Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Mapping Ideals A mapping ideal O assigns O ( X , Y ) ⊆ CB ( X , Y ) to each ( X , Y ) s.t. ∀ ϕ ∈ M n ( O ( X , Y )), 1 � ϕ � cb ≤ � ϕ � O , and 2 for any linear mappings r : V → X and s : Y → W , � s n ◦ ϕ ◦ r � O ≤ � s � cb � ϕ � O � r � cb . Examples : CB ( X , Y ), Γ 2 , r ( X , Y ). When range is a dual space: CB ( X , Y ∗ ) = ( X � ⊗ Y ) ∗
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Mapping Ideals A mapping ideal O assigns O ( X , Y ) ⊆ CB ( X , Y ) to each ( X , Y ) s.t. ∀ ϕ ∈ M n ( O ( X , Y )), 1 � ϕ � cb ≤ � ϕ � O , and 2 for any linear mappings r : V → X and s : Y → W , � s n ◦ ϕ ◦ r � O ≤ � s � cb � ϕ � O � r � cb . Examples : CB ( X , Y ), Γ 2 , r ( X , Y ). When range is a dual space: CB ( X , Y ∗ ) = ( X � ⊗ Y ) ∗ = ( X ⊗ h Y ) ∗ Γ 2 , r ( X , Y ∗ ) ∼
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Mapping Ideals A mapping ideal O assigns O ( X , Y ) ⊆ CB ( X , Y ) to each ( X , Y ) s.t. ∀ ϕ ∈ M n ( O ( X , Y )), 1 � ϕ � cb ≤ � ϕ � O , and 2 for any linear mappings r : V → X and s : Y → W , � s n ◦ ϕ ◦ r � O ≤ � s � cb � ϕ � O � r � cb . Examples : CB ( X , Y ), Γ 2 , r ( X , Y ). When range is a dual space: CB ( X , Y ∗ ) = ( X � ⊗ Y ) ∗ = ( X ⊗ h Y ) ∗ ∼ = X ∗ ⊗ w ∗ h Y ∗ Γ 2 , r ( X , Y ∗ ) ∼
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Mapping Ideals Completely nuclear mappings N ( X , Y ) is the image of ⊗ Y → X ∗ ⊗ ∨ Y ⊆ CB ( X , Y ) X ∗ � with the quotient norm ν ( ϕ ).
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Mapping Ideals Completely nuclear mappings N ( X , Y ) is the image of ⊗ Y → X ∗ ⊗ ∨ Y ⊆ CB ( X , Y ) X ∗ � with the quotient norm ν ( ϕ ). ϕ : X → Y is completely integral if ι ( ϕ ) := sup { ν ( ϕ | E ) | E ⊆ X finite dimensional } < ∞ . We let I ( X , Y ) denote the completely integral mappings.
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Mapping Ideals Completely nuclear mappings N ( X , Y ) is the image of ⊗ Y → X ∗ ⊗ ∨ Y ⊆ CB ( X , Y ) X ∗ � with the quotient norm ν ( ϕ ). ϕ : X → Y is completely integral if ι ( ϕ ) := sup { ν ( ϕ | E ) | E ⊆ X finite dimensional } < ∞ . We let I ( X , Y ) denote the completely integral mappings. = ( X ⊗ ∨ Y ) ∗ . Y is locally reflexive iff I ( X , Y ∗ ) ∼
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Mapping Ideals Completely nuclear mappings N ( X , Y ) is the image of ⊗ Y → X ∗ ⊗ ∨ Y ⊆ CB ( X , Y ) X ∗ � with the quotient norm ν ( ϕ ). ϕ : X → Y is completely integral if ι ( ϕ ) := sup { ν ( ϕ | E ) | E ⊆ X finite dimensional } < ∞ . We let I ( X , Y ) denote the completely integral mappings. = ( X ⊗ ∨ Y ) ∗ . Y is locally reflexive iff I ( X , Y ∗ ) ∼ N ( X , Y ) ⊆ I ( X , Y ) ⊆ Γ 2 , r ( X , Y ) ⊆ CB ( X , Y )
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Mapping Ideals CB L 1 ( G ) ( L 1 ( G ) , L ∞ ( G )) L ∞ ( G ) ∪ ∪ Γ 2 , r L 1 ( G ) ( L 1 ( G ) , L ∞ ( G )) M cb A ( G )
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Mapping Ideals CB L 1 ( G ) ( L 1 ( G ) , L ∞ ( G )) L ∞ ( G ) ∪ ∪ Γ 2 , r L 1 ( G ) ( L 1 ( G ) , L ∞ ( G )) M cb A ( G ) || || I L 1 ( G ) ( L 1 ( G ) , L ∞ ( G )) M cb A ( G ) (Racher ’94)
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Mapping Ideals CB L 1 ( G ) ( L 1 ( G ) , L ∞ ( G )) L ∞ ( G ) ∪ ∪ Γ 2 , r L 1 ( G ) ( L 1 ( G ) , L ∞ ( G )) M cb A ( G ) || || I L 1 ( G ) ( L 1 ( G ) , L ∞ ( G )) M cb A ( G ) (Racher ’94) ∪ ∪ N L 1 ( G ) ( L 1 ( G ) , L ∞ ( G )) B ( G ) ∩ AP ( G ) = A ( bG ) (R. ’94)
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Mapping Ideals: quantum group perspective CB L 1 ( G ) ( L 1 ( G ) , L ∞ ( G )) L ∞ ( G ) ∪ ∪ Γ 2 , r M cb A ( G ) = M cb ( � L 1 ( G ) ( L 1 ( G ) , L ∞ ( G )) L 1 ( G )) || || I L 1 ( G ) ( L 1 ( G ) , L ∞ ( G )) M cb A ( G ) = M cb ( � L 1 ( G )) ∪ ∪ A ( bG ) = � N L 1 ( G ) ( L 1 ( G ) , L ∞ ( G )) L 1 ( bG )
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Mapping Ideals: quantum group perspective CB L 1 ( G ) ( L 1 ( G ) , L ∞ ( G )) L ∞ ( G ) ∪ ∪ Γ 2 , r M cb A ( G ) = M cb ( � L 1 ( G ) ( L 1 ( G ) , L ∞ ( G )) L 1 ( G )) || || I L 1 ( G ) ( L 1 ( G ) , L ∞ ( G )) M cb A ( G ) = M cb ( � L 1 ( G )) ∪ ∪ A ( bG ) = � N L 1 ( G ) ( L 1 ( G ) , L ∞ ( G )) L 1 ( bG ) Special case of quantum group phenomena?
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Locally Compact Quantum Groups Definition (Kustermans–Vaes ’00) A LCQG G = ( M , ∆ , ϕ, ψ ) M is a von Neumann algebra ; ∆ : M → M ⊗ M is a co-multiplication : normal, unital, isometric ∗ -homomorphism that is co-associative (∆ ⊗ id ) ◦ ∆ = ( id ⊗ ∆) ◦ ∆; ϕ is a left Haar weight on M: ϕ (( ω ⊗ id )∆( x )) = ω (1) ϕ ( x ) , x ∈ M ϕ , ω ∈ M ∗ ; ψ is a right Haar weight on M: ψ (( id ⊗ ω )∆( x )) = ω (1) ψ ( x ) , x ∈ M ψ , ω ∈ M ∗ .
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Locally Compact Quantum Groups Notation : L ∞ ( G ) := M , L 1 ( G ) := M ∗ , L 2 ( G ) := L 2 ( M , ϕ ).
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Locally Compact Quantum Groups Notation : L ∞ ( G ) := M , L 1 ( G ) := M ∗ , L 2 ( G ) := L 2 ( M , ϕ ). ∆ ∗ := ⋆ : L 1 ( G ) � ⊗ L 1 ( G ) → L 1 ( G ) ccBa .
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Locally Compact Quantum Groups Notation : L ∞ ( G ) := M , L 1 ( G ) := M ∗ , L 2 ( G ) := L 2 ( M , ϕ ). ∆ ∗ := ⋆ : L 1 ( G ) � ⊗ L 1 ( G ) → L 1 ( G ) ccBa . Bimodule structure on L ∞ ( G ): � f ⋆ x , g � = � x , g ⋆ f � and � x ⋆ f , g � = � x , f ⋆ g � .
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Locally Compact Quantum Groups Notation : L ∞ ( G ) := M , L 1 ( G ) := M ∗ , L 2 ( G ) := L 2 ( M , ϕ ). ∆ ∗ := ⋆ : L 1 ( G ) � ⊗ L 1 ( G ) → L 1 ( G ) ccBa . Bimodule structure on L ∞ ( G ): � f ⋆ x , g � = � x , g ⋆ f � and � x ⋆ f , g � = � x , f ⋆ g � . The anitpode S = R ◦ τ i / 2 .
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Locally Compact Quantum Groups Notation : L ∞ ( G ) := M , L 1 ( G ) := M ∗ , L 2 ( G ) := L 2 ( M , ϕ ). ∆ ∗ := ⋆ : L 1 ( G ) � ⊗ L 1 ( G ) → L 1 ( G ) ccBa . Bimodule structure on L ∞ ( G ): � f ⋆ x , g � = � x , g ⋆ f � and � x ⋆ f , g � = � x , f ⋆ g � . The anitpode S = R ◦ τ i / 2 . Commutative: G a = ( L ∞ ( G ) , ∆ a , ϕ, ψ ): ∆ a ( f )( s , t ) = f ( st ), S a ( f )( s ) = f ( s − 1 ) ϕ and ψ are Haar integrals. ( L 1 ( G a ) , ⋆ a ) = ( L 1 ( G ) , ∗ ).
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Locally Compact Quantum Groups Notation : L ∞ ( G ) := M , L 1 ( G ) := M ∗ , L 2 ( G ) := L 2 ( M , ϕ ). ∆ ∗ := ⋆ : L 1 ( G ) � ⊗ L 1 ( G ) → L 1 ( G ) ccBa . Bimodule structure on L ∞ ( G ): � f ⋆ x , g � = � x , g ⋆ f � and � x ⋆ f , g � = � x , f ⋆ g � . The anitpode S = R ◦ τ i / 2 . Commutative: G a = ( L ∞ ( G ) , ∆ a , ϕ, ψ ): ∆ a ( f )( s , t ) = f ( st ), S a ( f )( s ) = f ( s − 1 ) ϕ and ψ are Haar integrals. ( L 1 ( G a ) , ⋆ a ) = ( L 1 ( G ) , ∗ ). Co-commutative: G s = ( VN ( G ) , ∆ s , ϕ ): ∆ s ( λ ( t )) = λ ( t ) ⊗ λ ( t ), S s ( λ ( t )) = λ ( t − 1 ) ϕ = ψ is the Plancherel weight. ( L 1 ( G s ) , ⋆ s ) = ( A ( G ) , · ).
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Completely bounded multipliers Regular Representation: λ : L 1 ( G ) → B ( L 2 ( G )) f ∈ L 1 ( G ) . λ ( f ) = ( f ⊗ id )( W ) , Dual Quantum Group: L ∞ ( � G ) := { λ ( f ) | f ∈ L 1 ( G ) } ′′ .
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Completely bounded multipliers Regular Representation: λ : L 1 ( G ) → B ( L 2 ( G )) f ∈ L 1 ( G ) . λ ( f ) = ( f ⊗ id )( W ) , Dual Quantum Group: L ∞ ( � G ) := { λ ( f ) | f ∈ L 1 ( G ) } ′′ . ˆ b ∈ L ∞ ( � G ) is a cb-left multiplier of L 1 ( G ) if b λ ( L 1 ( G )) ⊆ λ ( L 1 ( G )) , f �→ λ − 1 ( b λ ( f )) is cb .
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Completely bounded multipliers Regular Representation: λ : L 1 ( G ) → B ( L 2 ( G )) f ∈ L 1 ( G ) . λ ( f ) = ( f ⊗ id )( W ) , Dual Quantum Group: L ∞ ( � G ) := { λ ( f ) | f ∈ L 1 ( G ) } ′′ . ˆ b ∈ L ∞ ( � G ) is a cb-left multiplier of L 1 ( G ) if b λ ( L 1 ( G )) ⊆ λ ( L 1 ( G )) , f �→ λ − 1 ( b λ ( f )) is cb . We let M l cb ( L 1 ( G )) denote the resulting ccBa.
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Completely bounded multipliers Regular Representation: λ : L 1 ( G ) → B ( L 2 ( G )) f ∈ L 1 ( G ) . λ ( f ) = ( f ⊗ id )( W ) , Dual Quantum Group: L ∞ ( � G ) := { λ ( f ) | f ∈ L 1 ( G ) } ′′ . ˆ b ∈ L ∞ ( � G ) is a cb-left multiplier of L 1 ( G ) if b λ ( L 1 ( G )) ⊆ λ ( L 1 ( G )) , f �→ λ − 1 ( b λ ( f )) is cb . We let M l cb ( L 1 ( G )) denote the resulting ccBa. Commutative: M cb ( L 1 ( G a )) = M cb ( L 1 ( G )) = M ( G ).
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Completely bounded multipliers Regular Representation: λ : L 1 ( G ) → B ( L 2 ( G )) f ∈ L 1 ( G ) . λ ( f ) = ( f ⊗ id )( W ) , Dual Quantum Group: L ∞ ( � G ) := { λ ( f ) | f ∈ L 1 ( G ) } ′′ . ˆ b ∈ L ∞ ( � G ) is a cb-left multiplier of L 1 ( G ) if b λ ( L 1 ( G )) ⊆ λ ( L 1 ( G )) , f �→ λ − 1 ( b λ ( f )) is cb . We let M l cb ( L 1 ( G )) denote the resulting ccBa. Commutative: M cb ( L 1 ( G a )) = M cb ( L 1 ( G )) = M ( G ). Co-commutative: M cb ( L 1 ( G s )) = M cb A ( G ).
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Examples A LCQG G is said to be: compact if ϕ (1) < ∞ ;
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Examples A LCQG G is said to be: compact if ϕ (1) < ∞ ; discrete if L 1 ( G ) is unital;
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Examples A LCQG G is said to be: compact if ϕ (1) < ∞ ; discrete if L 1 ( G ) is unital; co-amenable if L 1 ( G ) has a bounded approximate identity (bai).
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Examples A LCQG G is said to be: compact if ϕ (1) < ∞ ; discrete if L 1 ( G ) is unital; co-amenable if L 1 ( G ) has a bounded approximate identity (bai). a Kac algebra if S is bounded and the modular element δη Z ( L ∞ ( G )).
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Quasi-SIN quantum groups A LC group G is said to quasi-SIN (QSIN) if ∃ ( ξ i ) in L 2 ( G ) �·� =1 satisfying � γ ( s ) ξ i − ξ i � → 0 , s ∈ G , supp ( ξ i ) → { e } , where γ ( s ) ξ ( t ) = ξ ( s − 1 ts ) δ ( s ) 1 / 2 is conjugation representation.
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Quasi-SIN quantum groups A LC group G is said to quasi-SIN (QSIN) if ∃ ( ξ i ) in L 2 ( G ) �·� =1 satisfying � γ ( s ) ξ i − ξ i � → 0 , s ∈ G , supp ( ξ i ) → { e } , where γ ( s ) ξ ( t ) = ξ ( s − 1 ts ) δ ( s ) 1 / 2 is conjugation representation. Losert–Rindler ’84: amenable ∪ discrete ⊂ QSIN
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Quasi-SIN quantum groups A LC group G is said to quasi-SIN (QSIN) if ∃ ( ξ i ) in L 2 ( G ) �·� =1 satisfying � γ ( s ) ξ i − ξ i � → 0 , s ∈ G , supp ( ξ i ) → { e } , where γ ( s ) ξ ( t ) = ξ ( s − 1 ts ) δ ( s ) 1 / 2 is conjugation representation. Losert–Rindler ’84: amenable ∪ discrete ⊂ QSIN Definition (ACN ’18) Let G be a locally compact quantum group. We say that G is quasi-SIN (or QSIN) if there exists a net ( ξ i ) of unit vectors in L 2 ( G ) such that 1 � W σ V ση ⊗ ξ i − η ⊗ ξ i � → 0 , η ∈ L 2 ( G ) ; 2 � � V ξ i ⊗ η − ξ i ⊗ η � → 0 , η ∈ L 2 ( G ) ;
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals QSIN examples 1 G a = L ∞ ( G ) is QSIN precisely when G is QSIN.
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals QSIN examples 1 G a = L ∞ ( G ) is QSIN precisely when G is QSIN. 2 G s = VN ( G ) is QSIN if and only if G is amenable.
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals QSIN examples 1 G a = L ∞ ( G ) is QSIN precisely when G is QSIN. 2 G s = VN ( G ) is QSIN if and only if G is amenable. 3 Any discrete Kac algebra.
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals QSIN examples 1 G a = L ∞ ( G ) is QSIN precisely when G is QSIN. 2 G s = VN ( G ) is QSIN if and only if G is amenable. 3 Any discrete Kac algebra. 4 Any co-amenable compact Kac algebra.
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals QSIN examples 1 G a = L ∞ ( G ) is QSIN precisely when G is QSIN. 2 G s = VN ( G ) is QSIN if and only if G is amenable. 3 Any discrete Kac algebra. 4 Any co-amenable compact Kac algebra. Proposition (ACN ’18) If ( G 1 , G 2 ) is a modular matched pair of LC groups s.t. G 2 is discrete. δ | G 1 = δ 1 . ∃ ( ξ i ) in L 2 ( G 1 ) �·� =1 satisfying � ρ ( s ) ξ i − ξ i � , � β ( s ) ξ i − ξ i � → 0 , s ∈ G . Then VN ( G 1 ) β ⊲ ⊳ α ℓ ∞ ( G 2 ) is QSIN.
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Quantum Gilbert representation: QSIN case Theorem (ACN ’18) Let G be a LCQG s.t. � G is QSIN. Then L 1 ( G ) ( L 1 ( G ) , L ∞ ( G )) ∼ Γ 2 , r cb ( L 1 ( � = M l G )) completely isometrically.
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Quantum Gilbert representation: QSIN case Theorem (ACN ’18) Let G be a LCQG s.t. � G is QSIN. Then L 1 ( G ) ( L 1 ( G ) , L ∞ ( G )) ∼ Γ 2 , r cb ( L 1 ( � = M l G )) completely isometrically. Corollary (ACN ’18) A ( G ) ( A ( G ) , VN ( G )) ∼ Let G be QSIN. Then Γ 2 , r = M ( G ) .
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Quantum Gilbert representation G s = � G a = VN ( G ) is QSIN iff G is amenable,
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Quantum Gilbert representation G s = � G a = VN ( G ) is QSIN iff G is amenable, so Corollary (ACN ’18) For G amenable, Γ 2 , r L 1 ( G ) ( L 1 ( G ) , L ∞ ( G )) ∼ = M cb A ( G ) = B ( G ) .
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Quantum Gilbert representation G s = � G a = VN ( G ) is QSIN iff G is amenable, so Corollary (ACN ’18) For G amenable, Γ 2 , r L 1 ( G ) ( L 1 ( G ) , L ∞ ( G )) ∼ = M cb A ( G ) = B ( G ) . Only recover Gilbert’s theorem for amenable groups.
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Quantum Gilbert representation G s = � G a = VN ( G ) is QSIN iff G is amenable, so Corollary (ACN ’18) For G amenable, Γ 2 , r L 1 ( G ) ( L 1 ( G ) , L ∞ ( G )) ∼ = M cb A ( G ) = B ( G ) . Only recover Gilbert’s theorem for amenable groups. We say that G has bounded degree if deg ( G ) := sup { dim ( U ) | U ∈ Irr ( G ) } < ∞ ,
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Quantum Gilbert representation G s = � G a = VN ( G ) is QSIN iff G is amenable, so Corollary (ACN ’18) For G amenable, Γ 2 , r L 1 ( G ) ( L 1 ( G ) , L ∞ ( G )) ∼ = M cb A ( G ) = B ( G ) . Only recover Gilbert’s theorem for amenable groups. We say that G has bounded degree if deg ( G ) := sup { dim ( U ) | U ∈ Irr ( G ) } < ∞ , Examples: 1 G s = VN ( G ) has bounded degree with deg ( G s ) = 1.
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Quantum Gilbert representation G s = � G a = VN ( G ) is QSIN iff G is amenable, so Corollary (ACN ’18) For G amenable, Γ 2 , r L 1 ( G ) ( L 1 ( G ) , L ∞ ( G )) ∼ = M cb A ( G ) = B ( G ) . Only recover Gilbert’s theorem for amenable groups. We say that G has bounded degree if deg ( G ) := sup { dim ( U ) | U ∈ Irr ( G ) } < ∞ , Examples: 1 G s = VN ( G ) has bounded degree with deg ( G s ) = 1. 2 G a = L ∞ ( G ) has bounded degree if and only if G is virtually abelian (Moore ’72).
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Quantum Gilbert representation G s = � G a = VN ( G ) is QSIN iff G is amenable, so Corollary (ACN ’18) For G amenable, Γ 2 , r L 1 ( G ) ( L 1 ( G ) , L ∞ ( G )) ∼ = M cb A ( G ) = B ( G ) . Only recover Gilbert’s theorem for amenable groups. We say that G has bounded degree if deg ( G ) := sup { dim ( U ) | U ∈ Irr ( G ) } < ∞ , Examples: 1 G s = VN ( G ) has bounded degree with deg ( G s ) = 1. 2 G a = L ∞ ( G ) has bounded degree if and only if G is virtually abelian (Moore ’72). 3 VN ( G 1 ) β ⊲ ⊳ α ℓ ∞ ( G 2 ) has bounded degree if G 1 is countable discrete and G 2 is finite (Fima–Mukherjee–Patri ’17).
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Quantum Gilbert representation Theorem (ACN ’18) Let G be Kac algebra such that � G has bounded degree. Then L 1 ( G ) ( L 1 ( G ) , L ∞ ( G )) ∼ Γ 2 , r cb ( L 1 ( � = M l G )) completely isomorphically, with deg ( � G ) − 1 � [ b ij ] � ≤ γ 2 , r ([ b ij ]) ≤ deg ( � G ) � [ b ij ] � , cb ( L 1 ( � for all [ b ij ] ∈ M n ( M l G )) .
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Quantum Gilbert representation Theorem (ACN ’18) Let G be Kac algebra such that � G has bounded degree. Then L 1 ( G ) ( L 1 ( G ) , L ∞ ( G )) ∼ Γ 2 , r cb ( L 1 ( � = M l G )) completely isomorphically, with deg ( � G ) − 1 � [ b ij ] � ≤ γ 2 , r ([ b ij ]) ≤ deg ( � G ) � [ b ij ] � , cb ( L 1 ( � for all [ b ij ] ∈ M n ( M l G )) . Since deg ( � G a ) = deg ( G s ) = 1, we obtain Γ 2 , r L 1 ( G ) ( L 1 ( G ) , L ∞ ( G )) ∼ = M cb A ( G ) completely isometrically .
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals A few words on the proof Proof uses two manifestations of quantum group duality
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals A few words on the proof Proof uses two manifestations of quantum group duality Theorem (Junge–Neufang–Ruan ’09) Let G be a LCQG. Then cb ( L 1 ( G ))) c ∩ CB σ � cb ( L 1 ( � Θ( M l G ))) = Θ( M l L ∞ ( G ) ( B ( L 2 ( G ))) .
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals A few words on the proof Proof uses two manifestations of quantum group duality Theorem (Junge–Neufang–Ruan ’09) Let G be a LCQG. Then cb ( L 1 ( G ))) c ∩ CB σ � cb ( L 1 ( � Θ( M l G ))) = Θ( M l L ∞ ( G ) ( B ( L 2 ( G ))) . Proposition (Kasprzak–Sołtan ’14) Let G be a LCQG. Then G ))(1 ⊗ U )) ′ ∩ L ∞ ( G ) ⊗ L ∞ ( G ) . ∆( L ∞ ( G )) = ((1 ⊗ U ∗ ) � ∆( L ∞ ( �
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals A few words on the proof Proof uses two manifestations of quantum group duality Theorem (Junge–Neufang–Ruan ’09) Let G be a LCQG. Then cb ( L 1 ( G ))) c ∩ CB σ � cb ( L 1 ( � Θ( M l G ))) = Θ( M l L ∞ ( G ) ( B ( L 2 ( G ))) . Proposition (Kasprzak–Sołtan ’14) Let G be a LCQG. Then G ))(1 ⊗ U )) ′ ∩ L ∞ ( G ) ⊗ L ∞ ( G ) . ∆( L ∞ ( G )) = ((1 ⊗ U ∗ ) � ∆( L ∞ ( � together with structure of subhomogeneous C ∗ -algebras.
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Mapping Ideals CB L 1 ( G ) ( L 1 ( G ) , L ∞ ( G )) L ∞ ( G ) ∪ ∪ Γ 2 , r L 1 ( G ) ( L 1 ( G ) , L ∞ ( G )) cb ( L 1 ( � M l G )) ∪ ∪ Other mapping ideals ????
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Completely integral multipliers Theorem (ACN ’18) Let G LCQG for which either � G is QSIN with trivial scaling group 1
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Completely integral multipliers Theorem (ACN ’18) Let G LCQG for which either � G is QSIN with trivial scaling group, or 1 � G is a Kac algebra with bounded degree. 2
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Completely integral multipliers Theorem (ACN ’18) Let G LCQG for which either � G is QSIN with trivial scaling group, or 1 � G is a Kac algebra with bounded degree. 2 Then I L 1 ( G ) ( L 1 ( G ) , L ∞ ( G )) ∼ cb ( L 1 ( � = M l G )) isomorphically
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Completely integral multipliers Theorem (ACN ’18) Let G LCQG for which either � G is QSIN with trivial scaling group, or 1 � G is a Kac algebra with bounded degree. 2 Then I L 1 ( G ) ( L 1 ( G ) , L ∞ ( G )) ∼ cb ( L 1 ( � = M l G )) isomorphically, with 1 � b � cb ≤ ι ( b ) ≤ 2 � b � cb
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Completely integral multipliers Theorem (ACN ’18) Let G LCQG for which either � G is QSIN with trivial scaling group, or 1 � G is a Kac algebra with bounded degree. 2 Then I L 1 ( G ) ( L 1 ( G ) , L ∞ ( G )) ∼ cb ( L 1 ( � = M l G )) isomorphically, with 1 � b � cb ≤ ι ( b ) ≤ 2 � b � cb 2 deg ( � G ) − 1 � b � cb ≤ ι ( b ) ≤ deg ( � G )(1 + deg ( � G ) 2 ) � b � cb
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Proof sketch: QSIN case Lemma (ACN ’18) cb ( L 1 ( � Same hypotheses, for every b ∈ M l G )) , ∃ ( a i ) , ( b i ) in L ∞ ( G ) such that � � � � � a i a ∗ a ∗ b i b ∗ b ∗ a i ⊗ b i , i b i < ∞ . ∆( b ) = i , i a i , i , i i i i i
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Proof sketch: QSIN case Lemma (ACN ’18) cb ( L 1 ( � Same hypotheses, for every b ∈ M l G )) , ∃ ( a i ) , ( b i ) in L ∞ ( G ) such that � � � � � a i a ∗ a ∗ b i b ∗ b ∗ a i ⊗ b i , i b i < ∞ . ∆( b ) = i , i a i , i , i i i i i Pisier–Shlyakhtenko ’02; Haagerup–Musat ’08: For ⊗ L ∞ ( G )) ∗ ∃ states f ∈ L 1 ( G ) ⊗ L 1 ( G ) ֒ → ( L ∞ ( G ) � ϕ 1 , ϕ 2 , ψ 1 , ψ 2 ∈ L ∞ ( G ) ∗ such that � f , a ⊗ b � ≤ � f � L 1 ( G ) ⊗ ∨ L 1 ( G ) ( ϕ 1 ( aa ∗ ) 1 / 2 ψ 1 ( b ∗ b ) 1 / 2 + ϕ 2 ( a ∗ a ) 1 / 2 ψ 2 ( bb ∗ ) 1 / 2 )
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Proof sketch: QSIN case Lemma (ACN ’18) cb ( L 1 ( � Same hypotheses, for every b ∈ M l G )) , ∃ ( a i ) , ( b i ) in L ∞ ( G ) such that � � � � � a i a ∗ a ∗ b i b ∗ b ∗ a i ⊗ b i , i b i < ∞ . ∆( b ) = i , i a i , i , i i i i i Pisier–Shlyakhtenko ’02; Haagerup–Musat ’08: For ⊗ L ∞ ( G )) ∗ ∃ states f ∈ L 1 ( G ) ⊗ L 1 ( G ) ֒ → ( L ∞ ( G ) � ϕ 1 , ϕ 2 , ψ 1 , ψ 2 ∈ L ∞ ( G ) ∗ such that � f , a ⊗ b � ≤ � f � L 1 ( G ) ⊗ ∨ L 1 ( G ) ( ϕ 1 ( aa ∗ ) 1 / 2 ψ 1 ( b ∗ b ) 1 / 2 + ϕ 2 ( a ∗ a ) 1 / 2 ψ 2 ( bb ∗ ) 1 / 2 ) Effros–Junge–Ruan ’00: L 1 ( G ) is locally reflexive.
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Completely nuclear multipliers Racher ’94: N L 1 ( G ) ( L 1 ( G ) , L ∞ ( G )) ∼ = B ( G ) ∩ AP ( G ).
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Completely nuclear multipliers Racher ’94: N L 1 ( G ) ( L 1 ( G ) , L ∞ ( G )) ∼ = B ( G ) ∩ AP ( G ). Runde–Spronk ’04: B ( G ) ∩ AP ( G ) = A ( bG ) - bG is the Bohr compactification of G
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Completely nuclear multipliers Racher ’94: N L 1 ( G ) ( L 1 ( G ) , L ∞ ( G )) ∼ = B ( G ) ∩ AP ( G ). Runde–Spronk ’04: B ( G ) ∩ AP ( G ) = A ( bG ) - bG is the Bohr compactification of G . Moreover, B ( G ) = A ( bG ) ⊕ 1 A ( bG ) ⊥ .
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Completely nuclear multipliers Racher ’94: N L 1 ( G ) ( L 1 ( G ) , L ∞ ( G )) ∼ = B ( G ) ∩ AP ( G ). Runde–Spronk ’04: B ( G ) ∩ AP ( G ) = A ( bG ) - bG is the Bohr compactification of G . Moreover, B ( G ) = A ( bG ) ⊕ 1 A ( bG ) ⊥ . Dual to M ( G ) = ℓ 1 ( G d ) ⊕ 1 M c ( G ).
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Completely nuclear multipliers Racher ’94: N L 1 ( G ) ( L 1 ( G ) , L ∞ ( G )) ∼ = B ( G ) ∩ AP ( G ). Runde–Spronk ’04: B ( G ) ∩ AP ( G ) = A ( bG ) - bG is the Bohr compactification of G . Moreover, B ( G ) = A ( bG ) ⊕ 1 A ( bG ) ⊥ . Dual to M ( G ) = ℓ 1 ( G d ) ⊕ 1 M c ( G ). Lemma (ACN ’18) G ) ∗ ∼ C u ( � = ℓ 1 ( � b G ) ⊕ 1 ℓ 1 ( � b G ) ⊥
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Completely nuclear multipliers Racher ’94: N L 1 ( G ) ( L 1 ( G ) , L ∞ ( G )) ∼ = B ( G ) ∩ AP ( G ). Runde–Spronk ’04: B ( G ) ∩ AP ( G ) = A ( bG ) - bG is the Bohr compactification of G . Moreover, B ( G ) = A ( bG ) ⊕ 1 A ( bG ) ⊥ . Dual to M ( G ) = ℓ 1 ( G d ) ⊕ 1 M c ( G ). Lemma (ACN ’18) G ) ∗ ∼ C u ( � = ℓ 1 ( � b G ) ⊕ 1 ℓ 1 ( � b G ) ⊥ b G is the quantum Bohr compactification (Sołtan ’05).
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Theorem (ACN ’18) Let G Kac algebra for which either � G is QSIN 1
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Theorem (ACN ’18) Let G Kac algebra for which either � G is QSIN, or 1 � G is has bounded degree. 2
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Theorem (ACN ’18) Let G Kac algebra for which either � G is QSIN, or 1 � G is has bounded degree. 2 Then TFAE for x ∈ L ∞ ( G ) : x ∈ ℓ 1 ( � cb ( L 1 ( � b G ) ⊆ M l G ))
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Theorem (ACN ’18) Let G Kac algebra for which either � G is QSIN, or 1 � G is has bounded degree. 2 Then TFAE for x ∈ L ∞ ( G ) : x ∈ ℓ 1 ( � cb ( L 1 ( � b G ) ⊆ M l G )) ; f �→ x ⋆ f ∈ N L 1 ( G ) ( L 1 ( G ) , L ∞ ( G ))
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Theorem (ACN ’18) Let G Kac algebra for which either � G is QSIN, or 1 � G is has bounded degree. 2 Then TFAE for x ∈ L ∞ ( G ) : x ∈ ℓ 1 ( � cb ( L 1 ( � b G ) ⊆ M l G )) ; f �→ x ⋆ f ∈ N L 1 ( G ) ( L 1 ( G ) , L ∞ ( G )) ; ∆( x ) ∈ L ∞ ( G ) ⊗ h L ∞ ( G )
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Theorem (ACN ’18) Let G Kac algebra for which either � G is QSIN, or 1 � G is has bounded degree. 2 Then TFAE for x ∈ L ∞ ( G ) : x ∈ ℓ 1 ( � cb ( L 1 ( � b G ) ⊆ M l G )) ; f �→ x ⋆ f ∈ N L 1 ( G ) ( L 1 ( G ) , L ∞ ( G )) ; ∆( x ) ∈ L ∞ ( G ) ⊗ h L ∞ ( G ) . Moreover, N L 1 ( G ) ( L 1 ( G ) , L ∞ ( G )) ∼ = ℓ 1 ( � b G ) isomorphically
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Theorem (ACN ’18) Let G Kac algebra for which either � G is QSIN, or 1 � G is has bounded degree. 2 Then TFAE for x ∈ L ∞ ( G ) : x ∈ ℓ 1 ( � cb ( L 1 ( � b G ) ⊆ M l G )) ; f �→ x ⋆ f ∈ N L 1 ( G ) ( L 1 ( G ) , L ∞ ( G )) ; ∆( x ) ∈ L ∞ ( G ) ⊗ h L ∞ ( G ) . Moreover, N L 1 ( G ) ( L 1 ( G ) , L ∞ ( G )) ∼ = ℓ 1 ( � b G ) isomorphically, with 1 � ˆ f � ≤ ν (ˆ f ) ≤ 2 � ˆ f �
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals Theorem (ACN ’18) Let G Kac algebra for which either � G is QSIN, or 1 � G is has bounded degree. 2 Then TFAE for x ∈ L ∞ ( G ) : x ∈ ℓ 1 ( � cb ( L 1 ( � b G ) ⊆ M l G )) ; f �→ x ⋆ f ∈ N L 1 ( G ) ( L 1 ( G ) , L ∞ ( G )) ; ∆( x ) ∈ L ∞ ( G ) ⊗ h L ∞ ( G ) . Moreover, N L 1 ( G ) ( L 1 ( G ) , L ∞ ( G )) ∼ = ℓ 1 ( � b G ) isomorphically, with 1 � ˆ f � ≤ ν (ˆ f ) ≤ 2 � ˆ f � 2 deg ( � G ) − 1 � ˆ f � ≤ ν (ˆ f ) ≤ 2 � ˆ f �
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals ∆( x ) ∈ L ∞ ( G ) ⊗ h L ∞ ( G ) ⇒ x ∈ ℓ 1 ( � b G ): QSIN case �·� cb ( L 1 ( � ⊆ L ∞ ( G ) Let A := M l G ))
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals ∆( x ) ∈ L ∞ ( G ) ⊗ h L ∞ ( G ) ⇒ x ∈ ℓ 1 ( � b G ): QSIN case �·� cb ( L 1 ( � ⊆ L ∞ ( G ). Then Let A := M l G )) ∆( x ) ∈ F ( A , A ; L ∞ ( G ) ⊗ h L ∞ ( G ))
Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals ∆( x ) ∈ L ∞ ( G ) ⊗ h L ∞ ( G ) ⇒ x ∈ ℓ 1 ( � b G ): QSIN case �·� cb ( L 1 ( � ⊆ L ∞ ( G ). Then Let A := M l G )) ∆( x ) ∈ F ( A , A ; L ∞ ( G ) ⊗ h L ∞ ( G )) = A ⊗ h A ( Smith ′ 91) .
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