Groupoid C -algebras and their canonical diagonal subalgebras - PowerPoint PPT Presentation

Motivation Groupoid construction Applications Morita Equivalence Groupoid C ∗ -algebras and their canonical diagonal subalgebras Efren Ruiz Work in progress with Toke Carlsen, Aidan Sims, and Mark Tomforde University of Hawai’i at Hilo APPLICATIONS OF MODEL THEORY TO OPERATOR ALGEBRAS

Motivation Groupoid construction Applications Morita Equivalence Objects of interest ( A , D ) A and D are separable C ∗ -algebras D is a commutative C ∗ -subalgebra of A

Motivation Groupoid construction Applications Morita Equivalence Objects of interest ( A , D ) A and D are separable C ∗ -algebras D is a commutative C ∗ -subalgebra of A Main Example A = C ∗ ( G ) D = C 0 ( G ( 0 ) ) and G is a second-countable, locally compact, Hausdorff, étale groupoid s : γ �→ γ − 1 γ r : γ �→ γγ − 1 and are local homeomorphisms.

Motivation Groupoid construction Applications Morita Equivalence C ∗ r ( G ) C c ( G ) : � ( f ⋆ g )( γ ) = f ( λ ) g ( β ) λβ = γ f ∗ ( γ ) = f ( γ − 1 ) π u λ : C c ( G ) → B ( ℓ 2 ( s − 1 ( u ))) , � ( π u λ ( f ) ξ )( γ ) = f ( λ ) ξ ( β ) λβ = γ � λ ( f ) � : u ∈ G ( 0 ) � � π u � f � r := sup �·� r C ∗ r ( G ) := C c ( G )

Motivation Groupoid construction Applications Morita Equivalence Motivating Examples

Motivation Groupoid construction Applications Morita Equivalence Motivating Examples Theorem (Tomiyama) Let X and Y be compact, Hausdorff spaces and let ( X , σ ) and ( Y , τ ) be topologically free dynamical systems. Then the following are equivalent: ( C ( X ) ⋊ σ Z , C ( X )) ∼ = ( C ( Y ) ⋊ τ Z , C ( Y )) and 1 ( X , σ ) and ( Y , τ ) are continuous orbit equivalent, i.e., there 2 exist a homeomorphism h : X → Y and continuous functions m , n : X → Z such that h ( σ ( x )) = τ m ( x ) ( h ( x )) τ ( h ( x )) = h ( σ n ( x ) ( x )) . and

Motivation Groupoid construction Applications Morita Equivalence Transformation groupoid: X ⋊ σ Z X ⋊ σ Z = X × Z (Product topology) 1 ( x , n )( y , m ) = ( x , n + m ) if and only if σ n ( x ) = y 2 ( n , x ) − 1 = ( σ n ( x ) , − n ) 3 ( X ⋊ σ Z ) ( 0 ) = X × { 0 } ∼ = X . 4

Motivation Groupoid construction Applications Morita Equivalence Transformation groupoid: X ⋊ σ Z X ⋊ σ Z = X × Z (Product topology) 1 ( x , n )( y , m ) = ( x , n + m ) if and only if σ n ( x ) = y 2 ( n , x ) − 1 = ( σ n ( x ) , − n ) 3 ( X ⋊ σ Z ) ( 0 ) = X × { 0 } ∼ = X . 4 Theorem ( C ( X ) ⋊ σ Z , C ( X )) ∼ = ( C ∗ r ( X ⋊ σ Z ) , C (( X ⋊ σ Z ) ( 0 ) ))

Motivation Groupoid construction Applications Morita Equivalence Theorem (Tomiyama and Renault) Let X and Y be second-countable, compact, Hausdorff spaces and let ( X , σ ) and ( Y , τ ) be topologically free dynamical systems. Then the following are equivalent: ( C ( X ) ⋊ σ Z , C ( X )) ∼ = ( C ( Y ) ⋊ τ Z , C ( Y )) , 1 ( X , σ ) and ( Y , τ ) are continuous orbit equivalent, i.e., there 2 exist a homeomorphism h : X → Y and continuous functions m , n : X → Z such that h ( σ ( x )) = τ m ( x ) ( h ( x )) τ ( h ( x )) = h ( σ n ( x ) ( x )) , and and X ⋊ σ Z ∼ = Y ⋊ τ Z . 3

Motivation Groupoid construction Applications Morita Equivalence Cuntz-Krieger algebras One-sided shift space Let A ∈ M N ( { 0 , 1 } ) . X A = { ( x n ) n ∈ N ∈ { 1 , 2 , . . . , N } N : A ( x n , x n + 1 ) = 1 } 1 σ A : X A → X A , [ σ A (( x n ) n ∈ N )] n = x n + 1 2

Motivation Groupoid construction Applications Morita Equivalence Theorem (Matsumoto-Matui, Brownlowe-Carlsen-Whittaker, Arklint-Eilers-R (Carlsen-Winger)) Let A ∈ M N ( { 0 , 1 } ) and let B ∈ M N ′ ( { 0 , 1 } ) . Then the following are equivalent: ( O A , C ( X A )) ∼ = ( O B , C ( X B )) and 1 ( X A , σ A ) and ( X B , σ B ) are continuous orbit equivalent, i.e., 2 there exist a homeomorphism h : X A → X B , and continuous functions k , l : X A → N and k ′ , l ′ : X B → N such that σ k ( x ) ( h ( σ A ( x ))) = σ l ( x ) ( h ( x )) B A σ k ′ ( y ) ( h − 1 ( σ B ( y ))) = σ l ′ ( y ) ( h − 1 ( y )) . A A

Motivation Groupoid construction Applications Morita Equivalence Theorem (Matsumoto-Matui, Carlsen-Eilers-Ortega-Restorff) Let A ∈ M N ( { 0 , 1 } ) and let B ∈ M N ′ ( { 0 , 1 } ) . Then the following are equivalent: ( O A ⊗ K , C ( X A ) ⊗ c 0 ( N )) ∼ = ( O B ⊗ K , C ( X B ) ⊗ c 0 ( N )) and 1 the two-sided shift spaces ( X A , σ A ) and ( X B , σ B ) are flow 2 equivalent.

Motivation Groupoid construction Applications Morita Equivalence The groupoid of a one-sided shift space Let A ∈ M N ( { 0 , 1 } ) . G A = { ( x , n − m , y ): x , y ∈ X A , n , m ∈ Z > 0 , σ n A ( x ) = σ m A ( y ) } 1 ( x , n − m , y )( x ′ , n ′ − m ′ , y ′ ) = ( x , n + n ′ − m − m ′ , y ′ ) 2 if and only if y = x ′ ( x , n − m , y ) − 1 = ( y , m − n , x ) 3 G ( 0 ) = { ( x , 0 , x ) : x ∈ X A } ∼ = X A 4 A Z ( U , n , m , V ) = 5 � � ( x , n − m , y ) : x ∈ U , y ∈ V , σ n A ( x ) = σ m A ( y ) , U , V are open subets of X A

Motivation Groupoid construction Applications Morita Equivalence Theorem ( O A , C ( X A )) ∼ r ( G A ) , C ( G ( 0 ) = ( C ∗ A ))

Motivation Groupoid construction Applications Morita Equivalence Theorem ( O A , C ( X A )) ∼ r ( G A ) , C ( G ( 0 ) = ( C ∗ A )) Theorem (Matsumoto-Matui, Brownlowe-Carlsen-Whittaker, Arklint-Eilers-R (Carlsen-Winger)) Let A ∈ M N ( { 0 , 1 } ) and let B ∈ M N ′ ( { 0 , 1 } ) . Then the following are equivalent: ( O A , C ( X A )) ∼ = ( O B , C ( X B )) , 1 there exists a continuous orbit equivalence between 2 ( X A , σ A ) and ( X B , σ B ) , and G A ∼ = G B . 3

Motivation Groupoid construction Applications Morita Equivalence Theorem (Renault, Brownlowe-Carlsen-Whittaker) Let G , H be second-countable, locally compact, Hausdorff, étale groupoids. Then the following are equivalent: r ( G ) , C 0 ( G ( 0 ) )) ∼ ( C ∗ = ( C ∗ r ( H ) , C 0 ( H ( 0 ) )) 1 G ∼ = H 2 whenever G , H are topologically principal groupoids or G , H are groupoids associated to one-sided shift spaces.

Motivation Groupoid construction Applications Morita Equivalence Theorem (Renault, Brownlowe-Carlsen-Whittaker) Let G , H be second-countable, locally compact, Hausdorff, étale groupoids. Then the following are equivalent: r ( G ) , C 0 ( G ( 0 ) )) ∼ ( C ∗ = ( C ∗ r ( H ) , C 0 ( H ( 0 ) )) 1 G ∼ = H 2 whenever G , H are topologically principal groupoids or G , H are groupoids associated to one-sided shift spaces. Key Idea Construct a groupoid H ( C ∗ r ( G ) , C 0 ( G ( 0 ) )) such that r ( G ) , C 0 ( G ( 0 ) )) ∼ H ( C ∗ = G .

Motivation Groupoid construction Applications Morita Equivalence Definition A semidiagonal pair of C ∗ -algebras is a pair ( A , D ) consisting of a separable C ∗ -algebra A and a subalgebra D of A such that D is abelian, 1 D contains an approximate identity for A , 2 D , the quotient D ′ / J φ of D ′ by the ideal for each φ ∈ � 3 J φ := ker ( φ ) D ′ is a unital C ∗ -algebra, and for each φ ∈ � D , there exist d ∈ D and an open 4 neighbourhood U of φ such that d + J ψ = 1 D ′ / J ψ for all ψ ∈ U .

Motivation Groupoid construction Applications Morita Equivalence Definition Let A be a C ∗ -algebra and D be a C ∗ -subalgebra of A . A normalizer of D is an element n ∈ A such that nDn ∗ ∪ n ∗ Dn ⊆ D .

Motivation Groupoid construction Applications Morita Equivalence Definition Let A be a C ∗ -algebra and D be a C ∗ -subalgebra of A . A normalizer of D is an element n ∈ A such that nDn ∗ ∪ n ∗ Dn ⊆ D . Theorem (Kumjian, Renault) Let A be a C ∗ -algebra and D an abelian C ∗ -subalgebra of A that contains an approximate unit for A. Suppose that n is a normalizer of D. Then there is a homeomorphism α n : { u ∈ � D : u ( n ∗ n ) > 0 } → { u ∈ � D : u ( nn ∗ ) > 0 } such that u ( n ∗ n ) α n ( u )( d ) = u ( n ∗ dn ) for all d ∈ D.

Motivation Groupoid construction Applications Morita Equivalence Lemma Let ( A , D ) be a semidiagonal pair, n , m be normalizers of D, and φ ∈ � D. Suppose there exists an open neighborhood U of φ such that U ⊆ supp ( n ∗ n ) ∩ supp ( m ∗ m ) . Then for any d ∈ D with supp ( d ) ⊆ U and φ ( d ) = 1 , we have that φ ( m ∗ nn ∗ m ) − 1 2 dn ∗ md is in D ′ and φ ( m ∗ nn ∗ m ) − 1 2 dn ∗ md + J φ is a unitary in D ′ / J φ that is independent of the choices of U and d.

Motivation Groupoid construction Applications Morita Equivalence � � ( n , φ ) ∈ N ( D ) × � D : φ ( n ∗ n ) > 0 S ( A , D ) = ( n , φ ) ∼ ( m , ψ ) if and only if φ = ψ , 1 there exists an open neighborhood of φ such that 2 α n | U = α m | U , and φ ( m ∗ nn ∗ m ) − 1 2 dn ∗ md + J φ ∈ U 0 ( D ′ / J φ ) . 3

Motivation Groupoid construction Applications Morita Equivalence The groupoid H ( A , D ) H ( A , D ) = { [( n , φ )] : ( n , φ ) ∈ S ( A , D ) }

Motivation Groupoid construction Applications Morita Equivalence The groupoid H ( A , D ) H ( A , D ) = { [( n , φ )] : ( n , φ ) ∈ S ( A , D ) } [( n , φ )][( m , ψ )] = [( nm , ψ )] if and only if φ = α m ( ψ ) 1 [( n , φ )] − 1 = [( n ∗ , α n ( φ ))] 2 Z ( n , U ) = { [( n , φ )] : φ ∈ U and φ ( n ∗ n ) > 0 } 3 U open subset of � D .