Mapping ideals of quantum group multipliers Jason Crann with M. - - PowerPoint PPT Presentation

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Mapping ideals of quantum group multipliers

Jason Crann

with M. Alaghmandan and M. Neufang arXiv:1803.08342 Carleton University

Quantum Groups and Their Analysis, University of Oslo August 8th, 2019

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Operator Spaces

Banach spaces: X ֒ → ℓ∞(S).

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Operator Spaces

Banach spaces: X ֒ → ℓ∞(S). Definition An operator space is a closed subspace of B(H).

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Operator Spaces

Banach spaces: X ֒ → ℓ∞(S). Definition An operator space is a closed subspace of B(H). Theorem (Ruan ’88) A vector space X is an operator space ⇔ ∃ a matrix norm ·n

• n Mn(X) ∀ n ≥ 1 satisfying

R1 x ⊕ ym+n = max{xm, yn}, R2 αxβn ≤ αxmβ. ∀ m, n ∈ N, x ∈ Mm(X), y ∈ Mn(Y ), α ∈ Mn,m(C), β ∈ Mm,n(C).

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Operator Spaces

Given Φ : X → Y , for each n ∈ N ∃ a linear map Φn : Mn(X) → Mn(Y ) given by Φn([xij]) = [Φ(xij)] for all [xij] ∈ Mn(X), called the nth amplification of Φ.

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Operator Spaces

Given Φ : X → Y , for each n ∈ N ∃ a linear map Φn : Mn(X) → Mn(Y ) given by Φn([xij]) = [Φ(xij)] for all [xij] ∈ Mn(X), called the nth amplification of Φ. Definition Let X and Y be operator spaces. A linear map Φ : X → Y is called: completely bounded if Φcb := sup{Φn : n ∈ N} < ∞; completely contractive if Φcb ≤ 1; a complete isometry if Φn is an isometry for all n ∈ N.

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Operator Space Tensor Products

Projective ⊗: linearizes bilinear jointly cb maps J CB(X × Y , Z) = CB(X ⊗Y , Z).

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Operator Space Tensor Products

Projective ⊗: linearizes bilinear jointly cb maps J CB(X × Y , Z) = CB(X ⊗Y , Z). Injective ⊗∨: if X ⊆ B(H) and Y ⊆ B(K), then X ⊗∨ Y ⊆ B(H ⊗ K)

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Operator Space Tensor Products

Projective ⊗: linearizes bilinear jointly cb maps J CB(X × Y , Z) = CB(X ⊗Y , Z). Injective ⊗∨: if X ⊆ B(H) and Y ⊆ B(K), then X ⊗∨ Y ⊆ B(H ⊗ K) Haagerup ⊗h: if X ⊆ B(H) and Y ⊆ B(K), then uh = inf{

• xix∗

i 1/2

• y∗

i yi1/2 | u =

• xi ⊗ yi}

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Operator Space Tensor Products

Projective ⊗: linearizes bilinear jointly cb maps J CB(X × Y , Z) = CB(X ⊗Y , Z). Injective ⊗∨: if X ⊆ B(H) and Y ⊆ B(K), then X ⊗∨ Y ⊆ B(H ⊗ K) Haagerup ⊗h: if X ⊆ B(H) and Y ⊆ B(K), then uh = inf{

• xix∗

i 1/2

• y∗

i yi1/2 | u =

• xi ⊗ yi}

·∨ ≤ ·h ≤ ·∧

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Completely bounded multipliers

Eymard ’64: The Fourier algebra of a LCG G is the space A(G) := {λ(·)ξ, η | ξ, η ∈ L2(G)}. where λ : G → B(L2(G)) is left regular representation: λ(s)ξ(t) = ξ(s−1t), s, t ∈ G, ξ ∈ L2(G).

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Completely bounded multipliers

Eymard ’64: The Fourier algebra of a LCG G is the space A(G) := {λ(·)ξ, η | ξ, η ∈ L2(G)}. where λ : G → B(L2(G)) is left regular representation: λ(s)ξ(t) = ξ(s−1t), s, t ∈ G, ξ ∈ L2(G). Under the norm uA(G) = inf{ξη | u = λ(·)ξ, η} A(G) is a Banach algebra under pointwise multiplication.

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Completely bounded multipliers

Eymard ’64: The Fourier algebra of a LCG G is the space A(G) := {λ(·)ξ, η | ξ, η ∈ L2(G)}. where λ : G → B(L2(G)) is left regular representation: λ(s)ξ(t) = ξ(s−1t), s, t ∈ G, ξ ∈ L2(G). Under the norm uA(G) = inf{ξη | u = λ(·)ξ, η} A(G) is a Banach algebra under pointwise multiplication. A(G)∗ ∼ = VN(G) := {λ(s) | s ∈ G}′′ – the group vN algebra.

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Completely bounded multipliers

Eymard ’64: The Fourier algebra of a LCG G is the space A(G) := {λ(·)ξ, η | ξ, η ∈ L2(G)}. where λ : G → B(L2(G)) is left regular representation: λ(s)ξ(t) = ξ(s−1t), s, t ∈ G, ξ ∈ L2(G). Under the norm uA(G) = inf{ξη | u = λ(·)ξ, η} A(G) is a Banach algebra under pointwise multiplication. A(G)∗ ∼ = VN(G) := {λ(s) | s ∈ G}′′ – the group vN algebra. Induces operator space structure on A(G)

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Completely bounded multipliers

Eymard ’64: The Fourier algebra of a LCG G is the space A(G) := {λ(·)ξ, η | ξ, η ∈ L2(G)}. where λ : G → B(L2(G)) is left regular representation: λ(s)ξ(t) = ξ(s−1t), s, t ∈ G, ξ ∈ L2(G). Under the norm uA(G) = inf{ξη | u = λ(·)ξ, η} A(G) is a Banach algebra under pointwise multiplication. A(G)∗ ∼ = VN(G) := {λ(s) | s ∈ G}′′ – the group vN algebra. Induces operator space structure on A(G) cb-multipliers McbA(G) := {ϕ : G → C | mϕ ∈ CB(A(G))}.

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Gilbert Representation

Gilbert ’73-’74: For ϕ : G → C, TFAE

1 ϕ ∈ McbA(G) 2 ∃ Hilbert space H, continuous ξ, η : G → H such that

ϕ(st−1) = η(t), ξ(s), s, t ∈ G. Moreover, ϕcb = inf{ξ∞η∞}.

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Gilbert Representation

Gilbert ’73-’74: For ϕ : G → C, TFAE

1 ϕ ∈ McbA(G) 2 ∃ Hilbert space H, continuous ξ, η : G → H such that

ϕ(st−1) = η(t), ξ(s), s, t ∈ G. Moreover, ϕcb = inf{ξ∞η∞}. Convolution: ϕ ∈ McbA(G) iff L1(G) ∋ f → ϕ ∗ f ∈ L∞(G) factors through a Hilbert space.

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Gilbert Representation

Gilbert ’73-’74: For ϕ : G → C, TFAE

1 ϕ ∈ McbA(G) 2 ∃ Hilbert space H, continuous ξ, η : G → H such that

ϕ(st−1) = η(t), ξ(s), s, t ∈ G. Moreover, ϕcb = inf{ξ∞η∞}. Convolution: ϕ ∈ McbA(G) iff L1(G) ∋ f → ϕ ∗ f ∈ L∞(G) factors through a Hilbert space. McbA(G) ∼ = Γ2,r

L1(G)(L1(G), L∞(G))

where Γ2,r

L1(G)(L1(G), L∞(G)) = {ϕ ∈ CBL1(G)(L1(G), L∞(G)) |

factor through a row Hilbert space}.

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Mapping Ideals

CBL1(G)(L1(G), L∞(G)) L∞(G)

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Mapping Ideals

CBL1(G)(L1(G), L∞(G)) ∪ Γ2,r

L1(G)(L1(G), L∞(G))

L∞(G) ∪ McbA(G)

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Mapping Ideals

CBL1(G)(L1(G), L∞(G)) ∪ Γ2,r

L1(G)(L1(G), L∞(G))

∪ Other mapping ideals L∞(G) ∪ McbA(G) ∪ ????

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 ∈ Mn(O(X, Y )),

1 ϕcb ≤ ϕO, and 2 for any linear mappings r : V → X and s : Y → W ,

sn ◦ ϕ ◦ rO ≤ scbϕOrcb.

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 ∈ Mn(O(X, Y )),

1 ϕcb ≤ ϕO, and 2 for any linear mappings r : V → X and s : Y → W ,

sn ◦ ϕ ◦ rO ≤ scbϕOrcb. Examples: CB(X, Y ), Γ2,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 ∈ Mn(O(X, Y )),

1 ϕcb ≤ ϕO, and 2 for any linear mappings r : V → X and s : Y → W ,

sn ◦ ϕ ◦ rO ≤ scbϕOrcb. Examples: CB(X, Y ), Γ2,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 ∈ Mn(O(X, Y )),

1 ϕcb ≤ ϕO, and 2 for any linear mappings r : V → X and s : Y → W ,

sn ◦ ϕ ◦ rO ≤ scbϕOrcb. Examples: CB(X, Y ), Γ2,r(X, Y ). When range is a dual space: CB(X, Y ∗) = (X ⊗Y )∗ Γ2,r(X, Y ∗) ∼ = (X ⊗h 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 ∈ Mn(O(X, Y )),

1 ϕcb ≤ ϕO, and 2 for any linear mappings r : V → X and s : Y → W ,

sn ◦ ϕ ◦ rO ≤ scbϕOrcb. Examples: CB(X, Y ), Γ2,r(X, Y ). When range is a dual space: CB(X, Y ∗) = (X ⊗Y )∗ Γ2,r(X, Y ∗) ∼ = (X ⊗h Y )∗ ∼ = X ∗ ⊗w∗h Y ∗

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Mapping Ideals

Completely nuclear mappings N(X, Y ) is the image of X ∗ ⊗Y → X ∗ ⊗∨ Y ⊆ CB(X, Y ) 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 X ∗ ⊗Y → X ∗ ⊗∨ Y ⊆ CB(X, Y ) 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 X ∗ ⊗Y → X ∗ ⊗∨ Y ⊆ CB(X, Y ) 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. Y is locally reflexive iff I(X, Y ∗) ∼ = (X ⊗∨ Y )∗.

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Mapping Ideals

Completely nuclear mappings N(X, Y ) is the image of X ∗ ⊗Y → X ∗ ⊗∨ Y ⊆ CB(X, Y ) 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. Y is locally reflexive iff I(X, Y ∗) ∼ = (X ⊗∨ Y )∗. N(X, Y ) ⊆ I(X, Y ) ⊆ Γ2,r(X, Y ) ⊆ CB(X, Y )

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Mapping Ideals

CBL1(G)(L1(G), L∞(G)) ∪ Γ2,r

L1(G)(L1(G), L∞(G))

L∞(G) ∪ McbA(G)

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Mapping Ideals

CBL1(G)(L1(G), L∞(G)) ∪ Γ2,r

L1(G)(L1(G), L∞(G))

|| IL1(G)(L1(G), L∞(G)) L∞(G) ∪ McbA(G) || McbA(G) (Racher ’94)

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Mapping Ideals

CBL1(G)(L1(G), L∞(G)) ∪ Γ2,r

L1(G)(L1(G), L∞(G))

|| IL1(G)(L1(G), L∞(G)) ∪ NL1(G)(L1(G), L∞(G)) L∞(G) ∪ McbA(G) || McbA(G) (Racher ’94) ∪ B(G) ∩ AP(G) = A(bG) (R. ’94)

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Mapping Ideals: quantum group perspective

CBL1(G)(L1(G), L∞(G)) ∪ Γ2,r

L1(G)(L1(G), L∞(G))

|| IL1(G)(L1(G), L∞(G)) ∪ NL1(G)(L1(G), L∞(G)) L∞(G) ∪ McbA(G) = Mcb( L1(G)) || McbA(G) = Mcb( L1(G)) ∪ A(bG) = L1(bG)

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Mapping Ideals: quantum group perspective

CBL1(G)(L1(G), L∞(G)) ∪ Γ2,r

L1(G)(L1(G), L∞(G))

|| IL1(G)(L1(G), L∞(G)) ∪ NL1(G)(L1(G), L∞(G)) L∞(G) ∪ McbA(G) = Mcb( L1(G)) || McbA(G) = Mcb( L1(G)) ∪ A(bG) = L1(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 ∈ Mϕ, ω ∈ M∗; ψ is a right Haar weight on M: ψ((id ⊗ ω)∆(x)) = ω(1)ψ(x), x ∈ Mψ, ω ∈ M∗.

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Locally Compact Quantum Groups

Notation: L∞(G) := M, L1(G) := M∗, L2(G) := L2(M, ϕ).

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Locally Compact Quantum Groups

Notation: L∞(G) := M, L1(G) := M∗, L2(G) := L2(M, ϕ). ∆∗ := ⋆ : L1(G) ⊗L1(G) → L1(G) ccBa.

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Locally Compact Quantum Groups

Notation: L∞(G) := M, L1(G) := M∗, L2(G) := L2(M, ϕ). ∆∗ := ⋆ : L1(G) ⊗L1(G) → L1(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, L1(G) := M∗, L2(G) := L2(M, ϕ). ∆∗ := ⋆ : L1(G) ⊗L1(G) → L1(G) ccBa. Bimodule structure on L∞(G): f ⋆ x, g = x, g ⋆ f and x ⋆ f , g = x, f ⋆ g. The anitpode S = R ◦ τi/2.

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Locally Compact Quantum Groups

Notation: L∞(G) := M, L1(G) := M∗, L2(G) := L2(M, ϕ). ∆∗ := ⋆ : L1(G) ⊗L1(G) → L1(G) ccBa. Bimodule structure on L∞(G): f ⋆ x, g = x, g ⋆ f and x ⋆ f , g = x, f ⋆ g. The anitpode S = R ◦ τi/2. Commutative: Ga = (L∞(G), ∆a, ϕ, ψ): ∆a(f )(s, t) = f (st), Sa(f )(s) = f (s−1) ϕ and ψ are Haar integrals. (L1(Ga), ⋆a) = (L1(G), ∗).

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Locally Compact Quantum Groups

Notation: L∞(G) := M, L1(G) := M∗, L2(G) := L2(M, ϕ). ∆∗ := ⋆ : L1(G) ⊗L1(G) → L1(G) ccBa. Bimodule structure on L∞(G): f ⋆ x, g = x, g ⋆ f and x ⋆ f , g = x, f ⋆ g. The anitpode S = R ◦ τi/2. Commutative: Ga = (L∞(G), ∆a, ϕ, ψ): ∆a(f )(s, t) = f (st), Sa(f )(s) = f (s−1) ϕ and ψ are Haar integrals. (L1(Ga), ⋆a) = (L1(G), ∗). Co-commutative: Gs = (VN(G), ∆s, ϕ): ∆s(λ(t)) = λ(t) ⊗ λ(t), Ss(λ(t)) = λ(t−1) ϕ = ψ is the Plancherel weight. (L1(Gs), ⋆s) = (A(G), ·).

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Completely bounded multipliers

Regular Representation: λ : L1(G) → B(L2(G)) λ(f ) = (f ⊗ id)(W ), f ∈ L1(G). Dual Quantum Group: L∞( G) := {λ(f ) | f ∈ L1(G)}′′.

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Completely bounded multipliers

Regular Representation: λ : L1(G) → B(L2(G)) λ(f ) = (f ⊗ id)(W ), f ∈ L1(G). Dual Quantum Group: L∞( G) := {λ(f ) | f ∈ L1(G)}′′. ˆ b ∈ L∞( G) is a cb-left multiplier of L1(G) if bλ(L1(G)) ⊆ λ(L1(G)), f → λ−1(bλ(f )) is cb.

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Completely bounded multipliers

Regular Representation: λ : L1(G) → B(L2(G)) λ(f ) = (f ⊗ id)(W ), f ∈ L1(G). Dual Quantum Group: L∞( G) := {λ(f ) | f ∈ L1(G)}′′. ˆ b ∈ L∞( G) is a cb-left multiplier of L1(G) if bλ(L1(G)) ⊆ λ(L1(G)), f → λ−1(bλ(f )) is cb. We let Ml

cb(L1(G)) denote the resulting ccBa.

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Completely bounded multipliers

Regular Representation: λ : L1(G) → B(L2(G)) λ(f ) = (f ⊗ id)(W ), f ∈ L1(G). Dual Quantum Group: L∞( G) := {λ(f ) | f ∈ L1(G)}′′. ˆ b ∈ L∞( G) is a cb-left multiplier of L1(G) if bλ(L1(G)) ⊆ λ(L1(G)), f → λ−1(bλ(f )) is cb. We let Ml

cb(L1(G)) denote the resulting ccBa.

Commutative: Mcb(L1(Ga)) = Mcb(L1(G)) = M(G).

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Completely bounded multipliers

Regular Representation: λ : L1(G) → B(L2(G)) λ(f ) = (f ⊗ id)(W ), f ∈ L1(G). Dual Quantum Group: L∞( G) := {λ(f ) | f ∈ L1(G)}′′. ˆ b ∈ L∞( G) is a cb-left multiplier of L1(G) if bλ(L1(G)) ⊆ λ(L1(G)), f → λ−1(bλ(f )) is cb. We let Ml

cb(L1(G)) denote the resulting ccBa.

Commutative: Mcb(L1(Ga)) = Mcb(L1(G)) = M(G). Co-commutative: Mcb(L1(Gs)) = McbA(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 L1(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 L1(G) is unital; co-amenable if L1(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 L1(G) is unital; co-amenable if L1(G) has a bounded approximate identity (bai). a Kac algebra if S is bounded and the modular element δηZ(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 ∃ (ξi) in L2(G)·=1 satisfying γ(s)ξi − ξi → 0, s ∈ G, supp(ξi) → {e}, where γ(s)ξ(t) = ξ(s−1ts)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 ∃ (ξi) in L2(G)·=1 satisfying γ(s)ξi − ξi → 0, s ∈ G, supp(ξi) → {e}, where γ(s)ξ(t) = ξ(s−1ts)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 ∃ (ξi) in L2(G)·=1 satisfying γ(s)ξi − ξi → 0, s ∈ G, supp(ξi) → {e}, where γ(s)ξ(t) = ξ(s−1ts)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 (ξi) of unit vectors in L2(G) such that

1 W σV ση ⊗ ξi − η ⊗ ξi → 0, η ∈ L2(G); 2

V ξi ⊗ η − ξi ⊗ η → 0, η ∈ L2(G);

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

QSIN examples

1 Ga = L∞(G) is QSIN precisely when G is QSIN.

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

QSIN examples

1 Ga = L∞(G) is QSIN precisely when G is QSIN. 2 Gs = VN(G) is QSIN if and only if G is amenable.

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

QSIN examples

1 Ga = L∞(G) is QSIN precisely when G is QSIN. 2 Gs = 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 Ga = L∞(G) is QSIN precisely when G is QSIN. 2 Gs = 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 Ga = L∞(G) is QSIN precisely when G is QSIN. 2 Gs = 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 (G1, G2) is a modular matched pair of LC groups s.t. G2 is discrete. δ|G1 = δ1. ∃ (ξi) in L2(G1)·=1 satisfying ρ(s)ξi − ξi, β(s)ξi − ξi → 0, s ∈ G. Then VN(G1)β ⊲ ⊳α ℓ∞(G2) 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 Γ2,r

L1(G)(L1(G), L∞(G)) ∼

= Ml

cb(L1(

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 Γ2,r

L1(G)(L1(G), L∞(G)) ∼

= Ml

cb(L1(

G)) completely isometrically. Corollary (ACN ’18) Let G be QSIN. Then Γ2,r

A(G)(A(G), VN(G)) ∼

= M(G).

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Quantum Gilbert representation

Gs = Ga = VN(G) is QSIN iff G is amenable,

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Quantum Gilbert representation

Gs = Ga = VN(G) is QSIN iff G is amenable, so Corollary (ACN ’18) For G amenable, Γ2,r

L1(G)(L1(G), L∞(G)) ∼

= McbA(G) = B(G).

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Quantum Gilbert representation

Gs = Ga = VN(G) is QSIN iff G is amenable, so Corollary (ACN ’18) For G amenable, Γ2,r

L1(G)(L1(G), L∞(G)) ∼

= McbA(G) = B(G). Only recover Gilbert’s theorem for amenable groups.

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Quantum Gilbert representation

Gs = Ga = VN(G) is QSIN iff G is amenable, so Corollary (ACN ’18) For G amenable, Γ2,r

L1(G)(L1(G), L∞(G)) ∼

= McbA(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

Gs = Ga = VN(G) is QSIN iff G is amenable, so Corollary (ACN ’18) For G amenable, Γ2,r

L1(G)(L1(G), L∞(G)) ∼

= McbA(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 Gs = VN(G) has bounded degree with deg(Gs) = 1.

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Quantum Gilbert representation

Gs = Ga = VN(G) is QSIN iff G is amenable, so Corollary (ACN ’18) For G amenable, Γ2,r

L1(G)(L1(G), L∞(G)) ∼

= McbA(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 Gs = VN(G) has bounded degree with deg(Gs) = 1. 2 Ga = 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

Gs = Ga = VN(G) is QSIN iff G is amenable, so Corollary (ACN ’18) For G amenable, Γ2,r

L1(G)(L1(G), L∞(G)) ∼

= McbA(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 Gs = VN(G) has bounded degree with deg(Gs) = 1. 2 Ga = L∞(G) has bounded degree if and only if G is virtually

abelian (Moore ’72).

3 VN(G1)β ⊲

⊳α ℓ∞(G2) has bounded degree if G1 is countable discrete and G2 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 Γ2,r

L1(G)(L1(G), L∞(G)) ∼

= Ml

cb(L1(

G)) completely isomorphically, with deg( G)−1[bij] ≤ γ2,r([bij]) ≤ deg( G)[bij], for all [bij] ∈ Mn(Ml

cb(L1(

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 Γ2,r

L1(G)(L1(G), L∞(G)) ∼

= Ml

cb(L1(

G)) completely isomorphically, with deg( G)−1[bij] ≤ γ2,r([bij]) ≤ deg( G)[bij], for all [bij] ∈ Mn(Ml

cb(L1(

G)). Since deg( Ga) = deg(Gs) = 1, we obtain Γ2,r

L1(G)(L1(G), L∞(G)) ∼

= McbA(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

• Θ(Ml

cb(L1(

G))) = Θ(Ml

cb(L1(G)))c ∩ CBσ L∞(G)(B(L2(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

• Θ(Ml

cb(L1(

G))) = Θ(Ml

cb(L1(G)))c ∩ CBσ L∞(G)(B(L2(G))).

Proposition (Kasprzak–Sołtan ’14) Let G be a LCQG. Then ∆(L∞(G)) = ((1 ⊗ U∗) ∆(L∞( G))(1 ⊗ U))′ ∩ L∞(G)⊗L∞(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

• Θ(Ml

cb(L1(

G))) = Θ(Ml

cb(L1(G)))c ∩ CBσ L∞(G)(B(L2(G))).

Proposition (Kasprzak–Sołtan ’14) Let G be a LCQG. Then ∆(L∞(G)) = ((1 ⊗ U∗) ∆(L∞( G))(1 ⊗ U))′ ∩ L∞(G)⊗L∞(G). together with structure of subhomogeneous C∗-algebras.

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Mapping Ideals

CBL1(G)(L1(G), L∞(G)) ∪ Γ2,r

L1(G)(L1(G), L∞(G))

∪ Other mapping ideals L∞(G) ∪ Ml

cb(L1(

G)) ∪ ????

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Completely integral multipliers

Theorem (ACN ’18) Let G LCQG for which either

1

• G is QSIN with trivial scaling group

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Completely integral multipliers

Theorem (ACN ’18) Let G LCQG for which either

1

• G is QSIN with trivial scaling group, or

2

• G is a Kac algebra with bounded degree.

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Completely integral multipliers

Theorem (ACN ’18) Let G LCQG for which either

1

• G is QSIN with trivial scaling group, or

2

• G is a Kac algebra with bounded degree.

Then IL1(G)(L1(G), L∞(G)) ∼ = Ml

cb(L1(

G)) isomorphically

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Completely integral multipliers

Theorem (ACN ’18) Let G LCQG for which either

1

• G is QSIN with trivial scaling group, or

2

• G is a Kac algebra with bounded degree.

Then IL1(G)(L1(G), L∞(G)) ∼ = Ml

cb(L1(

G)) isomorphically, with

1 bcb ≤ ι(b) ≤ 2bcb

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Completely integral multipliers

Theorem (ACN ’18) Let G LCQG for which either

1

• G is QSIN with trivial scaling group, or

2

• G is a Kac algebra with bounded degree.

Then IL1(G)(L1(G), L∞(G)) ∼ = Ml

cb(L1(

G)) isomorphically, with

1 bcb ≤ ι(b) ≤ 2bcb 2 deg(

G)−1bcb ≤ ι(b) ≤ deg( G)(1 + deg( G)2)bcb

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Proof sketch: QSIN case

Lemma (ACN ’18) Same hypotheses, for every b ∈ Ml

cb(L1(

G)), ∃ (ai), (bi) in L∞(G) such that ∆(b) =

• i

ai ⊗ bi,

• i

aia∗

i ,

• i

a∗

i ai,

• i

bib∗

i ,

• i

b∗

i bi < ∞.

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Proof sketch: QSIN case

Lemma (ACN ’18) Same hypotheses, for every b ∈ Ml

cb(L1(

G)), ∃ (ai), (bi) in L∞(G) such that ∆(b) =

• i

ai ⊗ bi,

• i

aia∗

i ,

• i

a∗

i ai,

• i

bib∗

i ,

• i

b∗

i bi < ∞.

Pisier–Shlyakhtenko ’02; Haagerup–Musat ’08: For f ∈ L1(G) ⊗ L1(G) ֒ → (L∞(G) ⊗L∞(G))∗ ∃ states ϕ1, ϕ2, ψ1, ψ2 ∈ L∞(G)∗ such that f , a ⊗ b ≤ f L1(G)⊗∨L1(G)(ϕ1(aa∗)1/2ψ1(b∗b)1/2 + ϕ2(a∗a)1/2ψ2(bb∗)1/2)

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Proof sketch: QSIN case

Lemma (ACN ’18) Same hypotheses, for every b ∈ Ml

cb(L1(

G)), ∃ (ai), (bi) in L∞(G) such that ∆(b) =

• i

ai ⊗ bi,

• i

aia∗

i ,

• i

a∗

i ai,

• i

bib∗

i ,

• i

b∗

i bi < ∞.

Pisier–Shlyakhtenko ’02; Haagerup–Musat ’08: For f ∈ L1(G) ⊗ L1(G) ֒ → (L∞(G) ⊗L∞(G))∗ ∃ states ϕ1, ϕ2, ψ1, ψ2 ∈ L∞(G)∗ such that f , a ⊗ b ≤ f L1(G)⊗∨L1(G)(ϕ1(aa∗)1/2ψ1(b∗b)1/2 + ϕ2(a∗a)1/2ψ2(bb∗)1/2) Effros–Junge–Ruan ’00: L1(G) is locally reflexive.

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Completely nuclear multipliers

Racher ’94: NL1(G)(L1(G), L∞(G)) ∼ = B(G) ∩ AP(G).

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Completely nuclear multipliers

Racher ’94: NL1(G)(L1(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: NL1(G)(L1(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: NL1(G)(L1(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) = ℓ1(Gd) ⊕1 Mc(G).

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Completely nuclear multipliers

Racher ’94: NL1(G)(L1(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) = ℓ1(Gd) ⊕1 Mc(G). Lemma (ACN ’18) Cu( G)∗ ∼ = ℓ1( bG) ⊕1 ℓ1( bG)⊥

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Completely nuclear multipliers

Racher ’94: NL1(G)(L1(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) = ℓ1(Gd) ⊕1 Mc(G). Lemma (ACN ’18) Cu( G)∗ ∼ = ℓ1( bG) ⊕1 ℓ1( bG)⊥ bG 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

1

• G is QSIN

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Theorem (ACN ’18) Let G Kac algebra for which either

1

• G is QSIN, or

2

• G is has bounded degree.

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Theorem (ACN ’18) Let G Kac algebra for which either

1

• G is QSIN, or

2

• G is has bounded degree.

Then TFAE for x ∈ L∞(G): x ∈ ℓ1( bG) ⊆ Ml

cb(L1(

G))

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Theorem (ACN ’18) Let G Kac algebra for which either

1

• G is QSIN, or

2

• G is has bounded degree.

Then TFAE for x ∈ L∞(G): x ∈ ℓ1( bG) ⊆ Ml

cb(L1(

G)); f → x ⋆ f ∈ NL1(G)(L1(G), L∞(G))

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Theorem (ACN ’18) Let G Kac algebra for which either

1

• G is QSIN, or

2

• G is has bounded degree.

Then TFAE for x ∈ L∞(G): x ∈ ℓ1( bG) ⊆ Ml

cb(L1(

G)); f → x ⋆ f ∈ NL1(G)(L1(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

1

• G is QSIN, or

2

• G is has bounded degree.

Then TFAE for x ∈ L∞(G): x ∈ ℓ1( bG) ⊆ Ml

cb(L1(

G)); f → x ⋆ f ∈ NL1(G)(L1(G), L∞(G)); ∆(x) ∈ L∞(G) ⊗h L∞(G). Moreover, NL1(G)(L1(G), L∞(G)) ∼ = ℓ1( bG) isomorphically

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Theorem (ACN ’18) Let G Kac algebra for which either

1

• G is QSIN, or

2

• G is has bounded degree.

Then TFAE for x ∈ L∞(G): x ∈ ℓ1( bG) ⊆ Ml

cb(L1(

G)); f → x ⋆ f ∈ NL1(G)(L1(G), L∞(G)); ∆(x) ∈ L∞(G) ⊗h L∞(G). Moreover, NL1(G)(L1(G), L∞(G)) ∼ = ℓ1( bG) isomorphically, with

1 ˆ

f ≤ ν(ˆ f ) ≤ 2ˆ f

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Theorem (ACN ’18) Let G Kac algebra for which either

1

• G is QSIN, or

2

• G is has bounded degree.

Then TFAE for x ∈ L∞(G): x ∈ ℓ1( bG) ⊆ Ml

cb(L1(

G)); f → x ⋆ f ∈ NL1(G)(L1(G), L∞(G)); ∆(x) ∈ L∞(G) ⊗h L∞(G). Moreover, NL1(G)(L1(G), L∞(G)) ∼ = ℓ1( bG) isomorphically, with

1 ˆ

f ≤ ν(ˆ f ) ≤ 2ˆ f

2 deg(

G)−1ˆ f ≤ ν(ˆ f ) ≤ 2ˆ f

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

∆(x) ∈ L∞(G) ⊗h L∞(G) ⇒ x ∈ ℓ1( bG): QSIN case

Let A := Ml

cb(L1(

G))

·

⊆ L∞(G)

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

∆(x) ∈ L∞(G) ⊗h L∞(G) ⇒ x ∈ ℓ1( bG): QSIN case

Let A := Ml

cb(L1(

G))

·

⊆ L∞(G). Then ∆(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 ∈ ℓ1( bG): QSIN case

Let A := Ml

cb(L1(

G))

·

⊆ L∞(G). Then ∆(x) ∈ F(A, A; L∞(G) ⊗h L∞(G)) = A ⊗h A (Smith ′91).

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

∆(x) ∈ L∞(G) ⊗h L∞(G) ⇒ x ∈ ℓ1( bG): QSIN case

Let A := Ml

cb(L1(

G))

·

⊆ L∞(G). Then ∆(x) ∈ F(A, A; L∞(G) ⊗h L∞(G)) = A ⊗h A (Smith ′91).

• G QSIN ⇒ Ml

cb(L1(

G)) = Cu( G)∗ = ℓ1( bG) ⊕1 ℓ1( bG)⊥.

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

∆(x) ∈ L∞(G) ⊗h L∞(G) ⇒ x ∈ ℓ1( bG): QSIN case

Let A := Ml

cb(L1(

G))

·

⊆ L∞(G). Then ∆(x) ∈ F(A, A; L∞(G) ⊗h L∞(G)) = A ⊗h A (Smith ′91).

• G QSIN ⇒ Ml

cb(L1(

G)) = Cu( G)∗ = ℓ1( bG) ⊕1 ℓ1( bG)⊥. Show Φ(Ml

cb(L1(

G))⊗Ml

cb(L1(

G))) = Φ(Ml

cb(L1(

G))⊗ℓ1( bG)) ⊆ ℓ1( bG)

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

∆(x) ∈ L∞(G) ⊗h L∞(G) ⇒ x ∈ ℓ1( bG): QSIN case

Let A := Ml

cb(L1(

G))

·

⊆ L∞(G). Then ∆(x) ∈ F(A, A; L∞(G) ⊗h L∞(G)) = A ⊗h A (Smith ′91).

• G QSIN ⇒ Ml

cb(L1(

G)) = Cu( G)∗ = ℓ1( bG) ⊕1 ℓ1( bG)⊥. Show Φ(Ml

cb(L1(

G))⊗Ml

cb(L1(

G))) = Φ(Ml

cb(L1(

G))⊗ℓ1( bG)) ⊆ ℓ1( bG) using quantum Eberlein compactifications (Das–Daws ’14).

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

∆(x) ∈ L∞(G) ⊗h L∞(G) ⇒ x ∈ ℓ1( bG): QSIN case

Let A := Ml

cb(L1(

G))

·

⊆ L∞(G). Then ∆(x) ∈ F(A, A; L∞(G) ⊗h L∞(G)) = A ⊗h A (Smith ′91).

• G QSIN ⇒ Ml

cb(L1(

G)) = Cu( G)∗ = ℓ1( bG) ⊕1 ℓ1( bG)⊥. Show Φ(Ml

cb(L1(

G))⊗Ml

cb(L1(

G))) = Φ(Ml

cb(L1(

G))⊗ℓ1( bG)) ⊆ ℓ1( bG) using quantum Eberlein compactifications (Das–Daws ’14). x = Φ(∆(x)) ∈ Φ(A ⊗h A) ⊆ ℓ1( bG).

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Mapping Ideals: quantum group perspective

If G is a Kac algebra with G either QSIN or bounded degree CBL1(G)(L1(G), L∞(G)) ∪ Γ2,r

L1(G)(L1(G), L∞(G))

|| IL1(G)(L1(G), L∞(G)) ∪ NL1(G)(L1(G), L∞(G)) L∞(G) ∪ Ml

cb(L1(

G)) || Ml

cb(L1(

G)) ∪ ℓ1( bG)

Introduction QSIN Quantum Groups Quantum Gilbert Representation Mapping Ideals

Thank you!