Introduction Hilbert modules equipped with orthogonal filtrations Quantum symmetry group of a Hilbert C*-module Quantum symmetry groups of Hilbert C*-modules equipped with orthogonal filtrations Manon Thibault de Chanvalon Universit´ e Blaise Pascal de Clermont-Ferrand Conference “Noncommutative Geometry and Applications” 20th June 2014, Frascati 1/23 Manon Thibault de Chanvalon Quantum symmetry groups of Hilbert C*-modules
Introduction Hilbert modules equipped with orthogonal filtrations Quantum symmetry group of a Hilbert C*-module D. Goswami - Quantum group of isometries in classical and noncommutative geometry. Comm. Math. Phys., 285(1):141–160 (2009) 2/23 Manon Thibault de Chanvalon Quantum symmetry groups of Hilbert C*-modules
Introduction Hilbert modules equipped with orthogonal filtrations Quantum symmetry group of a Hilbert C*-module D. Goswami - Quantum group of isometries in classical and noncommutative geometry. Comm. Math. Phys., 285(1):141–160 (2009) T. Banica & A. Skalski - Quantum symmetry groups of C ∗ -algebras equipped with orthogonal filtrations. Proc. Amer. Math. Soc., 106(5):980–1004 (2013) 2/23 Manon Thibault de Chanvalon Quantum symmetry groups of Hilbert C*-modules
Introduction Hilbert modules equipped with orthogonal filtrations Quantum symmetry group of a Hilbert C*-module The symmetry group of a given space is the final object in the category of groups acting on this space. 3/23 Manon Thibault de Chanvalon Quantum symmetry groups of Hilbert C*-modules
Introduction Hilbert modules equipped with orthogonal filtrations Quantum symmetry group of a Hilbert C*-module The symmetry group of a given space is the final object in the category of groups acting on this space. In other words: The symmetry group Sym( X ) of a given space X is the group satisfying the universal property: for each group G acting on X , there exists a unique morphism G → Sym ( X ) . It is given by: G → Sym ( X ) g �→ ( x �→ x . g ) 3/23 Manon Thibault de Chanvalon Quantum symmetry groups of Hilbert C*-modules
Introduction Hilbert modules equipped with orthogonal filtrations Quantum symmetry group of a Hilbert C*-module Analogously, the quantum symmetry group of a given space is defined as the final object in the category of quantum groups acting on this space. 4/23 Manon Thibault de Chanvalon Quantum symmetry groups of Hilbert C*-modules
Introduction Hilbert modules equipped with orthogonal filtrations Quantum symmetry group of a Hilbert C*-module Analogously, the quantum symmetry group of a given space is defined as the final object in the category of quantum groups acting on this space. Steps for defining the quantum symmetry group of a given object: Define the category of its quantum transformation groups. 4/23 Manon Thibault de Chanvalon Quantum symmetry groups of Hilbert C*-modules
Introduction Hilbert modules equipped with orthogonal filtrations Quantum symmetry group of a Hilbert C*-module Analogously, the quantum symmetry group of a given space is defined as the final object in the category of quantum groups acting on this space. Steps for defining the quantum symmetry group of a given object: Define the category of its quantum transformation groups. Check that this category admits a final object. 4/23 Manon Thibault de Chanvalon Quantum symmetry groups of Hilbert C*-modules
Introduction Hilbert modules equipped with orthogonal filtrations Quantum symmetry group of a Hilbert C*-module Definition Let A be a C ∗ -algebra and let E be a Hilbert A -module. An orthogonal filtration ( τ, ( V i ) i ∈I , J , W ) of E consists of: a faithful state τ on A , a family ( V i ) i ∈I of finite-dimensional subspaces of E such that: 1 for all i , j ∈ I with i � = j , ∀ ξ ∈ V i and ∀ η ∈ V j , τ ( � ξ | η � A ) = 0 , 2 the space E 0 = � V i is dense in ( E , � · � A ) , i ∈I a one-to-one antilinear operator J : E 0 → E 0 , a finite-dimensional subspace W of E . 5/23 Manon Thibault de Chanvalon Quantum symmetry groups of Hilbert C*-modules
Introduction Hilbert modules equipped with orthogonal filtrations Quantum symmetry group of a Hilbert C*-module Example Let M be a compact Riemannian manifold. The space of continuous sections of the bundle of exterior forms on M , Γ(Λ ∗ M ) , is a Hilbert C ( M ) -module. A natural orthogonal filtration of Γ(Λ ∗ M ) is given by: ( V i ) i ∈ N is the family of eigenspaces of the de Rham operator D = d + d ∗ , � τ = · d vol , W = C . ( m �→ 1 Λ ∗ m M ) , J : Γ(Λ ∗ M ) → Γ(Λ ∗ M ) is the canonical involution. 6/23 Manon Thibault de Chanvalon Quantum symmetry groups of Hilbert C*-modules
Introduction Hilbert modules equipped with orthogonal filtrations Quantum symmetry group of a Hilbert C*-module Definition A spectral triple ( A , H , D ) is said to be finitely summable if there exists p ∈ N such that | D | − p admits a Dixmier trace, which is nonzero. 7/23 Manon Thibault de Chanvalon Quantum symmetry groups of Hilbert C*-modules
Introduction Hilbert modules equipped with orthogonal filtrations Quantum symmetry group of a Hilbert C*-module Definition A spectral triple ( A , H , D ) is said to be finitely summable if there exists p ∈ N such that | D | − p admits a Dixmier trace, which is nonzero. ( A , H , D ) is said to be regular if for all a ∈ A and all n ∈ N , a and [ D , a ] are in the domain of the unbounded operator δ n on L ( H ) , where δ = [ | D | , · ] . 7/23 Manon Thibault de Chanvalon Quantum symmetry groups of Hilbert C*-modules
Introduction Hilbert modules equipped with orthogonal filtrations Quantum symmetry group of a Hilbert C*-module Definition ( A , H , D ) satisfies the finiteness and absolute continuity condition if furthermore, the space H ∞ = � dom( D k ) is a finitely k ∈ N generated projective left A -module, and if there exists q ∈ M n ( A ) with q = q 2 = q ∗ such that: 1 H ∞ ∼ = A n q , 2 the left A -scalar product A �·|·� induced on H ∞ by the previous isomorphism satisfies: Tr ω ( A � ξ | η �| D | − p ) = ( η | ξ ) H . Tr ω ( | D | − p ) 8/23 Manon Thibault de Chanvalon Quantum symmetry groups of Hilbert C*-modules
Introduction Hilbert modules equipped with orthogonal filtrations Quantum symmetry group of a Hilbert C*-module Example Setting: A = closure of A in L ( H ) , E = completion of H ∞ (for the A -norm), ( V i ) i ∈ N = eigenspaces of D , τ = a �→ Tr ω ( a | D | − p ) Tr ω ( | D | − p ) If we assume furthermore that τ is faithful and E 0 = � V i is i ∈I dense in E , we get an orthogonal filtration of E (with J : E 0 → E 0 any one-to-one antilinear map and e.g. W = (0) ). 9/23 Manon Thibault de Chanvalon Quantum symmetry groups of Hilbert C*-modules
Introduction Definition Hilbert modules equipped with orthogonal filtrations Idea of proof Quantum symmetry group of a Hilbert C*-module Definition A Woronowicz C ∗ -algebra is a couple ( C ( G ) , ∆) , where C ( G ) is a C ∗ -algebra and ∆ : C ( G ) → C ( G ) ⊗ C ( G ) is a ∗ -morphism such that: (∆ ⊗ id ) ◦ ∆ = ( id ⊗ ∆) ◦ ∆ , the spaces span { ∆( C ( G )) . ( C ( G ) ⊗ 1) } and span { ∆( C ( G )) . (1 ⊗ C ( G )) } are both dense in C ( G ) ⊗ C ( G ) . 10/23 Manon Thibault de Chanvalon Quantum symmetry groups of Hilbert C*-modules
Introduction Definition Hilbert modules equipped with orthogonal filtrations Idea of proof Quantum symmetry group of a Hilbert C*-module Definition Let ( C ( G ) , ∆) be a Woronowicz C ∗ -algebra, and A be a C ∗ -algebra. A coaction of C ( G ) on A is a ∗ -morphism α : A → A ⊗ C ( G ) satisfying: ( α ⊗ id C ( G ) ) ◦ α = ( id A ⊗ ∆) ◦ α α ( A ) . (1 ⊗ C ( G )) is dense in A ⊗ C ( G ) . 11/23 Manon Thibault de Chanvalon Quantum symmetry groups of Hilbert C*-modules
Introduction Definition Hilbert modules equipped with orthogonal filtrations Idea of proof Quantum symmetry group of a Hilbert C*-module Definition Let ( C ( G ) , ∆ , α ) be a Woronowicz C ∗ -algebra coacting on a C ∗ -algebra A , and let E be a Hilbert A -module. A coaction of C ( G ) on E is a linear map β : E → E ⊗ C ( G ) satisfying: ( β ⊗ id ) ◦ β = ( id ⊗ ∆) ◦ β β ( E ) . ( A ⊗ C ( G )) is dense in E ⊗ C ( G ) ∀ ξ, η ∈ E , � β ( ξ ) | β ( η ) � A ⊗ C ( G ) = α ( � ξ | η � A ) ∀ ξ ∈ E , ∀ a ∈ A , β ( ξ. a ) = β ( ξ ) .α ( a ) 12/23 Manon Thibault de Chanvalon Quantum symmetry groups of Hilbert C*-modules
Introduction Definition Hilbert modules equipped with orthogonal filtrations Idea of proof Quantum symmetry group of a Hilbert C*-module Definition Let ( C ( G ) , ∆ , α ) be a Woronowicz C ∗ -algebra coacting on a C ∗ -algebra A , and let E be a Hilbert A -module. A coaction of C ( G ) on E is a linear map β : E → E ⊗ C ( G ) satisfying: ( β ⊗ id ) ◦ β = ( id ⊗ ∆) ◦ β β ( E ) . ( A ⊗ C ( G )) is dense in E ⊗ C ( G ) ∀ ξ, η ∈ E , � β ( ξ ) | β ( η ) � A ⊗ C ( G ) = α ( � ξ | η � A ) ∀ ξ ∈ E , ∀ a ∈ A , β ( ξ. a ) = β ( ξ ) .α ( a ) We say that the coaction ( α, β ) of C ( G ) on E is faithful if there exists no nontrivial Woronowicz C ∗ -subalgebra C ( H ) of C ( G ) such that β ( E ) ⊂ E ⊗ C ( H ) . 12/23 Manon Thibault de Chanvalon Quantum symmetry groups of Hilbert C*-modules
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