Spectral theory of weighted Fourier algebras of (locally) compact - PowerPoint PPT Presentation

Motivations The case of compact quantum groups and SU q (2) The case of non-compact (quantum) groups Spectral theory of weighted Fourier algebras of (locally) compact quantum groups Hun Hee Lee ( 이 훈 희 ) Seoul National University Based on joint works with U. Franz (Besancon), M. Ghandehari (Delaware)/J. Ludwig (Lorraine)/N. Spronk (Waterloo)/L. Turowska (Gothenberg) August 9, 2019

Motivations The case of compact quantum groups and SU q (2) The case of non-compact (quantum) groups Overview Motivations The case of compact quantum groups and SU q (2) The case of non-compact (quantum) groups

Motivations The case of compact quantum groups and SU q (2) The case of non-compact (quantum) groups Fourier algebras on locally compact groups • The Fourier algebra A ( G ) of a locally compact group G is 1. A ( G ) = VN ( G ) ∗ , where VN ( G ) ⊆ B ( L 2 ( G )) is the group von Neumann algebra OR 2. A ( G ) = L 1 ( � G ), where � G is the dual quantum group OR g : f , g ∈ L 2 ( G ) } ⊆ C 0 ( G ), where 3. A ( G ) = { f ∗ ˇ g ( x ) = g ( x − 1 ). ˇ • A ( G ) is a (non-closed) subalgebra of C 0 ( G ), which is still a commutative Banach algebra under its own norm. • ( Prop, Eymard ‘64 ) We have a homeomorphism Spec A ( G ) ∼ = G , ϕ x �→ x , where ϕ x is the evaluation at the point x . Here, Spec A ( G ) is the Gelfand spectrum, i.e. all (bounded) non-zero multiplicative linear maps from A ( G ) into T .

Motivations The case of compact quantum groups and SU q (2) The case of non-compact (quantum) groups Fourier algebras of compact groups and their weighted versions • G: a compact group. � d π � ˆ A ( G ) = { f ∈ C ( G ) : � f � A ( G ) = f ( π ) � 1 < ∞} , π ∈ Irr ( G ) � where ˆ G f ( x ) π ( x ) ∗ dx ∈ M d π is the Fourier coefficient f ( π ) = of f at π . • For a weight function w : Irr ( G ) → [1 , ∞ ) we can define the weighted space A ( G , w ) with the norm � w ( π ) d π � ˆ � f � A ( G , w ) = f ( π ) � 1 . π ∈ Irr ( G ) • When w satisfies a “sub-multiplicativity” we have A ( G , w ) ⊆ A ( G ) ⊆ C ( G ) , which are commutative Banach algebras under their own norms.

Motivations The case of compact quantum groups and SU q (2) The case of non-compact (quantum) groups The spectrum of weighted Fourier algebras on compact groups • ( Q ) G: a compact group, w : Irr ( G ) → (0 , ∞ ) a weight function Spec A ( G , w ) =? • ( A ) For a compact (Lie) group G we have G ⊆ Spec A ( G , w ) ⊆ G C , where G C is the complexification of G . • ( Why? ) We have Pol( G ) ⊆ A ( G , w ) densely and (by Chevalley) Spec Pol( G ) ∼ = G C .

Motivations The case of compact quantum groups and SU q (2) The case of non-compact (quantum) groups Examples • ( Ex ) G = T and w β : Z → (0 , ∞ ) , n �→ β | n | , β ≥ 1: = { c ∈ C : 1 β ≤ | c | ≤ β } ⊆ C ∗ = T C . Spec A ( T , w β ) ∼ Moreover, we have � = C ∗ = T C . Spec A ( T , w β ) ∼ β ≥ 1 • ( Ex, Ludwig/Spronk/Turowska, ‘12 ) G = SU (2) with w β : Irr ( SU (2)) = 1 2 Z + → (0 , ∞ ) , s �→ β 2 s , β ≥ 1. � c � 0 V : U , V ∈ SU (2) , 1 Spec A ( SU (2) , w β ) ∼ = { U β ≤ | c | ≤ β } c − 1 0 and � Spec A ( SU (2) , w β ) ∼ = SL 2 ( C ) = SU (2) C . β ≥ 1 • ( Rem ) Bounded weights are not interesting!

Motivations The case of compact quantum groups and SU q (2) The case of non-compact (quantum) groups Quantum extensions • ( Q ) How about the case of a compact quantum group G ? • ( Preparations ) • Discrete dual quantum group � G : � � c 0 ( � M n s , ℓ ∞ ( � G ) = ℓ ∞ - G ) = c 0 - M n s . s ∈ Irr( G ) s ∈ Irr( G ) • c 00 ( � G ): the subalgebra of c 0 ( � G ) consisting of finitely supported elements. • The right Haar weight � h R on ℓ ∞ ( � G ) is given by � � d s Tr ( X s Q − 1 h R ( X ) = ) s s ∈ Irr ( G ) for X = ( X s ) s ∈ Irr ( G ) ∈ c 00 ( � G ), where Q s is the deformation matrix for Schur orthogonality and d s is the quantum dimension.

Motivations The case of compact quantum groups and SU q (2) The case of non-compact (quantum) groups Preparations: continued • ( Fourier transform on � G ): G : c 00 ( � � G ) → Pol ( G ) , X �→ ( X · � h R ⊗ id ) U , F = F where U = ⊕ s ∈ Irr( G ) u ( s ) is the multiplicative unitary for a choice of mutually inequivalent irreducible unitary representations of G , ( u ( s ) ) s ∈ Irr ( G ) . • ( The Fourier algebra A ( G )) We define A ( G ) = ℓ ∞ ( � G ) ∗ = VN ( G ) ∗ equipped with the multiplication � ∆ ∗ , which is the preadjoint of � ∆, the canonical co-multiplication on ℓ ∞ ( � G ).

Motivations The case of compact quantum groups and SU q (2) The case of non-compact (quantum) groups Preparations: continued 2 • We have a natural embedding c 00 ( � → A ( G ) , X �→ X · � G ) ֒ h R , which allows to extend the Fourier transform F to A ( G ) as follows. F : A ( G ) → C r ( G ) , ψ �→ ( ψ ⊗ id ) U . • For the element X · � h R ∈ A ( G ) with X = ( X s ) ∈ c 00 ( � G ) we get the concrete norm formula as follows. � || X · � d s · || X s Q − 1 h R || A ( G ) = || 1 . s s ∈ Irr ( G )

Motivations The case of compact quantum groups and SU q (2) The case of non-compact (quantum) groups Weighted Fourier algebras on compact quantum groups • ( Def/Prop ) For a weight function w : Irr ( G ) → [1 , ∞ ) satisfying a “sub-multiplicativity” we define � � X · � w ( s ) d s · || X s Q − 1 h R � A ( G , w ) = || 1 s s ∈ Irr ( G ) and we have contractive inclusions of Banach algebras A ( G , w ) ⊆ A ( G ) ⊆ C r ( G ) .

Motivations The case of compact quantum groups and SU q (2) The case of non-compact (quantum) groups Spectral theory for A ( G , w ): Scenario 1 • ( Q ) Spec A ( G , w ) = ? Any connection to “complexification”? • ( A ) We can see the complexification of the maximal classical closed subgroup of G . • ( Why? ) Pol( G ) ⊆ A ( G , w ) densely and Spec Pol( G ) is actually the (abstract) complexification of � G = Spec A ( G ), which is the maximal classical closed subgroup of G . • ( Thm ) Let w β ( s ) = β 2 s , s ∈ 1 2 Z + , then we have = { ρ ∈ C \{ 0 } : 1 Spec A ( SU q (2) , w β ) ∼ β ≤ | ρ | ≤ β } � ρ � 0 ∼ ∈ M 2 ( C ) : || V || ∞ ≤ β } . = { V = ρ − 1 0

Motivations The case of compact quantum groups and SU q (2) The case of non-compact (quantum) groups Spectral theory for A ( G , w ): Scenario 2 • An immediate limitation of Spec A ( G , w ) comes from the fact that the algebra A ( G , w ) is non-commutative. • G : a compact Lie group Spec Pol( G ) ∼ = G C ∼ = Spec C 0 ( G C ) ∼ = sp C 0 ( G C ) , where sp C 0 ( G C ) is the C ∗ -algebra spectrum, which is the set of equivalence classes of all irreducible ∗ -representation π : C 0 ( G C ) → B ( H ) for some Hilbert space H . • π ∈ sp C 0 ( G C ), π : C 0 ( G C ) → B ( H ) ⇒ ∃ x ∈ G C such that π = ϕ x : C 0 ( G C ) → C ⇒ ϕ x : H ( G C ) → C , where H ( G C ) is the algebra of holomorphic functions on G C . ⇒ ϕ x : Pol( G ) → C , a homomorphism since Pol( G ) ⊆ H ( G C ).

Motivations The case of compact quantum groups and SU q (2) The case of non-compact (quantum) groups Quantum double • For a compact quantum group G we have the quantum ⋉ � double G ⋊ G by Podles/Woronowicz. • The associated C ∗ -algebra is given by ⋉ � G ) := C ( G ) ⊗ c 0 ( � C 0 ( G ⋊ G ) with the co-multiplication ∆ C = ( id ⊗ Σ U ⊗ id )(∆ ⊗ � ∆) , where Σ U is the ∗ -isomorphism given by Σ U : C ( G ) ⊗ c 0 ( � G ) → c 0 ( � G ) ⊗ C ( G ) , a ⊗ x �→ U ( x ⊗ a ) U ∗ . ⋉ � • The left (and right) Haar weight on C 0 ( G ⋊ G ) is given by h ⊗ � h R , where h is the Haar state on C ( G ).

Motivations The case of compact quantum groups and SU q (2) The case of non-compact (quantum) groups The case of G = SU q (2), 0 < q < 1 ⋉ � • Our choice of complexification G C is G ⋊ G , which we write SL q (2 , C ). • sp C 0 ( SL q (2 , C )) ∼ = sp C ( SU q (2)) × sp c 0 ( � SU q (2)). • A : the ∗ -algebra of all elements affilliated to C 0 ( SL q (2 , C )) A hol : a subalgebra of A generated by the coefficient “function”s α, β, γ, δ of SL q (2 , C ) Q : A hol → Pol( SU q (2)) a bijective homomorphism given by Q ( α ) = a q , Q ( β ) = − qc ∗ q , Q ( γ ) = c q and Q ( δ ) = a ∗ q , where a q and c q are canonical SU q (2) generators. • From π : C 0 ( SL q (2 , C )) → B ( H ) ⇒ π : A → B ( H ), the canonical extension ⇒ ϕ = π ◦ Q − 1 : Pol( SU q (2)) → A hol ⊆ A → B ( H ), homomorphism ⇒ v ∈ � s ∈ Irr ( G ) ( M n s ⊗ B ( H )) associated element.

Motivations The case of compact quantum groups and SU q (2) The case of non-compact (quantum) groups The case of G = SU q (2): continued • We begin with π ∈ sp C 0 ( SL q (2 , C )) �→ ( π c , π d ) ∈ sp C ( SU q (2)) × sp c 0 ( � SU q (2)) with the associated elements v , v c , v d ∈ � s ∈ Irr ( G ) ( M n s ⊗ B ( H )) respectively. • ( Prop ) We have v = v c v d and v c is a unitary (no contribution to norm). • For the above reason we may focus on the case SU q (2)) = { A s : s ∈ 1 π = π d ∈ sp c 0 ( � 2 Z + } , where A s , s ∈ 1 2 Z + are irreducible AN q -matrices by Podles/Woronowicz.