C -correspondence functoriality of Cuntz-Pimsner algebras Menevs - PowerPoint PPT Presentation

C ∗ -correspondence functoriality of Cuntz-Pimsner algebras Menevs ¸e Ery¨ uzl¨ u September, 2020

What/Why are we doing? Menevs ¸e Ery¨ uzl¨ u C ∗ -correspondence functoriality of Cuntz-Pimsner algebras

What/Why are we doing? A X A O X Menevs ¸e Ery¨ uzl¨ u C ∗ -correspondence functoriality of Cuntz-Pimsner algebras

What/Why are we doing? A X A O X Cross products by automorphisms Menevs ¸e Ery¨ uzl¨ u C ∗ -correspondence functoriality of Cuntz-Pimsner algebras

What/Why are we doing? A X A O X Cross products by automorphisms Cuntz algebras Menevs ¸e Ery¨ uzl¨ u C ∗ -correspondence functoriality of Cuntz-Pimsner algebras

What/Why are we doing? A X A O X Cross products by automorphisms Cuntz algebras Cuntz-Krieger algebras Menevs ¸e Ery¨ uzl¨ u C ∗ -correspondence functoriality of Cuntz-Pimsner algebras

What/Why are we doing? A X A O X Graph algebras of Cross products by automorphisms graphs with sinks Cuntz algebras Cuntz-Krieger algebras Menevs ¸e Ery¨ uzl¨ u C ∗ -correspondence functoriality of Cuntz-Pimsner algebras

What/Why are we doing? A X A O X Graph algebras of Cross products by automorphisms graphs with sinks Cuntz algebras Crossed products by Cuntz-Krieger algebras partial automorphisms Menevs ¸e Ery¨ uzl¨ u C ∗ -correspondence functoriality of Cuntz-Pimsner algebras

The Motivation A X A and B Y B are called Morita equivalent if there is an imprimitivity bimodule A M B such that A X ⊗ A M B ∼ = A M ⊗ B Y B . Menevs ¸e Ery¨ uzl¨ u C ∗ -correspondence functoriality of Cuntz-Pimsner algebras

The Motivation A X A and B Y B are called Morita equivalent if there is an imprimitivity bimodule A M B such that A X ⊗ A M B ∼ = A M ⊗ B Y B . If the injective C ∗ -correspondences A X A and B Y B are Morita equivalent then the corresponding Cuntz-Pimsner algebras O X and O Y are Morita equivalent. Menevs ¸e Ery¨ uzl¨ u C ∗ -correspondence functoriality of Cuntz-Pimsner algebras

Tensor Product Let A X B and B Y C be C ∗ -correspondences. The algebraic tensor product X ⊙ Y has the A − C bimodule structure: a ( x ⊗ y ) c = ax ⊗ yc for a ∈ A , x ∈ X , y ∈ Y , c ∈ C , and the unique C -valued sesquilinear form � x ⊗ y , u ⊗ v � C = � y , � x , u � B v � C for x , u ∈ X , y , v ∈ Y . The Hausdorff completion is an A − C correspondence X ⊗ B Y (with the left action a �→ ϕ X ( a ) ⊗ A 1 Y ). Menevs ¸e Ery¨ uzl¨ u C ∗ -correspondence functoriality of Cuntz-Pimsner algebras

Definition 0.1 A representation of A X A on a C ∗ -algebra B is a pair consisting of a homomorphism π : A → B and a linear map t : X → B satisfying 1 t ( x ) ∗ t ( y ) = π ( � x , y � A ) , 2 π ( a ) t ( x ) = t ( ϕ ( a ) x ) , 3 t ( x ) π ( a ) = t ( xa ). Also, there is a homomorphism ψ t : K ( X ) → B with ψ t ( θ x , y ) = t ( x ) t ( y ) ∗ . X t Denote C ∗ ( π, t ) the C ∗ - K ( X ) ψ algebra generated by the B images of π and t in B . π A Menevs ¸e Ery¨ uzl¨ u C ∗ -correspondence functoriality of Cuntz-Pimsner algebras

C ∗ -algebra generated by representations Consider A X A with the associated hom ϕ and a representa- tion ( π, t ). Define X ⊗ 0 = A , X ⊗ 1 = X , X ⊗ n = X ⊗ A X ⊗ n − 1 . Each X ⊗ n is a C ∗ - correspondence over A with the left action defined by , ϕ n ( a )( x 1 ⊗ A x 2 ⊗ A .... x n ) := ϕ ( a ) x 1 ⊗ A .... ⊗ A x n . Now, set t 0 = π and t 1 = t . For n = 2 , 3 , ..., define a linear map t n : X ⊗ n → C ∗ ( π, t ) by t n ( x ⊗ A y ) = t ( x ) t n − 1 ( y ) for x ∈ X and y ∈ X ⊗ n − 1 . Each ( t n , π ) is a representation of the C ∗ -correspondence X ⊗ n and we have that C ∗ ( π, t )= span { t n ( x ) t m ( y ) ∗ | x ∈ X ⊗ n , y ∈ X ⊗ m , n , m ∈ N } . Menevs ¸e Ery¨ uzl¨ u C ∗ -correspondence functoriality of Cuntz-Pimsner algebras

Fock Space!!! ∞ X ⊗ n = A ⊕ X ⊕ X ⊗ 2 ⊕ X ⊗ 3 ⊕ .... F ( X ) = � n =0 n X ⊗ n : � = { x = ( x ( n )) ∈ � n � x ( n ) , x ( n ) � A converges . } For any y ∈ X , define t ∞ ( y ) ∈ L ( F ( X )) by t ∞ ( y )( a , x 1 , x 2 , .... )=(0 , ya , y ⊗ A x 1 , y ⊗ A x 2 , .... ). Define the *-homomorphism ϕ ∞ : A → L ( F ( X )) by ϕ ∞ ( a ) = diag ( a , ϕ ( a ) , ϕ 2 ( a ) , .... ) Menevs ¸e Ery¨ uzl¨ u C ∗ -correspondence functoriality of Cuntz-Pimsner algebras

Here is how the adjoint of t ∞ ( y ) behaves : t ∞ ( y ) ∗ ( a ) = 0 , t ∞ ( y ) ∗ ( z ⊗ A x n ) = ϕ n ( � y , z � A )( x n ), for x n ∈ X ⊗ n . So, t ∞ ( y ) ∗ ( a , x 1 , x 2 ⊗ A x ′ 2 , x 3 ⊗ A x ′ 3 ⊗ A x ′′ 3 , .... ) = ( � y , x 1 � A , ϕ ( � y , x 2 � ( x ′ 2 ) , ϕ 2 ( � y , x 3 � ( x ′ 3 ⊗ A x ′′ 3 ) , ...... ). Now, ( t ∞ , ϕ ∞ ) is a representation of A X A on L ( F ( X )), which is called the Fock representation . Notice that the Fock space can be viewed as a full A − A correspondence with the non-degenerate homomorphism ϕ ∞ defined as above. Menevs ¸e Ery¨ uzl¨ u C ∗ -correspondence functoriality of Cuntz-Pimsner algebras

Here is how the adjoint of t ∞ ( y ) behaves : t ∞ ( y ) ∗ ( a ) = 0 , t ∞ ( y ) ∗ ( z ⊗ A x n ) = ϕ n ( � y , z � A )( x n ), for x n ∈ X ⊗ n . So, t ∞ ( y ) ∗ ( a , x 1 , x 2 ⊗ A x ′ 2 , x 3 ⊗ A x ′ 3 ⊗ A x ′′ 3 , .... ) = ( � y , x 1 � A , ϕ ( � y , x 2 � ( x ′ 2 ) , ϕ 2 ( � y , x 3 � ( x ′ 3 ⊗ A x ′′ 3 ) , ...... ). Now, ( t ∞ , ϕ ∞ ) is a representation of A X A on L ( F ( X )), which is called the Fock representation . Notice that the Fock space can be viewed as a full A − A correspondence with the non-degenerate homomorphism ϕ ∞ defined as above. ∞ ( y m ) ∗ : x n ∈ X ⊗ n , y m ∈ X ⊗ m , n , m ∈ N } , C ∗ ( ϕ ∞ , t ∞ )= span { t n ∞ ( x n ) t m which is isomorphic to T X !!! Menevs ¸e Ery¨ uzl¨ u C ∗ -correspondence functoriality of Cuntz-Pimsner algebras

Here is how the adjoint of t ∞ ( y ) behaves : t ∞ ( y ) ∗ ( a ) = 0 , t ∞ ( y ) ∗ ( z ⊗ A x n ) = ϕ n ( � y , z � A )( x n ), for x n ∈ X ⊗ n . So, t ∞ ( y ) ∗ ( a , x 1 , x 2 ⊗ A x ′ 2 , x 3 ⊗ A x ′ 3 ⊗ A x ′′ 3 , .... ) = ( � y , x 1 � A , ϕ ( � y , x 2 � ( x ′ 2 ) , ϕ 2 ( � y , x 3 � ( x ′ 3 ⊗ A x ′′ 3 ) , ...... ). Now, ( t ∞ , ϕ ∞ ) is a representation of A X A on L ( F ( X )), which is called the Fock representation . Notice that the Fock space can be viewed as a full A − A correspondence with the non-degenerate homomorphism ϕ ∞ defined as above. ∞ ( y m ) ∗ : x n ∈ X ⊗ n , y m ∈ X ⊗ m , n , m ∈ N } , C ∗ ( ϕ ∞ , t ∞ )= span { t n ∞ ( x n ) t m which is isomorphic to T X !!! Side Note: By using ϕ ∞ : A → T X , we get A ( T X ) T X . Menevs ¸e Ery¨ uzl¨ u C ∗ -correspondence functoriality of Cuntz-Pimsner algebras

Here is how the adjoint of t ∞ ( y ) behaves : t ∞ ( y ) ∗ ( a ) = 0 , t ∞ ( y ) ∗ ( z ⊗ A x n ) = ϕ n ( � y , z � A )( x n ), for x n ∈ X ⊗ n . So, t ∞ ( y ) ∗ ( a , x 1 , x 2 ⊗ A x ′ 2 , x 3 ⊗ A x ′ 3 ⊗ A x ′′ 3 , .... ) = ( � y , x 1 � A , ϕ ( � y , x 2 � ( x ′ 2 ) , ϕ 2 ( � y , x 3 � ( x ′ 3 ⊗ A x ′′ 3 ) , ...... ). Now, ( t ∞ , ϕ ∞ ) is a representation of A X A on L ( F ( X )), which is called the Fock representation . Notice that the Fock space can be viewed as a full A − A correspondence with the non-degenerate homomorphism ϕ ∞ defined as above. ∞ ( y m ) ∗ : x n ∈ X ⊗ n , y m ∈ X ⊗ m , n , m ∈ N } , C ∗ ( ϕ ∞ , t ∞ )= span { t n ∞ ( x n ) t m which is isomorphic to T X !!! Side Note: By using ϕ ∞ : A → T X , we get A ( T X ) T X . Menevs ¸e Ery¨ uzl¨ u C ∗ -correspondence functoriality of Cuntz-Pimsner algebras

Katsura Ideal Definition 0.2 For a C ∗ -correspondence X over A , define an ideal J X of A to be ϕ − 1 ( K ( X )) ∩ ( Ker ϕ ) ⊥ . Note that J X is the largest ideal on which the restriction of ϕ is an injection into K ( X ). F ( X ) J X is a Hilbert J X -module, and we have K ( F ( X ) J X ) = span { θ ζ a ,η ∈ K ( F ( X )) : ζ, η ∈ F ( X ) , a ∈ J X } , which is an ideal of L ( F ( X )). In fact, it is an ideal of T X !!! Menevs ¸e Ery¨ uzl¨ u C ∗ -correspondence functoriality of Cuntz-Pimsner algebras

There comes the Cuntz-Pimsner algebra... Let ρ : L ( F ( X )) → L ( F ( X ) / K ( F ( X ) J X ) be the quotient map and set φ = ρ ◦ ϕ ∞ and t = ρ ◦ t ∞ . Then, 1 ( φ, t ) is a covariant representation of A X A on L ( F ( X ) / K ( F ( X ) J X )) 2 This representation is injective. Menevs ¸e Ery¨ uzl¨ u C ∗ -correspondence functoriality of Cuntz-Pimsner algebras

There comes the Cuntz-Pimsner algebra... Let ρ : L ( F ( X )) → L ( F ( X ) / K ( F ( X ) J X ) be the quotient map and set φ = ρ ◦ ϕ ∞ and t = ρ ◦ t ∞ . Then, 1 ( φ, t ) is a covariant representation of A X A on L ( F ( X ) / K ( F ( X ) J X )) 2 This representation is injective. Katsura proves that C ∗ ( φ, t ) ∼ = O X . Hence, O X is the quotient C ∗ -algebra T X / K ( F ( X ) J X ). Menevs ¸e Ery¨ uzl¨ u C ∗ -correspondence functoriality of Cuntz-Pimsner algebras