A Brief Overview of Bi-Free Probability
Paul Skoufranis
TAMU
May 25, 2016
Paul Skoufranis
A Brief Overview of Bi-Free Probability
May 25, 2016
1 / 32
The Left Regular Representation
Let F2 denote the free group on two elements a and b. The left regular
representation of F2 is the group homomorphism λ : F2 → U(`2 (F2 ))
defined for all g , h ∈ F2 by
λ(g )δh = δgh .
If L(F2 ) = λ(F2 )00 , then the linear map τ : L(F2 ) → C defined by
τ (T ) = hT δe , δe i
is a tracial state on L(F2 ). Indeed if nj , mj ∈ Z \ {0}, then
τ (λ(a)n1 λ(b)m1 λ(a)n2 λ(b)m2 · · · λ(a)nk λ(b)mk ) = 0.
Paul Skoufranis
A Brief Overview of Bi-Free Probability
May 25, 2016
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Free Independence
Definition
A non-commutative probability space is a pair (A, ϕ) where A is a unital
algebra over C and ϕ : A → C is a linear map with ϕ(IA ) = 1.
Definition
Let (A, ϕ) be a non-commutative probability space. A collection {Ak }k∈K
of unital subalgebras of A are said to be freely independent with respect to
ϕ if
ϕ(Z1 · · · Zn ) = 0
for all n ≥ 1, for all k1 , . . . , kn ∈ K with km 6= km+1 for all m, and for all
Zm ∈ Akm with ϕ(Zm ) = 0.
For example, alg(λ(a)) and alg(λ(b)) are freely independent with respect
to τ .
Paul Skoufranis
A Brief Overview of Bi-Free Probability
May 25, 2016
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Free Independence via Free Products
Theorem (Voiculescu; 1985)
Let (A, ϕ) be a non-commutative probability space. Unital subalgebras A1
and A2 of A are freely independent if and only if there exist vector spaces
Xk and unital homomorphisms αk : Ak → L(Xk ) such that the following
diagram commutes:
A1 ∗ A2
i
A
ϕ
α1 ∗ α2
L(X1 ) ∗ L(X2 )
Paul Skoufranis
C
λ1 ∗ λ2
L(X1 ∗ X2 )
A Brief Overview of Bi-Free Probability
ψ
May 25, 2016
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Central Limit Distributions
Theorem (Central Limit Theorem)
For each N ∈ N, let X1 , . . . , XN be independent, identically distributed
real-valued random variables with expectation zero and variance one. Then
N
1 X
Xk
SN = √
N k=1
distribution
−→
N→∞
x2
1
√ e − 2 dx.
2π
Theorem (Free Central Limit Theorem; Voiculescu)
For each N ∈ N, let X1 , . . . , XN be freely independent, identically
distributed real-valued random variables with expectation zero and
variance one. Then
N
1 X
SN = √
Xk
N k=1
Paul Skoufranis
distribution
−→
N→∞
1 p
√
4 − x 2 dx.
2π
A Brief Overview of Bi-Free Probability
May 25, 2016
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Bi-Free Independence
What about the right regular representation?
Definition (Voiculescu; 2013)
Let (A, ϕ) be a non-commutative probability space. Pairs of unital
algebras (A`,1 , Ar ,1 ) and (A`,2 , Ar ,2 ) of A are said to be bi-freely
independent if there exist vector spaces Xk and unital homomorphisms
unital homomorphisms αk : A`,k → L(Xk ) and βk : Ar ,k → L(Xk ) such
that the following diagram commutes:
A`,1 ∗ Ar ,1 ∗ A`,2 ∗ Ar ,2
i
ϕ
A
α1 ∗ β1 ∗ α2 ∗ β2
ψ
L(X1 ) ∗ L(X1 ) ∗ L(X2 ) ∗ L(X2 )
Paul Skoufranis
C
λ1 ∗ ρ1 ∗ λ2 ∗ ρ2
A Brief Overview of Bi-Free Probability
L(X1 ∗ X2 )
May 25, 2016
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A Simple Characterization of Bi-Freeness?
Freely Independent
⇐⇒
Alternating Centred Moments Vanish
Bi-Freely Independent
⇐⇒
???
Mastnak and Nica attempted to develop a bi-free analogue of non-crossing
partitions.
Paul Skoufranis
A Brief Overview of Bi-Free Probability
May 25, 2016
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Non-Crossing Partitions
In 1994, Speicher developed a combinatorial approach to free probability
via non-crossing partitions.
Definition
An partition π of {1, . . . , n} is said to be non-crossing if for all U, V ∈ π
with U 6= V , for all a, b ∈ U, and for all c, d ∈ V , it is not the case that
a < c < b < d. The set of non-crossing partitions on {1, . . . , n} is
denoted NC (n).
{{1, 2}, {3, 4}}
{{1, 3}, {2, 4}}
{{1, 4}, {2, 3}}
1
1
1
2
Paul Skoufranis
3
4
2
3
4
A Brief Overview of Bi-Free Probability
2
3
4
May 25, 2016
8 / 32
Moment and Cumulant Functions
The free cumulant of Z1 , . . . , Zn is a Möbius inversion of the moments.
Theorem (Speicher; 1994)
Let (A, ϕ) be a non-commutative probability space. A collection {Ak }k∈K
of unital subalgebras of A is freely independent if and only if
κ1n (Z1 , . . . , Zn ) = 0
for all non-constant : {1, . . . , n} → K and for all Zk ∈ A(k) .
Paul Skoufranis
A Brief Overview of Bi-Free Probability
May 25, 2016
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The Permutation
Let χ : {1, . . . , n} → {`, r } designate whether the k th operator is considered
a left operator (χ(k) = `) or a right operator (χ(k) = r ). If
χ−1 ({`}) = {k1 < k2 < · · · < km }
χ−1 ({r }) = {km+1 > km+2 > · · · > kn }
define the permutation sχ of {1, . . . , n} via sχ (t) = kt .
Paul Skoufranis
A Brief Overview of Bi-Free Probability
May 25, 2016
10 / 32
The Permutation
Let χ : {1, . . . , n} → {`, r } designate whether the k th operator is considered
a left operator (χ(k) = `) or a right operator (χ(k) = r ). If
χ−1 ({`}) = {k1 < k2 < · · · < km }
χ−1 ({r }) = {km+1 > km+2 > · · · > kn }
define the permutation sχ of {1, . . . , n} via sχ (t) = kt .
Consider χ : {1, . . . , 7} → {`, r } with χ−1 ({`}) = {1, 4, 6, 7}.
Suppose Z1 , . . . , Z7 are operators (either left or right based on χ) for which
we want to consider Z1 · · · Z7 ξ.
Paul Skoufranis
A Brief Overview of Bi-Free Probability
May 25, 2016
10 / 32
The Permutation
Let χ : {1, . . . , n} → {`, r } designate whether the k th operator is considered
a left operator (χ(k) = `) or a right operator (χ(k) = r ). If
χ−1 ({`}) = {k1 < k2 < · · · < km }
χ−1 ({r }) = {km+1 > km+2 > · · · > kn }
define the permutation sχ of {1, . . . , n} via sχ (t) = kt .
Z1 , L
Z2 , R
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Z3 , R
Z4 , L
Z5 , R
A Brief Overview of Bi-Free Probability
Z6 , L
Z7 , L ξ
May 25, 2016
10 / 32
The Permutation
Let χ : {1, . . . , n} → {`, r } designate whether the k th operator is considered
a left operator (χ(k) = `) or a right operator (χ(k) = r ). If
χ−1 ({`}) = {k1 < k2 < · · · < km }
χ−1 ({r }) = {km+1 > km+2 > · · · > kn }
define the permutation sχ of {1, . . . , n} via sχ (t) = kt .
Z1 , L
Z2 , R
Z3 , R
Z4 , L
Z5 , R
Z6 , L
Z7 , L
Paul Skoufranis
A Brief Overview of Bi-Free Probability
May 25, 2016
10 / 32
The Permutation
Let χ : {1, . . . , n} → {`, r } designate whether the k th operator is considered
a left operator (χ(k) = `) or a right operator (χ(k) = r ). If
χ−1 ({`}) = {k1 < k2 < · · · < km }
χ−1 ({r }) = {km+1 > km+2 > · · · > kn }
define the permutation sχ of {1, . . . , n} via sχ (t) = kt .
Z1 , L
Paul Skoufranis
Z2 , R
Z3 , R
Z6 , L
Z7 , L
Z4 , L
A Brief Overview of Bi-Free Probability
Z5 , R
May 25, 2016
10 / 32
The Permutation
Let χ : {1, . . . , n} → {`, r } designate whether the k th operator is considered
a left operator (χ(k) = `) or a right operator (χ(k) = r ). If
χ−1 ({`}) = {k1 < k2 < · · · < km }
χ−1 ({r }) = {km+1 > km+2 > · · · > kn }
define the permutation sχ of {1, . . . , n} via sχ (t) = kt .
Z1 , L
Z2 , R
Z6 , L
Paul Skoufranis
Z3 , R
Z7 , L
Z4 , L
Z5 , R
A Brief Overview of Bi-Free Probability
May 25, 2016
10 / 32
The Permutation
Let χ : {1, . . . , n} → {`, r } designate whether the k th operator is considered
a left operator (χ(k) = `) or a right operator (χ(k) = r ). If
χ−1 ({`}) = {k1 < k2 < · · · < km }
χ−1 ({r }) = {km+1 > km+2 > · · · > kn }
define the permutation sχ of {1, . . . , n} via sχ (t) = kt .
Z4 , L
Paul Skoufranis
Z1 , L
Z2 , R
Z3 , R
Z6 , L
Z7 , L
Z5 , R
A Brief Overview of Bi-Free Probability
May 25, 2016
10 / 32
The Permutation
Let χ : {1, . . . , n} → {`, r } designate whether the k th operator is considered
a left operator (χ(k) = `) or a right operator (χ(k) = r ). If
χ−1 ({`}) = {k1 < k2 < · · · < km }
χ−1 ({r }) = {km+1 > km+2 > · · · > kn }
define the permutation sχ of {1, . . . , n} via sχ (t) = kt .
Z1 , L
Paul Skoufranis
Z4 , L
Z6 , L
Z7 , L
Z5 , R
A Brief Overview of Bi-Free Probability
Z3 , R
Z2 , R
May 25, 2016
10 / 32
The Permutation
Let χ : {1, . . . , n} → {`, r } designate whether the k th operator is considered
a left operator (χ(k) = `) or a right operator (χ(k) = r ). If
χ−1 ({`}) = {k1 < k2 < · · · < km }
χ−1 ({r }) = {km+1 > km+2 > · · · > kn }
define the permutation sχ of {1, . . . , n} via sχ (t) = kt .
Z1 , L
Z2 , R
Z1 , L
Paul Skoufranis
Z4 , L
Z3 , R
Z6 , L
Z4 , L
Z7 , L
Z5 , R
Z5 , R
A Brief Overview of Bi-Free Probability
Z6 , L
Z3 , R
Z7 , L ξ
Z2 , R
May 25, 2016
10 / 32
The Permutation
Let χ : {1, . . . , n} → {`, r } designate whether the k th operator is considered
a left operator (χ(k) = `) or a right operator (χ(k) = r ). If
χ−1 ({`}) = {k1 < k2 < · · · < km }
χ−1 ({r }) = {km+1 > km+2 > · · · > kn }
define the permutation sχ of {1, . . . , n} via sχ (t) = kt .
Definition (Mastnak, Nica; 2013)
Given χ : {1, . . . , n} → {`, r }, a partition π of {1, . . . , n} is said to be
bi-non-crossing with respect to χ if the partition sχ−1 · π (the partition
obtained by applying sχ−1 to each block of π) is non-crossing.
Paul Skoufranis
A Brief Overview of Bi-Free Probability
May 25, 2016
10 / 32
Bi-Non-Crossing Partitions
Let χ−1 ({`}) = {1, 2, 4}, χ−1 ({r }) = {3, 5}, and
n
o
n
o
π = {1, 3}, {2, 4, 5} = sχ · {1, 5}, {2, 3, 4} .
1
2
3
4
5
1
2
4
5
3
1
2
3
4
5
µBNC (π, σ) = µNC (sχ−1 · π, sχ−1 · σ)
Paul Skoufranis
A Brief Overview of Bi-Free Probability
May 25, 2016
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Bi-Freeness and Mixed Cumulants
Theorem (Charlesworth, Nelson, Skoufranis; 2014)
Let (A, ϕ) be a ncps and let {(A`,k , Ar ,k )}k∈K be pairs of unital
subalgebras of A. Then the following are equivalent:
{(A`,k , Ar ,k )}k∈K are bi-freely independent.
For all χ : {1, . . . , n} → {`, r }, : {1, . . . , n} → K , and
Zm ∈ Aχ(m),(m) ,

ϕ(Z1 · · · Zm ) =
X
π∈BNC (χ)




X
σ∈BNC (χ)
π≤σ≤

µBNC (π, σ)
 ϕπ (Z1 , . . . , Zm )
For all χ : {1, . . . , n} → {`, r }, : {1, . . . , n} → K non-constant, and
Zm ∈ Aχ(m),(m) ,
κχ (Z1 , . . . , Zn ) = 0.
Paul Skoufranis
A Brief Overview of Bi-Free Probability
May 25, 2016
12 / 32
Bi-Free Independence from Free Independence
Theorem (Charlesworth, Nelson, Skoufranis; 2014)
Let (A, ϕ) be a ncps and let {(A`,k , Ar ,k )}k∈K be pairs of unital
subalgebras of A. Suppose
1
for all k1 , k2 ∈ K , ϕ(Z1 XYZ2 ) = ϕ(Z1 YXZ2 ) for all Z1 , Z2 ∈ A,
X ∈ A`,k1 , and Y ∈ Ar ,k2 , and
2
for each k ∈ K and Y ∈ Ar ,k there exists a X ∈ A`,k such that
ϕ(ZY ) = ϕ(ZX ) for all Z ∈ A.
Then {(A`,k , Ar ,k )}k∈K are bi-freely independent if and only if {A`,k }k∈K
are freely independent.
Example
If M = M1 ∗τ M2 and λ, ρ : M → L(L2 (M, τ )) are the left and right
actions of M, then (λ(M1 ), ρ(M1 )) and (λ(M2 ), ρ(M2 )) are bi-freely
independent.
Paul Skoufranis
A Brief Overview of Bi-Free Probability
May 25, 2016
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Types of Independence
If (A`,1 , Ar ,1 ) and (A`,2 , Ar ,2 ) are bi-freely independent, then
A`,1 and A`,2 are freely independent,
Ar ,1 and Ar ,2 are freely independent,
A`,1 and Ar ,2 are classically independent,
Ar ,1 and A`,2 are classically independent,
under additional assumptions on the algebras,
alg(A`,1 Ar ,1 ) and alg(A`,2 Ar ,2 ) are Boolean independent, and
under additional assumptions on the algebras,
alg(A`,1 Ar ,1 ) and A`,2 are monotone independent.
Bi-freeness contains the five universal notions of independent of Speicher
and Muraki.
Paul Skoufranis
A Brief Overview of Bi-Free Probability
May 25, 2016
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Bi-Free Central Limit Distributions
Definition (Voiculescu; 2013)
A pair ({Zi }i∈I , {Zj }j∈J ) is said to be a bi-free central limit distribution if
all cumulants of order at least three vanish.
Theorem (Voiculescu; 2013)
Given non-empty disjoint sets I and J, for each matrix
C = [Ck1 ,k2 ]k1 ,k2 ∈I tJ
there exists exactly one centred bi-free central limit distribution with
κχ (Zk1 , Zk2 ) = Ck1 ,k2 . In particular, if H is a Hilbert space and
h, h∗ : I t J → H are such that Ck1 ,k2 = hh(k2 ), h∗ (k1 )iH , then
Zi = l(h(i)) + l ∗ (h∗ (i))
and
Zj = r (h(j)) + r ∗ (h∗ (j))
is a realization of said bi-free central limit distribution.
Paul Skoufranis
A Brief Overview of Bi-Free Probability
May 25, 2016
15 / 32
The R-Transform
If (A, ϕ) is a non-commutative probability space and X ∈ A, the Cauchy
transform of X is
GX (z) = ϕ((zIA − X )−1 ).
The R-Transform of X is
RX (z) =
X
κn+1 (X )z n .
n≥0
It is possible to show that GX RX (z) +
1
z
= z.
Theorem (Voiculescu; 1985)
If X1 and X2 are freely independent, then RX1 +X2 = RX1 + RX2 .
There exists a combinatorial approach due to Speicher.
Paul Skoufranis
A Brief Overview of Bi-Free Probability
May 25, 2016
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The Bi-Free Partial R-Transform
Given X , Y ∈ A, define the two-variable Green’s function by
X ϕ(X n Y m )
1
GX ,Y (z, w ) = ϕ((zIA − X )−1 (wIA − Y )−1 ) =
+
,
zw
z n+1 w m+1
n,m≥0
n+m≥1
The bi-free partial R-transform of (X , Y ) is
X
RX ,Y (z, w ) =
κn,m (X , Y )z n w m .
n,m≥0
n+m≥1
If (X1 , Y1 ) and (X2 , Y2 ) are bi-freely independent, then
RX1 +X2 ,Y1 +Y2 = RX1 ,Y1 + RX2 ,Y2 .
Theorem (Voiculescu; 2013), combinatorial proof (Skoufranis, 2014)
As holomorphic functions near (0, 0),
RX ,Y (z, w ) = 1 + zRX (z) + wRY (w ) −
Paul Skoufranis
GX ,Y
zw
RX (z) + z1 , RY (w ) +
A Brief Overview of Bi-Free Probability
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1
w
.
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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 0 ) = bp(ξ)b 0 .
For b ∈ B, define Lb , Rb ∈ L(X ) by Lb (ξ) = b · ξ and Rb (ξ) = ξ · b.
Define E : L(X ) → B by E (T ) = p(T (1B ⊕ 0)).
E (Lb Rb0 T ) = p(Lb Rb0 (E (T ) ⊕ η)) = p(bE (T )b 0 ⊕ η 0 ) = bE (T )b 0 .
E (TLb ) = p(T (b ⊕ 0)) = E (TRb ).
Paul Skoufranis
A Brief Overview of Bi-Free Probability
May 25, 2016
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B-B-Non-Commutative Probability Space
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 unital linear map
such that
E (ε(b1 ⊗ b2 )T ) = b1 E (T )b2
and
E (T ε(b ⊗ 1B )) = E (T ε(1B ⊗ b)).
Denote Lb = ε(b ⊗ 1B ) and Rb = ε(1B ⊗ b).
Every B-B-non-commutative probability space can be embedded into
L(X ) for some B-B-bimodule X .
Paul Skoufranis
A Brief Overview of Bi-Free Probability
May 25, 2016
19 / 32
B-NCPS via B-B-NCPS
Definition
Let (A, E , ε) be a B-B-ncps. The unital subalgebras of A defined by
A` := {Z ∈ A | ZRb = Rb Z for all b ∈ B} and
Ar := {Z ∈ A | ZLb = Lb Z for all b ∈ B}
are called the left and right algebras of A respectively. A pair of algebras
(A1 , A2 ) is said to be a pair of B-faces if
{Lb }b∈B ⊆ A1 ⊆ A`
Paul Skoufranis
and {Rb }b∈B op ⊆ A2 ⊆ Ar .
A Brief Overview of Bi-Free Probability
May 25, 2016
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Bi-Free Independence with Amalgamation
Definition
Let (A, EA , ε) be a B-B-ncps. Pairs of B-faces (A`,1 , Ar ,1 ) and (A`,2 , Ar ,2 )
of A are said to be bi-freely independent with amalgamation over B if
there exist B-B-bimodules Xk and unital B-homomorphisms
αk : A`,k → L(Xk )` and βk : Ar ,k → L(Xk )r such that the following
diagram commutes:
A`,1 ∗ Ar ,1 ∗ A`,2 ∗ Ar ,2
i
A
EA
EL(X1 ∗X2 )
α1 ∗ β1 ∗ α2 ∗ β2
L(X1 )` ∗ L(X1 )r ∗ L(X2 )` ∗ L(X2 )r
Paul Skoufranis
B
λ1 ∗ ρ1 ∗ λ2 ∗ ρ2
A Brief Overview of Bi-Free Probability
L(X1 ∗ X2 )
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Operator-Valued Bi-Freeness and Mixed Cumulants
Theorem (Charlesworth, Nelson, Skoufranis; 2014)
Let (A, E , ε) be a B-B-ncps and let {(A`,k , Ar ,k )}k∈K be pairs of B-faces.
Then the following are equivalent:
{(A`,k , Ar ,k )}k∈K are bi-free over B.
For all χ : {1, . . . , n} → {`, r }, : {1, . . . , n} → K , and
Zm ∈ Aχ(m),(m) ,

E (Z1 · · · Zm ) =
X
π∈BNC (χ)




X
σ∈BNC (χ)
π≤σ≤

µBNC (π, σ)
 Eπ (Z1 , . . . , Zm )
For all χ : {1, . . . , n} → {`, r }, : {1, . . . , n} → K non-constant, and
Zm ∈ Aχ(m),(m) ,
κχ (Z1 , . . . , Zn ) = 0.
Paul Skoufranis
A Brief Overview of Bi-Free Probability
May 25, 2016
22 / 32
Amalgamating Over Matrices
Let (A, ϕ) be a non-commutative probability space.
MN (A) is naturally a MN (C)-ncps where the expectation map
ϕN : MN (A) → MN (C) is defined via
ϕN ([Ai,j ]) = [ϕ(Ai,j )].
If A1 , A2 are unital subalgebras of A that are free with respect to ϕ,
then MN (A1 ) and MN (A2 ) are free with amalgamation over MN (C)
with respect to ϕN .
Is there a bi-free analogue of this result?
Is MN (A) a MN (C)-MN (C)-ncps?
Paul Skoufranis
A Brief Overview of Bi-Free Probability
May 25, 2016
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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
Lb (a ⊗ b 0 ) = a ⊗ bb 0 ,
and
Rb (a ⊗ b 0 ) = a ⊗ b 0 b.
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 (1A ⊗ 1B )).
If X , Y ∈ A, defined L(X ⊗ b) ∈ L(A ⊗ B)` and R(Y ⊗ b) ∈ L(A ⊗ B)r
via
L(X ⊗ b)(a ⊗ b 0 ) = Xa ⊗ bb 0
Paul Skoufranis
and
R(Y ⊗ b)(a ⊗ b 0 ) = Ya ⊗ b 0 b.
A Brief Overview of Bi-Free Probability
May 25, 2016
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Bi-Freeness Preserved Under Tensoring
Theorem (Skoufranis; 2015)
Let (A, ϕ) be a non-commutative probability space and let
{(A`,k , Ar ,k )}k∈K be bi-free pairs of faces with respect to ϕ. If B is a
unital algebra, then {(L(A`,k ⊗ B), R(Ar ,k ⊗ B))}k∈K are bi-free over B
with respect to some E .
Paul Skoufranis
A Brief Overview of Bi-Free Probability
May 25, 2016
25 / 32
Bi-Matrix Models - q-Deformed Fock Space
Let q ∈ [−1, 1], let H be a Hilbert space, let Fq (H) be the q-deformed
Fock space, and, for h ∈ H, let lq (h), rq (h), lq∗ (h), and rq∗ (h) denote the
left/right q-deformed creation and annihilation operators on Fq (H).
Given an index set K , an N ∈ N, and an orthonormal set of vectors
k | i, j ∈ {1, . . . , N}, k ∈ K } ⊆ H, let
{hi,j
N
1 X
k
) ⊗ Ei,j ),
L(lq (hi,j
Lk (N) := √
N i,j=1
N
1 X
k
L∗k (N) := √
) ⊗ Ei,j )
L(lq∗ (hj,i
N i,j=1
N
N
1 X
1 X
k
k
Rk (N) := √
R(rq (hi,j
) ⊗ Ei,j ), Rk∗ (N) := √
R(rq∗ (hj,i
) ⊗ Ei,j ).
N i,j=1
N i,j=1
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A Brief Overview of Bi-Free Probability
May 25, 2016
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Bi-Matrix Models - q-Deformed Fock Space
Theorem (Skoufranis; 2015)
If E : L(L(Fq (H)) ⊗ MN (C)) → MN (C) is the expectation, the joint
distribution of
{Lk (N), L∗k (N), Rk (N), Rk∗ (N)}k∈K
with respect to
1
N Tr
◦ E is asymptotically equal the joint distribution of
{l0 (hk ), l0∗ (hk ), r0 (hk ), r0∗ (hk )}k∈K
with respect to ϕ where {hk }k∈K ⊆ H is an orthonormal set.
Paul Skoufranis
A Brief Overview of Bi-Free Probability
May 25, 2016
27 / 32
Bi-R-Cyclic Families
Definition
Let I and J be disjoint index sets and let
{[Zk;i,j ]}k∈I ∪ {[Zk;i,j ]}k∈J ⊆ MN (A).
The pair
({[Zk;i,j ]}k∈I , {[Zk;i,j ]}k∈J )
is said to be R-cyclic if for every n ≥ 1, ω : {1, . . . , n} → I t J, and
1 ≤ i1 , . . . , in , j1 , . . . , jn ≤ d,
κC
χω (Zω(1);i1 ,j1 , Zω(2);i2 ,j2 , . . . , Zω(n);in ,jn ) = 0
whenever at least one of
jsχ (1) = isχ (2) , jsχ (2) = isχ (3) , . . . , jsχ (n−1) = isχ(n) , jsχ (n) = isχ (1)
fail.
Paul Skoufranis
A Brief Overview of Bi-Free Probability
May 25, 2016
28 / 32
Bi-R-Cyclic Families and Bi-Free over the Diagonal
k | i, j ∈ {1, . . . , N}, k ∈ K } ⊆ H
For example, if K is an index set and {hi,j
is an orthonormal set of vectors, then
k
k
k
k
)]})k∈K
({[l(hi,j
)], [l ∗ (hj,i
)]}, {[r (hi,j
)], [r ∗ (hj,i
is an R-cyclic family.
Theorem (Skoufranis; 2015)
Let (A, ϕ) be a non-commutative probability space and let
{[Zk;i,j ]}k∈I ∪ {[Zk;i,j ]}k∈J ⊆ MN (A).
Then the following are equivalent:
({[Zk;i,j ]}k∈I , {[Zk;i,j ]}k∈J ) is R-cyclic.
({L([Zk;i,j ])}k∈I , {R([Zk;i,j ])}k∈J ) is bi-free from
(L(MN (C)), R(MN (C)op )) with amalgamation over DN with respect
to F ◦ EN where F : MN (C) → DN is the expectation onto the
diagonal.
Paul Skoufranis
A Brief Overview of Bi-Free Probability
May 25, 2016
29 / 32
Operator-Valued Bi-Free Distributions
Suppose {Zi }i∈I ⊆ A` and {Zj }j∈J ⊆ Ar . Suppose we wanted to describe
all B-valued moments involving Zi1 , Zj1 , Zi2 , and Zj2 each occurring once
in that order.
Zi1
Zj1
Zi2
Zj2
Paul Skoufranis
A Brief Overview of Bi-Free Probability
May 25, 2016
30 / 32
Operator-Valued Bi-Free Distributions
Suppose {Zi }i∈I ⊆ A` and {Zj }j∈J ⊆ Ar . Suppose we wanted to describe
all B-valued moments involving Zi1 , Zj1 , Zi2 , and Zj2 each occurring once
in that order.
Zi1
Zj1
Zi2
Zj2
Paul Skoufranis
A Brief Overview of Bi-Free Probability
May 25, 2016
30 / 32
Operator-Valued Bi-Free S-Transform Formula
Theorem (Skoufranis; 2015)
If (X1 , Y1 ) and (X2 , Y2 ) are bi-free over a unital algebra B, then
SX1 X2 ,Y1 Y2 (b, c, d)
equals
Z` SX1 ,Y1 Z`−1 bZ` , Z`−1 SX2 ,Y2 (b, c, d)Zr−1 , Zr dZr−1 Zr
where Z` = SX` 2 (b) and Zr = SYr 2 (d).
Paul Skoufranis
A Brief Overview of Bi-Free Probability
May 25, 2016
31 / 32
Thanks for Listening!
Paul Skoufranis
A Brief Overview of Bi-Free Probability
May 25, 2016
32 / 32