Groupoids and Pseudodifferential calculus I. D. & Skandalis - - PowerPoint PPT Presentation

Groupoids and Pseudodifferential calculus I. D. & Skandalis - Adiabatic groupoid, crossed product by R ∗ + and Pseudodifferential calculus - Adv. Math 2014 http://math.univ-bpclermont.fr/ ∼ debord/ NGA - Frascati 2014 NGA - Frascati (2014) Groupoids and Pseudodifferential calculus I. 1 / 16

Motivations Let G ⇒ G (0) be a smooth groupoid and denote by A G its Lie algebroid. One gets exact sequences of C ∗ -algebras: NGA - Frascati (2014) Groupoids and Pseudodifferential calculus I. 2 / 16

Motivations Let G ⇒ G (0) be a smooth groupoid and denote by A G its Lie algebroid. One gets exact sequences of C ∗ -algebras: From Analysis : The pseudodifferential operators exact sequence 0 → C ∗ ( G ) − → Ψ ∗ → C ( S ∗ A G ) → 0 0 ( G ) − (PDO) NGA - Frascati (2014) Groupoids and Pseudodifferential calculus I. 2 / 16

Motivations Let G ⇒ G (0) be a smooth groupoid and denote by A G its Lie algebroid. One gets exact sequences of C ∗ -algebras: From Analysis : The pseudodifferential operators exact sequence 0 → C ∗ ( G ) − → Ψ ∗ → C ( S ∗ A G ) → 0 0 ( G ) − (PDO) which is a generalization, for a smooth compact manifold M , of σ 0 0 → K ( L 2 ( M )) − → Ψ ∗ → C ( S ∗ TM ) → 0 0 ( M ) − NGA - Frascati (2014) Groupoids and Pseudodifferential calculus I. 2 / 16

Motivations Let G ⇒ G (0) be a smooth groupoid and denote by A G its Lie algebroid. One gets exact sequences of C ∗ -algebras: From Analysis : The pseudodifferential operators exact sequence 0 → C ∗ ( G ) − → Ψ ∗ → C ( S ∗ A G ) → 0 0 ( G ) − (PDO) which is a generalization, for a smooth compact manifold M , of σ 0 0 → K ( L 2 ( M )) − → Ψ ∗ → C ( S ∗ TM ) → 0 0 ( M ) − From Geometry : The Gauge adiabatic groupoid short exact sequence : 0 → C ∗ ( G ) ⊗ K − → J ( G ) ⋊ R ∗ → C ( S ∗ A G ) ⊗ K → 0 + − (GAG) Where J ( G ) ⊂ C ∗ ( G ad ) is an ideal of the C ∗ -algebra of the adiabatic groupoid G ad of G , and the natural action of R ∗ + on G ad is considered. NGA - Frascati (2014) Groupoids and Pseudodifferential calculus I. 2 / 16

Main Results Theorem (D. & Skandalis) There is an ideal J ( G ) ⊂ C ∞ c ( G ad ) such that : ⋆ The order 0 pseudo differential operators on G are multipliers of � ∞ dt C ∞ c ( G ) of the form f t t where f = ( f t ) t ∈ R + ∈ J ( G ) . 0 NGA - Frascati (2014) Groupoids and Pseudodifferential calculus I. 3 / 16

Main Results Theorem (D. & Skandalis) There is an ideal J ( G ) ⊂ C ∞ c ( G ad ) such that : ⋆ The order 0 pseudo differential operators on G are multipliers of � ∞ dt C ∞ c ( G ) of the form f t t where f = ( f t ) t ∈ R + ∈ J ( G ) . 0 � One can make a completion of J ( G ) into a bimodule E which leads to a Morita equivalence between Ψ ∗ 0 ( G ) and J ( G ) ⋊ R ∗ + . NGA - Frascati (2014) Groupoids and Pseudodifferential calculus I. 3 / 16

Main Results Theorem (D. & Skandalis) There is an ideal J ( G ) ⊂ C ∞ c ( G ad ) such that : ⋆ The order 0 pseudo differential operators on G are multipliers of � ∞ dt C ∞ c ( G ) of the form f t t where f = ( f t ) t ∈ R + ∈ J ( G ) . 0 � One can make a completion of J ( G ) into a bimodule E which leads to a Morita equivalence between Ψ ∗ 0 ( G ) and J ( G ) ⋊ R ∗ + . Today, in this talk : Describe the short exact sequence ( PDO ). NGA - Frascati (2014) Groupoids and Pseudodifferential calculus I. 3 / 16

Main Results Theorem (D. & Skandalis) There is an ideal J ( G ) ⊂ C ∞ c ( G ad ) such that : ⋆ The order 0 pseudo differential operators on G are multipliers of � ∞ dt C ∞ c ( G ) of the form f t t where f = ( f t ) t ∈ R + ∈ J ( G ) . 0 � One can make a completion of J ( G ) into a bimodule E which leads to a Morita equivalence between Ψ ∗ 0 ( G ) and J ( G ) ⋊ R ∗ + . Today, in this talk : Describe the short exact sequence ( PDO ). Describe the short exact sequence ( GAG ). NGA - Frascati (2014) Groupoids and Pseudodifferential calculus I. 3 / 16

Main Results Theorem (D. & Skandalis) There is an ideal J ( G ) ⊂ C ∞ c ( G ad ) such that : ⋆ The order 0 pseudo differential operators on G are multipliers of � ∞ dt C ∞ c ( G ) of the form f t t where f = ( f t ) t ∈ R + ∈ J ( G ) . 0 � One can make a completion of J ( G ) into a bimodule E which leads to a Morita equivalence between Ψ ∗ 0 ( G ) and J ( G ) ⋊ R ∗ + . Today, in this talk : Describe the short exact sequence ( PDO ). Describe the short exact sequence ( GAG ). Describe the ideal J ( G ) and give a precise statement of ⋆ . NGA - Frascati (2014) Groupoids and Pseudodifferential calculus I. 3 / 16

s r G (0) Lie algebroid and exponential map of G ⇒ NGA - Frascati (2014) Groupoids and Pseudodifferential calculus I. 4 / 16

s r G (0) Lie algebroid and exponential map of G ⇒ For x ∈ G (0) denote G x = s − 1 ( x ) and G x = r − 1 ( x ). NGA - Frascati (2014) Groupoids and Pseudodifferential calculus I. 4 / 16

s r G (0) Lie algebroid and exponential map of G ⇒ For x ∈ G (0) denote G x = s − 1 ( x ) and G x = r − 1 ( x ). The Lie algebroid π : A G → G (0) of G is the normal bundle of the inclusion of units G (0) → G it can be identified with the restriction to G (0) of Ker ( ds ) : � A G = TG / TG (0) ≃ Ker ( ds ) | G (0) = T x G x x ∈ G (0) The differential map dr of r leads to the anchor map : ♯ : A G → TG (0) . NGA - Frascati (2014) Groupoids and Pseudodifferential calculus I. 4 / 16

s r G (0) Lie algebroid and exponential map of G ⇒ For x ∈ G (0) denote G x = s − 1 ( x ) and G x = r − 1 ( x ). The Lie algebroid π : A G → G (0) of G is the normal bundle of the inclusion of units G (0) → G it can be identified with the restriction to G (0) of Ker ( ds ) : � A G = TG / TG (0) ≃ Ker ( ds ) | G (0) = T x G x x ∈ G (0) The differential map dr of r leads to the anchor map : ♯ : A G → TG (0) . An exponential map θ : V ′ → V for G is a diffeomorphism where G (0) ⊂ V ′ ⊂ A G , G (0) ⊂ V ⊂ G , V and V ′ being open and such that : θ | G (0) = Id and r ◦ θ = π , For x ∈ G (0) , d θ ( x , 0) is the ”identity” on the normal direction of the inclusion of G (0) : A G x ≃ T ( x , 0) A G / T x G (0) → A G x . NGA - Frascati (2014) Groupoids and Pseudodifferential calculus I. 4 / 16

The (PDO) exact sequence Given the groupoid G , one can define : A convolution ∗ -algebra C ∞ c ( G ) which leads to a C ∗ -algebra C ∗ ( G ) after choosing a norm (J. Renault). NGA - Frascati (2014) Groupoids and Pseudodifferential calculus I. 5 / 16

The (PDO) exact sequence Given the groupoid G , one can define : A convolution ∗ -algebra C ∞ c ( G ) which leads to a C ∗ -algebra C ∗ ( G ) after choosing a norm (J. Renault). The multiplier algebra M ( C ∞ c ( G )) of C ∞ c ( G ). NGA - Frascati (2014) Groupoids and Pseudodifferential calculus I. 5 / 16

The (PDO) exact sequence Given the groupoid G , one can define : A convolution ∗ -algebra C ∞ c ( G ) which leads to a C ∗ -algebra C ∗ ( G ) after choosing a norm (J. Renault). The multiplier algebra M ( C ∞ c ( G )) of C ∞ c ( G ). Pseudodifferential calculus (A. Connes, B. Monthubert & F. Pierrot, V. Nistor , A. Weinstein & P. Xu) NGA - Frascati (2014) Groupoids and Pseudodifferential calculus I. 5 / 16

The (PDO) exact sequence Given the groupoid G , one can define : A convolution ∗ -algebra C ∞ c ( G ) which leads to a C ∗ -algebra C ∗ ( G ) after choosing a norm (J. Renault). The multiplier algebra M ( C ∞ c ( G )) of C ∞ c ( G ). Pseudodifferential calculus (A. Connes, B. Monthubert & F. Pierrot, V. Nistor , A. Weinstein & P. Xu) . For any m ∈ Z , the set S m ( A ∗ G ) ⊂ C ∞ ( A ∗ G ) of order m polyhomogeneous symbols : NGA - Frascati (2014) Groupoids and Pseudodifferential calculus I. 5 / 16

The (PDO) exact sequence Given the groupoid G , one can define : A convolution ∗ -algebra C ∞ c ( G ) which leads to a C ∗ -algebra C ∗ ( G ) after choosing a norm (J. Renault). The multiplier algebra M ( C ∞ c ( G )) of C ∞ c ( G ). Pseudodifferential calculus (A. Connes, B. Monthubert & F. Pierrot, V. Nistor , A. Weinstein & P. Xu) . For any m ∈ Z , the set S m ( A ∗ G ) ⊂ C ∞ ( A ∗ G ) of order m polyhomogeneous symbols : ϕ ∈ C ∞ ( A ∗ G ) belongs to S m ( A ∗ G ) if there exists ( a j ) j ∈ � m , ∞ � , where a j ∈ C ∞ ( A ∗ G ) is homogeneous of order j : a j ( x , λξ ) = λ j a j ( x , ξ ) and ∞ � ϕ ∼ a m − k k =0 NGA - Frascati (2014) Groupoids and Pseudodifferential calculus I. 5 / 16