Semidirect Product Fell Bundles L. Hall*, S. Kaliszewski, J. Quigg, - PowerPoint PPT Presentation

References Semidirect Product Fell Bundles L. Hall*, S. Kaliszewski, J. Quigg, D. Williams *H*KQW Semidirect Products 1 / 15

References Groupoid A set G and a subset G ( 2 ) ⊂ G × G together with maps Composition: G ( 2 ) → G , denoted ( γ, η ) �→ γ · η i. Inversion: G → G , denoted γ �→ γ − 1 ii. *H*KQW Semidirect Products 2 / 15

References Groupoid A set G and a subset G ( 2 ) ⊂ G × G together with maps Composition: G ( 2 ) → G , denoted ( γ, η ) �→ γ · η i. Inversion: G → G , denoted γ �→ γ − 1 ii. Associative composition Involutive inversion Cancellative products γ − 1 · γ · η = η and γ · η · η − 1 = γ *H*KQW Semidirect Products 2 / 15

References Groupoid A set G and a subset G ( 2 ) ⊂ G × G together with maps Composition: G ( 2 ) → G , denoted ( γ, η ) �→ γ · η i. Inversion: G → G , denoted γ �→ γ − 1 ii. Associative composition Involutive inversion Cancellative products γ − 1 · γ · η = η and γ · η · η − 1 = γ G ( 0 ) *H*KQW Semidirect Products 2 / 15

References Groupoid A set G and a subset G ( 2 ) ⊂ G × G together with maps Composition: G ( 2 ) → G , denoted ( γ, η ) �→ γ · η i. Inversion: G → G , denoted γ �→ γ − 1 ii. Associative composition Involutive inversion Cancellative products γ − 1 · γ · η = η and γ · η · η − 1 = γ G ( 0 ) Locally compact groupoids, with Haar systems. *H*KQW Semidirect Products 2 / 15

References Examples Groups *H*KQW Semidirect Products 3 / 15

References Examples Groups T opological Spaces *H*KQW Semidirect Products 3 / 15

References Examples Groups T opological Spaces Group Bundles *H*KQW Semidirect Products 3 / 15

References Examples Groups T opological Spaces Group Bundles Equivalence Relations R ⊂ S × S associative composition ( t, s ) · ( s, r ) = ( t, r ) involutive inversion ( t, s ) − 1 = ( s, t ) cancellative products ( t, s ) · ( s, r ) · ( r, s ) = ( t, s ) · ( s, s ) = ( t, s ) . *H*KQW Semidirect Products 3 / 15

References Examples Groups T opological Spaces Group Bundles Equivalence Relations R ⊂ S × S associative composition ( t, s ) · ( s, r ) = ( t, r ) involutive inversion ( t, s ) − 1 = ( s, t ) cancellative products ( t, s ) · ( s, r ) · ( r, s ) = ( t, s ) · ( s, s ) = ( t, s ) . Group Transformation Groupoids Group Γ acting on a space T . Define Γ ⋊ T by Composition ( h, y ) · ( g, x ) = ( hg, x ) whenever y = gx Inverse ( g, x ) − 1 = ( g − 1 , gx ) *H*KQW Semidirect Products 3 / 15

References Haar Systems Definition A Haar system[Wil19] on a locally compact groupoid G is a family λ = { λ u } u ∈ G ( 0 ) of radon measures on G such that *H*KQW Semidirect Products 4 / 15

References Haar Systems Definition A Haar system[Wil19] on a locally compact groupoid G is a family λ = { λ u } u ∈ G ( 0 ) of radon measures on G such that supp λ u = r − 1 ( u ) 1 *H*KQW Semidirect Products 4 / 15

References Haar Systems Definition A Haar system[Wil19] on a locally compact groupoid G is a family λ = { λ u } u ∈ G ( 0 ) of radon measures on G such that supp λ u = r − 1 ( u ) 1 For γ ∈ G , γ · λ s ( γ ) = λ r ( γ ) 2 *H*KQW Semidirect Products 4 / 15

References Haar Systems Definition A Haar system[Wil19] on a locally compact groupoid G is a family λ = { λ u } u ∈ G ( 0 ) of radon measures on G such that supp λ u = r − 1 ( u ) 1 For γ ∈ G , γ · λ s ( γ ) = λ r ( γ ) 2 � G f dλ u Continuous assembly: For f ∈ C c ( G ) , the map u �→ 3 is continuous. *H*KQW Semidirect Products 4 / 15

References Haar Systems Definition A Haar system[Wil19] on a locally compact groupoid G is a family λ = { λ u } u ∈ G ( 0 ) of radon measures on G such that supp λ u = r − 1 ( u ) 1 For γ ∈ G , γ · λ s ( γ ) = λ r ( γ ) 2 � G f dλ u Continuous assembly: For f ∈ C c ( G ) , the map u �→ 3 is continuous. Define a convolution product on C c ( G ) : G f ( γ ) g ( γ − 1 · η ) dλ r ( η ) ( γ ) � f ∗ g ( η ) = Involution: f ∗ ( γ ) = f ( γ − 1 ) *H*KQW Semidirect Products 4 / 15

References Haar Systems Definition A Haar system[Wil19] on a locally compact groupoid G is a family λ = { λ u } u ∈ G ( 0 ) of radon measures on G such that supp λ u = r − 1 ( u ) 1 For γ ∈ G , γ · λ s ( γ ) = λ r ( γ ) 2 � G f dλ u Continuous assembly: For f ∈ C c ( G ) , the map u �→ 3 is continuous. Define a convolution product on C c ( G ) : G f ( γ ) g ( γ − 1 · η ) dλ r ( η ) ( γ ) � f ∗ g ( η ) = Involution: f ∗ ( γ ) = f ( γ − 1 ) Complete to get a C ∗ -algebra! *H*KQW Semidirect Products 4 / 15

References Groupoid Actions Groupoid actions[Wil19]: For a space T , begin with moment map ρ : T → G ( 0 ) . *H*KQW Semidirect Products 5 / 15

References Groupoid Actions Groupoid actions[Wil19]: For a space T , begin with moment map ρ : T → G ( 0 ) . Fibre product G ∗ T = { ( γ, t ) : ρ ( t ) = s ( γ ) } . proj G ∗ T T ρ proj s G ( 0 ) G *H*KQW Semidirect Products 5 / 15

References Groupoid Actions Groupoid actions[Wil19]: For a space T , begin with moment map ρ : T → G ( 0 ) . Fibre product G ∗ T = { ( γ, t ) : ρ ( t ) = s ( γ ) } . proj G ∗ T T ρ proj s G ( 0 ) G The action: map G ∗ T → T , styled ( γ, t ) �→ γt with ρ ( t ) · t = t For ( η, γ ) ∈ G ( 2 ) , ( γ, t ) ∈ G ∗ T , η · γ t = ηγt . *H*KQW Semidirect Products 5 / 15

References Groupoid Bundle Actions Let p : T → B be a bundle ( p is continuous, open, surjective). *H*KQW Semidirect Products 6 / 15

References Groupoid Bundle Actions Let p : T → B be a bundle ( p is continuous, open, surjective). G acts on the bundle provided G acts on T , B , and p intertwines the actions: γp ( t ) = p ( γt ) . *H*KQW Semidirect Products 6 / 15

References Groupoid Bundle Actions Let p : T → B be a bundle ( p is continuous, open, surjective). G acts on the bundle provided G acts on T , B , and p intertwines the actions: γp ( t ) = p ( γt ) . Subbundle isomorphism: G, T, B as above. For x ∈ G t �→ xt T | B s ( x ) T | B r ( x ) p | p | B s ( x ) B r ( x ) b �→ xb *H*KQW Semidirect Products 6 / 15

References Actions by Isomorphisms G acting on H a LCH groupoid. Definition G acts by isomorphisms provided *H*KQW Semidirect Products 7 / 15

References Actions by Isomorphisms G acting on H a LCH groupoid. Definition G acts by isomorphisms provided fibres over G ( 0 ) are subgroupoids of H ; *H*KQW Semidirect Products 7 / 15

References Actions by Isomorphisms G acting on H a LCH groupoid. Definition G acts by isomorphisms provided fibres over G ( 0 ) are subgroupoids of H ; for x ∈ G , the map h �→ xh is an isomorphism of the fibres. *H*KQW Semidirect Products 7 / 15

References Actions by Isomorphisms G acting on H a LCH groupoid. Definition G acts by isomorphisms provided fibres over G ( 0 ) are subgroupoids of H ; for x ∈ G , the map h �→ xh is an isomorphism of the fibres. ♥ ♥ H ♠ ♣ ♠ ♣ ρ G ( 0 ) • • *H*KQW Semidirect Products 7 / 15

References Actions on H Definition (Semidirect Product Groupoid) Denote by H ≀ G the fibre product H ∗ G = { ( h, x ) : ρ ( h ) = r ( x ) } with operations ( h, x ) − 1 = ( x − 1 h − 1 , x − 1 ) ( h, x ) · ( k, y ) = ( h · xk, x · y ) *H*KQW Semidirect Products 8 / 15

References Actions on H Definition (Semidirect Product Groupoid) Denote by H ≀ G the fibre product H ∗ G = { ( h, x ) : ρ ( h ) = r ( x ) } with operations ( h, x ) − 1 = ( x − 1 h − 1 , x − 1 ) ( h, x ) · ( k, y ) = ( h · xk, x · y ) ♥ ♥ h h · xk ♠ ♣ ♠ ♣ xk k y x · y • • x *H*KQW Semidirect Products 8 / 15

References Fell Bundles Definition A Fell Bundle over H is a Banach bundle p : A → H which linearly r eflects the structure of H . A ( u ) a C ∗ -algebra A ( η ) an A ( r ( η )) − A ( s ( η )) imprimitivity bimodule *H*KQW Semidirect Products 9 / 15

References Fell Bundles Definition A Fell Bundle over H is a Banach bundle p : A → H which linearly r eflects the structure of H . A ( u ) a C ∗ -algebra A ( η ) an A ( r ( η )) − A ( s ( η )) imprimitivity bimodule It is profitable (though abusive) to think of a Fell bundle as a functor into H . *H*KQW Semidirect Products 9 / 15

References Fell Bundle Algebras Given a Fell bundle p : A → H , mimic the construction of C ∗ ( H ) : *H*KQW Semidirect Products 10 / 15

References Fell Bundle Algebras Given a Fell bundle p : A → H , mimic the construction of C ∗ ( H ) : A 0 = C c ( H, A ) compactly supported sections of p (so p ◦ f = id H ) *H*KQW Semidirect Products 10 / 15

References Fell Bundle Algebras Given a Fell bundle p : A → H , mimic the construction of C ∗ ( H ) : A 0 = C c ( H, A ) compactly supported sections of p (so p ◦ f = id H ) Multiplication: vector valued integral � f ( γ ) g ( γ − 1 η ) dλ r ( η ) ( γ ) f ∗ g ( η ) = H *H*KQW Semidirect Products 10 / 15