On Operator-Valued Bi-Free Distributions

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 b1 TL b2 ) = E(L b1 TR b2 ) = E(L b1 R b2 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: A l,1 A r,1 A l,2 A i r,2 A B E A α 1 β 1 α 2 β 2 E L(X1 X 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}, ɛ : {1,..., n} K, and Z m A χ(m),ɛ(m), E(Z 1 Z m ) = π BNC(χ) σ BNC(χ) π σ ɛ µ BNC (π, σ) E π(z 1,..., Z m ) For all χ : {1,..., n} {l, r}, ɛ : {1,..., n} K non-constant, and Z m A χ(m),ɛ(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 1χ (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 1χ (Z 1 L b1, Z 2 R b2, Z 3 L b3, Z 4 ) = Φ 1χ (Z 1, Z 2, L b1 Z 3, R b2 Z 4 R b3 ). Z 1 L b1 Z 2 R b2 Z 3 L b3 Z 4 Z 1 L b1 Z 2 Z 3 R b2 Z 4 R b3 Paul Skoufranis On Operator-Valued Bi-Free Distributions March 22, 2016 8 / 30

Property 1 of Bi-Multiplicative Functions Φ 1χ (Z 1 L b1, Z 2 R b2, Z 3 L b3, Z 4 ) = Φ 1χ (Z 1, Z 2, L b1 Z 3, R b2 Z 4 R b3 ). Z 1 L b1 Z 2 R b2 Z 3 L b3 Z 4 Z 1 L b1 Z 2 Z 3 R b2 Z 4 R b3 Paul Skoufranis On Operator-Valued Bi-Free Distributions March 22, 2016 8 / 30

Property 2 of Bi-Multiplicative Functions Φ 1χ (L b1 Z 1, Z 2, R b2 Z 3, Z 4 ) = b 1 Φ 1χ (Z 1, Z 2, Z 3, Z 4 )b 2. L b1 Z 1 Z 2 R b2 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 1χ1 (Z 1, Z 3, Z 4 )Φ 1χ2 (Z 5, Z 7, Z 8 )Φ 1χ3 (Z 2, Z 6 ). 1 3 4 5 8 2 6 7 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 1χ1 (Z 1, Z 3, L Φ1χ2 (Z 4,Z 5 )Z 6, R Φ1χ3 (Z 2,Z 8 )Z 9 R Φ1χ4 (Z 7,Z 10 ) ) Z 1 Z 1 Z 2 Z 3 Z 4 Z 5 Z 6 Z 7 Z 3 L Φ1χ2 (Z 4,Z 5 ) Z 6 R Φ1χ3 (Z 2,Z 8 ) Z 8 Z 9 Z 9 Z 10 Φ 1χ4 (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 1χ1 (Z 1, Z 3, L Φ1χ2 (Z 4,Z 5 )Z 6, R Φ1χ3 (Z 2,Z 8 )Z 9 R Φ1χ4 (Z 7,Z 10 ) ) Z 1 Z 1 Z 2 Z 3 Z 4 Z 5 Z 6 Z 7 Z 3 L Φ1χ2 (Z 4,Z 5 ) Z 6 R Φ1χ3 (Z 2,Z 8 ) Z 8 Z 9 Z 9 Z 10 Φ 1χ4 (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, and R b (a b ) = a b b. If p : A B B is defined by then L(A B) is a B-B-ncps with p(a b) = ϕ(a)b, 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 and R(Y b)(a b ) = Ya b b. 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 sχ(1) b sχ(n) Also E π (Z 1 Z n ) = ϕ π (X 1,..., X n ) b sχ(1) b sχ(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 L k (N) := 1 N R k (N) := 1 N N i,j=1 N i,j=1 L(l(hi,j) k E i,j ), L 1 k (N) := N R(r(hi,j) k E i,j ), Rk 1 (N) := N N i,j=1 N i,j=1 L(l(h k j,i) E i,j ) R(r(h k j,i) E i,j ). 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

Bi-Matrix Models - q-deformed Fock Space Moreover (L(I F(H) M N (C)), R(I F(H) M N (C))) and {(L k (N), L k (N)), (R k(n), R k (N))} k K are bi-free. Considering the q-deformed Fock space, the joint distribution of the q-deformed versions {(L k (N), L k (N), Lt k (N), L,t k (N)), (R k(n), Rk (N), Rt k (N), R,t k (N))} k K with respect to 1 N Tr E asymptotically equals the joint distribution of {(l(h k ), l (h k ), l(h k 0), l (h k 0)), (r(h k ), r (h k ), r(h k 0), r (h k 0))} k K with respect to ϕ where {h k, h k 0 } k K H is an orthonormal set, and are asymptotically bi-free from (L(I Fq(H) M N (C)), R(I Fq(H) M N (C))). Paul Skoufranis On Operator-Valued Bi-Free Distributions March 22, 2016 16 / 30

Amalgamating over a Smaller Subalgebra Suppose (A, E, ε) is a B-B-ncps. Let D be a unital subalgebra of B, and let F : B D be such that F (1 B ) = 1 D and F (d 1 bd 2 ) = d 1 F (b)d 2 for all d 1, d 2 D and b B. Note (A, F E, ε D D op) is a D-D-ncps since F (E(L d R d Z)) = F (de(z)d ) = df (E(Z))d F (E(ZL d )) = F (E(ZR d )) for all d, d D and Z A. Note A l,b A l,d and A r,b A r,d. How do the B-valued and D-valued distributions interact? How can one described said distributions? Paul Skoufranis On Operator-Valued Bi-Free Distributions March 22, 2016 17 / 30

Operator-Valued Bi-Free Distributions Suppose {Z i } i I A l and {Z j } j J A r. Suppose we wanted to describe all B-valued moments involving Z i1, Z j1, Z i2, and Z j2 each occurring once in that order. Z i1 Z j1 Z i2 Z j2 Paul Skoufranis On Operator-Valued Bi-Free Distributions March 22, 2016 18 / 30

Operator-Valued Bi-Free Distributions Suppose {Z i } i I A l and {Z j } j J A r. Suppose we wanted to describe all B-valued moments involving Z i1, Z j1, Z i2, and Z j2 each occurring once in that order. Z i1 Z j1 Z i2 Z j2 Paul Skoufranis On Operator-Valued Bi-Free Distributions March 22, 2016 18 / 30

Operator-Valued Bi-Free Distributions Suppose {Z i } i I A l and {Z j } j J A r. Let Z = {Z i } i I {Z j } j J. For n 1, ω : {1,..., n} I J, and b 1,..., b n 1 B, let µ B Z,ω (b 1,..., b n 1 ) = Expectation of Z ω(1),..., Z ω(n) in that order κ B Z,ω (b 1,..., b n 1 ) = with b 1,..., b n 1 in-between gaps with respect to the χ-ordering. Cumulant of Z ω(1),..., Z ω(n) in that order with b 1,..., b n 1 in-between gaps with respect to the χ-ordering. Similarly, we can define µ D Z,ω (d 1,..., d n 1 ) and κ D Z,ω (d 1,..., d n 1 ). Paul Skoufranis On Operator-Valued Bi-Free Distributions March 22, 2016 19 / 30

D-Valued Cumulants from B-Valued Cumulants Theorem (Skoufranis; 2015) If κ B Z,ω (d 1,..., d n 1 ) D for all n 1, ω : {1,..., n} I J, and d 1,..., d n 1 D, then κ D Z,ω (d 1,..., d n 1 ) = κ B Z,ω (d 1,..., d n 1 ) for all n 1, ω : {1,..., n} I J, and d 1,..., d n 1 D. Paul Skoufranis On Operator-Valued Bi-Free Distributions March 22, 2016 20 / 30

Bi-Free from B over D Theorem (Skoufranis; 2015) Assume that F : B D satisfies the following faithfulness condition: if b 1 B and F (b 2 b 1 ) = 0 for all b 2 B, then b 1 = 0. Then (alg(ε(d 1 D ), {Z i } i I ), alg(ε(1 D D op ), {Z j } j J )) is bi-free from (ε(b 1 B ), ε(1 B B op )) with amalgamation over D if and only if ( ) κ B Z,ω (b 1,..., b n 1 ) = F κ B Z,ω (F (b 1),..., F (b n 1 )) (1) for all n 1, ω : {1,..., n} I J, and b 1,..., b n 1 B. Alternatively, equation (1) is equivalent to κ B Z,ω (b 1,..., b n 1 ) = κ D Z,ω (F (b 1),..., F (b n 1 )). (2) This is a bi-free analogue of a result of Nica, Shlyakhtenko, and Speicher. Paul Skoufranis On Operator-Valued Bi-Free Distributions March 22, 2016 21 / 30

Bi-R-Cyclic Families Definition Let I and J be disjoint index sets and let The pair {[Z k;i,j ]} k I {[Z k;i,j ]} k J M N (A). ({[Z k;i,j ]} k I, {[Z k;i,j ]} k J ) is said to be R-cyclic if for every n 1, ω : {1,..., n} I J, and 1 i 1,..., i n, j 1,..., j n d, whenever at least one of κ C χ ω (Z ω(1);i1,j 1, Z ω(2);i2,j 2,..., Z ω(n);in,j n ) = 0 j sχ(1) = i sχ(2), j sχ(2) = i sχ(3),..., j sχ(n 1) = i sχ(n), j sχ(n) = i sχ(1) fail. Paul Skoufranis On Operator-Valued Bi-Free Distributions March 22, 2016 22 / 30

Bi-R-Cyclic Families and Bi-Free over the Diagonal Theorem (Skoufranis; 2015) Let (A, ϕ) be a non-commutative probability space and let Then the following are equivalent: {[Z k;i,j ]} k I {[Z k;i,j ]} k J M N (A). ({[Z k;i,j ]} k I, {[Z k;i,j ]} k J ) is R-cyclic. ({L([Z k;i,j ])} k I, {R([Z k;i,j ])} k J ) is bi-free from (L(M N (C)), R(M N (C) op )) with amalgamation over D N with respect to F E N. This is a bi-free analogue of a result of Nica, Shlyakhtenko, and Speicher. One of the first non-trivial, concretely constructed examples of bi-freeness with amalgamation. Paul Skoufranis On Operator-Valued Bi-Free Distributions March 22, 2016 23 / 30

Bi-Free Partial R-Transform - Operator-Valued If (A, E, ε) is a Banach B-B-ncps, b, d B, X A l, and Y A r, let M l X (b) = 1 + n 1 E((L b X ) n ) M r Y (d) = 1 + n 1 E((R d Y ) n ) C l X (b) = 1 + n 1 κ B χ n,0 (L b X,..., L b X ) C r Y (d) = 1 + n 1 κ B χ 0,n (R d Y,..., R d Y ) Paul Skoufranis On Operator-Valued Bi-Free Distributions March 22, 2016 24 / 30

Bi-Free Partial R-Transform - Operator-Valued If (A, E, ε) is a Banach B-B-ncps, b, db, d B, X A l, and Y A r, let M X,Y (b, db, d) := E((L b X ) n (R d Y ) m R db ) and n,m 0 C X,Y (b, db, d) := db + κ B χ n,0 (L b X,..., L b X, L }{{} b XL db ) n 1 n 1 entries + κ χn,m (L b X,..., L b X, R }{{} d Y,..., R d Y, R }{{} d YR db ). m 1 n entries m 1 entries n 0 Theorem (Skoufranis; 2015) With the above notation, MX l (b)m X,Y (b, db, d) + M X,Y (b, db, d)my r (d) = MX l (b)dbmr Y (d) + C X,Y (MX l (b)b, M X,Y (b, db, d), dmy r (d)). Paul Skoufranis On Operator-Valued Bi-Free Distributions March 22, 2016 25 / 30

Operator-Valued Free S-Transform If E(X ) and E(Y ) are invertible, let Ψ l,x (b) = M l X (b) 1 = n 1 E((L b X ) n ) Ψ r,y (d) = M r Y (d) 1 = n 1 E((R d Y ) n ) Φ l,x (b) = C l X (b) 1 = n 1 κ B χ n,0 (L b X,..., L b X ) Φ r,y (d) = C r Y (d) 1 = n 1 κ B χ 0,n (R d Y,..., R d Y ) S l X (b) = b 1 (b + 1)Ψ 1 l,x S r Y (b) = b 1 Φ 1 (b) l,x (d) = Φ 1 r,y (d)(d + 1)d 1 = Φ 1 r,y (d)d 1. Paul Skoufranis On Operator-Valued Bi-Free Distributions March 22, 2016 26 / 30

Operator-Valued Free S-Transform Theorem (Dykema; 2006) Let (A, E, ε) be a Banach B-B-non-commutative probability space, let (X 1, Y 1 ) and (X 2, Y 2 ) be bi-free over B. Assume that E(X k ) and E(Y k ) are invertible. Then S l X 1 X 2 (b) = S l X 2 (b)s l X 1 (S l X 2 (b) 1 bs X2 (b)) and S r Y 1 Y 2 (d) = S r Y 1 (S r Y 2 (d)ds Y2 (d) 1 )S r Y 2 (d) each on a neighbourhood of zero. Paul Skoufranis On Operator-Valued Bi-Free Distributions March 22, 2016 27 / 30

Operator-Valued Bi-Free S-Transform Let K X,Y (b, db, d) = κ B χ n,m (L b X,..., L b X, R }{{} d Y,..., R d Y, R }{{} d YR db ) n,m 1 n entries m 1 entries ( ) Υ X,Y (b, db, d) = K X,Y bsx l (b), db, S Y r (d)d. Definition (Skoufranis; 2015) The operator-valued bi-free partial S-transform of (X, Y ), denoted S X,Y (b, db, d), is the analytic function db + b 1 Υ X,Y (b, db, d) + Υ X,Y (b, db, d)d 1 + b 1 Υ X,Y (b, db, d)d 1 for any bounded collection of db provided b and d sufficiently small. Paul Skoufranis On Operator-Valued Bi-Free Distributions March 22, 2016 28 / 30

Operator-Valued Bi-Free S-Transform Formula Theorem (Skoufranis; 2015) If (X 1, Y 1 ) and (X 2, Y 2 ) are bi-free over a unital algebra B, then equals Z l S X1,Y 1 ( Z 1 l S X1 X 2,Y 1 Y 2 (b, db, d) bz l, Z 1 l where Z l = S l X 2 (b) and Z r = S r Y 2 (d). S X2,Y 2 (b, db, d)z 1 r, Z r dz 1 r ) Zr Paul Skoufranis On Operator-Valued Bi-Free Distributions March 22, 2016 29 / 30

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