On Operator-Valued Bi-Free Distributions Paul Skoufranis TAMU - PowerPoint PPT Presentation

On Operator-Valued Bi-Free Distributions Paul Skoufranis TAMU March 22, 2016 Paul Skoufranis On Operator-Valued Bi-Free Distributions March 22, 2016 1 / 30

Bi-Free with Amalgamation Let B be a unital algebra. Let X be a B - B -bimodule that may be decomposed as X = B ⊕ X ⊥ . The projection map p : X → B is given by p ( b ⊕ η ) = b . Thus p ( b · ξ · b ′ ) = bp ( ξ ) b ′ . For b ∈ B , define L b , R b ∈ L ( X ) by L b ( ξ ) = b · ξ and R b ( ξ ) = ξ · b . Define E : L ( X ) → B by E ( T ) = p ( T (1 B ⊕ 0)). E ( L b R b ′ T ) = p ( L b R b ′ ( E ( T ) ⊕ η )) = p ( bE ( T ) b ′ ⊕ η ′ ) = bE ( T ) b ′ . E ( TL b ) = p ( T ( b ⊕ 0)) = E ( TR b ). Paul Skoufranis On Operator-Valued Bi-Free Distributions March 22, 2016 2 / 30

B - B -Non-Commutative Probability Space E ( L b R b ′ T ) = bE ( T ) b ′ and E ( TL b ) = E ( TR b ). Definition A B-B-non-commutative probability space is a triple ( A , E , ε ) where A is a unital algebra over C , ε : B ⊗ B op → A is a unital homomorphism such that ε | B ⊗ I and ε | I ⊗ B op are injective, and E : A → B is a linear map such that E ( ε ( b 1 ⊗ b 2 ) T ) = b 1 E ( T ) b 2 and E ( T ε ( b ⊗ 1 B )) = E ( T ε (1 B ⊗ b )) . Denote L b = ε ( b ⊗ 1 B ) and R b = ε (1 B ⊗ b ). Every B - B -non-commutative probability space can be embedded into L ( X ) for some B - B -bimodule X . Paul Skoufranis On Operator-Valued Bi-Free Distributions March 22, 2016 3 / 30

B -NCPS via B - B -NCPS Definition Let ( A , E , ε ) be a B - B -ncps. The unital subalgebras of A defined by A ℓ := { Z ∈ A | ZR b = R b Z for all b ∈ B } and A r := { Z ∈ A | ZL b = L b Z for all b ∈ B } are called the left and right algebras of A respectively. A pair of algebras ( A 1 , A 2 ) is said to be a pair of B-faces if { L b } b ∈ B ⊆ A 1 ⊆ A ℓ and { R b } b ∈ B op ⊆ A 2 ⊆ A r . Note ( A ℓ , E ) is a B -ncps where { L b } b ∈ B is the copy of B . Indeed for T ∈ A ℓ and b 1 , b 2 ∈ B , E ( L b 1 TL b 2 ) = E ( L b 1 TR b 2 ) = E ( L b 1 R b 2 T ) = b 1 E ( T ) b 2 . Similarly ( A r , E ) is as B op -ncps where { R b } b ∈ B op is the copy of B op . Paul Skoufranis On Operator-Valued Bi-Free Distributions March 22, 2016 4 / 30

Bi-Free Independence with Amalgamation Definition Let ( A , E A , ε ) be a B - B -ncps. Pairs of B -faces ( A ℓ, 1 , A r , 1 ) and ( A ℓ, 2 , A r , 2 ) of A are said to be bi-freely independent with amalgamation over B if there exist B - B -bimodules X k and unital B -homomorphisms α k : A ℓ, k → L ( X k ) ℓ and β k : A r , k → L ( X k ) r such that the following diagram commutes: E A i A ℓ, 1 ∗ A r , 1 ∗ A ℓ, 2 ∗ A r , 2 A B E L ( X 1 ∗X 2 ) α 1 ∗ β 1 ∗ α 2 ∗ β 2 λ 1 ∗ ρ 1 ∗ λ 2 ∗ ρ 2 L ( X 1 ) ℓ ∗ L ( X 1 ) r ∗ L ( X 2 ) ℓ ∗ L ( X 2 ) r L ( X 1 ∗ X 2 ) Paul Skoufranis On Operator-Valued Bi-Free Distributions March 22, 2016 5 / 30

Operator-Valued Bi-Freeness and Mixed Cumulants Theorem (Charlesworth, Nelson, Skoufranis; 2015) Let ( A , E , ε ) be a B-B-ncps and let { ( A ℓ, k , A r , k ) } k ∈ K be pairs of B-faces. Then the following are equivalent: { ( A ℓ, k , A r , k ) } k ∈ K are bi-free over B. For all χ : { 1 , . . . , n } → { ℓ, r } , ǫ : { 1 , . . . , n } → K, and Z m ∈ A χ ( m ) ,ǫ ( m ) ,   � �   E ( Z 1 · · · Z m ) = µ BNC ( π, σ )  E π ( Z 1 , . . . , Z m )    π ∈ BNC ( χ ) σ ∈ BNC ( χ ) π ≤ σ ≤ ǫ For all χ : { 1 , . . . , n } → { ℓ, r } , ǫ : { 1 , . . . , n } → K non-constant, and Z m ∈ A χ ( m ) ,ǫ ( m ) , κ χ ( Z 1 , . . . , Z n ) = 0 . Paul Skoufranis On Operator-Valued Bi-Free Distributions March 22, 2016 6 / 30

Bi-Multiplicative Functions κ and E are special functions where E 1 χ ( Z 1 , . . . , Z n ) = E ( Z 1 · · · Z n ). Given ( A , E , ε ), a bi-multiplicative function Φ is a map � � Φ : BNC ( χ ) × A χ (1) × · · · × A χ ( n ) → B n ≥ 1 χ : { 1 ,..., n }→{ ℓ, r } whose properties are described as follows: Paul Skoufranis On Operator-Valued Bi-Free Distributions March 22, 2016 7 / 30

Property 1 of Bi-Multiplicative Functions Φ 1 χ ( Z 1 L b 1 , Z 2 R b 2 , Z 3 L b 3 , Z 4 ) = Φ 1 χ ( Z 1 , Z 2 , L b 1 Z 3 , R b 2 Z 4 R b 3 ) . Z 1 Z 1 L b 1 Z 2 Z 2 R b 2 L b 1 Z 3 Z 3 L b 3 R b 2 Z 4 Z 4 R b 3 Paul Skoufranis On Operator-Valued Bi-Free Distributions March 22, 2016 8 / 30

Property 1 of Bi-Multiplicative Functions Φ 1 χ ( Z 1 L b 1 , Z 2 R b 2 , Z 3 L b 3 , Z 4 ) = Φ 1 χ ( Z 1 , Z 2 , L b 1 Z 3 , R b 2 Z 4 R b 3 ) . Z 1 Z 1 L b 1 Z 2 Z 2 R b 2 L b 1 Z 3 Z 3 L b 3 R b 2 Z 4 Z 4 R b 3 Paul Skoufranis On Operator-Valued Bi-Free Distributions March 22, 2016 8 / 30

Property 2 of Bi-Multiplicative Functions Φ 1 χ ( L b 1 Z 1 , Z 2 , R b 2 Z 3 , Z 4 ) = b 1 Φ 1 χ ( Z 1 , Z 2 , Z 3 , Z 4 ) b 2 . L b 1 Z 1 Z 2 R b 2 Z 3 Z 4 Paul Skoufranis On Operator-Valued Bi-Free Distributions March 22, 2016 9 / 30

Property 3 of Bi-Multiplicative Functions Φ π ( Z 1 , . . . , Z 8 ) = Φ 1 χ 1 ( Z 1 , Z 3 , Z 4 )Φ 1 χ 2 ( Z 5 , Z 7 , Z 8 )Φ 1 χ 3 ( Z 2 , Z 6 ) . 1 2 3 4 5 6 7 8 Paul Skoufranis On Operator-Valued Bi-Free Distributions March 22, 2016 10 / 30

Property 4 of Bi-Multiplicative Functions � � Φ π ( Z 1 , . . . , Z 10 ) = Φ 1 χ 1 Z 1 , Z 3 , L Φ 1 χ 2 ( Z 4 , Z 5 ) Z 6 , R Φ 1 χ 3 ( Z 2 , Z 8 ) Z 9 R Φ 1 χ 4 ( Z 7 , Z 10 ) Z 1 Z 1 Z 2 Z 3 Z 3 Z 4 L Φ 1 χ 2 ( Z 4 , Z 5 ) R Φ 1 χ 3 ( Z 2 , Z 8 ) Z 5 Z 6 Z 6 Z 7 Z 8 Z 9 Z 9 Z 10 Φ 1 χ 4 ( Z 7 , Z 10 ) Paul Skoufranis On Operator-Valued Bi-Free Distributions March 22, 2016 11 / 30

Property 4 of Bi-Multiplicative Functions � � Φ π ( Z 1 , . . . , Z 10 ) = Φ 1 χ 1 Z 1 , Z 3 , L Φ 1 χ 2 ( Z 4 , Z 5 ) Z 6 , R Φ 1 χ 3 ( Z 2 , Z 8 ) Z 9 R Φ 1 χ 4 ( Z 7 , Z 10 ) Z 1 Z 1 Z 2 Z 3 Z 3 Z 4 L Φ 1 χ 2 ( Z 4 , Z 5 ) R Φ 1 χ 3 ( Z 2 , Z 8 ) Z 5 Z 6 Z 6 Z 7 Z 8 Z 9 Z 9 Z 10 Φ 1 χ 4 ( Z 7 , Z 10 ) Paul Skoufranis On Operator-Valued Bi-Free Distributions March 22, 2016 11 / 30

Amalgamating Over Matrices Let ( A , ϕ ) be a non-commutative probability space. M N ( A ) is naturally a M N ( C )-ncps where the expectation map ϕ N : M N ( A ) → M N ( C ) is defined via ϕ N ([ A i , j ]) = [ ϕ ( A i , j )] . If A 1 , A 2 are unital subalgebras of A that are free with respect to ϕ , then M N ( A 1 ) and M N ( A 2 ) are free with amalgamation over M N ( C ) with respect to ϕ N . Is there a bi-free analogue of this result? Is M N ( A ) a M N ( C )- M N ( C )-ncps? Paul Skoufranis On Operator-Valued Bi-Free Distributions March 22, 2016 12 / 30

B - B -NCPS Associated to A Let ( A , ϕ ) be a non-commutative probability space and let B be a unital algebra. Then A ⊗ B is a B - B -bi-module where L b ( a ⊗ b ′ ) = a ⊗ bb ′ , R b ( a ⊗ b ′ ) = a ⊗ b ′ b . and If p : A ⊗ B → B is defined by p ( a ⊗ b ) = ϕ ( a ) b , then L ( A ⊗ B ) is a B - B -ncps with E ( Z ) = p ( Z (1 A ⊗ 1 B )) . If X , Y ∈ A , defined L ( X ⊗ b ) ∈ L ( A ⊗ B ) ℓ and R ( Y ⊗ b ) ∈ L ( A ⊗ B ) r via L ( X ⊗ b )( a ⊗ b ′ ) = Xa ⊗ bb ′ R ( Y ⊗ b )( a ⊗ b ′ ) = Ya ⊗ b ′ b . and Paul Skoufranis On Operator-Valued Bi-Free Distributions March 22, 2016 13 / 30

Bi-Freeness Preserved Under Tensoring Theorem (Skoufranis; 2015) Let ( A , ϕ ) be a non-commutative probability space and let { ( A ℓ, k , A r , k ) } k ∈ K be bi-free pairs of faces with respect to ϕ . If B is a unital algebra, then { ( L ( A ℓ, k ⊗ B ) , R ( A r , k ⊗ B )) } k ∈ K are bi-free over B with respect to E as described above. Proof Sketch. If χ : { 1 , . . . , n } → { ℓ, r } , Z m = L ( X m ⊗ b m ) if χ ( m ) = ℓ , and Z m = R ( X m ⊗ b m ) if χ ( m ) = r , then E ( Z 1 · · · Z n ) = ϕ ( X 1 · · · X n ) ⊗ b s χ (1) · · · b s χ ( n ) Also E π ( Z 1 · · · Z n ) = ϕ π ( X 1 , . . . , X n ) ⊗ b s χ (1) · · · b s χ ( n ) . Paul Skoufranis On Operator-Valued Bi-Free Distributions March 22, 2016 14 / 30

Bi-Matrix Models - Creation/Annihilation on a Fock Space Theorem (Skoufranis; 2015) Given an index set K, an N ∈ N , and an orthonormal set of vectors { h k i , j | i , j ∈ { 1 , . . . , N } , k ∈ K } ⊆ H , let N N 1 1 � � j , i ) ∗ ⊗ E i , j ) L ( l ( h k L ∗ L ( l ( h k L k ( N ) := √ i , j ) ⊗ E i , j ) , k ( N ) := √ N N i , j =1 i , j =1 N N 1 1 � � j , i ) ∗ ⊗ E i , j ) . R ( r ( h k R ∗ R ( r ( h k R k ( N ) := √ i , j ) ⊗ E i , j ) , k ( N ) := √ N N i , j =1 i , j =1 If E : L ( L ( F ( H )) ⊗ M N ( C )) → M N ( C ) is the expectation, the joint distribution of { L k ( N ) , L ∗ k ( N ) , R k ( N ) , R ∗ k ( N ) } k ∈ K with respect to 1 N Tr ◦ E is equal the joint distribution of { l ( h k ) , l ∗ ( h k ) , r ( h k ) , r ∗ ( h k ) } k ∈ K with respect to ϕ where { h k } k ∈ K ⊆ H is an orthonormal set. Paul Skoufranis On Operator-Valued Bi-Free Distributions March 22, 2016 15 / 30