( λ red ⊗ λ red ) ◦ α = α red ◦ λ red . ◮ O G red ( X red ) = O G ( X ) : faithfulness E Y on O G ( X ) . ◮ Hence C 0 ( Y red ) = C 0 ( Y ) .