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 L 8 p Ω q ` ˘ ` ˘ ` ˘ X , Y independent ð ñ E f p X q g p Y q “ E f p X q g p Y q E ð ñ P p X , Y q “ P X b P Y 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 L 8 p Ω q ` ˘ ` ˘ ` ˘ X , Y independent ð ñ E f p X q g p Y q “ E f p X q g p Y q E ð ñ P p X , Y q “ P X b P Y Non-commutative situations a i hermitian random matrices, independent entries „ N p 0 , 1 q ˆ a 1 ` ¨ ¨ ¨ ` a k ˙ n ? tr p a 1 a 2 a 1 a 2 q “ ? , tr “ ? k T 1 , T 2 rooted trees, T 1 ˛ T 2 : “glued together at root” x e 0 , p A T 1 ˛ T 2 q n e 0 y “ x e 0 , p A T 1 b p 0 ` p 0 b A T 2 q n e 0 y “ ? 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 L 8 p Ω q ` ˘ ` ˘ ` ˘ X , Y independent ð ñ E f p X q g p Y q “ E f p X q g p Y q E ð ñ P p X , Y q “ P X b P Y Non-commutative situations a i hermitian random matrices, independent entries „ N p 0 , 1 q ˆ a 1 ` ¨ ¨ ¨ ` a k ˙ n ? tr p a 1 a 2 a 1 a 2 q “ ? , tr “ ? k T 1 , T 2 rooted trees, T 1 Ź T 2 : “glue T 2 to every vertex” x e 0 , p A T 1 Ź T 2 q n e 0 y “ x e 0 , p A T 1 b p 0 ` id b A T 2 q n e 0 y “ ? 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 Multifaced random variables 1 Moments and cumulants 2 Example: bi-monotone independence 3 Universal product Partitions Moment cumulant formula Central limit theorem A 4-faced independence 4 More examples 5 Malte Gerhold Multifaced noncommutative stochastic independence 5 / 34

Multifaced random variables Moments and cumulants Bi-monotone independence A 4-faced independence More examples Overview Multifaced random variables 1 Moments and cumulants 2 Example: bi-monotone independence 3 Universal product Partitions Moment cumulant formula Central limit theorem A 4-faced independence 4 More examples 5 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 p A , Φ q with unital ˚ -algebra A state Φ on A Definition (random variable) ˚ -homomorphism j : B Ñ A ( B is ˚ -algebra) B unitization of B , r r j unital extension of j Φ ˝ r j is called distribution of j selfadjoint a P A � j a : C r x s 0 Ñ A , x ÞÑ a ` ˘ Φ p a k q distribution of j a ú collection of moments k P N ˚ -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 p A , Φ q with unital ˚ -algebra A state Φ on A Definition (random variable) ˚ -homomorphism j : B Ñ A ( B is ˚ -algebra) B unitization of B , r r j unital extension of j Φ ˝ r j is called distribution of j selfadjoint a P A � j a : C r x s 0 Ñ A , x ÞÑ a ` ˘ Φ p a k q distribution of j a ú collection of moments k P N ˚ -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 p A , Φ q with unital ˚ -algebra A state Φ on A Definition (random variable) ˚ -homomorphism j : B Ñ A ( B is ˚ -algebra) B unitization of B , r r j unital extension of j Φ ˝ r j is called distribution of j selfadjoint a P A � j a : C r x s 0 Ñ A , x ÞÑ a ` ˘ Φ p a k q distribution of j a ú collection of moments k P N ˚ -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 p A , Φ q with unital ˚ -algebra A state Φ on A Definition (random variable) ˚ -homomorphism j : B Ñ A ( B is ˚ -algebra) B unitization of B , r r j unital extension of j Φ ˝ r j is called distribution of j selfadjoint a P A � j a : C r x s 0 Ñ A , x ÞÑ a ` ˘ Φ p a k q distribution of j a ú collection of moments k P N ˚ -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 p A , Φ q with unital ˚ -algebra A state Φ on A Definition (random variable) ˚ -homomorphism j : B Ñ A ( B is ˚ -algebra) B unitization of B , r r j unital extension of j Φ ˝ r j is called distribution of j selfadjoint a P A � j a : C r x s 0 Ñ A , x ÞÑ a ` ˘ Φ p a k q distribution of j a ú collection of moments k P N ˚ -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 p A , Φ q with unital ˚ -algebra A state Φ on A Definition (random variable) ˚ -homomorphism j : B Ñ A ( B is ˚ -algebra) B unitization of B , r r j unital extension of j Φ ˝ r j is called distribution of j selfadjoint a P A � j a : C r x s 0 Ñ A , x ÞÑ a ` ˘ Φ p a k q distribution of j a ú collection of moments k P N ˚ -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 p A , Φ q with unital ˚ -algebra A state Φ on A Definition (random variable) ˚ -homomorphism j : B Ñ A ( B is ˚ -algebra) B unitization of B , r r j unital extension of j Φ ˝ r j is called distribution of j selfadjoint a P A � j a : C r x s 0 Ñ A , x ÞÑ a ` ˘ Φ p a k q distribution of j a ú collection of moments k P N ˚ -subalgebra B � embedding ι : B ã Ñ A Malte Gerhold Multifaced noncommutative stochastic independence 7 / 34