Shuffle algebra perspective on operator valued probability theory - PowerPoint PPT Presentation

Shuffle algebra perspective on operator valued probability theory 30 mars 2020 1/25

Operator valued probability theory 2/25

Operator valued probability theory Definition (Operator valued space) An (algebraic) operator valued probability consist of : A complex unital algebra B endowed with an involution ⋆ , 3/25

Operator valued probability theory Definition (Operator valued space) An (algebraic) operator valued probability consist of : A complex unital algebra B endowed with an involution ⋆ , A ⋆ -algeba ( A , ⋆ ) , which is a B - B bimodule over B : b 1 · ( a · b 2 ) = ( b 1 · a ) · b 2 , ( a 1 · b ) a 2 = a 1 ( b · a 2 ) . 3/25

Operator valued probability theory Definition (Operator valued space) An (algebraic) operator valued probability consist of : A complex unital algebra B endowed with an involution ⋆ , A ⋆ -algeba ( A , ⋆ ) , which is a B - B bimodule over B : b 1 · ( a · b 2 ) = ( b 1 · a ) · b 2 , ( a 1 · b ) a 2 = a 1 ( b · a 2 ) . A positive B - B module map E : A → B : E ( b 1 ab 2 ) = b 1 E ( a ) b 2 , E ( aa ⋆ ) ∈ BB ⋆ . 3/25

Operator valued probability theory Definition (Operator valued space) An (algebraic) operator valued probability consist of : A complex unital algebra B endowed with an involution ⋆ , A ⋆ -algeba ( A , ⋆ ) , which is a B - B bimodule over B : b 1 · ( a · b 2 ) = ( b 1 · a ) · b 2 , ( a 1 · b ) a 2 = a 1 ( b · a 2 ) . A positive B - B module map E : A → B : E ( b 1 ab 2 ) = b 1 E ( a ) b 2 , E ( aa ⋆ ) ∈ BB ⋆ . � Speicher, R. Combinatorial theory of the free product with amalgamation and operator-valued free probability theory. 3/25

Operator valued probability theory Definition (Operator valued space) An (algebraic) operator valued probability consist of : A complex unital algebra B endowed with an involution ⋆ , A ⋆ -algeba ( A , ⋆ ) , which is a B - B bimodule over B : b 1 · ( a · b 2 ) = ( b 1 · a ) · b 2 , ( a 1 · b ) a 2 = a 1 ( b · a 2 ) . A positive B - B module map E : A → B : E ( b 1 ab 2 ) = b 1 E ( a ) b 2 , E ( aa ⋆ ) ∈ BB ⋆ . � Speicher, R. Combinatorial theory of the free product with amalgamation and operator-valued free probability theory. � Speicher, R. Operator-valued free probability and block random matrices. 3/25

Operator valued probability theory Definition (Distribution of random variables) Let a 1 , . . . , a n ∈ A . The distribution of a 1 , . . . , a n is the collection of elements in B defined by : E ( b 1 a 1 b 2 · · · a n b n + 1 ) , b 1 , . . . , b n + 1 ∈ B . 4/25

Operator valued probability theory Definition (Distribution of random variables) Let a 1 , . . . , a n ∈ A . The distribution of a 1 , . . . , a n is the collection of elements in B defined by : E ( b 1 a 1 b 2 · · · a n b n + 1 ) , b 1 , . . . , b n + 1 ∈ B . Definition (Free Multiplicative extension on NC .) E π ( b 1 , . . . , b 10 ) = E ( b 1 ab 2 aE ( b 3 aE ( b 4 ab 5 aE ( b 6 ab 7 )) ab 8 ) ab 9 ab 10 ) 4/25

Operator valued probability theory Definition (Boolean multiplicative extension) Let IP be the poset of interval partitions, and write I = I 1 · · · I p for I = { I 1 , . . . , I p } ∈ IP . � E I ( b 1 , . . . , b | I | ) = E ( b ··· + I j − 1 + 1 · · · b ··· + I j ) i ∈ 1 ,..., p 5/25

Operator valued probability theory Definition (Boolean multiplicative extension) Let IP be the poset of interval partitions, and write I = I 1 · · · I p for I = { I 1 , . . . , I p } ∈ IP . � E I ( b 1 , . . . , b | I | ) = E ( b ··· + I j − 1 + 1 · · · b ··· + I j ) i ∈ 1 ,..., p Definition (Boolean and Free cumulants) � � E ( b 1 , . . . , b n + 1 ) = κ π ( b 1 , . . . , b n + 1 ) = β π ( b 1 , . . . , b n + 1 ) . π ∈ NC ( n ) β ∈ IP ( n ) 5/25

Operator valued probability theory Definition (Boolean multiplicative extension) Let IP be the poset of interval partitions, and write I = I 1 · · · I p for I = { I 1 , . . . , I p } ∈ IP . � E I ( b 1 , . . . , b | I | ) = E ( b ··· + I j − 1 + 1 · · · b ··· + I j ) i ∈ 1 ,..., p Definition (Boolean and Free cumulants) � � E ( b 1 , . . . , b n + 1 ) = κ π ( b 1 , . . . , b n + 1 ) = β π ( b 1 , . . . , b n + 1 ) . π ∈ NC ( n ) β ∈ IP ( n ) Free and Boolean cumulants linearize Free and Boolean operator valued independance. 5/25

Shuffle approach to scalar probability theory 6/25

✁ ✁ Double bar construction H = ¯ T ( T ( A )) . n 1 | a 2 a 1 1 · · · a 1 1 · · · a 2 ∅ , a 1 · · · a n , m 1 ∆ ✁ ( · ) = ∅ ⊗ · + · ⊗ ∅ + ¯ ∆( · ) = ∅ ⊗ · + · ⊗ ∅ + ∆ ≺ ( · ) + ∆ ≻ ( · ) . 7/25

Double bar construction H = ¯ T ( T ( A )) . n 1 | a 2 a 1 1 · · · a 1 1 · · · a 2 ∅ , a 1 · · · a n , m 1 ∆ ✁ ( · ) = ∅ ⊗ · + · ⊗ ∅ + ¯ ∆( · ) = ∅ ⊗ · + · ⊗ ∅ + ∆ ≺ ( · ) + ∆ ≻ ( · ) . Proposition Hom Vect C ( H , ✁ ) is a monoid and G = Hom Alg ( H , ✁ ) is a group. 7/25

Double bar construction H = ¯ T ( T ( A )) . n 1 | a 2 a 1 1 · · · a 1 1 · · · a 2 ∅ , a 1 · · · a n , m 1 ∆ ✁ ( · ) = ∅ ⊗ · + · ⊗ ∅ + ¯ ∆( · ) = ∅ ⊗ · + · ⊗ ∅ + ∆ ≺ ( · ) + ∆ ≻ ( · ) . Proposition Hom Vect C ( H , ✁ ) is a monoid and G = Hom Alg ( H , ✁ ) is a group. � � k ≺ n , k ≻ n exp ≺ ( k ) = 1 ⋆ + exp ≻ ( k ) = 1 ⋆ + n ≥ 1 n ≥ 1 7/25

Double bar construction H = ¯ T ( T ( A )) . n 1 | a 2 a 1 1 · · · a 1 1 · · · a 2 ∅ , a 1 · · · a n , m 1 ∆ ✁ ( · ) = ∅ ⊗ · + · ⊗ ∅ + ¯ ∆( · ) = ∅ ⊗ · + · ⊗ ∅ + ∆ ≺ ( · ) + ∆ ≻ ( · ) . Proposition Hom Vect C ( H , ✁ ) is a monoid and G = Hom Alg ( H , ✁ ) is a group. � � k ≺ n , k ≻ n exp ≺ ( k ) = 1 ⋆ + exp ≻ ( k ) = 1 ⋆ + n ≥ 1 n ≥ 1 exp ≺ ( k ) − 1 = exp ≻ ( − k ) . 7/25

Shuffle and non-commutative probability theory A ⋆ -algebra A and an expectation E : A → C . M ∈ G , M ( a 1 ⊗ · · · ⊗ a n ) = E ( a 1 · A · · · · A a n ) k ∈ Lie ( G ) , k ( a 1 ⊗ · · · ⊗ a n ) = κ ( a 1 , . . . , a n ) b ∈ Lie ( G ) , b ( a 1 , . . . , a n ) = β ( a 1 ⊗ · · · ⊗ a n ) 8/25

Shuffle and non-commutative probability theory A ⋆ -algebra A and an expectation E : A → C . M ∈ G , M ( a 1 ⊗ · · · ⊗ a n ) = E ( a 1 · A · · · · A a n ) k ∈ Lie ( G ) , k ( a 1 ⊗ · · · ⊗ a n ) = κ ( a 1 , . . . , a n ) b ∈ Lie ( G ) , b ( a 1 , . . . , a n ) = β ( a 1 ⊗ · · · ⊗ a n ) M = ε + k ≺ M , M = ε + M ≻ b M = exp ≺ ( k ) = exp ≻ ( b ) 8/25

Shuffle and non-commutative probability theory A ⋆ -algebra A and an expectation E : A → C . M ∈ G , M ( a 1 ⊗ · · · ⊗ a n ) = E ( a 1 · A · · · · A a n ) k ∈ Lie ( G ) , k ( a 1 ⊗ · · · ⊗ a n ) = κ ( a 1 , . . . , a n ) b ∈ Lie ( G ) , b ( a 1 , . . . , a n ) = β ( a 1 ⊗ · · · ⊗ a n ) M = ε + k ≺ M , M = ε + M ≻ b M = exp ≺ ( k ) = exp ≻ ( b ) � Ebrahimi-Fard, K., Patras, F. Cumulants, free cumulants and half-shuffles. � Ebrahimi-Fard, K., Patras, F. Monotone, free, and boolean cumulants : a shuffle algebra approach. 8/25

Relation between Möbius inversion and Shuffles � Ebrahimi-Fard, K., Foissy, L., Kock, J., Patras, F. Operads of (noncrossing) partitions, interacting bialgebras, and moment-cumulant relations. ⇒ Gap insertion operad of non-crossing Shuffle Approach = partitions Operad NC − → incidence bi-algebra ( N , ∆) on words on non-crossing partitions : � q ⊗ ( p 1 ⊗ . . . ⊗ p n ) = ∆ + ≺ ( π ) + ∆ + ∆( π ) = ≻ ( π ) . π = q ◦ ( p 1 ,..., p n ) f = ( E ( a n )) n ≥ 1 F : NC → C , multiplicative F : N → C , F = ε N + f ≺ F . 9/25

Relation between Möbius inversion and Shuffles � Ebrahimi-Fard, K., Foissy, L., Kock, J., Patras, F. Operads of (noncrossing) partitions, interacting bialgebras, and moment-cumulant relations. ⇒ Block substitution operad Möbius inversion = 9/25

Shuffle operadic approach to operator valued cumulants and moments 10/25

⊠ Express multiplicativity of { E π , π ∈ NC } . Define a decomposition map ∆ that presevers linear order of the "legs" of a non-crossing partition. ⊗ ⊗ ⊗ Give a Lie theoretic perspective, with a group of morphisms and a Lie algebra of infinitesimal morphisms and a fixed point equation for { E π , π ∈ NC } . 11/25

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Duoidal Category of bigraded collections n , m ≥ 0 , C n , m ∈ Vect C , C C C = ( C n , m ) n , m 13/25

Duoidal Category of bigraded collections n , m ≥ 0 , C n , m ∈ Vect C , C C C = ( C n , m ) n , m ⊗ Horizontal product ⊗ ⊗ and Vertical product ⊠ 13/25

Duoidal Category of bigraded collections n , m ≥ 0 , C n , m ∈ Vect C , C C C = ( C n , m ) n , m ⊗ Horizontal product ⊗ ⊗ and Vertical product ⊠ � � ⊗ ( C ⊗ ⊗ D ) n , m = C n c , m c ⊗ D n d , m d , ( C ⊠ D ) n , m = C n , k ⊗ D k , m n c + n d = n k m c + m d = m 13/25

Duoidal Category of bigraded collections n , m ≥ 0 , C n , m ∈ Vect C , C C C = ( C n , m ) n , m ⊗ Horizontal product ⊗ ⊗ and Vertical product ⊠ � � ⊗ ( C ⊗ ⊗ D ) n , m = C n c , m c ⊗ D n d , m d , ( C ⊠ D ) n , m = C n , k ⊗ D k , m n c + n d = n k m c + m d = m ⊗ ⊗ ⊗ ⊠ 13/25