Multifaced noncommutative stochastic independence Malte Gerhold - - PowerPoint PPT Presentation

Multifaced noncommutative stochastic independence

Malte Gerhold

University of Greifswald

22 May 2020 ACPMS Trondheim – Moeckow

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Motivation

Classical stochastic independence

X, Y P L8pΩq X, Y independent ð ñ E ` f pXqgpY q ˘ “ E ` f pXq ˘ E ` gpY q ˘ ð ñ PpX,Y q “ PX b PY

Malte Gerhold Multifaced noncommutative stochastic independence 2 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Motivation

Classical stochastic independence

X, Y P L8pΩq X, Y independent ð ñ E ` f pXqgpY q ˘ “ E ` f pXq ˘ E ` gpY q ˘ ð ñ PpX,Y q “ PX b PY

Non-commutative situations

ai hermitian random matrices, independent entries „ Np0, 1q trpa1a2a1a2q “?, tr ˆa1 ` ¨ ¨ ¨ ` ak ? k ˙n “? T1, T2 rooted trees, T1 ˛ T2: “glued together at root” xe0, pAT1˛T2qne0y “ xe0, pAT1 b p0 ` p0 b AT2qne0y “?

Malte Gerhold Multifaced noncommutative stochastic independence 2 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Motivation

Classical stochastic independence

X, Y P L8pΩq X, Y independent ð ñ E ` f pXqgpY q ˘ “ E ` f pXq ˘ E ` gpY q ˘ ð ñ PpX,Y q “ PX b PY

Non-commutative situations

ai hermitian random matrices, independent entries „ Np0, 1q trpa1a2a1a2q “?, tr ˆa1 ` ¨ ¨ ¨ ` ak ? k ˙n “? T1, T2 rooted trees, T1 Ź T2: “glue T2 to every vertex” xe0, pAT1ŹT2qne0y “ xe0, pAT1 b p0 ` id b AT2qne0y “?

Malte Gerhold Multifaced noncommutative stochastic independence 3 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Motivation

Scheme (works for tensor, Boolean, free, monotone)

Independence CLT vacuum-distr. of Fock space operators Important tool: Cumulants

Malte Gerhold Multifaced noncommutative stochastic independence 4 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Motivation

Scheme (works for tensor, Boolean, free, monotone)

Independence CLT vacuum-distr. of Fock space operators Important tool: Cumulants

Bi-freeness (Voiculescu 2014)

free Fock space left & right free creation/annihilation More examples followed and still do. Aim: Understanding of independences for pairs

Malte Gerhold Multifaced noncommutative stochastic independence 4 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Motivation

Scheme (works for tensor, Boolean, free, monotone)

Independence CLT vacuum-distr. of Fock space operators Important tool: Cumulants

Bi-freeness (Voiculescu 2014)

free Fock space left & right free creation/annihilation bi-freeness: independence for pairs of operators More examples followed and still do. Aim: Understanding of independences for pairs

Malte Gerhold Multifaced noncommutative stochastic independence 4 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Motivation

Scheme (works for tensor, Boolean, free, monotone)

Independence CLT vacuum-distr. of Fock space operators Important tool: Cumulants

Bi-freeness (Voiculescu 2014)

free Fock space left & right free creation/annihilation bi-freeness: independence for pairs of operators Scheme works! More examples followed and still do. Aim: Understanding of independences for pairs

Malte Gerhold Multifaced noncommutative stochastic independence 4 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Overview

1

Multifaced random variables

2

Moments and cumulants

3

Example: bi-monotone independence Universal product Partitions Moment cumulant formula Central limit theorem

4

A 4-faced independence

5

More examples

Malte Gerhold Multifaced noncommutative stochastic independence 5 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Overview

1

Multifaced random variables

2

Moments and cumulants

3

Example: bi-monotone independence Universal product Partitions Moment cumulant formula Central limit theorem

4

A 4-faced independence

5

More examples

Malte Gerhold Multifaced noncommutative stochastic independence 6 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Non-commutative probability: ˚-algebraic setting

Definition (non-commutative probability space)

pair pA, Φq with unital ˚-algebra A state Φ on A

Definition (random variable)

˚-homomorphism j : B Ñ A (B is ˚-algebra) r B unitization of B, r j unital extension of j Φ ˝ r j is called distribution of j selfadjoint a P A ja : Crxs0 Ñ A, x ÞÑ a distribution of ja ú collection of moments ` Φpakq ˘

kPN

˚-subalgebra B embedding ι: B ã Ñ A

Malte Gerhold Multifaced noncommutative stochastic independence 7 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Non-commutative probability: ˚-algebraic setting

Definition (non-commutative probability space)

pair pA, Φq with unital ˚-algebra A state Φ on A

Definition (random variable)

˚-homomorphism j : B Ñ A (B is ˚-algebra) r B unitization of B, r j unital extension of j Φ ˝ r j is called distribution of j selfadjoint a P A ja : Crxs0 Ñ A, x ÞÑ a distribution of ja ú collection of moments ` Φpakq ˘

kPN

˚-subalgebra B embedding ι: B ã Ñ A

Malte Gerhold Multifaced noncommutative stochastic independence 7 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Non-commutative probability: ˚-algebraic setting

Definition (non-commutative probability space)

pair pA, Φq with unital ˚-algebra A state Φ on A

Definition (random variable)

˚-homomorphism j : B Ñ A (B is ˚-algebra) r B unitization of B, r j unital extension of j Φ ˝ r j is called distribution of j selfadjoint a P A ja : Crxs0 Ñ A, x ÞÑ a distribution of ja ú collection of moments ` Φpakq ˘

kPN

˚-subalgebra B embedding ι: B ã Ñ A

Malte Gerhold Multifaced noncommutative stochastic independence 7 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Non-commutative probability: ˚-algebraic setting

Definition (non-commutative probability space)

pair pA, Φq with unital ˚-algebra A state Φ on A

Definition (random variable)

˚-homomorphism j : B Ñ A (B is ˚-algebra) r B unitization of B, r j unital extension of j Φ ˝ r j is called distribution of j selfadjoint a P A ja : Crxs0 Ñ A, x ÞÑ a distribution of ja ú collection of moments ` Φpakq ˘

kPN

˚-subalgebra B embedding ι: B ã Ñ A

Malte Gerhold Multifaced noncommutative stochastic independence 7 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Non-commutative probability: ˚-algebraic setting

Definition (non-commutative probability space)

pair pA, Φq with unital ˚-algebra A state Φ on A

Definition (random variable)

˚-homomorphism j : B Ñ A (B is ˚-algebra) r B unitization of B, r j unital extension of j Φ ˝ r j is called distribution of j selfadjoint a P A ja : Crxs0 Ñ A, x ÞÑ a distribution of ja ú collection of moments ` Φpakq ˘

kPN

˚-subalgebra B embedding ι: B ã Ñ A

Malte Gerhold Multifaced noncommutative stochastic independence 7 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Non-commutative probability: ˚-algebraic setting

Definition (non-commutative probability space)

pair pA, Φq with unital ˚-algebra A state Φ on A

Definition (random variable)

˚-homomorphism j : B Ñ A (B is ˚-algebra) r B unitization of B, r j unital extension of j Φ ˝ r j is called distribution of j selfadjoint a P A ja : Crxs0 Ñ A, x ÞÑ a distribution of ja ú collection of moments ` Φpakq ˘

kPN

˚-subalgebra B embedding ι: B ã Ñ A

Malte Gerhold Multifaced noncommutative stochastic independence 7 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Non-commutative probability: ˚-algebraic setting

Definition (non-commutative probability space)

pair pA, Φq with unital ˚-algebra A state Φ on A

Definition (random variable)

˚-homomorphism j : B Ñ A (B is ˚-algebra) r B unitization of B, r j unital extension of j Φ ˝ r j is called distribution of j selfadjoint a P A ja : Crxs0 Ñ A, x ÞÑ a distribution of ja ú collection of moments ` Φpakq ˘

kPN

˚-subalgebra B embedding ι: B ã Ñ A

Malte Gerhold Multifaced noncommutative stochastic independence 7 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Augmented algebras and unitization

Already for Boolean, monotone and anti-monotone product: Take care with units!

Definition/Notation (augmented algebras)

unital algebra with character (non-zero homomorphism to C) every augmented algebra is the unitization of its augmentation ideal denote the augmentation ideal simply by B, the augmented algebra as r B “ C1 ‘ B.

Malte Gerhold Multifaced noncommutative stochastic independence 8 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Augmented algebras and unitization

Already for Boolean, monotone and anti-monotone product: Take care with units!

Definition/Notation (augmented algebras)

unital algebra with character (non-zero homomorphism to C) every augmented algebra is the unitization of its augmentation ideal denote the augmentation ideal simply by B, the augmented algebra as r B “ C1 ‘ B.

Malte Gerhold Multifaced noncommutative stochastic independence 8 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Augmented algebras and unitization

Already for Boolean, monotone and anti-monotone product: Take care with units!

Definition/Notation (augmented algebras)

unital algebra with character (non-zero homomorphism to C) every augmented algebra is the unitization of its augmentation ideal denote the augmentation ideal simply by B, the augmented algebra as r B “ C1 ‘ B.

Malte Gerhold Multifaced noncommutative stochastic independence 8 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Augmented algebras and unitization

Already for Boolean, monotone and anti-monotone product: Take care with units!

Definition/Notation (augmented algebras)

unital algebra with character (non-zero homomorphism to C) every augmented algebra is the unitization of its augmentation ideal denote the augmentation ideal simply by B, the augmented algebra as r B “ C1 ‘ B.

Malte Gerhold Multifaced noncommutative stochastic independence 8 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Non-commutative independence

Fix product operation for states on unital (augmented) ˚-algebras ą

i

r B1

i Q pϕiqi ÞÑ

ä

i

ϕi P ´ Ć ğ

i

Bi ¯1

Definition (d-independence of random variables ji : Bi Ñ A)

Φ ˝ Ą ğ

i

ji “ ä

i

pΦ ˝ r jiq joint distribution “ product of marginals

Examples

tensor, free, monotone, anti-monotone, Boolean

Malte Gerhold Multifaced noncommutative stochastic independence 9 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Non-commutative independence

Fix product operation for states on unital (augmented) ˚-algebras ą

i

r B1

i Q pϕiqi ÞÑ

ä

i

ϕi P ´ Ć ğ

i

Bi ¯1

Definition (d-independence of random variables ji : Bi Ñ A)

Φ ˝ Ą ğ

i

ji “ ä

i

pΦ ˝ r jiq joint distribution “ product of marginals

Examples

tensor, free, monotone, anti-monotone, Boolean

Malte Gerhold Multifaced noncommutative stochastic independence 9 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Non-commutative independence

Fix product operation for states on unital (augmented) ˚-algebras ą

i

r B1

i Q pϕiqi ÞÑ

ä

i

ϕi P ´ Ć ğ

i

Bi ¯1

Definition (d-independence of random variables ji : Bi Ñ A)

Φ ˝ Ą ğ

i

ji “ ä

i

pΦ ˝ r jiq joint distribution “ product of marginals

Examples

tensor, free, monotone, anti-monotone, Boolean

Malte Gerhold Multifaced noncommutative stochastic independence 9 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Non-commutative independence

Fix product operation for states on unital (augmented) ˚-algebras ą

i

r B1

i Q pϕiqi ÞÑ

ä

i

ϕi P ´ Ć ğ

i

Bi ¯1

Definition (d-independence of random variables ji : Bi Ñ A)

Φ ˝ Ą ğ

i

ji “ ä

i

pΦ ˝ r jiq joint distribution “ product of marginals

Theorem (Muraki)

Tensor, free, monotone, anti-monotone, Boolean

Malte Gerhold Multifaced noncommutative stochastic independence 10 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Non-commutative probability: ˚-algebraic setting

Definition (non-commutative probability space)

pair pA, Φq with unital ˚-algebra A state Φ on A

Definition (m-faced random variable)

˚-homomorphism j : B Ñ A (B “ Bp1q \ ¨ ¨ ¨ \ Bpmq is m-faced ˚-algebra) m-tuple of selfadjoint elements a “ pap1q, . . . , apmqq P Am ja : Cxxp1q, . . . , xpmqy0 Ñ A, xpkq ÞÑ apkq aδ :“ apδ1q ¨ ¨ ¨ apδmq, δ P rms˚ distribution of ja ú collection of moments pΦpaδqqδPrms˚ m-tuple of ˚-subalgebras pB1, . . . , Bmq ι1 \ ¨ ¨ ¨ \ ιm

Malte Gerhold Multifaced noncommutative stochastic independence 11 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Non-commutative probability: ˚-algebraic setting

Definition (non-commutative probability space)

pair pA, Φq with unital ˚-algebra A state Φ on A

Definition (m-faced random variable)

˚-homomorphism j : B Ñ A (B “ Bp1q \ ¨ ¨ ¨ \ Bpmq is m-faced ˚-algebra) m-tuple of selfadjoint elements a “ pap1q, . . . , apmqq P Am ja : Cxxp1q, . . . , xpmqy0 Ñ A, xpkq ÞÑ apkq aδ :“ apδ1q ¨ ¨ ¨ apδmq, δ P rms˚ distribution of ja ú collection of moments pΦpaδqqδPrms˚ m-tuple of ˚-subalgebras pB1, . . . , Bmq ι1 \ ¨ ¨ ¨ \ ιm

Malte Gerhold Multifaced noncommutative stochastic independence 11 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Non-commutative probability: ˚-algebraic setting

Definition (non-commutative probability space)

pair pA, Φq with unital ˚-algebra A state Φ on A

Definition (m-faced random variable)

˚-homomorphism j : B Ñ A (B “ Bp1q \ ¨ ¨ ¨ \ Bpmq is m-faced ˚-algebra) m-tuple of selfadjoint elements a “ pap1q, . . . , apmqq P Am ja : Cxxp1q, . . . , xpmqy0 Ñ A, xpkq ÞÑ apkq aδ :“ apδ1q ¨ ¨ ¨ apδmq, δ P rms˚ distribution of ja ú collection of moments pΦpaδqqδPrms˚ m-tuple of ˚-subalgebras pB1, . . . , Bmq ι1 \ ¨ ¨ ¨ \ ιm

Malte Gerhold Multifaced noncommutative stochastic independence 11 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Non-commutative probability: ˚-algebraic setting

Definition (non-commutative probability space)

pair pA, Φq with unital ˚-algebra A state Φ on A

Definition (m-faced random variable)

˚-homomorphism j : B Ñ A (B “ Bp1q \ ¨ ¨ ¨ \ Bpmq is m-faced ˚-algebra) m-tuple of selfadjoint elements a “ pap1q, . . . , apmqq P Am ja : Cxxp1q, . . . , xpmqy0 Ñ A, xpkq ÞÑ apkq aδ :“ apδ1q ¨ ¨ ¨ apδmq, δ P rms˚ distribution of ja ú collection of moments pΦpaδqqδPrms˚ m-tuple of ˚-subalgebras pB1, . . . , Bmq ι1 \ ¨ ¨ ¨ \ ιm

Malte Gerhold Multifaced noncommutative stochastic independence 11 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Non-commutative probability: ˚-algebraic setting

Definition (non-commutative probability space)

pair pA, Φq with unital ˚-algebra A state Φ on A

Definition (m-faced random variable)

˚-homomorphism j : B Ñ A (B “ Bp1q \ ¨ ¨ ¨ \ Bpmq is m-faced ˚-algebra) m-tuple of selfadjoint elements a “ pap1q, . . . , apmqq P Am ja : Cxxp1q, . . . , xpmqy0 Ñ A, xpkq ÞÑ apkq aδ :“ apδ1q ¨ ¨ ¨ apδmq, δ P rms˚ distribution of ja ú collection of moments pΦpaδqqδPrms˚ m-tuple of ˚-subalgebras pB1, . . . , Bmq ι1 \ ¨ ¨ ¨ \ ιm

Malte Gerhold Multifaced noncommutative stochastic independence 11 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

m-Independence

Fix product operation for states on unital (augmented) m-faced ˚-algebras ą

i

r B1

i Q pϕiqi ÞÑ

ä

i

ϕi P ´ Ć ğ

i

Bi ¯1

Definition (d-independence of m-faced rv’s ji : Bi Ñ A)

Φ ˝ Ą ğ

i

ji “ ä

i

pΦ ˝ r jiq joint distribution “ product of marginals

Examples

bi-freeness, ??

Malte Gerhold Multifaced noncommutative stochastic independence 12 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

m-Independence

Fix product operation for states on unital (augmented) m-faced ˚-algebras ą

i

r B1

i Q pϕiqi ÞÑ

ä

i

ϕi P ´ Ć ğ

i

Bi ¯1

Definition (d-independence of m-faced rv’s ji : Bi Ñ A)

Φ ˝ Ą ğ

i

ji “ ä

i

pΦ ˝ r jiq joint distribution “ product of marginals

Examples

bi-freeness, ??

Malte Gerhold Multifaced noncommutative stochastic independence 12 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

m-Independence

Fix product operation for states on unital (augmented) m-faced ˚-algebras ą

i

r B1

i Q pϕiqi ÞÑ

ä

i

ϕi P ´ Ć ğ

i

Bi ¯1

Definition (d-independence of m-faced rv’s ji : Bi Ñ A)

Φ ˝ Ą ğ

i

ji “ ä

i

pΦ ˝ r jiq joint distribution “ product of marginals

Examples

bi-freeness, ??

Malte Gerhold Multifaced noncommutative stochastic independence 12 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

m-Independence

Fix product operation for states on unital (augmented) m-faced ˚-algebras ą

i

r B1

i Q pϕiqi ÞÑ

ä

i

ϕi P ´ Ć ğ

i

Bi ¯1

Definition (d-independence of m-faced rv’s ji : Bi Ñ A)

Φ ˝ Ą ğ

i

ji “ ä

i

pΦ ˝ r jiq joint distribution “ product of marginals

Theorem

???

Malte Gerhold Multifaced noncommutative stochastic independence 13 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Overview

1

Multifaced random variables

2

Moments and cumulants

3

Example: bi-monotone independence Universal product Partitions Moment cumulant formula Central limit theorem

4

A 4-faced independence

5

More examples

Malte Gerhold Multifaced noncommutative stochastic independence 14 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

uau-products

Definition (m-1-uau-product, cf. Manzel & Sch¨ urmann 2017)

B1

1 ˆ B1 2 Q pϕ1, ϕ2q ÞÑ ϕ1 d ϕ2 P pB1 \ B2q1

product operation (for arbitrary m-faced algebras B1, B2) which is unital in the sense that 1 d ϕ “ ϕ “ ϕ d 1 associative universal in the sense that (for ˚-hom’s ji : Bi Ñ Ai) pϕ1 d ϕ2q ˝ Č pj1 \ j2q “ pϕ1 ˝ r j1q d pϕ2 ˝ r j2q d positive if ϕ1, ϕ2 states ù ñ ϕ1 d ϕ2 state Note: Unital linear functionals on augmented m-faced algebras can be identified with linear functionals on m-faced algebras.

Malte Gerhold Multifaced noncommutative stochastic independence 15 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

uau-products

Definition (m-1-uau-product, cf. Manzel & Sch¨ urmann 2017)

B1

1 ˆ B1 2 Q pϕ1, ϕ2q ÞÑ ϕ1 d ϕ2 P pB1 \ B2q1

product operation (for arbitrary m-faced algebras B1, B2) which is unital in the sense that 1 d ϕ “ ϕ “ ϕ d 1 associative universal in the sense that (for ˚-hom’s ji : Bi Ñ Ai) pϕ1 d ϕ2q ˝ Č pj1 \ j2q “ pϕ1 ˝ r j1q d pϕ2 ˝ r j2q d positive if ϕ1, ϕ2 states ù ñ ϕ1 d ϕ2 state Note: Unital linear functionals on augmented m-faced algebras can be identified with linear functionals on m-faced algebras.

Malte Gerhold Multifaced noncommutative stochastic independence 15 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

uau-products

Definition (m-1-uau-product, cf. Manzel & Sch¨ urmann 2017)

B1

1 ˆ B1 2 Q pϕ1, ϕ2q ÞÑ ϕ1 d ϕ2 P pB1 \ B2q1

product operation (for arbitrary m-faced algebras B1, B2) which is unital in the sense that 1 d ϕ “ ϕ “ ϕ d 1 associative universal in the sense that (for ˚-hom’s ji : Bi Ñ Ai) pϕ1 d ϕ2q ˝ Č pj1 \ j2q “ pϕ1 ˝ r j1q d pϕ2 ˝ r j2q d positive if ϕ1, ϕ2 states ù ñ ϕ1 d ϕ2 state Note: Unital linear functionals on augmented m-faced algebras can be identified with linear functionals on m-faced algebras.

Malte Gerhold Multifaced noncommutative stochastic independence 15 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Convolution exponentials

Consider A “ Cxxp1q, . . . , xpmqy

Definition

convolution for ϕ1, ϕ2 P A1, ϕ1p1q “ ϕ2p1q “ 1 ϕ1 ˚ ϕ2pxδq :“ ϕ1 d ϕ2ppx1 ` x2qδq linearised convolution for ψ1, . . . , ψn P A1, ψkp1q “ 0 ψ1 ‘ ¨ ¨ ¨ ‘ ψn :“ Bn Bt1 ¨ ¨ ¨ Btn pt1ψ1q ˚ ¨ ¨ ¨ ˚ ptnψnq ˇ ˇ ˇ ˇ convolution exponential of ψ P A1 with ψp1q “ 0 expd ψ :“ ε `

8

ÿ

k“1

1 k!ψ‘k

Malte Gerhold Multifaced noncommutative stochastic independence 16 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Convolution exponentials

Consider A “ Cxxp1q, . . . , xpmqy

Definition

convolution for ϕ1, ϕ2 P A1, ϕ1p1q “ ϕ2p1q “ 1 ϕ1 ˚ ϕ2pxδq :“ ϕ1 d ϕ2ppx1 ` x2qδq linearised convolution for ψ1, . . . , ψn P A1, ψkp1q “ 0 ψ1 ‘ ¨ ¨ ¨ ‘ ψn :“ Bn Bt1 ¨ ¨ ¨ Btn pt1ψ1q ˚ ¨ ¨ ¨ ˚ ptnψnq ˇ ˇ ˇ ˇ convolution exponential of ψ P A1 with ψp1q “ 0 expd ψ :“ ε `

8

ÿ

k“1

1 k!ψ‘k

Malte Gerhold Multifaced noncommutative stochastic independence 16 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Convolution exponentials

Consider A “ Cxxp1q, . . . , xpmqy

Definition

convolution for ϕ1, ϕ2 P A1, ϕ1p1q “ ϕ2p1q “ 1 ϕ1 ˚ ϕ2pxδq :“ ϕ1 d ϕ2ppx1 ` x2qδq linearised convolution for ψ1, . . . , ψn P A1, ψkp1q “ 0 ψ1 ‘ ¨ ¨ ¨ ‘ ψn :“ Bn Bt1 ¨ ¨ ¨ Btn pt1ψ1q ˚ ¨ ¨ ¨ ˚ ptnψnq ˇ ˇ ˇ ˇ convolution exponential of ψ P A1 with ψp1q “ 0 expd ψ :“ ε `

8

ÿ

k“1

1 k!ψ‘k

Malte Gerhold Multifaced noncommutative stochastic independence 16 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Cumulants I

Theorem (cf. Manzel & Sch¨ urmann 2017, Lachs 2015, Ben Ghorbal & Sch¨ urmann 2005)

If d is a uau-product, then expd : tψ P A1 | ψp1q “ 0u Ñ tϕ P A1 | ϕp1q “ 1u is a well-defined bijection.

Idea of proof.

From ϕ “ expdpψq we get the recursive formula ψpxδq “ ϕpxδq ´

n

ÿ

k“2

1 k!ψ‘kpxδq with ψ‘kpxδq determined by ψætp P A | degppq ă |δ|u.

Malte Gerhold Multifaced noncommutative stochastic independence 17 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Cumulants I

Theorem (cf. Manzel & Sch¨ urmann 2017, Lachs 2015, Ben Ghorbal & Sch¨ urmann 2005)

If d is a uau-product, then expd : tψ P A1 | ψp1q “ 0u Ñ tϕ P A1 | ϕp1q “ 1u is a well-defined bijection.

Idea of proof.

From ϕ “ expdpψq we get the recursive formula ψpxδq “ ϕpxδq ´

n

ÿ

k“2

1 k!ψ‘kpxδq with ψ‘kpxδq determined by ψætp P A | degppq ă |δ|u.

Malte Gerhold Multifaced noncommutative stochastic independence 17 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Cumulants II

Definition

For pA, Φq be a ncps and a P Am put ϕa :“ Φ ˝ r ja P Cxxp1q, . . . , xpmqy1 ψa :“ logd ϕa, i.e. ψa is determined by expd ψa “ ϕa Then ψapxδq, δ P rms˚ are called d-cumulants of a.

Remarks

formula for expd is also called moment-cumulant-formula even if convolution is not commutative, we have logdpϕ1 ˚ ϕ2q “ CBHplogd ϕ1, logd ϕ2q, where rψ1, ψ2s :“ ψ1 ‘ ψ2 ´ ψ2 ‘ ψ1

Malte Gerhold Multifaced noncommutative stochastic independence 18 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Cumulants II

Definition

For pA, Φq be a ncps and a P Am put ϕa :“ Φ ˝ r ja P Cxxp1q, . . . , xpmqy1 ψa :“ logd ϕa, i.e. ψa is determined by expd ψa “ ϕa Then ψapxδq, δ P rms˚ are called d-cumulants of a.

Remarks

formula for expd is also called moment-cumulant-formula even if convolution is not commutative, we have logdpϕ1 ˚ ϕ2q “ CBHplogd ϕ1, logd ϕ2q, where rψ1, ψ2s :“ ψ1 ‘ ψ2 ´ ψ2 ‘ ψ1

Malte Gerhold Multifaced noncommutative stochastic independence 18 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Cumulants II

Definition

For pA, Φq be a ncps and a P Am put ϕa :“ Φ ˝ r ja P Cxxp1q, . . . , xpmqy1 ψa :“ logd ϕa, i.e. ψa is determined by expd ψa “ ϕa Then ψapxδq, δ P rms˚ are called d-cumulants of a.

Remarks

formula for expd is also called moment-cumulant-formula even if convolution is not commutative, we have logdpϕ1 ˚ ϕ2q “ CBHplogd ϕ1, logd ϕ2q, where rψ1, ψ2s :“ ψ1 ‘ ψ2 ´ ψ2 ‘ ψ1

Malte Gerhold Multifaced noncommutative stochastic independence 18 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Cumulants II

Definition

For pA, Φq be a ncps and a P Am put ϕa :“ Φ ˝ r ja P Cxxp1q, . . . , xpmqy1 ψa :“ logd ϕa, i.e. ψa is determined by expd ψa “ ϕa Then ψapxδq, δ P rms˚ are called d-cumulants of a.

Remarks

formula for expd is also called moment-cumulant-formula even if convolution is not commutative, we have logdpϕ1 ˚ ϕ2q “ CBHplogd ϕ1, logd ϕ2q, where rψ1, ψ2s :“ ψ1 ‘ ψ2 ´ ψ2 ‘ ψ1

Malte Gerhold Multifaced noncommutative stochastic independence 18 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Cumulants II

Definition

For pA, Φq be a ncps and a P Am put ϕa :“ Φ ˝ r ja P Cxxp1q, . . . , xpmqy1 ψa :“ logd ϕa, i.e. ψa is determined by expd ψa “ ϕa Then ψapxδq, δ P rms˚ are called d-cumulants of a.

Remarks

formula for expd is also called moment-cumulant-formula even if convolution is not commutative, we have logdpϕ1 ˚ ϕ2q “ CBHplogd ϕ1, logd ϕ2q, where rψ1, ψ2s :“ ψ1 ‘ ψ2 ´ ψ2 ‘ ψ1

Malte Gerhold Multifaced noncommutative stochastic independence 18 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Overview

1

Multifaced random variables

2

Moments and cumulants

3

Example: bi-monotone independence Universal product Partitions Moment cumulant formula Central limit theorem

4

A 4-faced independence

5

More examples

Malte Gerhold Multifaced noncommutative stochastic independence 19 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Bi-monotone product I

Definition (bi-monotone product of pointed rep’s)

For pπi : Bi Ñ LadjpHiq, Ωiq pointed rep’s on pre-HS define pπ1 ’ π2 : B1 \ B2 Ñ LadjpH1 b H2q, Ω1 b Ω2q π1 ’ π2pbq :“ # π1 Ÿ π2pbq b P Bℓ π1 Ź π2pbq b P Br “ $ ’ ’ ’ ’ & ’ ’ ’ ’ % π1pbq b id b P Bℓ

1

PΩ b π2pbq b P Bℓ

2

π1pbq b PΩ b P Br

1

id b π2pbq b P Br

2

Definition (bi-monotone product of states on r Bi)

ϕ1 ’ ϕ2pbq :“ xΩ, πϕ1 ’ πϕ2pbqΩy for all b P B1 \ B2 whenever ϕi “ xΩi, πip¨qΩiy.

Malte Gerhold Multifaced noncommutative stochastic independence 20 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Bi-monotone product I

Definition (bi-monotone product of pointed rep’s)

For pπi : Bi Ñ LadjpHiq, Ωiq pointed rep’s on pre-HS define pπ1 ’ π2 : B1 \ B2 Ñ LadjpH1 b H2q, Ω1 b Ω2q π1 ’ π2pbq :“ # π1 Ÿ π2pbq b P Bℓ π1 Ź π2pbq b P Br “ $ ’ ’ ’ ’ & ’ ’ ’ ’ % π1pbq b id b P Bℓ

1

PΩ b π2pbq b P Bℓ

2

π1pbq b PΩ b P Br

1

id b π2pbq b P Br

2

Definition (bi-monotone product of states on r Bi)

ϕ1 ’ ϕ2pbq :“ xΩ, πϕ1 ’ πϕ2pbqΩy for all b P B1 \ B2 whenever ϕi “ xΩi, πip¨qΩiy.

Malte Gerhold Multifaced noncommutative stochastic independence 20 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Bi-monotone product II

Theorem

The bi-monotone product of states (linear functionals) is unital in the sense that: 1 ’ ϕ “ ϕ “ ϕ ’ 1 associative universal in the sense that (for ˚-hom’s ji : Bi Ñ Ai) pϕ1 ’ ϕ2q ˝ Č pj1 \ j2q “ pϕ1 ˝ r j1q ’ pϕ2 ˝ r j2q In short: ’ is a positive 2-1-uau-product in the sense of Manzel & Sch¨ urmann (2017)

Malte Gerhold Multifaced noncommutative stochastic independence 21 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Bi-monotone product II

Theorem

The bi-monotone product of states (linear functionals) is unital in the sense that: 1 ’ ϕ “ ϕ “ ϕ ’ 1 associative universal in the sense that (for ˚-hom’s ji : Bi Ñ Ai) pϕ1 ’ ϕ2q ˝ Č pj1 \ j2q “ pϕ1 ˝ r j1q ’ pϕ2 ˝ r j2q In short: ’ is a positive 2-1-uau-product in the sense of Manzel & Sch¨ urmann (2017)

Malte Gerhold Multifaced noncommutative stochastic independence 21 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Bi-monotone product II

Theorem

The bi-monotone product of states (linear functionals) is unital in the sense that: 1 ’ ϕ “ ϕ “ ϕ ’ 1 associative universal in the sense that (for ˚-hom’s ji : Bi Ñ Ai) pϕ1 ’ ϕ2q ˝ Č pj1 \ j2q “ pϕ1 ˝ r j1q ’ pϕ2 ˝ r j2q In short: ’ is a positive 2-1-uau-product in the sense of Manzel & Sch¨ urmann (2017)

Malte Gerhold Multifaced noncommutative stochastic independence 21 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Bi-monotone product II

Theorem

The bi-monotone product of states (linear functionals) is unital in the sense that: 1 ’ ϕ “ ϕ “ ϕ ’ 1 associative universal in the sense that (for ˚-hom’s ji : Bi Ñ Ai) pϕ1 ’ ϕ2q ˝ Č pj1 \ j2q “ pϕ1 ˝ r j1q ’ pϕ2 ˝ r j2q In short: ’ is a positive 2-1-uau-product in the sense of Manzel & Sch¨ urmann (2017)

Malte Gerhold Multifaced noncommutative stochastic independence 21 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Bi-monotone product II

Theorem

The bi-monotone product of states (linear functionals) is unital in the sense that: 1 ’ ϕ “ ϕ “ ϕ ’ 1 associative universal in the sense that (for ˚-hom’s ji : Bi Ñ Ai) pϕ1 ’ ϕ2q ˝ Č pj1 \ j2q “ pϕ1 ˝ r j1q ’ pϕ2 ˝ r j2q In short: ’ is a positive 2-1-uau-product in the sense of Manzel & Sch¨ urmann (2017)

Malte Gerhold Multifaced noncommutative stochastic independence 21 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Overview

1

Multifaced random variables

2

Moments and cumulants

3

Example: bi-monotone independence Universal product Partitions Moment cumulant formula Central limit theorem

4

A 4-faced independence

5

More examples

Malte Gerhold Multifaced noncommutative stochastic independence 22 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Bi-monotone partitions

Definition (bi-partition, ordered bi-partition)

bi-partition of a set X: set partition of X together with a map δ: X Ñ tℓ, ru

• rdered bi-partition of X:

bi-partition with total order between blocks

Definition (bi-monotone partition)

bi-monotone partition of a totally ordered set X:

• rdered bi-partition π s.t. for elements a ă b ă c of X and blocks

V , W P π: a, c P V (outer block) & b P W (inner block) ó V ď W if δpbq “ r and V ě W if δpbq “ ℓ

Malte Gerhold Multifaced noncommutative stochastic independence 23 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Bi-monotone partitions

Definition (bi-partition, ordered bi-partition)

bi-partition of a set X: set partition of X together with a map δ: X Ñ tℓ, ru

• rdered bi-partition of X:

bi-partition with total order between blocks

Definition (bi-monotone partition)

bi-monotone partition of a totally ordered set X:

• rdered bi-partition π s.t. for elements a ă b ă c of X and blocks

V , W P π: a, c P V (outer block) & b P W (inner block) ó V ď W if δpbq “ r and V ě W if δpbq “ ℓ

Malte Gerhold Multifaced noncommutative stochastic independence 23 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Visualization

Three bi-monotone partitions

1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6

horizontal lines indicate blocks of π height indicates block order vertical lines indicate δ (ℓˆ “downwards, r ˆ “upwards)

Malte Gerhold Multifaced noncommutative stochastic independence 24 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Overview

1

Multifaced random variables

2

Moments and cumulants

3

Example: bi-monotone independence Universal product Partitions Moment cumulant formula Central limit theorem

4

A 4-faced independence

5

More examples

Malte Gerhold Multifaced noncommutative stochastic independence 25 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Calculating Mixed moments

Lemma

A1, . . . , An bi-monotonely independent ai P Aδi

εi

π ordered bi-partition corresponding to pε, δq ρ partition obtained from π by dividing blocks at each crossing ù ñ Φpa1 ¨ ¨ ¨ amq “ ź

V Pρ

ΦpaV q

Idea of proof

V-line at j crossing H-line at h πpajq has PΩ at leg h V-line at j not crossing H-line at h πpajq has id at leg h V-line at j touches H-line at h πpajq has πεjpajq at leg h

Malte Gerhold Multifaced noncommutative stochastic independence 26 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Calculating Mixed moments

Lemma

A1, . . . , An bi-monotonely independent ai P Aδi

εi

π ordered bi-partition corresponding to pε, δq ρ partition obtained from π by dividing blocks at each crossing ù ñ Φpa1 ¨ ¨ ¨ amq “ ź

V Pρ

ΦpaV q

Idea of proof

V-line at j crossing H-line at h πpajq has PΩ at leg h V-line at j not crossing H-line at h πpajq has id at leg h V-line at j touches H-line at h πpajq has πεjpajq at leg h

Malte Gerhold Multifaced noncommutative stochastic independence 26 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Moment Cumulant Formula

Theorem

Given cumulants pcδqδPtℓ,ru˚: mδ “ ÿ

πPP’pδq

1 |π|! ź

βPπ

cβ

Proof.

mδ “ exp’pψqpxδq “ εpxδq `

8

ÿ

k“1

1 k!ψ‘kpxδq ψ‘kpxδq “ Bk Bt1 ¨ ¨ ¨ Btk pt1ψq ˚ ¨ ¨ ¨ ˚ ptkψqpxδq ˇ ˇ ˇ ˇ “ ÿ

πPPpkqpδq

Bk Bt1 ¨ ¨ ¨ Btk pt1ψq ’ ¨ ¨ ¨ ’ ptkψqpxπq ˇ ˇ ˇ ˇ

Malte Gerhold Multifaced noncommutative stochastic independence 27 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Overview

1

Multifaced random variables

2

Moments and cumulants

3

Example: bi-monotone independence Universal product Partitions Moment cumulant formula Central limit theorem

4

A 4-faced independence

5

More examples

Malte Gerhold Multifaced noncommutative stochastic independence 28 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Bi-monotone CLT

Theorem

pbiqiPN, bi “ pbpℓq

i

, bprq

i

q sequence of pairs in ncps A s.t. Φpbpjq

i q “ 0 for all i, j (centered)

pbiqiPN bi-monotonely independent bi identically distributed Φpbppq

i

bpqq

i

q “ 1 for p, q P tℓ, ru Then for sN :“

řN

n“1 bn

N1{2

lim

NÑ8 Φpsδ Nq “ #PP’pδq

k! for all δ P tℓ, ru2k.

Tools for proof.

MCF + CLT of Accardi, Hashimoto, Obata (’98)

Malte Gerhold Multifaced noncommutative stochastic independence 29 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Bi-monotone CLT

Theorem

pbiqiPN, bi “ pbpℓq

i

, bprq

i

q sequence of pairs in ncps A s.t. Φpbpjq

i q “ 0 for all i, j (centered)

pbiqiPN bi-monotonely independent bi identically distributed Φpbppq

i

bpqq

i

q “ 1 for p, q P tℓ, ru Then for sN :“

řN

n“1 bn

N1{2

lim

NÑ8 Φpsδ Nq “ #PP’pδq

k! for all δ P tℓ, ru2k.

Tools for proof.

MCF + CLT of Accardi, Hashimoto, Obata (’98)

Malte Gerhold Multifaced noncommutative stochastic independence 29 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Bi-monotone CLT

Theorem

pbiqiPN, bi “ pbpℓq

i

, bprq

i

q sequence of pairs in ncps A s.t. Φpbpjq

i q “ 0 for all i, j (centered)

pbiqiPN bi-monotonely independent bi identically distributed Φpbppq

i

bpqq

i

q “ 1 for p, q P tℓ, ru Then for sN :“

řN

n“1 bn

N1{2

lim

NÑ8 Φpsδ Nq “ #PP’pδq

k! for all δ P tℓ, ru2k.

Tools for proof.

MCF + CLT of Accardi, Hashimoto, Obata (’98)

Malte Gerhold Multifaced noncommutative stochastic independence 29 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Bi-monotone CLT

Theorem

pbiqiPN, bi “ pbpℓq

i

, bprq

i

q sequence of pairs in ncps A s.t. Φpbpjq

i q “ 0 for all i, j (centered)

pbiqiPN bi-monotonely independent bi identically distributed Φpbppq

i

bpqq

i

q “ 1 for p, q P tℓ, ru Then for sN :“

řN

n“1 bn

N1{2

lim

NÑ8 Φpsδ Nq “ #PP’pδq

k! for all δ P tℓ, ru2k.

Tools for proof.

MCF + CLT of Accardi, Hashimoto, Obata (’98)

Malte Gerhold Multifaced noncommutative stochastic independence 29 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Bi-monotone CLT

Theorem

pbiqiPN, bi “ pbpℓq

i

, bprq

i

q sequence of pairs in ncps A s.t. Φpbpjq

i q “ 0 for all i, j (centered)

pbiqiPN bi-monotonely independent bi identically distributed Φpbppq

i

bpqq

i

q “ 1 for p, q P tℓ, ru Then for sN :“

řN

n“1 bn

N1{2

lim

NÑ8 Φpsδ Nq “ #PP’pδq

k! for all δ P tℓ, ru2k.

Tools for proof.

MCF + CLT of Accardi, Hashimoto, Obata (’98)

Malte Gerhold Multifaced noncommutative stochastic independence 29 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Bi-monotone CLT

Theorem

pbiqiPN, bi “ pbpℓq

i

, bprq

i

q sequence of pairs in ncps A s.t. Φpbpjq

i q “ 0 for all i, j (centered)

pbiqiPN bi-monotonely independent bi identically distributed Φpbppq

i

bpqq

i

q “ 1 for p, q P tℓ, ru Then for sN :“

řN

n“1 bn

N1{2

lim

NÑ8 Φpsδ Nq “ #PP’pδq

k! for all δ P tℓ, ru2k.

Tools for proof.

MCF + CLT of Accardi, Hashimoto, Obata (’98)

Malte Gerhold Multifaced noncommutative stochastic independence 29 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Bi-monotone CLT

Theorem

pbiqiPN, bi “ pbpℓq

i

, bprq

i

q sequence of pairs in ncps A s.t. Φpbpjq

i q “ 0 for all i, j (centered)

pbiqiPN bi-monotonely independent bi identically distributed Φpbppq

i

bpqq

i

q “ 1 for p, q P tℓ, ru Then for sN :“

řN

n“1 bn

N1{2

lim

NÑ8 Φpsδ Nq “ #PP’pδq

k! for all δ P tℓ, ru2k.

Tools for proof.

MCF + CLT of Accardi, Hashimoto, Obata (’98)

Malte Gerhold Multifaced noncommutative stochastic independence 29 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

tbma-product

Product of rep’s

For pπi : Bi Ñ LadjpHiq, Ωiq pointed rep’s on pre-HS define pπ1 b π2 : B1 \ B2 Ñ LadjpH1 b H2q, Ω1 b Ω2q π1 b π2pbq :“ $ ’ ’ ’ ’ & ’ ’ ’ ’ % π1 b π2pbq b P Bb π1 ˛ π2pbq b P B˛ π1 Ÿ π2pbq b P BŸ π1 Ź π2pbq b P BŹ

Definition (product of states)

ϕ1 b ϕ2pbq :“ xΩ, πϕ1 b πϕ2pbqΩy for all b P B1 \ B2 whenever ϕi “ xΩi, πip¨qΩiy.

Malte Gerhold Multifaced noncommutative stochastic independence 30 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

tbma-product

Product of rep’s

For pπi : Bi Ñ LadjpHiq, Ωiq pointed rep’s on pre-HS define pπ1 b π2 : B1 \ B2 Ñ LadjpH1 b H2q, Ω1 b Ω2q π1 b π2pbq :“ $ ’ ’ ’ ’ & ’ ’ ’ ’ % π1 b π2pbq b P Bb π1 ˛ π2pbq b P B˛ π1 Ÿ π2pbq b P BŸ π1 Ź π2pbq b P BŹ

Definition (product of states)

ϕ1 b ϕ2pbq :“ xΩ, πϕ1 b πϕ2pbqΩy for all b P B1 \ B2 whenever ϕi “ xΩi, πip¨qΩiy.

Malte Gerhold Multifaced noncommutative stochastic independence 30 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Calculating mixed moments

tbma-partitions

Visualize 4-faced ordered partitions pπ, δq with faces pb, ˛, Ź, Ÿq as two-faced partitions but such that b ˆ “ nothing, ˛ ˆ “ top-to-bottom, Ź ˆ “ upwards, Ÿ ˆ “ downwards

Three tbma-partitions

1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6

Malte Gerhold Multifaced noncommutative stochastic independence 31 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Moment cumulant formula

Lemma

A1, . . . , An tbma-independent ai P Aδi

εi

pπ, δq ordered 4-partition corresponding to pε, δq ρ partition obtained from π by dividing blocks at each crossing ù ñ Φpa1 ¨ ¨ ¨ amq “ ź

V Pρ

ΦpaV q

Theorem

Given cumulants pcδqδPtℓ,ru˚: mδ “ ÿ

πPPtbmapδq

1 |π|! ź

βPπ

cβ

Malte Gerhold Multifaced noncommutative stochastic independence 32 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Moment cumulant formula

Lemma

A1, . . . , An tbma-independent ai P Aδi

εi

pπ, δq ordered 4-partition corresponding to pε, δq ρ partition obtained from π by dividing blocks at each crossing ù ñ Φpa1 ¨ ¨ ¨ amq “ ź

V Pρ

ΦpaV q

Theorem

Given cumulants pcδqδPtℓ,ru˚: mδ “ ÿ

πPPtbmapδq

1 |π|! ź

βPπ

cβ

Malte Gerhold Multifaced noncommutative stochastic independence 32 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

More examples

Similar construction on free product of Hilbert spaces (Liu)

free-Boolean 2-faced independence free-free-Boolean 3-faced independence “full-featured”

Examples built by generalizing “c-freeness” (Gu & Skoufranis)

c-bi-free independence; defined via a (2,2) uau produkt bi-Boolean independence bi-monotone independence (& Hasebe) do not preserve Hermitianity / positivity!

Malte Gerhold Multifaced noncommutative stochastic independence 33 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

More examples

Similar construction on free product of Hilbert spaces (Liu)

free-Boolean 2-faced independence free-free-Boolean 3-faced independence “full-featured”

Examples built by generalizing “c-freeness” (Gu & Skoufranis)

c-bi-free independence; defined via a (2,2) uau produkt bi-Boolean independence bi-monotone independence (& Hasebe) do not preserve Hermitianity / positivity!

Malte Gerhold Multifaced noncommutative stochastic independence 33 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

More examples

Similar construction on free product of Hilbert spaces (Liu)

free-Boolean 2-faced independence free-free-Boolean 3-faced independence “full-featured”

Examples built by generalizing “c-freeness” (Gu & Skoufranis)

c-bi-free independence; defined via a (2,2) uau produkt bi-Boolean independence bi-monotone independence (& Hasebe) do not preserve Hermitianity / positivity!

Malte Gerhold Multifaced noncommutative stochastic independence 33 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

More examples

Similar construction on free product of Hilbert spaces (Liu)

free-Boolean 2-faced independence free-free-Boolean 3-faced independence “full-featured”

Examples built by generalizing “c-freeness” (Gu & Skoufranis)

c-bi-free independence; defined via a (2,2) uau produkt bi-Boolean independence bi-monotone independence (& Hasebe) do not preserve Hermitianity / positivity!

Malte Gerhold Multifaced noncommutative stochastic independence 33 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

More examples

Similar construction on free product of Hilbert spaces (Liu)

free-Boolean 2-faced independence free-free-Boolean 3-faced independence “full-featured”

Examples built by generalizing “c-freeness” (Gu & Skoufranis)

c-bi-free independence; defined via a (2,2) uau produkt bi-Boolean independence bi-monotone independence (& Hasebe) do not preserve Hermitianity / positivity!

Malte Gerhold Multifaced noncommutative stochastic independence 33 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

More examples

Similar construction on free product of Hilbert spaces (Liu)

free-Boolean 2-faced independence free-free-Boolean 3-faced independence “full-featured”

Examples built by generalizing “c-freeness” (Gu & Skoufranis)

c-bi-free independence; defined via a (2,2) uau produkt bi-Boolean independence bi-monotone independence (& Hasebe) do not preserve Hermitianity / positivity!

Malte Gerhold Multifaced noncommutative stochastic independence 33 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

More examples

Similar construction on free product of Hilbert spaces (Liu)

free-Boolean 2-faced independence free-free-Boolean 3-faced independence “full-featured”

Examples built by generalizing “c-freeness” (Gu & Skoufranis)

c-bi-free independence; defined via a (2,2) uau produkt bi-Boolean independence bi-monotone independence (& Hasebe) do not preserve Hermitianity / positivity!

Malte Gerhold Multifaced noncommutative stochastic independence 33 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

More examples

Similar construction on free product of Hilbert spaces (Liu)

free-Boolean 2-faced independence free-free-Boolean 3-faced independence “full-featured”

Examples built by generalizing “c-freeness” (Gu & Skoufranis)

c-bi-free independence; defined via a (2,2) uau produkt bi-Boolean independence bi-monotone independence (& Hasebe) do not preserve Hermitianity / positivity!

Malte Gerhold Multifaced noncommutative stochastic independence 33 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

More examples

Similar construction on free product of Hilbert spaces (Liu)

free-Boolean 2-faced independence free-free-Boolean 3-faced independence “full-featured”

Examples built by generalizing “c-freeness” (Gu & Skoufranis)

c-bi-free independence; defined via a (2,2) uau produkt bi-Boolean independence bi-monotone independence (& Hasebe) do not preserve Hermitianity / positivity!

Malte Gerhold Multifaced noncommutative stochastic independence 33 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples

Takk for din oppmerksomhet!

Malte Gerhold Multifaced noncommutative stochastic independence 34 / 34