Experimental Evidence of a non-Globally Trace-Preserving Positive Operator Valued Measure
Operator-valued free probability theory and the self ... · Operator-valued free probability theory...
Transcript of Operator-valued free probability theory and the self ... · Operator-valued free probability theory...
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Operator-valued free probability theory and the
self-adjoint linearization trick
Tobias Mai
Saarland University
Workshop on Analytic, Stochastic, and Operator Algebraic Aspects
of Noncommutative Distributions and Free Probability
Fields Institute, Toronto � July 25, 2013
Tobias Mai (Saarland University) The self-adjoint linearization trick July 25, 2013 1 / 14
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Contents
1 A quick reminder on...
...Anderson's self-adjoint version of the linearization trick
...operator-valued free probability theory
2 First application: Polynomials in free random variables
3 Second application: Multivariate free Berry-Esseen
Tobias Mai (Saarland University) The self-adjoint linearization trick July 25, 2013 2 / 14
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A quick reminder on... ...Anderson's self-adjoint version of the linearization trick
De�nition
Let p ∈ C〈X1, . . . , Xn〉 be given. A matrix
Lp :=
[0 uv Q
]∈ MN (C〈X1, . . . , Xn〉),
of dimension N ∈ N, whereu and v are row and column vectors, respectively, both of dimension
N − 1 with entries in C〈X1, . . . , Xn〉 andQ ∈ MN−1(C〈X1, . . . , Xn〉) is invertible,
is called a linearization of p, if the following conditions are satis�ed:
(i) There are matrices b0, . . . , bn ∈ MN (C), such that
Lp = b0 ⊗ 1 + b1 ⊗X1 + · · ·+ bn ⊗Xn ∈ MN (C)⊗ C〈X1, . . . , Xn〉.
(ii) It holds true that p = −uQ−1v.
Tobias Mai (Saarland University) The self-adjoint linearization trick July 25, 2013 3 / 14
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A quick reminder on... ...Anderson's self-adjoint version of the linearization trick
De�nition
Let p ∈ C〈X1, . . . , Xn〉 be given. A matrix
Lp :=
[0 uv Q
]∈ MN (C〈X1, . . . , Xn〉),
of dimension N ∈ N, whereu and v are row and column vectors, respectively, both of dimension
N − 1 with entries in C〈X1, . . . , Xn〉 andQ ∈ MN−1(C〈X1, . . . , Xn〉) is invertible,
is called a linearization of p, if the following conditions are satis�ed:
(i) There are matrices b0, . . . , bn ∈ MN (C), such that
Lp = b0 ⊗ 1 + b1 ⊗X1 + · · ·+ bn ⊗Xn ∈ MN (C)⊗ C〈X1, . . . , Xn〉.
(ii) It holds true that p = −uQ−1v.
Tobias Mai (Saarland University) The self-adjoint linearization trick July 25, 2013 3 / 14
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A quick reminder on... ...Anderson's self-adjoint version of the linearization trick
Theorem (Anderson, 2012)
(i) Consider a polynomial p ∈ C〈X1, . . . , Xn〉 with linearization
Lp = b0 ⊗ 1 + b1 ⊗X1 + · · ·+ bn ⊗Xn ∈ MN (C)⊗ C〈X1, . . . , Xn〉.
Let A be a complex unital algebra and let x1, . . . , xn ∈ A be given.
Put P := p(x1, . . . , xn) and
LP := b0 ⊗ 1 + b1 ⊗ x1 + · · ·+ bn ⊗ xn ∈ MN (C)⊗A.
By using the notation Λ(z) := diag(z, 0, . . . , 0) ∈ MN (C), we get
z − P is invertible in A ⇐⇒ Λ(z)− LP is invertible in MN (C)⊗A
and moreover: [(Λ(z)− LP )−1]1,1 = (z − P )−1
(ii) Any polynomial p ∈ C〈X1, . . . , Xn〉 has a linearization Lp.If p is self-adjoint, Lp can be chosen to be self-adjoint as well.
Tobias Mai (Saarland University) The self-adjoint linearization trick July 25, 2013 4 / 14
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A quick reminder on... ...Anderson's self-adjoint version of the linearization trick
Theorem (Anderson, 2012)
(i) Consider a polynomial p ∈ C〈X1, . . . , Xn〉 with linearization
Lp = b0 ⊗ 1 + b1 ⊗X1 + · · ·+ bn ⊗Xn ∈ MN (C)⊗ C〈X1, . . . , Xn〉.
Let A be a complex unital algebra and let x1, . . . , xn ∈ A be given.
Put P := p(x1, . . . , xn) and
LP := b0 ⊗ 1 + b1 ⊗ x1 + · · ·+ bn ⊗ xn ∈ MN (C)⊗A.
By using the notation Λ(z) := diag(z, 0, . . . , 0) ∈ MN (C), we get
z − P is invertible in A ⇐⇒ Λ(z)− LP is invertible in MN (C)⊗A
and moreover: [(Λ(z)− LP )−1]1,1 = (z − P )−1
(ii) Any polynomial p ∈ C〈X1, . . . , Xn〉 has a linearization Lp.If p is self-adjoint, Lp can be chosen to be self-adjoint as well.
Tobias Mai (Saarland University) The self-adjoint linearization trick July 25, 2013 4 / 14
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A quick reminder on... ...Anderson's self-adjoint version of the linearization trick
Theorem (Anderson, 2012)
(i) Consider a polynomial p ∈ C〈X1, . . . , Xn〉 with linearization
Lp = b0 ⊗ 1 + b1 ⊗X1 + · · ·+ bn ⊗Xn ∈ MN (C)⊗ C〈X1, . . . , Xn〉.
Let A be a complex unital algebra and let x1, . . . , xn ∈ A be given.
Put P := p(x1, . . . , xn) and
LP := b0 ⊗ 1 + b1 ⊗ x1 + · · ·+ bn ⊗ xn ∈ MN (C)⊗A.
By using the notation Λ(z) := diag(z, 0, . . . , 0) ∈ MN (C), we get
z − P is invertible in A ⇐⇒ Λ(z)− LP is invertible in MN (C)⊗A
and moreover: [(Λ(z)− LP )−1]1,1 = (z − P )−1
(ii) Any polynomial p ∈ C〈X1, . . . , Xn〉 has a linearization Lp.If p is self-adjoint, Lp can be chosen to be self-adjoint as well.
Tobias Mai (Saarland University) The self-adjoint linearization trick July 25, 2013 4 / 14
![Page 8: Operator-valued free probability theory and the self ... · Operator-valued free probability theory I De nition An operator-valued C-probability space (A;E;B) consists of a unital](https://reader036.fdocuments.net/reader036/viewer/2022071004/5fc1418c99a9c97ebb54a22d/html5/thumbnails/8.jpg)
A quick reminder on... ...Anderson's self-adjoint version of the linearization trick
Theorem (Anderson, 2012)
(i) Consider a polynomial p ∈ C〈X1, . . . , Xn〉 with linearization
Lp = b0 ⊗ 1 + b1 ⊗X1 + · · ·+ bn ⊗Xn ∈ MN (C)⊗ C〈X1, . . . , Xn〉.
Let A be a complex unital algebra and let x1, . . . , xn ∈ A be given.
Put P := p(x1, . . . , xn) and
LP := b0 ⊗ 1 + b1 ⊗ x1 + · · ·+ bn ⊗ xn ∈ MN (C)⊗A.
By using the notation Λ(z) := diag(z, 0, . . . , 0) ∈ MN (C), we get
z − P is invertible in A ⇐⇒ Λ(z)− LP is invertible in MN (C)⊗A
and moreover: [(Λ(z)− LP )−1]1,1 = (z − P )−1
(ii) Any polynomial p ∈ C〈X1, . . . , Xn〉 has a linearization Lp.If p is self-adjoint, Lp can be chosen to be self-adjoint as well.
Tobias Mai (Saarland University) The self-adjoint linearization trick July 25, 2013 4 / 14
![Page 9: Operator-valued free probability theory and the self ... · Operator-valued free probability theory I De nition An operator-valued C-probability space (A;E;B) consists of a unital](https://reader036.fdocuments.net/reader036/viewer/2022071004/5fc1418c99a9c97ebb54a22d/html5/thumbnails/9.jpg)
A quick reminder on... ...Anderson's self-adjoint version of the linearization trick
Theorem (Anderson, 2012)
(i) Consider a polynomial p ∈ C〈X1, . . . , Xn〉 with linearization
Lp = b0 ⊗ 1 + b1 ⊗X1 + · · ·+ bn ⊗Xn ∈ MN (C)⊗ C〈X1, . . . , Xn〉.
Let A be a complex unital algebra and let x1, . . . , xn ∈ A be given.
Put P := p(x1, . . . , xn) and
LP := b0 ⊗ 1 + b1 ⊗ x1 + · · ·+ bn ⊗ xn ∈ MN (C)⊗A.
By using the notation Λ(z) := diag(z, 0, . . . , 0) ∈ MN (C), we get
z − P is invertible in A ⇐⇒ Λ(z)− LP is invertible in MN (C)⊗A
and moreover: [(Λ(z)− LP )−1]1,1 = (z − P )−1
(ii) Any polynomial p ∈ C〈X1, . . . , Xn〉 has a linearization Lp.
If p is self-adjoint, Lp can be chosen to be self-adjoint as well.
Tobias Mai (Saarland University) The self-adjoint linearization trick July 25, 2013 4 / 14
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A quick reminder on... ...Anderson's self-adjoint version of the linearization trick
Theorem (Anderson, 2012)
(i) Consider a polynomial p ∈ C〈X1, . . . , Xn〉 with linearization
Lp = b0 ⊗ 1 + b1 ⊗X1 + · · ·+ bn ⊗Xn ∈ MN (C)⊗ C〈X1, . . . , Xn〉.
Let A be a complex unital algebra and let x1, . . . , xn ∈ A be given.
Put P := p(x1, . . . , xn) and
LP := b0 ⊗ 1 + b1 ⊗ x1 + · · ·+ bn ⊗ xn ∈ MN (C)⊗A.
By using the notation Λ(z) := diag(z, 0, . . . , 0) ∈ MN (C), we get
z − P is invertible in A ⇐⇒ Λ(z)− LP is invertible in MN (C)⊗A
and moreover: [(Λ(z)− LP )−1]1,1 = (z − P )−1
(ii) Any polynomial p ∈ C〈X1, . . . , Xn〉 has a linearization Lp.If p is self-adjoint, Lp can be chosen to be self-adjoint as well.
Tobias Mai (Saarland University) The self-adjoint linearization trick July 25, 2013 4 / 14
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A quick reminder on... ...operator-valued free probability theory
Operator-valued free probability theory I
De�nition
An operator-valued C∗-probability space (A, E,B) consists of
a unital C∗-algebra A,a unital C∗-subalgebra B of A and
a conditional expectation E : A → B, i.e. a positive and unital mapsatisfying
I E[b] = b for all b ∈ B andI E[b1ab2] = b1E[a]b2 for all a ∈ A, b1, b2 ∈ A.
Example
Let (A0, φ) be a C∗-probability space. Then
A := MN (C)⊗A0, B := MN (C) and E := idMN (C)⊗φ
gives an operator-valued C∗-probability space (A, E,B).
Tobias Mai (Saarland University) The self-adjoint linearization trick July 25, 2013 5 / 14
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A quick reminder on... ...operator-valued free probability theory
Operator-valued free probability theory I
De�nition
An operator-valued C∗-probability space (A, E,B) consists of
a unital C∗-algebra A,a unital C∗-subalgebra B of A and
a conditional expectation E : A → B, i.e. a positive and unital mapsatisfying
I E[b] = b for all b ∈ B andI E[b1ab2] = b1E[a]b2 for all a ∈ A, b1, b2 ∈ A.
Example
Let (A0, φ) be a C∗-probability space. Then
A := MN (C)⊗A0, B := MN (C) and E := idMN (C)⊗φ
gives an operator-valued C∗-probability space (A, E,B).
Tobias Mai (Saarland University) The self-adjoint linearization trick July 25, 2013 5 / 14
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A quick reminder on... ...operator-valued free probability theory
Operator-valued free probability theory I
De�nition
An operator-valued C∗-probability space (A, E,B) consists of
a unital C∗-algebra A,a unital C∗-subalgebra B of A and
a conditional expectation E : A → B, i.e. a positive and unital mapsatisfying
I E[b] = b for all b ∈ B andI E[b1ab2] = b1E[a]b2 for all a ∈ A, b1, b2 ∈ A.
Example
Let (A0, φ) be a C∗-probability space. Then
A := MN (C)⊗A0, B := MN (C) and E := idMN (C)⊗φ
gives an operator-valued C∗-probability space (A, E,B).
Tobias Mai (Saarland University) The self-adjoint linearization trick July 25, 2013 5 / 14
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A quick reminder on... ...operator-valued free probability theory
Operator-valued free probability theory II
De�nition
Let (A, E,B) be an operator-valued C∗-probability space. We call
H±(B) := {b ∈ B| ∃ε > 0 : ±=(b) ≥ ε1}
the upper and lower half-plane, respectively, where we use the notation
=(b) :=1
2i(b− b∗).
For x = x∗ ∈ A, we introduce
the Cauchy transform Gx : H+(B)→ H−(B), Gx(b) := E[(b−x)−1],
the F -transform Fx : H+(B)→ H+(B), Fx(b) := Gx(b)−1 and
the h-transform hx : H+(B)→ H+(B), hx(b) := Fx(b)− b.
Tobias Mai (Saarland University) The self-adjoint linearization trick July 25, 2013 6 / 14
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A quick reminder on... ...operator-valued free probability theory
Operator-valued free probability theory II
De�nition
Let (A, E,B) be an operator-valued C∗-probability space. We call
H±(B) := {b ∈ B| ∃ε > 0 : ±=(b) ≥ ε1}
the upper and lower half-plane, respectively, where we use the notation
=(b) :=1
2i(b− b∗).
For x = x∗ ∈ A, we introduce
the Cauchy transform Gx : H+(B)→ H−(B), Gx(b) := E[(b−x)−1],
the F -transform Fx : H+(B)→ H+(B), Fx(b) := Gx(b)−1 and
the h-transform hx : H+(B)→ H+(B), hx(b) := Fx(b)− b.
Tobias Mai (Saarland University) The self-adjoint linearization trick July 25, 2013 6 / 14
![Page 16: Operator-valued free probability theory and the self ... · Operator-valued free probability theory I De nition An operator-valued C-probability space (A;E;B) consists of a unital](https://reader036.fdocuments.net/reader036/viewer/2022071004/5fc1418c99a9c97ebb54a22d/html5/thumbnails/16.jpg)
A quick reminder on... ...operator-valued free probability theory
Operator-valued free probability theory II
De�nition
Let (A, E,B) be an operator-valued C∗-probability space. We call
H±(B) := {b ∈ B| ∃ε > 0 : ±=(b) ≥ ε1}
the upper and lower half-plane, respectively, where we use the notation
=(b) :=1
2i(b− b∗).
For x = x∗ ∈ A, we introduce
the Cauchy transform Gx : H+(B)→ H−(B), Gx(b) := E[(b−x)−1],
the F -transform Fx : H+(B)→ H+(B), Fx(b) := Gx(b)−1 and
the h-transform hx : H+(B)→ H+(B), hx(b) := Fx(b)− b.
Tobias Mai (Saarland University) The self-adjoint linearization trick July 25, 2013 6 / 14
![Page 17: Operator-valued free probability theory and the self ... · Operator-valued free probability theory I De nition An operator-valued C-probability space (A;E;B) consists of a unital](https://reader036.fdocuments.net/reader036/viewer/2022071004/5fc1418c99a9c97ebb54a22d/html5/thumbnails/17.jpg)
A quick reminder on... ...operator-valued free probability theory
Operator-valued free probability theory II
De�nition
Let (A, E,B) be an operator-valued C∗-probability space. We call
H±(B) := {b ∈ B| ∃ε > 0 : ±=(b) ≥ ε1}
the upper and lower half-plane, respectively, where we use the notation
=(b) :=1
2i(b− b∗).
For x = x∗ ∈ A, we introduce
the Cauchy transform Gx : H+(B)→ H−(B), Gx(b) := E[(b−x)−1],
the F -transform Fx : H+(B)→ H+(B), Fx(b) := Gx(b)−1
and
the h-transform hx : H+(B)→ H+(B), hx(b) := Fx(b)− b.
Tobias Mai (Saarland University) The self-adjoint linearization trick July 25, 2013 6 / 14
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A quick reminder on... ...operator-valued free probability theory
Operator-valued free probability theory II
De�nition
Let (A, E,B) be an operator-valued C∗-probability space. We call
H±(B) := {b ∈ B| ∃ε > 0 : ±=(b) ≥ ε1}
the upper and lower half-plane, respectively, where we use the notation
=(b) :=1
2i(b− b∗).
For x = x∗ ∈ A, we introduce
the Cauchy transform Gx : H+(B)→ H−(B), Gx(b) := E[(b−x)−1],
the F -transform Fx : H+(B)→ H+(B), Fx(b) := Gx(b)−1 and
the h-transform hx : H+(B)→ H+(B), hx(b) := Fx(b)− b.
Tobias Mai (Saarland University) The self-adjoint linearization trick July 25, 2013 6 / 14
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First application: Polynomials in free random variables
First application: Polynomials in free random variables
Let (A, φ) be a C∗-probability space and let x1, . . . , xn ∈ A be self-adjoint
and freely independent.
Question
Given a self-adjoint non-commutative polynomial p ∈ C〈X1, . . . , Xn〉.How can we calculate the distribution of p(x1, . . . , xn) out of the given
distributions of x1, . . . , xn?
Solution (Belinschi, M., Speicher, 2013)
Combine the linearization trick in its self-adjoint version by Anderson with
results about the operator-valued free additive convolution in the setting of
operator-valued C∗-probability spaces.
This gives an algorithmic solution for the question above, which is
moreover easily accessible for numerical computations!
Tobias Mai (Saarland University) The self-adjoint linearization trick July 25, 2013 7 / 14
![Page 20: Operator-valued free probability theory and the self ... · Operator-valued free probability theory I De nition An operator-valued C-probability space (A;E;B) consists of a unital](https://reader036.fdocuments.net/reader036/viewer/2022071004/5fc1418c99a9c97ebb54a22d/html5/thumbnails/20.jpg)
First application: Polynomials in free random variables
First application: Polynomials in free random variables
Let (A, φ) be a C∗-probability space and let x1, . . . , xn ∈ A be self-adjoint
and freely independent.
Question
Given a self-adjoint non-commutative polynomial p ∈ C〈X1, . . . , Xn〉.How can we calculate the distribution of p(x1, . . . , xn) out of the given
distributions of x1, . . . , xn?
Solution (Belinschi, M., Speicher, 2013)
Combine the linearization trick in its self-adjoint version by Anderson with
results about the operator-valued free additive convolution in the setting of
operator-valued C∗-probability spaces.
This gives an algorithmic solution for the question above, which is
moreover easily accessible for numerical computations!
Tobias Mai (Saarland University) The self-adjoint linearization trick July 25, 2013 7 / 14
![Page 21: Operator-valued free probability theory and the self ... · Operator-valued free probability theory I De nition An operator-valued C-probability space (A;E;B) consists of a unital](https://reader036.fdocuments.net/reader036/viewer/2022071004/5fc1418c99a9c97ebb54a22d/html5/thumbnails/21.jpg)
First application: Polynomials in free random variables
First application: Polynomials in free random variables
Let (A, φ) be a C∗-probability space and let x1, . . . , xn ∈ A be self-adjoint
and freely independent.
Question
Given a self-adjoint non-commutative polynomial p ∈ C〈X1, . . . , Xn〉.How can we calculate the distribution of p(x1, . . . , xn) out of the given
distributions of x1, . . . , xn?
Solution (Belinschi, M., Speicher, 2013)
Combine the linearization trick in its self-adjoint version by Anderson with
results about the operator-valued free additive convolution in the setting of
operator-valued C∗-probability spaces.
This gives an algorithmic solution for the question above, which is
moreover easily accessible for numerical computations!
Tobias Mai (Saarland University) The self-adjoint linearization trick July 25, 2013 7 / 14
![Page 22: Operator-valued free probability theory and the self ... · Operator-valued free probability theory I De nition An operator-valued C-probability space (A;E;B) consists of a unital](https://reader036.fdocuments.net/reader036/viewer/2022071004/5fc1418c99a9c97ebb54a22d/html5/thumbnails/22.jpg)
First application: Polynomials in free random variables
First application: Polynomials in free random variables
Let (A, φ) be a C∗-probability space and let x1, . . . , xn ∈ A be self-adjoint
and freely independent.
Question
Given a self-adjoint non-commutative polynomial p ∈ C〈X1, . . . , Xn〉.How can we calculate the distribution of p(x1, . . . , xn) out of the given
distributions of x1, . . . , xn?
Solution (Belinschi, M., Speicher, 2013)
Combine the linearization trick in its self-adjoint version by Anderson with
results about the operator-valued free additive convolution in the setting of
operator-valued C∗-probability spaces.
This gives an algorithmic solution for the question above, which is
moreover easily accessible for numerical computations!
Tobias Mai (Saarland University) The self-adjoint linearization trick July 25, 2013 7 / 14
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First application: Polynomials in free random variables
Linearization leads to an operator-valued problem
Let (A, φ) be a C∗-probability space.
Consider P := p(x1, . . . , xn), where x1, . . . , xn ∈ A are self-adjoint and
freely independent and p ∈ C〈X1, . . . , Xn〉 is a self-adjoint polynomial.
Anderson's linearization trick shows that there is an self-adjoint operator
LP := b0 ⊗ 1 + b1 ⊗ x1 + · · ·+ bn ⊗ xn ∈ MN (C)⊗A,
such that we have with respect to E = idMN (C)⊗φ for all z ∈ C+
GP (z) =[E[(Λ(z)− LP )−1
]]1,1
= limε↘0
[E[(Λε(z)− LP )−1
]]1,1,
where Λε(z) := diag(z, iε, . . . , iε) ∈ H+(MN (C)).
Observation
b1 ⊗ x1, . . . , bn ⊗ xn are free with amalgamation over MN (C).
Tobias Mai (Saarland University) The self-adjoint linearization trick July 25, 2013 8 / 14
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First application: Polynomials in free random variables
Linearization leads to an operator-valued problem
Let (A, φ) be a C∗-probability space.
Consider P := p(x1, . . . , xn), where x1, . . . , xn ∈ A are self-adjoint and
freely independent and p ∈ C〈X1, . . . , Xn〉 is a self-adjoint polynomial.
Anderson's linearization trick shows that there is an self-adjoint operator
LP := b0 ⊗ 1 + b1 ⊗ x1 + · · ·+ bn ⊗ xn ∈ MN (C)⊗A,
such that we have with respect to E = idMN (C)⊗φ for all z ∈ C+
GP (z) =[E[(Λ(z)− LP )−1
]]1,1
= limε↘0
[E[(Λε(z)− LP )−1
]]1,1,
where Λε(z) := diag(z, iε, . . . , iε) ∈ H+(MN (C)).
Observation
b1 ⊗ x1, . . . , bn ⊗ xn are free with amalgamation over MN (C).
Tobias Mai (Saarland University) The self-adjoint linearization trick July 25, 2013 8 / 14
![Page 25: Operator-valued free probability theory and the self ... · Operator-valued free probability theory I De nition An operator-valued C-probability space (A;E;B) consists of a unital](https://reader036.fdocuments.net/reader036/viewer/2022071004/5fc1418c99a9c97ebb54a22d/html5/thumbnails/25.jpg)
First application: Polynomials in free random variables
Linearization leads to an operator-valued problem
Let (A, φ) be a C∗-probability space.
Consider P := p(x1, . . . , xn), where x1, . . . , xn ∈ A are self-adjoint and
freely independent and p ∈ C〈X1, . . . , Xn〉 is a self-adjoint polynomial.
Anderson's linearization trick shows that there is an self-adjoint operator
LP := b0 ⊗ 1 + b1 ⊗ x1 + · · ·+ bn ⊗ xn ∈ MN (C)⊗A,
such that we have with respect to E = idMN (C)⊗φ for all z ∈ C+
GP (z) =[E[(Λ(z)− LP )−1
]]1,1
= limε↘0
[E[(Λε(z)− LP )−1
]]1,1,
where Λε(z) := diag(z, iε, . . . , iε) ∈ H+(MN (C)).
Observation
b1 ⊗ x1, . . . , bn ⊗ xn are free with amalgamation over MN (C).
Tobias Mai (Saarland University) The self-adjoint linearization trick July 25, 2013 8 / 14
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First application: Polynomials in free random variables
Linearization leads to an operator-valued problem
Let (A, φ) be a C∗-probability space.
Consider P := p(x1, . . . , xn), where x1, . . . , xn ∈ A are self-adjoint and
freely independent and p ∈ C〈X1, . . . , Xn〉 is a self-adjoint polynomial.
Anderson's linearization trick shows that there is an self-adjoint operator
LP := b0 ⊗ 1 + b1 ⊗ x1 + · · ·+ bn ⊗ xn ∈ MN (C)⊗A,
such that we have with respect to E = idMN (C)⊗φ for all z ∈ C+
GP (z) =[E[(Λ(z)− LP )−1
]]1,1
= limε↘0
[E[(Λε(z)− LP )−1
]]1,1,
where Λε(z) := diag(z, iε, . . . , iε) ∈ H+(MN (C)).
Observation
b1 ⊗ x1, . . . , bn ⊗ xn are free with amalgamation over MN (C).
Tobias Mai (Saarland University) The self-adjoint linearization trick July 25, 2013 8 / 14
![Page 27: Operator-valued free probability theory and the self ... · Operator-valued free probability theory I De nition An operator-valued C-probability space (A;E;B) consists of a unital](https://reader036.fdocuments.net/reader036/viewer/2022071004/5fc1418c99a9c97ebb54a22d/html5/thumbnails/27.jpg)
First application: Polynomials in free random variables
Theorem (Belinschi, M., Speicher, 2013)
Assume that (A, E,B) is an operator-valued C∗-probability space.
If x, y ∈ A are free with respect to E, then there exists a unique pair of
(Fréchet-)holomorphic maps
ω1, ω2 : H+(B)→ H+(B)
such that
Gx(ω1(b)) = Gy(ω2(b)) = Gx+y(b), b ∈ H+(B).
Moreover, ω1 and ω2 can easily be calculated via the following �xed point
iterations on H+(B):
w 7→ hy(b+ hx(w)) + b for ω1(b)
w 7→ hx(b+ hy(w)) + b for ω2(b)
Tobias Mai (Saarland University) The self-adjoint linearization trick July 25, 2013 9 / 14
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First application: Polynomials in free random variables
Theorem (Belinschi, M., Speicher, 2013)
Assume that (A, E,B) is an operator-valued C∗-probability space.
If x, y ∈ A are free with respect to E, then there exists a unique pair of
(Fréchet-)holomorphic maps
ω1, ω2 : H+(B)→ H+(B)
such that
Gx(ω1(b)) = Gy(ω2(b)) = Gx+y(b), b ∈ H+(B).
Moreover, ω1 and ω2 can easily be calculated via the following �xed point
iterations on H+(B):
w 7→ hy(b+ hx(w)) + b for ω1(b)
w 7→ hx(b+ hy(w)) + b for ω2(b)
Tobias Mai (Saarland University) The self-adjoint linearization trick July 25, 2013 9 / 14
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First application: Polynomials in free random variables
Appetizer (coming from free stochastic integrals)
Consider
p(X1, X2) =1
2(X3
1 +X1X2X1 +X2X1X2 +X32 )− (X1 +X2).
Let s1, s2 be two free (0, 1)-semicircular elements.
The density of the distribution of p(s1, s2) looks like:
Tobias Mai (Saarland University) The self-adjoint linearization trick July 25, 2013 10 / 14
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First application: Polynomials in free random variables
Appetizer (coming from free stochastic integrals)
Consider
p(X1, X2) =1
2(X3
1 +X1X2X1 +X2X1X2 +X32 )− (X1 +X2).
Let s1, s2 be two free (0, 1)-semicircular elements.
The density of the distribution of p(s1, s2) looks like:
Tobias Mai (Saarland University) The self-adjoint linearization trick July 25, 2013 10 / 14
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First application: Polynomials in free random variables
Appetizer (coming from free stochastic integrals)
Consider
p(X1, X2) =1
2(X3
1 +X1X2X1 +X2X1X2 +X32 )− (X1 +X2).
Let s1, s2 be two free (0, 1)-semicircular elements.
The density of the distribution of p(s1, s2) looks like:
Tobias Mai (Saarland University) The self-adjoint linearization trick July 25, 2013 10 / 14
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First application: Polynomials in free random variables
Appetizer (coming from free stochastic integrals)
Consider
p(X1, X2) =1
2(X3
1 +X1X2X1 +X2X1X2 +X32 )− (X1 +X2).
Let s1, s2 be two free (0, 1)-semicircular elements.
The density of the distribution of p(s1, s2) looks like:
Tobias Mai (Saarland University) The self-adjoint linearization trick July 25, 2013 10 / 14
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Second application: Multivariate free Berry-Esseen
Second application: Multivariate free Berry-Esseen
Let (A, φ) be a C∗-probability space with faithful state φ.
Let {xj,n| j = 1, . . . , d} for n ∈ N be sets of self-adjoint elements inA such that the following conditions are satis�ed:
I The (x1,n, . . . , xd,n)'s have the same distribution with respect to φand they satisfy φ(xj,n) = 0 for j = 1, . . . , d.
I The sets {xj,n| j = 1, . . . , d} are free with respect to φ.I sup
n∈Nmax
j=1,...,d‖xj,n‖ <∞.
We put σj,N :=1√N
N∑n=1
xj,n.
⇒ (σ1,N , . . . , σd,N )dist−−−−→
N→∞(s1, . . . , sd),
where (s1, . . . , sd) is a semicircular family.
⇒ p(σ1,N , . . . , σd,N )dist−−−−→
N→∞p(s1, . . . , sd),
for any (self-adjoint) polynomial p ∈ C〈X1, . . . , Xd〉.
Tobias Mai (Saarland University) The self-adjoint linearization trick July 25, 2013 11 / 14
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Second application: Multivariate free Berry-Esseen
Second application: Multivariate free Berry-Esseen
Let (A, φ) be a C∗-probability space with faithful state φ.
Let {xj,n| j = 1, . . . , d} for n ∈ N be sets of self-adjoint elements inA such that the following conditions are satis�ed:
I The (x1,n, . . . , xd,n)'s have the same distribution with respect to φand they satisfy φ(xj,n) = 0 for j = 1, . . . , d.
I The sets {xj,n| j = 1, . . . , d} are free with respect to φ.I sup
n∈Nmax
j=1,...,d‖xj,n‖ <∞.
We put σj,N :=1√N
N∑n=1
xj,n.
⇒ (σ1,N , . . . , σd,N )dist−−−−→
N→∞(s1, . . . , sd),
where (s1, . . . , sd) is a semicircular family.
⇒ p(σ1,N , . . . , σd,N )dist−−−−→
N→∞p(s1, . . . , sd),
for any (self-adjoint) polynomial p ∈ C〈X1, . . . , Xd〉.
Tobias Mai (Saarland University) The self-adjoint linearization trick July 25, 2013 11 / 14
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Second application: Multivariate free Berry-Esseen
Second application: Multivariate free Berry-Esseen
Let (A, φ) be a C∗-probability space with faithful state φ.
Let {xj,n| j = 1, . . . , d} for n ∈ N be sets of self-adjoint elements inA such that the following conditions are satis�ed:
I The (x1,n, . . . , xd,n)'s have the same distribution with respect to φand they satisfy φ(xj,n) = 0 for j = 1, . . . , d.
I The sets {xj,n| j = 1, . . . , d} are free with respect to φ.I sup
n∈Nmax
j=1,...,d‖xj,n‖ <∞.
We put σj,N :=1√N
N∑n=1
xj,n.
⇒ (σ1,N , . . . , σd,N )dist−−−−→
N→∞(s1, . . . , sd),
where (s1, . . . , sd) is a semicircular family.
⇒ p(σ1,N , . . . , σd,N )dist−−−−→
N→∞p(s1, . . . , sd),
for any (self-adjoint) polynomial p ∈ C〈X1, . . . , Xd〉.
Tobias Mai (Saarland University) The self-adjoint linearization trick July 25, 2013 11 / 14
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Second application: Multivariate free Berry-Esseen
Second application: Multivariate free Berry-Esseen
Let (A, φ) be a C∗-probability space with faithful state φ.
Let {xj,n| j = 1, . . . , d} for n ∈ N be sets of self-adjoint elements inA such that the following conditions are satis�ed:
I The (x1,n, . . . , xd,n)'s have the same distribution with respect to φand they satisfy φ(xj,n) = 0 for j = 1, . . . , d.
I The sets {xj,n| j = 1, . . . , d} are free with respect to φ.I sup
n∈Nmax
j=1,...,d‖xj,n‖ <∞.
We put σj,N :=1√N
N∑n=1
xj,n.
⇒ (σ1,N , . . . , σd,N )dist−−−−→
N→∞(s1, . . . , sd),
where (s1, . . . , sd) is a semicircular family.
⇒ p(σ1,N , . . . , σd,N )dist−−−−→
N→∞p(s1, . . . , sd),
for any (self-adjoint) polynomial p ∈ C〈X1, . . . , Xd〉.
Tobias Mai (Saarland University) The self-adjoint linearization trick July 25, 2013 11 / 14
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Second application: Multivariate free Berry-Esseen
Second application: Multivariate free Berry-Esseen
Let (A, φ) be a C∗-probability space with faithful state φ.
Let {xj,n| j = 1, . . . , d} for n ∈ N be sets of self-adjoint elements inA such that the following conditions are satis�ed:
I The (x1,n, . . . , xd,n)'s have the same distribution with respect to φand they satisfy φ(xj,n) = 0 for j = 1, . . . , d.
I The sets {xj,n| j = 1, . . . , d} are free with respect to φ.
I supn∈N
maxj=1,...,d
‖xj,n‖ <∞.
We put σj,N :=1√N
N∑n=1
xj,n.
⇒ (σ1,N , . . . , σd,N )dist−−−−→
N→∞(s1, . . . , sd),
where (s1, . . . , sd) is a semicircular family.
⇒ p(σ1,N , . . . , σd,N )dist−−−−→
N→∞p(s1, . . . , sd),
for any (self-adjoint) polynomial p ∈ C〈X1, . . . , Xd〉.
Tobias Mai (Saarland University) The self-adjoint linearization trick July 25, 2013 11 / 14
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Second application: Multivariate free Berry-Esseen
Second application: Multivariate free Berry-Esseen
Let (A, φ) be a C∗-probability space with faithful state φ.
Let {xj,n| j = 1, . . . , d} for n ∈ N be sets of self-adjoint elements inA such that the following conditions are satis�ed:
I The (x1,n, . . . , xd,n)'s have the same distribution with respect to φand they satisfy φ(xj,n) = 0 for j = 1, . . . , d.
I The sets {xj,n| j = 1, . . . , d} are free with respect to φ.I sup
n∈Nmax
j=1,...,d‖xj,n‖ <∞.
We put σj,N :=1√N
N∑n=1
xj,n.
⇒ (σ1,N , . . . , σd,N )dist−−−−→
N→∞(s1, . . . , sd),
where (s1, . . . , sd) is a semicircular family.
⇒ p(σ1,N , . . . , σd,N )dist−−−−→
N→∞p(s1, . . . , sd),
for any (self-adjoint) polynomial p ∈ C〈X1, . . . , Xd〉.
Tobias Mai (Saarland University) The self-adjoint linearization trick July 25, 2013 11 / 14
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Second application: Multivariate free Berry-Esseen
Second application: Multivariate free Berry-Esseen
Let (A, φ) be a C∗-probability space with faithful state φ.
Let {xj,n| j = 1, . . . , d} for n ∈ N be sets of self-adjoint elements inA such that the following conditions are satis�ed:
I The (x1,n, . . . , xd,n)'s have the same distribution with respect to φand they satisfy φ(xj,n) = 0 for j = 1, . . . , d.
I The sets {xj,n| j = 1, . . . , d} are free with respect to φ.I sup
n∈Nmax
j=1,...,d‖xj,n‖ <∞.
We put σj,N :=1√N
N∑n=1
xj,n.
⇒ (σ1,N , . . . , σd,N )dist−−−−→
N→∞(s1, . . . , sd),
where (s1, . . . , sd) is a semicircular family.
⇒ p(σ1,N , . . . , σd,N )dist−−−−→
N→∞p(s1, . . . , sd),
for any (self-adjoint) polynomial p ∈ C〈X1, . . . , Xd〉.
Tobias Mai (Saarland University) The self-adjoint linearization trick July 25, 2013 11 / 14
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Second application: Multivariate free Berry-Esseen
Second application: Multivariate free Berry-Esseen
Let (A, φ) be a C∗-probability space with faithful state φ.
Let {xj,n| j = 1, . . . , d} for n ∈ N be sets of self-adjoint elements inA such that the following conditions are satis�ed:
I The (x1,n, . . . , xd,n)'s have the same distribution with respect to φand they satisfy φ(xj,n) = 0 for j = 1, . . . , d.
I The sets {xj,n| j = 1, . . . , d} are free with respect to φ.I sup
n∈Nmax
j=1,...,d‖xj,n‖ <∞.
We put σj,N :=1√N
N∑n=1
xj,n.
⇒ (σ1,N , . . . , σd,N )dist−−−−→
N→∞(s1, . . . , sd),
where (s1, . . . , sd) is a semicircular family.
⇒ p(σ1,N , . . . , σd,N )dist−−−−→
N→∞p(s1, . . . , sd),
for any (self-adjoint) polynomial p ∈ C〈X1, . . . , Xd〉.
Tobias Mai (Saarland University) The self-adjoint linearization trick July 25, 2013 11 / 14
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Second application: Multivariate free Berry-Esseen
Second application: Multivariate free Berry-Esseen
Let (A, φ) be a C∗-probability space with faithful state φ.
Let {xj,n| j = 1, . . . , d} for n ∈ N be sets of self-adjoint elements inA such that the following conditions are satis�ed:
I The (x1,n, . . . , xd,n)'s have the same distribution with respect to φand they satisfy φ(xj,n) = 0 for j = 1, . . . , d.
I The sets {xj,n| j = 1, . . . , d} are free with respect to φ.I sup
n∈Nmax
j=1,...,d‖xj,n‖ <∞.
We put σj,N :=1√N
N∑n=1
xj,n.
⇒ (σ1,N , . . . , σd,N )dist−−−−→
N→∞(s1, . . . , sd),
where (s1, . . . , sd) is a semicircular family.
⇒ p(σ1,N , . . . , σd,N )dist−−−−→
N→∞p(s1, . . . , sd),
for any (self-adjoint) polynomial p ∈ C〈X1, . . . , Xd〉.
Tobias Mai (Saarland University) The self-adjoint linearization trick July 25, 2013 11 / 14
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Second application: Multivariate free Berry-Esseen
Linearization leads again to an operator-valued problem
Take a self-adjoint linearization Lp = b0 ⊗ 1 + b1 ⊗X1 + · · ·+ bd ⊗Xd of
the self-adjoint polynomial p ∈ C〈X1, . . . , Xd〉. We put
Xn := b1 ⊗ x1,n + · · ·+ bd ⊗ xd,n, n ∈ N.
Then (Xn)n∈N are self-adjoint elements in Mk(C)⊗A, which are
identically distributed and free with respect to E = idMk(C)⊗φ with
E[Xn] = 0 for all n ∈ N. Hence,
ΣN :=1√N
N∑n=1
Xn = b1 ⊗ σ1,N + · · ·+ bd ⊗ σd,N
converges as N →∞ in distribution (with respect to E) to the
operator-valued semicircular element
S := b1 ⊗ s1 + · · ·+ bd ⊗ sd,Note that we have
Lp(s1, . . . , sn) = b0⊗1+S and Lp(σ1,N , . . . , σd,N ) = b0⊗1+ΣN .
Tobias Mai (Saarland University) The self-adjoint linearization trick July 25, 2013 12 / 14
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Second application: Multivariate free Berry-Esseen
Linearization leads again to an operator-valued problemTake a self-adjoint linearization Lp = b0 ⊗ 1 + b1 ⊗X1 + · · ·+ bd ⊗Xd of
the self-adjoint polynomial p ∈ C〈X1, . . . , Xd〉.
We put
Xn := b1 ⊗ x1,n + · · ·+ bd ⊗ xd,n, n ∈ N.
Then (Xn)n∈N are self-adjoint elements in Mk(C)⊗A, which are
identically distributed and free with respect to E = idMk(C)⊗φ with
E[Xn] = 0 for all n ∈ N. Hence,
ΣN :=1√N
N∑n=1
Xn = b1 ⊗ σ1,N + · · ·+ bd ⊗ σd,N
converges as N →∞ in distribution (with respect to E) to the
operator-valued semicircular element
S := b1 ⊗ s1 + · · ·+ bd ⊗ sd,Note that we have
Lp(s1, . . . , sn) = b0⊗1+S and Lp(σ1,N , . . . , σd,N ) = b0⊗1+ΣN .
Tobias Mai (Saarland University) The self-adjoint linearization trick July 25, 2013 12 / 14
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Second application: Multivariate free Berry-Esseen
Linearization leads again to an operator-valued problemTake a self-adjoint linearization Lp = b0 ⊗ 1 + b1 ⊗X1 + · · ·+ bd ⊗Xd of
the self-adjoint polynomial p ∈ C〈X1, . . . , Xd〉. We put
Xn := b1 ⊗ x1,n + · · ·+ bd ⊗ xd,n, n ∈ N.
Then (Xn)n∈N are self-adjoint elements in Mk(C)⊗A, which are
identically distributed and free with respect to E = idMk(C)⊗φ with
E[Xn] = 0 for all n ∈ N. Hence,
ΣN :=1√N
N∑n=1
Xn = b1 ⊗ σ1,N + · · ·+ bd ⊗ σd,N
converges as N →∞ in distribution (with respect to E) to the
operator-valued semicircular element
S := b1 ⊗ s1 + · · ·+ bd ⊗ sd,Note that we have
Lp(s1, . . . , sn) = b0⊗1+S and Lp(σ1,N , . . . , σd,N ) = b0⊗1+ΣN .
Tobias Mai (Saarland University) The self-adjoint linearization trick July 25, 2013 12 / 14
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Second application: Multivariate free Berry-Esseen
Linearization leads again to an operator-valued problemTake a self-adjoint linearization Lp = b0 ⊗ 1 + b1 ⊗X1 + · · ·+ bd ⊗Xd of
the self-adjoint polynomial p ∈ C〈X1, . . . , Xd〉. We put
Xn := b1 ⊗ x1,n + · · ·+ bd ⊗ xd,n, n ∈ N.
Then (Xn)n∈N are self-adjoint elements in Mk(C)⊗A, which are
identically distributed and free with respect to E = idMk(C)⊗φ with
E[Xn] = 0 for all n ∈ N.
Hence,
ΣN :=1√N
N∑n=1
Xn = b1 ⊗ σ1,N + · · ·+ bd ⊗ σd,N
converges as N →∞ in distribution (with respect to E) to the
operator-valued semicircular element
S := b1 ⊗ s1 + · · ·+ bd ⊗ sd,Note that we have
Lp(s1, . . . , sn) = b0⊗1+S and Lp(σ1,N , . . . , σd,N ) = b0⊗1+ΣN .
Tobias Mai (Saarland University) The self-adjoint linearization trick July 25, 2013 12 / 14
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Second application: Multivariate free Berry-Esseen
Linearization leads again to an operator-valued problemTake a self-adjoint linearization Lp = b0 ⊗ 1 + b1 ⊗X1 + · · ·+ bd ⊗Xd of
the self-adjoint polynomial p ∈ C〈X1, . . . , Xd〉. We put
Xn := b1 ⊗ x1,n + · · ·+ bd ⊗ xd,n, n ∈ N.
Then (Xn)n∈N are self-adjoint elements in Mk(C)⊗A, which are
identically distributed and free with respect to E = idMk(C)⊗φ with
E[Xn] = 0 for all n ∈ N. Hence,
ΣN :=1√N
N∑n=1
Xn
= b1 ⊗ σ1,N + · · ·+ bd ⊗ σd,N
converges as N →∞ in distribution (with respect to E) to the
operator-valued semicircular element
S := b1 ⊗ s1 + · · ·+ bd ⊗ sd,
Note that we have
Lp(s1, . . . , sn) = b0⊗1+S and Lp(σ1,N , . . . , σd,N ) = b0⊗1+ΣN .
Tobias Mai (Saarland University) The self-adjoint linearization trick July 25, 2013 12 / 14
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Second application: Multivariate free Berry-Esseen
Linearization leads again to an operator-valued problemTake a self-adjoint linearization Lp = b0 ⊗ 1 + b1 ⊗X1 + · · ·+ bd ⊗Xd of
the self-adjoint polynomial p ∈ C〈X1, . . . , Xd〉. We put
Xn := b1 ⊗ x1,n + · · ·+ bd ⊗ xd,n, n ∈ N.
Then (Xn)n∈N are self-adjoint elements in Mk(C)⊗A, which are
identically distributed and free with respect to E = idMk(C)⊗φ with
E[Xn] = 0 for all n ∈ N. Hence,
ΣN :=1√N
N∑n=1
Xn = b1 ⊗ σ1,N + · · ·+ bd ⊗ σd,N
converges as N →∞ in distribution (with respect to E) to the
operator-valued semicircular element
S := b1 ⊗ s1 + · · ·+ bd ⊗ sd,
Note that we have
Lp(s1, . . . , sn) = b0⊗1+S and Lp(σ1,N , . . . , σd,N ) = b0⊗1+ΣN .
Tobias Mai (Saarland University) The self-adjoint linearization trick July 25, 2013 12 / 14
![Page 48: Operator-valued free probability theory and the self ... · Operator-valued free probability theory I De nition An operator-valued C-probability space (A;E;B) consists of a unital](https://reader036.fdocuments.net/reader036/viewer/2022071004/5fc1418c99a9c97ebb54a22d/html5/thumbnails/48.jpg)
Second application: Multivariate free Berry-Esseen
Linearization leads again to an operator-valued problemTake a self-adjoint linearization Lp = b0 ⊗ 1 + b1 ⊗X1 + · · ·+ bd ⊗Xd of
the self-adjoint polynomial p ∈ C〈X1, . . . , Xd〉. We put
Xn := b1 ⊗ x1,n + · · ·+ bd ⊗ xd,n, n ∈ N.
Then (Xn)n∈N are self-adjoint elements in Mk(C)⊗A, which are
identically distributed and free with respect to E = idMk(C)⊗φ with
E[Xn] = 0 for all n ∈ N. Hence,
ΣN :=1√N
N∑n=1
Xn = b1 ⊗ σ1,N + · · ·+ bd ⊗ σd,N
converges as N →∞ in distribution (with respect to E) to the
operator-valued semicircular element
S := b1 ⊗ s1 + · · ·+ bd ⊗ sd,Note that we have
Lp(s1, . . . , sn) = b0⊗1+S and Lp(σ1,N , . . . , σd,N ) = b0⊗1+ΣN .
Tobias Mai (Saarland University) The self-adjoint linearization trick July 25, 2013 12 / 14
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Second application: Multivariate free Berry-Esseen
Theorem (M., Speicher, 2013)
Let (A, E,B) be an operator-valued C∗-probability space with faithful
conditional expectation E and let (Xn)n∈N be a sequence of identically
distributed and self-adjoint elements in A, satisfying E[Xn] = 0, which are
free with respect to E. Then
ΣN :=1√N
N∑n=1
Xn
converges as N →∞ in distribution to an operator-valued semicircular
element S and we have for all b ∈ H+(B)
‖GΣN(b)−GS(b)‖ ≤ 2√
N‖=(b)−1‖4
√(2‖mXn
2 ‖2 + ‖mXn4 ‖)‖m
Xn2 ‖.
We use the notation mXn (b1, . . . , bn−1) := E[Xb1X . . .Xbn−1X] and
‖mXn ‖ := sup
‖b1‖≤1,...,‖bn−1‖≤1‖mX
n (b1, . . . , bn−1)‖ ≤ ‖X‖n.
Tobias Mai (Saarland University) The self-adjoint linearization trick July 25, 2013 13 / 14
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Second application: Multivariate free Berry-Esseen
Theorem (M., Speicher, 2013)
Let (A, E,B) be an operator-valued C∗-probability space with faithful
conditional expectation E and let (Xn)n∈N be a sequence of identically
distributed and self-adjoint elements in A, satisfying E[Xn] = 0, which are
free with respect to E. Then
ΣN :=1√N
N∑n=1
Xn
converges as N →∞ in distribution to an operator-valued semicircular
element S and we have for all b ∈ H+(B)
‖GΣN(b)−GS(b)‖ ≤ 2√
N‖=(b)−1‖4
√(2‖mXn
2 ‖2 + ‖mXn4 ‖)‖m
Xn2 ‖.
We use the notation mXn (b1, . . . , bn−1) := E[Xb1X . . .Xbn−1X] and
‖mXn ‖ := sup
‖b1‖≤1,...,‖bn−1‖≤1‖mX
n (b1, . . . , bn−1)‖ ≤ ‖X‖n.
Tobias Mai (Saarland University) The self-adjoint linearization trick July 25, 2013 13 / 14
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Second application: Multivariate free Berry-Esseen
Theorem (M., Speicher, 2013)
Let (A, E,B) be an operator-valued C∗-probability space with faithful
conditional expectation E and let (Xn)n∈N be a sequence of identically
distributed and self-adjoint elements in A, satisfying E[Xn] = 0, which are
free with respect to E. Then
ΣN :=1√N
N∑n=1
Xn
converges as N →∞ in distribution to an operator-valued semicircular
element S and we have for all b ∈ H+(B)
‖GΣN(b)−GS(b)‖ ≤ 2√
N‖=(b)−1‖4
√(2‖mXn
2 ‖2 + ‖mXn4 ‖)‖m
Xn2 ‖.
We use the notation mXn (b1, . . . , bn−1) := E[Xb1X . . .Xbn−1X] and
‖mXn ‖ := sup
‖b1‖≤1,...,‖bn−1‖≤1‖mX
n (b1, . . . , bn−1)‖ ≤ ‖X‖n.
Tobias Mai (Saarland University) The self-adjoint linearization trick July 25, 2013 13 / 14
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Second application: Multivariate free Berry-Esseen
Back to the multivariate case
Corollary
For each self-adjoint p ∈ C〈X1, . . . , Xd〉, there are M,C > 0 such that
|Gp(σ1,N ,...,σd,N )(z)−Gp(s1,...,sd)(z)| ≤ N−110
(M +
C
=(z)2
)holds for all z ∈ C+ and all N ∈ N with N > 1
=(z)10.
Remark
This is ongoing work: The order of convergence N−110 is (surely) not
optimal. There are many promising possibilities to improve the result.
The main statement here is: "Linearization is the right tool!"
Thank you!
Tobias Mai (Saarland University) The self-adjoint linearization trick July 25, 2013 14 / 14
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Second application: Multivariate free Berry-Esseen
Back to the multivariate case
Corollary
For each self-adjoint p ∈ C〈X1, . . . , Xd〉, there are M,C > 0 such that
|Gp(σ1,N ,...,σd,N )(z)−Gp(s1,...,sd)(z)| ≤ N−110
(M +
C
=(z)2
)holds for all z ∈ C+ and all N ∈ N with N > 1
=(z)10.
Remark
This is ongoing work: The order of convergence N−110 is (surely) not
optimal. There are many promising possibilities to improve the result.
The main statement here is: "Linearization is the right tool!"
Thank you!
Tobias Mai (Saarland University) The self-adjoint linearization trick July 25, 2013 14 / 14
![Page 54: Operator-valued free probability theory and the self ... · Operator-valued free probability theory I De nition An operator-valued C-probability space (A;E;B) consists of a unital](https://reader036.fdocuments.net/reader036/viewer/2022071004/5fc1418c99a9c97ebb54a22d/html5/thumbnails/54.jpg)
Second application: Multivariate free Berry-Esseen
Back to the multivariate case
Corollary
For each self-adjoint p ∈ C〈X1, . . . , Xd〉, there are M,C > 0 such that
|Gp(σ1,N ,...,σd,N )(z)−Gp(s1,...,sd)(z)| ≤ N−110
(M +
C
=(z)2
)holds for all z ∈ C+ and all N ∈ N with N > 1
=(z)10.
Remark
This is ongoing work: The order of convergence N−110 is (surely) not
optimal. There are many promising possibilities to improve the result.
The main statement here is: "Linearization is the right tool!"
Thank you!
Tobias Mai (Saarland University) The self-adjoint linearization trick July 25, 2013 14 / 14
![Page 55: Operator-valued free probability theory and the self ... · Operator-valued free probability theory I De nition An operator-valued C-probability space (A;E;B) consists of a unital](https://reader036.fdocuments.net/reader036/viewer/2022071004/5fc1418c99a9c97ebb54a22d/html5/thumbnails/55.jpg)
Second application: Multivariate free Berry-Esseen
Back to the multivariate case
Corollary
For each self-adjoint p ∈ C〈X1, . . . , Xd〉, there are M,C > 0 such that
|Gp(σ1,N ,...,σd,N )(z)−Gp(s1,...,sd)(z)| ≤ N−110
(M +
C
=(z)2
)holds for all z ∈ C+ and all N ∈ N with N > 1
=(z)10.
Remark
This is ongoing work: The order of convergence N−110 is (surely) not
optimal. There are many promising possibilities to improve the result.
The main statement here is: "Linearization is the right tool!"
Thank you!
Tobias Mai (Saarland University) The self-adjoint linearization trick July 25, 2013 14 / 14
![Page 56: Operator-valued free probability theory and the self ... · Operator-valued free probability theory I De nition An operator-valued C-probability space (A;E;B) consists of a unital](https://reader036.fdocuments.net/reader036/viewer/2022071004/5fc1418c99a9c97ebb54a22d/html5/thumbnails/56.jpg)
Second application: Multivariate free Berry-Esseen
Back to the multivariate case
Corollary
For each self-adjoint p ∈ C〈X1, . . . , Xd〉, there are M,C > 0 such that
|Gp(σ1,N ,...,σd,N )(z)−Gp(s1,...,sd)(z)| ≤ N−110
(M +
C
=(z)2
)holds for all z ∈ C+ and all N ∈ N with N > 1
=(z)10.
Remark
This is ongoing work: The order of convergence N−110 is (surely) not
optimal. There are many promising possibilities to improve the result.
The main statement here is: "Linearization is the right tool!"
Thank you!
Tobias Mai (Saarland University) The self-adjoint linearization trick July 25, 2013 14 / 14