On noncommutative distributional symmetries and de Finetti theorems - PowerPoint PPT Presentation

On noncommutative distributional symmetries and de Finetti theorems associated with them Weihua Liu University of California at Berkeley weihualiu@math.berkeley.edu FPLNL V March 26, 2016 Weihua Liu (UC Berkeley) Characterizations of independences FPLNL V March 26, 2016 1 / 43

Definitions Probability space ( A , φ ) A is a von Neumann algebra. φ is a normal state not necessarily faithful, but the GNS representation associated with φ is faithful. x ∈ A random variable. Joint distribution of { x i | i ∈ I } , µ : C � X i | i ∈ I � → C defined by µ ( X k 1 i 1 X k 2 i 2 · · · X k n i n ) = φ ( x k 1 i 1 x k 2 i 2 · · · x k n i n ) , An operator valued probability space ( A , B , E : A → B ) consists of an algebra A , a subalgebra B of A and a B − B bimodule linear map E : A → B , i.e. E [ b 1 ab 2 ] = b 1 E [ a ] b 2 , E [ b ] = b for all b 1 , b 2 , b ∈ B and a ∈ A . Weihua Liu (UC Berkeley) Characterizations of independences FPLNL V March 26, 2016 2 / 43

Definitions Definition For an algebra B , B� X � is freely generated by B and the indeterminant X and B� X � 0 is a subalgebra of B� X � which does not contain a constant term in B . Definition { x i } i ∈ I ⊂ ( A , B , E : A → B ) is said to be conditional independent over B if E [ p 1 ( x i 1 ) p 2 ( x i 2 ) · · · p n ( x i n )] = E [ p 1 ( x i 1 )] E [ p 2 ( x i 2 )] · · · E [ p n ( x i n )] whenever i 1 , · · · , i n are pairwisely different and p 1 , · · · , p n ∈ B� X � . Weihua Liu (UC Berkeley) Characterizations of independences FPLNL V March 26, 2016 3 / 43

Classical symmetries A finite sequence of random variables ( ξ 1 , ξ 2 , ..., ξ n ) is said to be exchangeable if ( ξ 1 , ..., ξ n ) d = ( ξ σ (1) , ..., ξ σ ( n ) ) , ∀ σ ∈ S n , where S n is the permutation group of n elements. Compare with exchangeability, there is a weaker condition of spreadability: ( ξ 1 , ..., ξ n ) is said to be spreadable if for any k < n , we have ( ξ 1 , ..., ξ k ) d = ( ξ l 1 , ..., ξ l k ) , 1 ≤ l 1 < l 2 < · · · < l k ≤ n Note that i.i.d ⇒ conditionally i.i.d ⇒ exchangeability ⇒ spreadability. Weihua Liu (UC Berkeley) Characterizations of independences FPLNL V March 26, 2016 4 / 43

de Finetti Theorem For infinite sequences of commutative random variables, we have Theorem (de Finetti 1930s) Infinite sequences of exchangeable random variables are conditionally i.i.d. Theorem (Ryll-Nardzewski 1957 ) Infinite sequences of spreadable random variables are conditionally i.i.d. Therefore, Conditionally i.i.d ⇐ ⇒ exchangeability ⇐ ⇒ spreadability. Weihua Liu (UC Berkeley) Characterizations of independences FPLNL V March 26, 2016 5 / 43

K¨ ostler Theorem In noncommutative probability, for infinite sequences, spreadability �⇒ exchangeability �⇒ any independence relation. Weihua Liu (UC Berkeley) Characterizations of independences FPLNL V March 26, 2016 6 / 43

Definitions Definition ( A , B , E : A → B ) such that A and B are unital. A family of ( x i ) i ∈ I is said to be freely independent over B , if E [ p 1 ( x i 1 ) p 2 ( x i 2 ) · · · p n ( x i n )] = 0 , whenever i 1 � = i 2 � = · · · � = i n , p 1 , ..., p n ∈ B� X � and E [ p k ( x i k )] = 0 for all k . Definition { x i } i ∈ I ⊂ ( A , B , E : A → B ) is said to be Boolean independent over B if E [ p 1 ( x i 1 ) p 2 ( x i 2 ) · · · p n ( x i n )] = E [ p 1 ( x i 1 )] E [ p 2 ( x i 2 )] · · · E [ p n ( x i n )] whenever i 1 , · · · , i n ∈ I , i 1 � = i 2 � = · · · � = i n and p 1 , · · · , p n ∈ B� X � 0 . Weihua Liu (UC Berkeley) Characterizations of independences FPLNL V March 26, 2016 7 / 43

Definition { x i } i ∈ I is said to be monotonically independent over B if E [ p 1 ( x i 1 ) · · · p k − 1 ( x i k − 1 ) p k ( x i k ) p k +1 ( x i k +1 ) · · · p n ( x i n )] = E [ p 1 ( x i 1 ) · · · p k − 1 ( x i k − 1 ) E [ p k ( x i k )] p k +1 ( x i k +1 ) · · · p n ( x i n )] whenever i 1 , · · · , i n ∈ I , i 1 � = i 2 � = · · · � = i n , i k − 1 < i k > i k +1 and p 1 , · · · , p n ∈ B� X � 0 . Weihua Liu (UC Berkeley) Characterizations of independences FPLNL V March 26, 2016 8 / 43

Quantum symmetries Definition A s ( n ) is the universal unital C ∗ -algebra generated by ( u i , j ) i , j =1 , ··· n : u ∗ i , j = u i , j = u 2 i , j for all i , j = 1 , · · · , n . For each i = 1 , · · · , n and k � = l we have u ik u il = 0 and u ki u li = 0; . for each i = 1 , · · · , n we have n n � � u ik = 1 = u ki . k =1 k =1 A s ( n ) is a compact quantum group in sense of Woronowicz. Weihua Liu (UC Berkeley) Characterizations of independences FPLNL V March 26, 2016 9 / 43

Quantum symmetries Right coaction of A s ( n ) on C � X 1 , ..., X n � is a unital homomorphism α n : C � X 1 , ..., X n � → C � X 1 , ..., X n � ⊗ A s ( n ) defined by n � α n ( X i ) = X j ⊗ u j , i j =1 ( x 1 , ..., x n ) ⊂ A is said to be quantum exchangeable if µ x 1 ,..., x n ( p )1 A s ( n ) = µ x 1 ,..., x n ⊗ id A s ( n ) ( α n ( p )) for all p ∈ C � X 1 , ..., X n � . An infinite sequence ( x i ) i ∈ N is quantum exchangeable if all its finite subsequences are quantum exchangeable. Weihua Liu (UC Berkeley) Characterizations of independences FPLNL V March 26, 2016 10 / 43

Free de Finetti Theorem Definition Let ( A , φ ) be W ∗ -probability space with a faithful state, A is generated by ( x i ) i ∈ N . The tail algebra of ( x i ) i ∈ N is ∞ � A tail = vN { x k | k ≥ n } , n =1 where vN { x k | k ≥ n } is the von Neumann algebra generated by { x k | k ≥ n } . Weihua Liu (UC Berkeley) Characterizations of independences FPLNL V March 26, 2016 11 / 43

Free de Finetti Theorem Theorem (K¨ ostler 2010) If ( x i ) i ∈ N are exchangeable, then ∃ a normal endomorphism α : A → A such that α ( x i ) = x i +1 for all i ∈ N . Moreover, n →∞ α n E = WOT − lim is a well defined conditional expectation from A onto A tail . Weihua Liu (UC Berkeley) Characterizations of independences FPLNL V March 26, 2016 12 / 43

Free de Finetti Theorem Theorem (K¨ ostler & speicher 2009) For infinite sequences, Quantum exchangeable ⇐ ⇒ free with respect to E : A → A tail . Weihua Liu (UC Berkeley) Characterizations of independences FPLNL V March 26, 2016 13 / 43

Boolean de Finetti Theorem Definition B s ( n ) is defined as the universal unital C ∗ -algebra generated by elements u i , j ( i , j = 1 , · · · n ) and a projection P such that we have i , j = u i , j = u 2 each u i , j is an orthogonal projection, i.e. u ∗ i , j for all i , j = 1 , · · · , n . u i , k u i , l = 0 and u k , i u l , i = 0 whenever k � = l . n For all 1 ≤ i ≤ n , P = � u k , i P . k =1 Weihua Liu (UC Berkeley) Characterizations of independences FPLNL V March 26, 2016 14 / 43

Boolean de Finetti Theorem Right coaction of B s ( n ) on C � X 1 , ..., X n � is a unital homomorphism α n : C � X 1 , ..., X n � → C � X 1 , ..., X n � ⊗ B s ( n ) defined by n � α n ( X i ) = X j ⊗ u j , i j =1 ( x 1 , ..., x n ) ⊂ A is said to be Boolean exchangeable if µ x 1 ,..., x n ( p ) P = P µ x 1 ,..., x n ⊗ id B s ( n ) ( α n ( p )) P for all p ∈ C � X 1 , ..., X n � . An infinite sequence ( x i ) i ∈ N is Boolean exchangeable if all its finite subsequences are Boolean exchangeable. Weihua Liu (UC Berkeley) Characterizations of independences FPLNL V March 26, 2016 15 / 43

Boolean de Finetti Theorem Remark There is no pair of Boolean independent random variables in probability spaces with faithful states. Therefore, in our framework, we just require the GNS representation associated with the state to be faithful. Tail algebra The tail algebra T of ( x i ) i ∈ N is defined by the following formula: ∞ � W ∗ { x k | k ≥ n } , T = n =1 where W ∗ { x k | k ≥ n } is the WOT closure of the non-unital algebra generated by { x k | k ≥ n } . We call T the non-unital tail algebra of ( x i ) i ∈ N Weihua Liu (UC Berkeley) Characterizations of independences FPLNL V March 26, 2016 16 / 43

Boolean de Finetti Theorem Theorem Let ( A , φ ) be a W ∗ -probability space and ( x i ) i ∈ N be an infinite sequence of selfadjoint random variables which generate A as a von Neumann algebra. Then the following are equivalent: a) The joint distribution of ( x i ) i ∈ N is Boolean exchangeable. b) The sequence ( x i ) i ∈ N is identically distributed and Boolean independent with respect to a φ -preserving conditional expectation E onto the tail algebra of the ( x i ) i ∈ N . Weihua Liu (UC Berkeley) Characterizations of independences FPLNL V March 26, 2016 17 / 43

Spreadability Rephrasing spreadability in words of quantum maps: I k , n set of increasing sequences I = (1 ≤ i 1 < · · · < i k ≤ n ). For 1 ≤ i ≤ n , 1 ≤ j ≤ k , define f i , j : I k , n → C by: � 1 , i j = i f i , j ( I ) = . 0 , i j � = i C ( I n , k ) generated by the functions f i , j . ∃ α : C [ X 1 , ..., X k ] → C [ X 1 , ..., X n ] ⊗ C ( I k , n ) define by: n � α : X j = X i ⊗ f i , j , α (1) = 1 C ( I k , n ) i =1 Weihua Liu (UC Berkeley) Characterizations of independences FPLNL V March 26, 2016 18 / 43