Extended de Finetti theorems for boolean independence and monotone independence
aa r X i v : . [ m a t h . OA ] M a y EXTENDED DE FINETTI THEOREMS FOR BOOLEANINDEPENDENCE AND MONOTONE INDEPENDENCE
WEIHUA LIU
Abstract.
We construct several new spaces of quantum sequences and their quantumfamilies of maps in sense of So ltan. Then, we introduce noncommutative distributionalsymmetries associated with these quantum maps and study simple relations betweenthem. We will focus on studying two kinds of noncommutative distributional symme-tries: monotone spreadability and boolean spreadability. We provide an example of aspreadable sequence of random variables for which the usual unilateral shift is an un-bounded map. As a result, it is natural to study bilateral sequences of random objects,which are indexed by integers, rather than unilateral sequences. In the end of the paper,we will show Ryll-Nardzewski type theorems for monotone independence and booleanindependence: Roughly speaking, an infinite bilateral sequence of random variables ismonotonically(boolean) spreadable if and only if the variables are identically distributedand monotone(boolean) with respect to the conditional expectation onto its tail algebra.For an infinite sequence of noncommutative random variables, boolean spreadability isequivalent to boolean exchangeability. Introduction
The characterization of random objects with distributional symmetries is an importantobject in modern probability and the recent context of Kallenberg [12] provides a compre-hensive treatment of distributional symmetries in classical probability. A finite sequenceof random variables ( ξ , ξ , ..., ξ n ) is said to be exchangeable if( ξ , ..., ξ n ) d = ( ξ σ (1) , ..., ξ σ ( n ) ) , ∀ σ ∈ S n , where S n is the permutation group of n elements and d = meas the joint distribution of thetwo sequences are the same. Compare with exchangeability, there is a weaker conditionof spreadability: ( ξ , ..., ξ n ) is said to be spreadable if for any k < n , we have(1) ( ξ , ..., ξ k ) d = ( ξ l , ..., ξ l k ) , l < l < · · · < l k An infinite sequence of random variables is said to be exchangeable or spreadable if allits finite subsequences have this property. In the study of distributional symmetries inclassical probability, one of the most important results is de Finetti’s theorem which statesthat an infinite sequence of random variables, whose joint distribution is invariant underall finite permutations, is conditionally independent and identically distributed. Later, in[22], Ryll-Nardzewski showed that de Finetti theorem hold under the weaker condition ofspreadability. Therefore, for infinite sequences of random variables in classical probability,spreadability is equivalent to exchangeability.
Recently, K¨oslter [14] studied three kinds of distributional symmetries, which are sta-tionarity, contractability and exchangeablity, in noncommutative probability. It wasshown that exchangeability and spreadability do not characterize any universal inde-pendent relation in his framework. In addition, for infinite sequences, exchangeability isstrictly stronger than spreadability in noncommutative probability. It should be pointedout that the framework in his paper is a W ∗ -probability space with a faithful state. Inthis paper, we will consider our problems in a more general framework.In the 1980’s, Voiculescu developed his free probability theory and introduced a univer-sal independent relation, namely free independence, via reduced free products of unital C ∗ -algebras[29]. For more details of free probability, the reader is referred to the mono-graph [28]. One can see that there is a deep parallel between classical probability and freeprobability. Recently, in [15], K¨oslter and Speicher extended this parallel to the aspectof distributional symmetries. In their work, by strengthening classical exchangeability toquantum exchangeability, they proved a de Finetti type theorem for free independence,i.e. for an infinite sequence of random variables, quantum exchangeability is equivalent tothe fact that the the random variables are identically distributed and free with respect tothe conditional expectation onto the tail algebra. The notion of quantum exchangeabilityis given by invariance conditions associated with quantum permutation groups A s ( n ) ofWang [31]. This noncommutative de Finetti type theorem is an instance that free indepen-dence plays in the noncommutative world the same role as classical independence plays inthe commutative world. It naturally raises a motivation for further study of noncommu-tative symmetries that “any result in classical probability should have an extension in freeprobability.” For applications of this philosophy, see[2], [5], [6]. Especially, in [5], Curranintroduced a quantum version of spreadability for free independence. It was shown thatquantum spreadability is weaker than quantum exchangeability and is a characterizationof free independence. More specifically, in a W ∗ -probability space with a tracial faithfulstate, for an infinite sequence of random variables, quantum spreadability is equivalent tothe fact that the the random variables are identically distributed and free with respect tothe conditional expectation onto the tail algebra. In other words, quantum spreadabilityis equivalent to quantum exchangeability for infinite sequences of random variables in tra-cial W ∗ -probability spaces. Another remarkable application of quantum exchangeabilitywas given by Freslon and Weber [10]. They characterize Voiculescu’s Bi-freeness [30] viacertain invariance conditions associated with Wang’s quantum groups A s ( n ).In [26], Speicher and Woroudi introduced another independence relation which is calledboolean independence. It was show that boolean independence is related to full freeproduct of algebras [4] and boolean product is the unique non-unital universal productin noncommutative probability [25]. The study of distributional symmetries for booleanindependence was started in [16]. We constructed a family of quantum semigroups inanalogue with Wang’s quantum permutation groups and defined their coactions on jointdistributions of sequences. It was shown that the distributional symmetries associatedthose coactions can be used to characterize boolean independence in a proper framework.For more details about boolean independence and universal products, see [25]. It inspiresus to study more distributional symmetries for boolean independence under the philos-ophy “any result in classical probability and free probability should have an extension ONCOMMUTATIVE SPREADABILITY 3 for boolean independence”. In analogue with easy quantum groups in [1], we construct“easy ”boolean semigroups and study their de Finetti type theorems in [17]. To apply ourphilosophy further, it is naturally to find an extended de Finetti type theorem for booleanindependence. Specifically, we need to find the “noncommutative version of spreadability”for boolean independence and prove an extended de Finetti type theorem associatedwith the noncommutative spreadability.The main purpose of this paper is to study noncommutative versions of spreadabilityand extended de Finetti type theorems associated with them.Some other objects come into our consideration when we study spreadable sequences ofrandom objects. It was shown in [19], there are two other universal products in noncom-mutative probability if people do not require the universal construction to be commuta-tive. We call the two universal products monotone and anti-monotone product. As tensorproduct, free product and boolean product, we can define monotone and anti-monotoneindependence associated with monotone and anti-monotone product. Monotone indepen-dence and anti-monotone independence are essentially the same but with different orders,i.e. if a is monotone with b , then b is anti-monotone with a . For more details of mono-tone independence, the reader is referred to [18], [21]. It is well known that a sequence ofmonotone random variables is not exchangeable but spreadable. Therefore, there shouldbe a noncommutative spreadability which can characterize conditionally monotone inde-pendence.The first several sections devote to defining noncommutative distributional symmetriesin analogue with spreadability and partial exchangeability. Recall that in [2] [5], noncom-mutative distributional symmetries are defined via invariance conditions associated withcertain quantum structures. For instance, Curran’s quantum spreadability is describedby a family of quantum increasing sequences and their quantum family of maps in senseof Soltan. The family of quantum increasing sequences are universal C ∗ -algebras A i ( n, k )generated by the entries of a n × k matrix which satisfy certain relations R . Followingthe idea in [16], to construct a boolean type of spaces of increasing sequences B i ( n, k ), wereplace the unit partition condition in R by an invariant projection condition. Recall thatin In [9], Franz studied relations between freeness, monotone independence and booleanindependence via Bo˙zejko, Marek and Speicher’s two-state free products[3]. In his con-struction, monotone product is something “between” free product and boolean product.Thereby, we construct the noncommutative spreadability for monotone independence bymodifying quantum spreadability and our boolean spreadability. We will study simplerelations between those distributional symmetries, i.e. which one is stronger.As the situation for boolean independence, there is no nontrivial pair of monotonicallyindependent random variables in W ∗ -probability spaces with faithful states. Therefore,the framework we use in this paper is a W ∗ -probability space with a non-degeneratednormal state which gives a faithful GNS representation of the probability space. In thisframework, we will see that spreadability is too weak to ensure the existence of a con-ditional expectation. Recall that, in W ∗ -probability spaces with faithful states, we candefine a normal shift on a unilateral infinite sequence of spreadable random variables. WEIHUA LIU
Here, “unilateral”means the sequence is indexed by natural numbers N . An importantproperty of this shift is that its norm is one. Therefore, given an operator, we can con-struct a WOT convergent sequence of bounded variables via shifts. The is the key stepto construct a normal conditional expectation in previous works. But, in W ∗ -probabilityspaces with non-degenerated normal states, the unilateral shift of spreadable random vari-ables is not necessarily norm one. An example is provided in the beginning of section 6.Actually, the sequence of random variables are monotonically spreadable which is an in-variance condition stronger than classical spreadability. Therefore, we can not construct aconditional expectation, for unilateral sequences, via shifts under the condition of spread-ability. To fix this issue, we will consider bilateral sequences of random variables insteadof unilateral sequences. “bilateral”means that the sequences are indexed by integers Z . Inthis framework, we will see that the shift of spreadable random variables is norm one sothat we can define a conditional expectation via shifts by following K¨ostler’s construction.Notice that the index set Z has two infinities, i.e. the positive infinity and the negativeinfinity. Therefore, we will have two tail algebras with respect to the two infinities andwill define two conditional expectations consequently. We denote by E + the conditionalexpectation which shifts indices to the positive infinity and E − the conditional expecta-tion which shifts indices to negative infinity. We will see that the two tail algebras aresubsets of fixed points of the shift and the conditional expectations may not be extendednormally to the whole algebra. In general, the two tail algebras are different and theconditional expectation may have different properties. To noncommutative spreadabilityfor monotone independence, we have the following: Theorem 1.1.
Let ( A , φ ) be a non degenerated W ∗ -probability space and ( x i ) i ∈ Z be a bilat-eral infinite sequence of selfadjoint random variables which generate A as a von Neumannalgebra. Let A + k be the WOT closure of the non-unital algebra generated by { x i | i ≥ k } .Then the following are equivalent: a) The joint distribution of ( x i ) i ∈ Z is monotonically spreadable. b) For all k ∈ Z , there exits a φ preserving conditional expectation E k : A + k →A + tail such that the sequence ( x i ) i ≥ k is identically distributed and monotonicallyindependent with respect E k . Moreover, E k | A k ′ = E k ′ when k ≥ k ′ . In general, we can not extend E + to the whole algebra A , but we have the following: Proposition 1.2.
Let ( A , φ ) be a non degenerated W ∗ -probability space and ( x i ) i ∈ Z bea bilateral infinite sequence of selfadjoint random variables which generate A as a vonNeumann algebra. If the joint distribution of ( x i ) i ∈ Z is monotonically spreadable, then E − can be extend to the whole algebra A normally. We will see that boolean spreadability implies monotone spreadability and anti-monotonespreadability. Therefore, both E + and E − can be extended normally to the whole alge-bra A . Moreover, for boolean spreadable sequences, E + = E − and the two algebras areidentical. In summary, we have Theorem 1.3.
Let ( A , φ ) be a non degenerated W ∗ -probability space and ( x i ) i ∈ Z be a bilat-eral infinite sequence of selfadjoint random variables which generate A as a von Neumannalgebra. Then the following are equivalent: ONCOMMUTATIVE SPREADABILITY 5 a) The joint distribution of ( x i ) i ∈ N is boolean spreadable. b) The sequence ( x i ) i ∈ Z is identically distributed and boolean independent with respectto the φ − preserving conditional expectation E onto the non unital tail algebra ofthe ( x i ) i ∈ Z The paper is organized as follows: In section 2, we will introduce preliminaries and no-tation from noncommutative probability and recall Wang’s quantum permutation groupsand boolean quantum semigroups. In section 3, we briefly review distributional symme-tries for finite sequences of random variables in classical probability and we restate thesesymmetries in words of quantum maps. Then, we introduce noncommutative versionsof these symmetries and their quantum maps. In the end of this section, we will definequantum spreadability, monotone spreadability and boolean spreadability for bilateral in-finite sequences of random variables. In section 4, we will study simple relations betweenour noncommutative symmetries. In particular, we will show boolean exchangeabilityis strictly stronger than boolean spreadability. Therefore, operator-valued boolean in-dependent random variables are boolean spreadable. In section 5, we will introduce anequivalence relation on the set of sequences of indices. With the help of the equivalencerelation, we will show that operator-valued monotone independent sequences of randomvariables are monotonically spreadable. In section 6, we first provide an example that amonotonically spreadable unilateral sequence of bounded random variables is unbounded.Therefore, we cannot define conditional expectation for unilateral spreadable sequencesvia shifts in a W ∗ -probability space with a non-degenerated normal state. Then we willturn to study bilateral sequences of random variables. We will introduce tail algebrasassociated with positive infinity and negative infinity and study elementary properties ofconditional expectations associated with the two tail algebras. In section 7, we will studyproperties of conditional expectations under the assumption that our bilateral sequencesare monotonically spreadable. In section 8, we will prove a Ryll-Nardzewski type theoremfor Monotone independence. In section 9, we will prove a Ryll-Nardzewski type theoremfor boolean independence. 2. Preliminaries and examples
We recall some necessary definitions and notions from noncommutative probability. Forfurther details, see contexts [15], [20], [28], [21].
Definition 2.1.
A non-commutative probability space ( A , φ ) consists of a unital algebra A and a linear functional φ : A → C such that φ (1 A ) = 1. ( A , φ ) is called a ∗ -probabilityspace if A is a ∗ -algebra and φ ( xx ∗ ) ≥ x ∈ A . ( A , φ ) is called a W ∗ - probabilityspace if A is a W ∗ -algebra and φ is a normal state on it. We will not assume that φ isfaithful. The elements of A are called random variables. Let x ∈ A be a random variable,then its distribution is a linear functional µ x on C [ X ]( the algebra of complex polynomialsin one variable), defined by µ x ( P ) = φ ( P ( x )). Definition 2.2.
Let A be a W ∗ -algebra, a normal state φ on A is said to be non-degenerated if x = 0 whenever φ ( axb ) = 0 for all a, b ∈ A . By proposition 7.1.15 in [11], if φ is a non-degenerated normal state on A then the GNSrepresentation associated to φ is faithful. In this paper, we will work with W ∗ -probability WEIHUA LIU space with a non-degenerated normal state. The reason is that there is no non-trivial pairof boolean or monotonically independent random variables in W ∗ -probability spaces withfaithful states. See [16]. Definition 2.3.
Let I be an index set. The algebra of noncommutative polynomialsin | I | variables, C h X i | i ∈ I i , is the linear span of 1 and noncommutative monomials ofthe form X k i X k i · · · X k n i n with i = i = · · · 6 = i n ∈ I and all k j ’s are positive integers.For convenience, we will denote by C h X i | i ∈ I i the set of noncommutative polynomialswithout a constant term. Let ( x i ) i ∈ I be a family of random variables in a noncommutativeprobability space ( A , φ ). Their joint distribution is the linear functional µ : C h X i | i ∈ I i → C defined by µ ( X k i X k i · · · X k n i n ) = φ ( x k i x k i · · · x k n i n ) , and µ (1) = 1.In general, the joint distribution depends on the order of the random variables, e.g µ x,y may not equal µ y,x . According to our notation, µ x,y ( X X ) = φ ( xy ), but µ y,x ( X X ) = φ ( yx ). In this paper, our index set I is always an ordered set with order “ > ”e.g. N , Z . Definition 2.4.
Let ( A , φ ) be a noncommutative probability space. A family of (notnecessarily unital) subalgebras {A i | i ∈ I } of A is said to be boolean independent if φ ( x x · · · x n ) = φ ( x ) φ ( x ) · · · φ ( x n )whenever x k ∈ A i ( k ) with i (1) = i (2) = · · · 6 = i ( n ). The family of subalgebras {A i | i ∈ I } is said to be monotonically independent if φ ( x · · · x k − x k x k +1 · · · x n ) = φ ( x k ) φ ( x · · · x k − x k +1 · · · x n )whenever x j ∈ A i j with i = i = · · · 6 = i n and i k − < i k > i k +1 . A set of random variables { x i ∈ A| i ∈ I } is said to be boolean(monotonically) independent if the family of non-unital subalgebras A i , which are generated by x i respectively, is boolean(monotonically)independent.One refers to [8] for more details of boolean product and monotone product of randomvariables. In general, the framework for boolean independence and monotone indepen-dence is a non-unital algebra. Thereby, we will use the following version of operator valuedprobability spaces: Definition 2.5. Operator valued probability space
An operator valued probabilityspace ( 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 ab ] = b E [ a ] b , E [ b ] = b for all b , b , b ∈ B and a ∈ A . According to the definition in [27], we call E a conditionalexpectation from A to B if E is onto, i.e. E [ A ] = B . The elements of A are called randomvariables. Remark 2.6.
In free probability theory, A and B are assumed to be unital and share thesame unit ONCOMMUTATIVE SPREADABILITY 7
Definition 2.7.
For an algebra B , we denote by Bh X i the algebra which is freely gen-erated by B and the indeterminant X . Let 1 X be the identity of C h X i , then Bh X i is set of linear combinations of the elements in B and the noncommutative monomials b Xb Xb · · · b n − Xb n where b k ∈ B ∪ { C X } and n ≥
0. The elements in Bh X i arecalled B -polynomials. In addition, Bh X i denotes the subalgebra of Bh X i which doesnot contain a constant term in B , i.e. the linear span of the noncommutative monomials b Xb Xb · · · b n − Xb n where b k ∈ B ∪ { C X } and n ≥ Definition 2.8.
Let { x i } i ∈ I be a family of random variables in an operator valued prob-ability space ( A , B , E : A → B ), where A and B are not necessarily unital. { x i } i ∈ I is saidto be boolean independent over B if E [ p ( x i ) p ( x i ) · · · p n ( x i n )] = E [ p ( x i )] E [ p ( x i )] · · · E [ p n ( x i n )]whenever i , · · · , i n ∈ I , i = i = · · · 6 = i n and p , · · · , p n ∈ Bh X i . { x i } i ∈ I is said to be monotonically independent over B if E [ p ( x i ) · · · p k − ( x i k − ) p k ( x i k ) p k +1 ( x i k +1 ) · · · p n ( x i n )]= E [ p ( x i ) · · · p k − ( x i k − ) E [ p k ( x i k )] p k +1 ( x i k +1 ) · · · p n ( x i n )]whenever i , · · · , i n ∈ I , i = i = · · · 6 = i n , i k − < i k > i k +1 and p , · · · , p n ∈ Bh X i .Notice that there is another natural order “ < ”on I , i.e. a < b if b > a . Therefore,we can define another noncommutative independence relation. { x i } i ∈ I is said to be anti-monotonically independent with respect to E and index order “ > ” if { x i } i ∈ I is said to beanti-monotonically independent with respect to E and index order “ < ”. See more detailsin [19].2.1. Noncommutative distributional symmetries.
Recall that, in [32], Wang intro-duced the following quantum analogue of permutation groups:
Definition 2.9. A s ( n ) is defined as the universal unital C ∗ -algebra generated by elements( u i,j ) i,j =1 , ··· n such that we have • Each u i,j is an orthogonal projection, i.e. u ∗ ij = u ij = u ij for all i, j = 1 , · · · , n . • The elements in each row and column of u = ( u ij ) ni,j =1 , ··· ,n form a partition of unit,i.e. are orthogonal and sum up to 1: for each i = 1 , · · · , n and k = l we have u ik u il = 0 and u ki u li = 0; . and for each i = 1 , · · · , n we have n X k =1 u ik = 1 = n X k =1 u ki .A s ( n ) is a compact quantum group in sense of Woronowicz [33], with comultiplication,counit and antipode given by the formulas:∆ u i,j = n X k =1 u i,k ⊗ u k,j WEIHUA LIU ǫ ( u i,j ) = δ i,j S ( u i,j ) = u j,i It was shown that the this quantum structure can be used to characterize conditionallyfree independence [15].In [16], we modify the universal conditions of Wang’s quantum permutation groups: Byreplacing the condition associated with partitions of the unit by a condition associatedwith an invariant projection, we get the following universal algebras:
Quantum semigroups ( B s ( n ) , ∆ ): The algebra 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 wehave • each u i,j is an orthogonal projection, i.e. u ∗ i,j = u i,j = u i,j for all i, j = 1 , · · · , n . • u i,k u i,l = 0 and u k,i u l,i = 0whenever k = l . • For all 1 ≤ i ≤ n , P = n P k =1 u k,i P .There is a natural comultiplication ∆ : B s ( n ) → B s ( n ) ⊗ min B s ( n ) defined by∆ u i,j = n X k =1 u i,k ⊗ u k,j , ∆( P ) = P ⊗ P , ∆( I ) = I ⊗ I, where ⊗ min stands for the reduced C ∗ -tensor product. The existence of of these maps isgiven by the universal property of B s ( n ). Therefore, ( B s ( n ) , ∆)’s are quantum semigroupsin sense of So ltan [24]. These quantum structures can characterize conditionally booleanindependence, see more details in [16].3. Distributional symmetries for finite sequences of random variables
In this section, we will review two kinds of distributional symmetries which are spread-ability and partial exchangeability, in classical probability. In [12], we see that the dis-tributional symmetries can be defined for either finite sequences or infinite sequences.Moreover, each kind of distributional symmetry for infinite sequences of random objectsis determined by distributional symmetries on all its finite subsequences. For example,an infinite sequence of random variables is exchangeable iff all its finite subsequences areexchangeable. We will present distributional symmetries for finite sequences and thenintroduce their counterparts in noncommutative probability. In the first subsection, werecall notions of spreadability and partial exchangeability in classical probability andrephrase these notions in words of quantum maps. In the second subsection, we willintroduce counterparts of spreadability and partial exchangeability in noncommutativeprobability. Even though there are many interesting properties of partial exchangeability,we are not going to study it too much in this paper because the main problem we concernis about extended de Finetti type theorems for noncommutative spreadable sequences ofrandom variables. We will discuss relations between those noncommutative distributionalsymmetries in the next section.
ONCOMMUTATIVE SPREADABILITY 9
Spreadability and partial exchangeability.
Recall that in [13], a finite sequenceof random variables ( x , ..., x n ) is said to be spreadable if for any k < n , we have(2) ( x , ..., x k ) d = ( x l , ..., x l k ) , l < l < · · · < l k For fixed natural numbers n > k , it is mentioned in [5], the above relation can be describedin words of quantum family of maps in sense of Soltan [23]: Considering the space I k,n of increasing sequences I = (1 ≤ i < · · · < i k ≤ n ). For 1 ≤ i ≤ n , 1 ≤ j ≤ k , define f i,j : I k,n → C by: f i,j ( I ) = (cid:26) , i j = i , i j = i . If we consider I n,k as a discrete space, then the functions f i,j generate C ( I n,k ) by the Stone-Weierstrass theorem. Let C [ X , ..., X m ] be the set of commutative polynomials in m vari-ables. The algebra C ( I n,k ) together with an algebraic homomorphism α : C [ X , ..., X k ] → C [ X , ..., X n ] ⊗ C ( I k,n ) define by: α : X j = n X i =1 X i ⊗ f i,j , α (1) = 1 C ( I k,n ) defines a quantum family of maps from { , ..., k } to { , ..., n } .We can use this family of quantum maps to rephrase equation (2): Let µ x ,....,x n be thejoint distribution of ( x , ..., x n ). For fixed natural numbers n > k ,(3) µ x ,...,x k ( p )1 C ( I n,k ) = µ x ,...,x n ⊗ id C ( I n,k ) ( α ( p ))for all p ∈ C [ x , ..., x k ].For completeness, we provide a sketch of proof here: Suppose equation (2) holds. Let p = X i j · · · X i m j m be a monomial in C [ X , ..., X k ] such that 1 ≤ j < j < · · · < j m ≤ k and i , ..., i m are positive integers. Let I = (1 ≤ l < · · · < l k ≤ n ) be a point in I k,n . Then,the I -th component of µ x ,...,x k ( p )1 C ( I n,k ) is E [ x i j · · · x i m j m ]. The I -th component µ x ,...,x n ⊗ id C ( I n,k ) ( α ( p )) is n X s ,...,s m =1 E [ x i s · · · x i m s m ]( f s ,j · · · f s m ,j m )( I ) . According the definition of f i,j , ( f s ,j · · · f s m ,j m )( I ) is not vanished only if s t = l j t for all1 ≤ t ≤ m . Therefore, n X s ,...,s m =1 E [ x i s · · · x i l s l ]( f s ,j · · · f s m ,j m )( I ) = E [ x i l j · · · x i m l jm ] . Since 1 ≤ j < j < · · · < j m ≤ k and I is an in creasing sequence, we have 1 ≤ l j < · · · < l j m ≤ n . Hence, the I -components of the two sides of equation (5) are equal to eachother. Since I is arbitrary, equation (5) holds. By checking the I component of equation5, we can also show that (5) implies (2). We will say that ( ξ , ..., ξ n ) is ( n, k )-spreadableif ( x , ..., x n ) satisfies equation (5). Remark 3.1.
We see that the above ( n, k )-spreadability describes limited relations be-tween the mixed moments of ( x , ..., x n ). Once we fix n, k , the ( n, k )-spreadability givesno information about mixed moments which involve k + 1 variables. For example, let n = 4, k = 2 and assume that ( x , ..., x ) is a (4 , E [ x x x ] and E [ x x x ].We will call this kind of distributional symmetries partial symmetries because they justprovide information of part of mixed moments but not all.By using the idea of partial symmetries, we can define another family of distributionalsymmetries which is stronger than ( n, k )-spreadability but weaker than exchangeability. Definition 3.2.
For fixed natural numbers n > k , we say a sequence of random variables ( x , ..., x n ) is ( n, k ) -exchangeable if ( x , ..., x k ) d = ( x σ (1) , ..., x σ ( k ) ) , ∀ σ ∈ S n , where S n is the permutation group of n elements. This kind of exchangeability is called partial exchangeability. For more details, see [7].As well as ( n, k )-spreadability, we can rephrase partial exchangeability in words of quan-tum family of maps: Considering the space E n,k of length k sequences {I = ( i , ..., i k ) | ≤ i , ..., i k ≤ n, i j = i j ′ for j = j ′ } . For 1 ≤ i ≤ n , 1 ≤ j ≤ k , define g i,j : I n,k → C by: g i,j ( I ) = (cid:26) , i j = i, , i j = i. Given two different sequences I = ( i , ..., i k ) and I ′ = ( i ′ , ..., i ′ k ), there must exists a num-ber j such that i j = i ′ j . Then, we have that g i,i j ( I ) = 1 = 0 = g i,i j ( I ). Therefore, the setof functions { g i,j | i = 1 , ..., n ; j = 1 , ..., k } separates E n,k . According to Stone Weierstrasstheorem, the functions g i,j generate C ( E n,k ). Again, we can define a homomorphism α ′ : C [ X , ..., X k ] → C [ X , ..., X n ] ⊗ C ( E n,k ) by the following formulas: α ′ : X j = n X i =1 X i ⊗ g i,j , α ′ (1) = 1 C ( I k,n ) . Lemma 3.3.
Let µ x ,....,x n be the joint distribution of x , ..., x n . Then µ x ,...,x k ( p )1 C ( I n,k ) = µ x ,...,x n ⊗ id C ( I n,k ) ( α ( p )) for all p ∈ C [ X , ..., X k ] if and only if x , ..., x n is ( n, k ) exchangeable. The proof is similar the proof of ( n, k )-spreadability, we just need to check the valuesat all components of E n,k .3.2. Noncommutative analogue of partial symmetries.
Now, we turn to introducenoncommutative versions of spreadability and partial exchangeability. The pioneeringwork was done by Curran [5]. He defined a quantum version of C ( I n,k ) in analogue ofWang’s quantum permutation groups as following: Definition 3.4.
For k, n ∈ N with k ≤ n , the quantum increasing space A ( n, k ) is theuniversal unital C ∗ − algebra generated by elements { u i,j | ≤ i ≤ n, ≤ j ≤ k } such that1. Each u i,j is an orthogonal projection: u i,j = u ∗ i,j = u i,j for all i = 1 , ..., n ; j =1 , ..., k . ONCOMMUTATIVE SPREADABILITY 11
2. Each column of the rectangular matrix u = ( u i,j ) i =1 ,...,n ; j =1 ,...,k forms a partition ofunity: for 1 ≤ j ≤ k we have n P i =1 u i,j = 1.3. Increasing sequence condition: u i,j u i ′ ,j ′ = 0 if j < j ′ and i ≥ i ′ . Remark 3.5.
Our notation is different from Curran’s, we use A i ( n, k ) instead of his A i ( k, n ) for our convenience.For any natural numbers k < n , in analogue of coactions of A s ( n ), there is a unital ∗ -homomorphism α n,k : C h X , ..., X k i → C h X , ..., X n i ⊗ A i ( n, k ) determined by: α n,k ( X j ) = n X i =1 X i ⊗ u i,j . The quantum spreadability of random variables is defined as the following:
Definition 3.6.
Let ( A , φ ) be a noncommutative probability space. A finite orderedsequence of random variables ( x i ) i =1 ,...,n in A is said to be A i ( n, k )-spreadable if theirjoint distribution µ x ,...,x n satisfies: µ x ,...,x n ( p )1 A i ( n,k ) = µ ⊗ id A i ( n,k ) ( α n,k ( p )) , for all p ∈ C h X , ..., X k i . ( x i ) i =1 ,...,n is said to be quantum spreadable if ( x i ) i =1 ,...,n is A i ( n, k )-spreadable for all k = 1 , ..., n − Remark 3.7.
In [5], Curran studied sequences of C ∗ -homomorphisms which are moregeneral than random variables. For consistency, we state his definitions in words ofrandom variables. It is a routine to extend our work to the framework of sequences of C ∗ -homomorphisms. Recall that in [16], by replacing the condition associated with partitions of the unity ofWang’s quantum permutation groups, we defined a family of quantum semigroups withinvariant projections. With a natural family of coactions, we defined invariance conditionswhich can characterize conditional boolean independence. Here, we can modify Curran’squantum increasing spaces in the same way:
Definition 3.8.
For k, n ∈ N with k ≤ n , the noncommutative increasing space B i ( k, n )is the unital universal C ∗ − algebra generated by elements { u ( b ) i,j | ≤ i ≤ n, ≤ j ≤ k } andan invariant projection P such that1. Each u ( b ) i,j is an orthogonal projection: u ( b ) i,j = ( u ( b ) i,j ) ∗ = ( u ( b ) i,j ) for all i = 1 , ..., n ; j =1 , ..., k .2. For 1 ≤ j ≤ k we have n P i =1 u ( b ) i,j P = P .3. Increasing sequence condition: u ( b ) i,j u ( b ) i ′ ,j ′ = 0 if j < j ′ and i ≥ i ′ .The same as A i ( n, k ), there is a unital ∗ -homomorphism α ( b ) n,k : C h X , ..., X k i → C h X , ..., X n i⊗ B i ( n, k ) determined by: α ( b ) n,k ( x j ) = n X i =1 x i ⊗ u ( b ) i,j As boolean exchangeability defined in [16], we have
Definition 3.9.
A finite ordered sequence of random variables ( x i ) i =1 ,...,n in ( A , φ ) is saidto be B i ( n, k )-spreadable if their joint distribution µ x ,...,x n satisfies: µ x ,...,x n ( p ) P = P µ ⊗ id B i ( n,k ) ( α ( b ) n,k ( p )) P , for all p ∈ C h X , ..., X k i . ( x i ) i =1 ,...,n is said to be boolean spreadable if ( x i ) i =1 ,...,n is B i ( n, k )-spreadable for all k = 1 , ..., n − B i ( k, n ) is an increasing space of boolean type, because we can derivean extended de Finetti type theorem for boolean independence.Recall that, in [9], Franz showed some relations between free independence, mono-tone independence and boolean independence via Bo˙zejko, Marek and Speicher’s two-states free products[3]. We can see that monotone product is “between” free product andboolean product. From this viewpoint of Franz’s work, we may hope to define a kindof “spreadability”for monotone independence by modifying quantum spreadability andboolean spreadability. Notice that there are at least two ways to get quotient algebrasof B i ( k, n )’s such that the P -invariance condition of the quotient algebras is equivalentquantum spreadability:1. Require P to be the unit of the algebra.2. Let P j = n P i =1 u i,j , require P j ′ u ij = u ij P j ′ for all 1 ≤ j, j ′ ≤ k and 1 ≤ i ≤ n .To define the monotone increasing spaces, we modify the second condition a little: Definition 3.10.
For fixed n, k ∈ N and k < n , a monotone increasing sequence space M i ( n, k ) is the universal unital C ∗ -algebra generated by elements { u ( m ) i,j } i =1 ,...,n ; j =1 ,...,k
1. Each u i,j is an orthogonal projection;2. Monotone condition: Let P j = n P i =1 u ( m ) i,j , P j u ( m ) i ′ j ′ = u i ′ j ′ if j ′ ≤ j. n P i =1 u ( m ) i,j P = P for all 1 ≤ j ≤ k.
4. Increasing condition: u ( m ) i,j u ( m ) i ′ ,j ′ = 0 if j < j ′ and i ≥ i ′ . We see that P plays the role as the invariant projection P in the boolean case. Forconsistency, we denote P by P . Then, we can define a P -invariance condition associatedwith M i ( n, k ) in analogy with B i ( n, k ): For fixed n, k ∈ N and k < n , there is a uniqueunital ∗ - isomorphism α ( m ) n,k : C h X , ..., X k i → C h X , ..., X n i ⊗ M i ( n, k ) such that α ( m ) n,k ( X j ) = n X i =1 X i ⊗ u ( m ) i,j . The existence of such a homomorphism is given by the universality of C h X , ..., X k i . Definition 3.11.
A finite ordered sequence of random variables ( x i ) i =1 ,...,n in ( A , φ ) issaid to be M i ( n, k )-invariant if their joint distribution µ x ,...,x n satisfies: µ x ,...,x k ( p ) P = P µ x ,...,x n ⊗ id M i ( n,k ) ( α ( m ) n,k ( p )) P , for all p ∈ C h X , ..., X k i . ( x i ) i =1 ,...,n is said to be monotonically spreadable if it is M i ( n, k )-invariant for all k = 1 , ..., n − . ONCOMMUTATIVE SPREADABILITY 13
We will see that these invariance conditions can characterize conditionally Monotoneindependence in a proper framework.As remark 2.3 in [5], a first question to our definitions is whether A i ( n, k ), B i ( n, k ), M i ( n, k ) exist. In [5], Curran has showed several nontrivial representations of A i ( n, k ).In the following, we provide a family of presentations of A i ( n, k ), B i ( n, k ), M i ( n, k ) for n > k : Fix natural numbers n > k , let l , ..., l k ∈ N such that l + · · · + l k = n. We denote by H i a l i -dimensional Hilbert spaces with orthonormal basis { e ( i ) j | j = 1 , ..., l i } .Let I l i be the unit of the algebra B ( H l i ), P e ( li ) j be the one dimensional orthogonal pro-jection onto C e ( l i ) j , P i be the one dimensional projection onto C P j e ( l i ) j . Consider thefollowing matrix: P , · · · P ,l · · · P , · · · P ,l · · ·
00 0 · · · · · · P k, ... ... . . . ...0 0 · · · P k,l k . We see that the entries of the matrix satisfy the increasing condition of spaces of increas-ing sequences. By choosing proper projections P i,j , we will get representations for ouruniversal algebras:Quantum family of increasing sequences: For each 1 ≤ j ≤ k , the algebra generated by { P e ij | i = 1 , ..., l j } is isomorphic to C ∗ ( Z l j ). The reduced free product ∗ kj =1 Z l i is a quotientalgebra of A i ( n, k ). One can define a C ∗ -homomorphism π from A i ( n, k ), such that π ( u i,j ) = the image of P e ( li ) j ′ in ∗ kj =1 C ∗ ( Z l j ) if 0 < j ′ = j − i − P l = m l m ≤ l i C ∗ -homomorphism π from B i ( n, k ) into B ( k N i =1 H i ) such that π ( u i,j ) = i − N m =1 P l m ⊗ P e ( li ) j ′ k N m = i +1 P l m if 0 < j ′ = j − i − P l = m l m ≤ l i Monotone family of increasing sequences: One can define a C ∗ -homomorphism π from M i ( n, k ) to B ( k N i =1 H i ) π ( u i,j ) = i − N m =1 I l m ⊗ P e ( li ) j ′ k N m = i +1 P l m if 0 < j ′ = j − i − P l = m l m ≤ l i A i ( n, k ), B i ( n, k ) and M i ( n, k ) respectively. Since the above representation of M i ( n, k ) plays animportant role in proving our main theorems, we summarize it as the following proposition: Proposition 3.12.
For fixed natural numbers n > k . Let l , ..., l k ∈ N such that l + · · · + l k = n. Let H i be a l i -dimensional Hilbert spaces with orthonormal basis { e ( i ) j | j = 1 , ..., l i } and I l i be the unit of the algebra B ( H l i ) , P e ( li ) j be the one dimensional orthogonal projectiononto C e ( l i ) j , P i be the one dimensional projection onto C P j e ( l i ) j . Then, there is a C ∗ -homomorphism π : M i ( n, k ) → B ( H ⊗ · · · ⊗ H k ) defined as follows: π ( u i,j ) = i − N m =1 I l m ⊗ P e ( li ) j ′ k N m = i +1 P l m if < j ′ = j − i − P l = m l m ≤ l i if otherwise Also, we need the following property in the future:
Lemma 3.13.
Given natural numbers n , n , n, k ∈ N such that n > k . Let ( u i,j ) i =1 ,...,n ; j =1 ,...,k be the standard generators of M i ( n, k ) and ( u ′ i,j ) i =1 ,...,n + n + n ; j =1 ,...,k + n + n be the standardgenerators of M i ( n + n + n , k + n + n ) . Then, there exists a C ∗ -homomorphism π : M i ( n + n + n , k + n + n ) → M i ( n, k ) such that π ( u ′ i,j ) = δ i,j P if ≤ i ≤ n δ i,j if n + 1 ≤ i ≤ n + n , n ≤ j ≤ n + k if n + 1 ≤ i ≤ n + n , j ≤ n or j > n + kδ i − n,j − k I if i ≥ n + n + 1 where P = P = n P i =1 u i, and I is the identity of M i ( n, k ) .Proof. We can see that the matrix form of ( π ( u ′ i,j )) i =1 ,...,n + n + n ; j =1 ,...,k + n + n is P · · · · · · · · · · · · P · · · · · · · · · u , · · · u ,k · · · · · · u n, · · · u n,k · · · · · · · · · I · · · · · · · · · · · · I ONCOMMUTATIVE SPREADABILITY 15
It is easy to check that the coordinates of the above matrix satisfy the universal conditionsof M i ( n + n + n , k + n + n ). The proof is complete. (cid:3) In analogue of the ( n, k )-partial exchangeability, we can define noncommutative versionsof partial exchangeability for free independence and boolean independence:
Definition 3.14.
For k, n ∈ N with k ≤ n , the quantum space A l ( n, k ) is the universalunital C ∗ -algebra generated by elements { u ij | ≤ i ≤ n, ≤ j ≤ k } such that1. Each u ij is an orthogonal projection: u ij = u ∗ ij = u ij .2. Each column of the rectangular matrix u = ( u ij ) forms a partition of unity: for1 ≤ j ≤ k we have n P i =1 u ij = 1. Remark 3.15. A i ( n, k ) is a quotient algebra of A l ( n, k ) , because the definition of A i ( n, k ) has one more restriction than A l ( n, k ) ’s. A l ( n, n ) is exactly Wang’s quantum permutationgroup A s ( n ) . There is a well defined unital algebraic homomorphism α ( fp ) n,k : C h X , ...X k i → C h X , ...X n i ⊗ A l ( n, k )such that α ( fp ) n,k X j = n X i =1 X i ⊗ u i,j where 1 ≤ j ≤ k . The distributional symmetry associated with this quantum structureis: Definition 3.16.
Let x , ..., x n ∈ ( A , φ ) be a sequence of n -noncommutative randomvariables, k ≤ n be a positive integer. We say the sequence is ( n, k )-quantum exchangeableif µ x ,...x k ( p ) = µ x ,...x n ⊗ id A l ( n,k ) ( α ( fp ) n,k ( p )) , for all p ∈ C h X , ..., X k i , where µ x ,...,x j is the joint distribution of x , ...x j with respect to φ for j = k, n .By modifying the second universal condition of A l ( n, k ), we can define a boolean versionof partial exchangeability: Definition 3.17.
For natural numbers k ≤ n , B l ( n, k ) is the non-unital universal C ∗ -algebra generated by the elements { u i,j } i =1 ,...,n ; j =1 ,...,k and an orthogonal projection P , suchthat (1) u i,j is an orthogonal projection, i.e. u i,j = u ∗ i,j = u i,j . (2) n P i =1 u i,j P = P for all ≤ j ≤ k . Remark 3.18. B l ( n, n ) is exactly the boolean exchangeable quantum semigroup B s ( n ) . There is a well defined unital algebraic homomorphism α ( bp ) n,k : C h X , ...X k i → C h X , ...X n i ⊗ B l ( n, k ) such that α ( bp ) n,k X j = n X i =1 X i ⊗ u i,j where 1 ≤ j ≤ k . The distributional symmetry associated with this quantum structureis: Definition 3.19.
Let x , ..., x n ∈ ( A , φ ) be a sequence of n -noncommutative randomvariables, k ≤ n be a positive integer. We say the sequence is ( n, k )-boolean exchangeableif µ x ,...x k ( p ) P = P µ x ,...x n ⊗ id B l ( n,k ) ( α ( bp ) n,k ( p )) P for all p ∈ C h X , ..., X k i , where µ x ,...,x j is the joint distribution of x , ...x j with respect to φ . Now, we turn to define our noncommutative distributional symmetries for infinite se-quences of random variables. In this paper, our infinite ordered index set I would beeither N or Z . Definition 3.20.
Let ( A , φ ) be a noncommutative probability space, I be an ordered in-dex set and ( x i ) i ∈ I a sequence of random variables in A . ( x i ) i ∈ I is said to be monotonically(boolean) spreadable if all its finite subsequences ( x i , ..., x i l ) are monotonically(boolean)spreadable. Lemma 3.21.
Let ( x , ..., x n +1 ) be a monotonically spreadable sequence of random vari-ables in ( A , φ ) . Then, all its subsequences are monotonically spreadable.Proof. It suffices to show that the subsequence ( x , ..., x l − , x l +1 , ..., x n +1 ) is monotonicallyspreadable for all 1 ≤ l ≤ n . If we denote ( x , ..., x l − , x l +1 , ..., x n +1 ) by ( y , ..., y n ), thenwe need to show that ( y , ..., y n ) is M i ( n, k )-spreadable for all k < n .Fix k < n , let { u i,j } i =1 ,...,n ; j =1 ,...,k be the set of generators of M i ( n, k ) and { P i,j } i =1 ,...,n +1; j =1 ,...,k +1 be an n + 1 by k + 1 matrix with entries in M i ( n, k ) such that P i,j = u ( m ) i,j if 1 ≤ i, j < lu ( m ) i − ,j if 1 ≤ j < l, i ≥ lu ( m ) i,j − if 1 ≤ i < l, j ≥ lu ( m ) i − ,j − if i, j ≥ l . It is a routine to check that the set { P i,j } i =1 ,...,n +1; j =1 ,...,k +1 satisfies the universal condi-tions of M i ( n +1 , k +1). Therefore, there exists a C ∗ -homomorphism ψ : M i ( n +1 , k +1) → M i ( n, k ) such that ψ ( u ′ i,j ) = P i,j where { u ′ i,j } is the set of generators of M i ( n + 1 , k + 1). Now, we need a convenientnotation: σ ( i ) = (cid:26) i if 1 ≤ i < li + 1 if i ≥ l ONCOMMUTATIVE SPREADABILITY 17
Then, P σ ( i ) ,σ ( j ) = u ( m ) i,j and y i = x σ ( i ) for all i = 1 , ..., n and j = 1 , ...., k + 1. For allmonomial X j · · · X j m ∈ C h X , ..., X k i , let P ′ = n P i =1 u ′ i, and P be the invariance projectionof M i ( n, k ), we have µ y ,...,y n ( X j · · · X j m ) P = P µ x ,...,x n +1 ( X σ ( j ) · · · X σ ( j m ) ) ψ ( P ′ ) P = P ψ ( µ x ,...,x n +1 ( X σ ( j ) · · · X σ ( j m ) ) P ′ ) P = P ψ ( µ x ,...,x n +1 ⊗ id M i ( n +1 ,k +1) ( n +1 P i ,...,i m =1 X i · · · X i m ⊗ u ′ i ,σ ( j ) · · · u ′ i m ,σ ( j m ) )) P Notice that u ′ l,σ ( j ) = 0 since σ ( j ) never equals l , the quality can be written as the following: µ y ,...,y n ( X j · · · X j m ) P = P ψ ( µ x ,...,x n +1 ⊗ id M i ( n +1 ,k +1) ( n P i ,...,i m =1 X σ ( i ) · · · X σ ( i m ) ⊗ u ′ σ ( i ) ,σ ( j ) · · · u ′ σ ( i m ) ,σ ( j m ) )) P = P n P i ,...,i m =1 µ x ,...,x n +1 ( X σ ( i ) · · · X σ ( i m ) ) ψ ( u ′ σ ( i ) ,σ ( j ) · · · u ′ σ ( i m ) ,σ ( j m ) ) P = n P i ,...,i m =1 µ y ,...,y n ( X i · · · X i m ) P u ( m ) i ,j · · · u ( m ) i m ,j m P which completes the proof. (cid:3) Then, we have
Proposition 3.22.
Let ( A , φ ) be a noncommutative probability space and ( x i ) i ∈ Z be asequence of random variables in A . Then, ( x i ) i ∈ Z is monotonically (quantum, boolean)spreadable if and only if ( x i ) i = − n, − n +1 ,...,.n − ,n is monotonically (quantum, boolean) spread-able for all n .Proof. It is sufficient to prove “ ⇐ ”. Given a subsequence ( x i , ..., x i l ) of ( x i ) i ∈ Z , thereexits an n such that − n < i , ..., i l < n . Since ( x i ) i = − n, − n +1 ,...,.n − ,n is monotonicallyspreadable, by Lemma 3.21, we have that ( x i , ..., x i l ) is monotonically spreadable. Thesame to quantum spreadability and boolean spreadability. (cid:3) Relations between noncommutative probabilistic symmetries
In this section, we will study relations between the noncommutative distributionalsymmetries which are introduced in the previous section.It is well know that every C ∗ -algebra admits a faithful representation. Fix n, k ∈ N ,such that 1 ≤ k ≤ n −
1. Let Φ be a faithful representation of B l ( n, k ) into B ( H ) forsome Hilbert space H . For convenience, we denote Φ( u i,j ) by u i,j and Φ( P ) by P .According to the definition of B l ( k, n ), u i,j ’s and P are orthogonal projections in B ( H ).Let Q i = k P j =1 u i,j for 1 ≤ i ≤ n . In [11], we know that the set P ( H ) of orthogonalprojections on H is a lattice with respect to the usual order ≤ on the set of selfadjointoperators, i.e. two selfadjoint operators A and B , A ≤ B iff B − A is a positive operator. Now, we need the following notation in our construction. Given two projections E and F , we denote by E ∨ F the minimal orthogonal projection in P ( H ), such that E ∨ F isgreater or equal to E and F . E ∨ F is well define and unique, we call it the supreme of E and F . It is easy to see that ( E ∨ F ) E = E and ( E ∨ F ) F = F We turn to define a sequence of orthogonal projections { P ′ i } i =1 ,...,n in P ( H ) as follows: P ′ = I − Q ,P ′ i = I − P ′ ∨ · · · ∨ P ′ i − ∨ Q i for 2 ≤ i ≤ n . Lemma 4.1.
Given a nonzero vector v ∈ H , E and F are two orthogonal projections on H . If ( E ∨ F ) x = x and Ex = 0 , then F x = x . According the construction of { P ′ i } ≤ i ≤ n , we have P ′ i P ′ j = δ i,j P ′ i and P ′ i u i,j = 0for all 1 ≤ i ≤ n and 1 ≤ j ≤ k . Lemma 4.2. n P i =1 P ′ i = I , where I is the identity in B ( H ) .Proof. Since the orthogonal projections P ′ i are orthogonal to each other, n P i =1 P ′ i is an or-thogonal projection which is less than or equal to the identity I . If n P i =1 P ′ i < I , then thereexists a nonzero vector v ∈ H such that n X i =1 P ′ i v = 0 . Then, we have 0 = P ′ i x = ( I − P ′ ∨ · · · ∨ P ′ i − ∨ P i ) x or say ( P ′ ∨ · · · ∨ P ′ i − ∨ P i ) x = x for all i . Since P ′ m x = 0 for all 1 ≤ m ≤ i −
1, by Lemma4.1, P i x = x . Then, we have nx = n P i =1 P i x = n P i =1 k P j =1 u i,j x = k P j =1 ( n P i =1 u i,j x ) , ONCOMMUTATIVE SPREADABILITY 19 which implies that n is in the spectrum of k P j =1 n P i =1 u i,j . Notice that, to every 1 ≤ j ≤ k , n P i =1 u i,j ≤ I since they are orthogonal projections and orthogonal to each other. Therefore,0 ≤ k X j =1 n X i =1 u i,j ≤ k X j =1 I ≤ kI. It contradicts to the implication above. The proof is complete. (cid:3)
Corollary 4.3. n P i =1 P ′ i P = P . Proposition 4.4.
Let ( A , φ ) be a noncommutative probability space, ( x i ) i =1 ,...,n is a finiteordered sequence of random variables in A . For fixed n > k , the joint distribution µ x ,...,x n is A l ( n, k ) -invariant if it is A l ( n, k + 1) -invariantProof. Let { u ij | ≤ i ≤ n, ≤ j ≤ k } the set of standard generators of A l ( n, k ), Φ be afaithful representation of A l ( n, k ) into B ( H ). With the above construction, we can define { u ′ i,j } i =1 ,...,n ; j =1 ,...,k +1 as following: u ′ i,j = (cid:26) Φ( u i,j ) if j ≤ kP ′ j if j = k By Lemma 4.2, { u ′ i,j } i =1 ,...,n ; j =1 ,...,k +1 satisfies the universal conditions for A l ( n, k + 1). Let { u ′′ ij | ≤ i ≤ n, ≤ j ≤ k + 1 } be the set of standard generators of A l ( n, k + 1). thenthere exists a C ∗ − homomorphism Φ : A l ( n, k + 1) → B ( H ) such that:Φ ′ ( u ′′ ij ) = u ′ i,j . Therefore, Φ − Φ ′ defines a unital C ∗ − homomorphismΦ − Φ ′ : C ∗ − alg { u ′ i,j | ≤ i ≤ n, ≤ j ≤ k } → A l ( n, k )such that Φ − Φ ′ ( u ′ i,j ) = u i,j for all 1 ≤ i ≤ n, ≤ j ≤ k .If µ x ,...,x n is A l ( n, k + 1)-invariant, then µ x ,...,x k +1 ( p ) = µ x ,...,x k ⊗ id A l ( n,k +1) ( α ( fp ) n,k +1 ( p ))for all p ∈ C h X , ..., X k +1 i . Let p = X j · · · X j l ∈ C h X , ..., X k i , then we have µ x ,...,x k ( P )1 A ( n,k ) Φ − Φ ′ ( µ x ,...,x k +1 ( P )1 A ( n,k +1) ))= Φ − Φ ′ ( µ x ,...,x n ⊗ id A l ( n,k +1) ( α ( fp ) n,k +1 ( X j · · · X j l ))= Φ − Φ ′ ( µ x ,...,x n ⊗ id A l ( n,k +1) ( n P i ,...,i l X i · · · X i l ⊗ u ′ i ,j · · · u ′ i l ,j l )= µ x ,...,x n ⊗ id A l ( n,k ) ( n P i ,...,i l X i · · · X i l ⊗ u i ,j · · · u i l ,j l )= µ x ,...,x n ⊗ id A l ( n,k ) ( α ( fp ) n,k ( P )) Since p is an arbitrary monomial, the proof is complete. (cid:3) The same, we can show that
Corollary 4.5. µ x ,...,x n is B l ( n, k ) -invariant if it is B l ( n, k + 1) -invariant Lemma 4.6. µ x ,...,x n is ( n, k ) -quantum spreadable if it is A l ( n, k ) -invariant.Proof. Let { u i,j } i =1 ,...,n ; j =1 ,...,k be generators of A i ( n, k ) and { u ′ i,j } i =1 ,...,n ; j =1 ,...,k be genera-tors of A l ( n, k ). Then, there is a well defined C ∗ -homomorphism β : A l ( n, k ) → A i ( n, k )such that β ( u i,j = u ′ i,j ). The existence of β is given by the universality of A l ( n, k ). Since µ x ,...,x n is A l ( n, k )-invariant, for all monomials p = X i · · · X i m ∈ C h X , ..., X k i , we have µ x ,...,x k ( p )1 A l ( n,k ) = µ x ,...x n ⊗ id A l ( n,k ) ( α ( fp ) n,k ( p )) = X j ,...,j m φ ( x j · · · x j m ) u j ,i · · · u j m ,i m . Apply β on both sides of the above equation, we have µ x ,...,x k ( p )1 A i ( n,k ) = X j ,...,j m φ ( x ′ j · · · x j m ) u ′ j ,i · · · u j m ,i m = µ x ,...x n ⊗ id A i ( n,k ) ( α n,k ( p )) . The proof is complete. (cid:3)
The same, we have
Corollary 4.7. µ x ,...,x n is ( n, k ) -boolean spreadable if it is B l ( n, k ) -invariant. Corollary 4.8. ( x , ..., x n ) is boolean spreadable if it is boolean exchangeable. ( x , ..., x n ) is quantum spreadable if it is quantum exchangeable. In summary, for fixed n, k ∈ N such that k < n , we have the following diagrams: B ( n, n ) inv / / (cid:15) (cid:15) B l ( n, k ) inv / / (cid:15) (cid:15) B i ( n, k ) inv (cid:15) (cid:15) M i ( n, k ) inv (cid:15) (cid:15) A ( n, n ) inv / / A l ( n, k ) inv / / A i ( n, k ) inv and Booolean exchangeability / / (cid:15) (cid:15) Boolean spreadability (cid:15) (cid:15)
M onotone spreadability (cid:15) (cid:15)
Quantum exchangeability / / quantum spreadability The arrow “condtion a) → condition b)” means that condition a) implies condition b). ONCOMMUTATIVE SPREADABILITY 21 Monotonically equivalent sequences
In order to study monotone spreadability, we need find relations between mixed mo-ments of monotonically spreadable sequences of random variables. Since all the mixedmoments can be denoted by finite sequences of indices, we will turn to study finite se-quences of ordered indices. In this section, we introduce an equivalent relation, which hasa deep relation with monotone spreadability, on finite sequences of ordered indices.
Definition 5.1.
Given two pairs of integers ( a, b ), ( c, d ) , we say these two pairs have thesame order if a − b, c − d are both positive or negative or 0.For example, (1 ,
2) and (3 ,
5) have the same order but (1 ,
2) and (5 ,
3) do not have thesame order.
Definition 5.2.
Let Z be the set of integers with natural order and Z L = Z × · · · × Z bethe set of finite sequences of length L . We define a partial relation ∼ m on Z L . Given twosequences of indices I = { i , ...., i L } , J = { j , ..., j L } ∈ Z L . If for all 1 ≤ l < l ≤ L suchthat i l > max { i l , i l } for all l < l < l , ( i l , i l ) and ( j l , j l ) have the same order, thenwe denote I ∼ m J . Example: (5 , , ∼ m (5 , ,
5) but (5 , , m (5 , , Remark 5.3.
In general, the relation can be defined on any ordered set but not only Z .We will show this partial relation is exactly an equivalence relation on the set of finitesequences of ordered indices. It follows the definition that ( i l , i l +1 ) and ( j l , j l +1 ) have the same order for all 1 ≤ l < L if I ∼ m J .Now we turn to show that ∼ m is actually an equivalent relation. To achieve it, we needto show that the relation ∼ m is reflexive, symmetric and transitive.(Reflexivity) First, reflexivity is obvious, because a pair ( i l , i l ) always has the same orderwith itself. Lemma 5.4. (Symmetry) Let I = { i , ...., i L } , J = { j , ..., j L } ∈ Z L such that I ∼ m J ,then J ∼ m I .Proof. Suppose that
J 6∼ m I . Then, there exist two natural numbers 1 ≤ l < l ≤ L such that j l > max { j l , j l } for all l < l < l , but ( j l , j l ) and ( i l , i l ) do not have the same order. Fix l , we choosethe smallest l which satisfies the above property. Notice that I ∼ m J , ( j l , j l +1 ) and( i l , i l +1 ) have the same order, then l = l + 1 . According to our assumption, we have j l ′ > max { j l , j l } for l < l ′ < l .Suppose that there exists an l between l and l such that i l ≤ max { i l , i l } . Without loss of generality, we assume that i l ≥ i l , then i l ≤ i l . Again, among these l , we choose the smallest one. Then, we have i l > i l ≥ i l for l < l < l . Since
I ∼ m J , ( i l , i l ) and ( j l , j l ) must have the same order, but i l ≥ i l and i l < j l .It contradicts the existence of our l . Hence, i l ′ > max { i l , i l } for all l < l ′ < l . Itfollows that ( i l , i l ) and ( j l , j l ) have the same order. But, it contradicts our originalassumption. Therefore, J ∼ m I . (cid:3) Lemma 5.5.
Given two sequences I = { i , ...., i L } , J = { j , ..., j L } ∈ Z L such that I ∼ m J . Let ≤ l < l ≤ L such that i l > max i l , i l for all l < l < l . Then, wehave j l > max { j l , j l } for all l < l < l .Proof. If the statement is false, then there exists l between l and l such that j l ≤ max { j l , j l } . Suppose j l ≥ j l , then j l ≤ j l . Among all these l , we take the smallest one. Then, we have j l > max { j l , j l } for all l < l < l . By Lemma5.4, J ∼ m I since I ∼ m J . Therefore, ( j l , j l ) and ( i l , i l )must have the same order which means i l ≥ i l . This is a contradiction. If we assume that j l < j l , then we just need to considerthe largest one among those l and we will get the same contradiction. The proof iscomplete. (cid:3) Lemma 5.6. (Transitivity)Given three sequences I = { i , ...., i L } , J = { j , ..., j L } , Q = { q , ...q L } ∈ Z L , such that I ∼ m J and J ∼ m Q , then I ∼ m J Proof.
Given 1 ≤ l < l ≤ L such that i l > max { i l , i l } for all l < l < l . By Lemma 5, we have j l > max { j l , j l } for all l < l < l . It follows the definition that ( i l , i l ), ( j l , j l ) have the same order and( j l , j l ), ( q l , q l ) have the same order. Therefore, ( i l , i l ), ( q l , q l ) have the same order.Since l , l are arbitrary, it completes the proof. (cid:3) ONCOMMUTATIVE SPREADABILITY 23
By now, we have shown that the relation ∼ m is reflexive, symmetric and transitive.Therefore, we have Proposition 5.7. ∼ m is an equivalence relation on Z L . As we mentioned before, Z can be replaced by any ordered set I . When there isno confusion, we always use ∼ m to denote the monotone equivalence relation on I L forordered set I and positive integers L . For example, I can be [ n ] = { , ..., n } . Definition 5.8.
Let I = ( i , ..., i L ) be a sequence of ordered indices. An ordered sub-sequence ( i l ′ , ..., i l ′ ) of I is called an interval if the sequence contains all the elements i l ′ whose position l ′ is between l ′ and l ′ . An interval ( i l ′ , ..., i l ′ ) of I is called a crest if i l ′ = i l ′ +1 · · · = i l ′ > max { i l ′ − , i l ′ +1 } . In addition , we assume that i < i and i L > i L +1 even though i , i L +1 are not in I . Example: (1 , , ,
4) has one crest of length 1, namely (4). (1 , , , , , , ,
5) has 3crests (2) , (4 , , (5) and 2 is the first peak of the sequence. (1 , , , ,
1) has one crest(1 , , , ,
1) which is the sequence itself, because we assumed i < i and i < i . Lemma 5.9.
Given I = ( i , ..., i L ) ∈ Z L , I has at least one crest.Proof. Since I consists of finite elements, it has a maximal one, i.e. i l such that i l ≥ i l ′ for 1 ≤ l ′ ≤ L . It is obvious that i l must be contained in an interval ( i l ′ , ..., i l ′ ) such that i l ′ = i l ′ +1 · · · = i l ′ = i l ′ and i l ′ > max { i l ′ − , i l ′ +1 } . Therefore, I contains a crest. (cid:3) Lemma 5.10.
Given two index sequences I , J ∈ Z L such that I ∼ m J . If ( i l ′ , ..., i l ′ ) isa crest of I , then ( j l ′ , ..., j l ′ ) is a crest of J Proof.
Since
I ∼ m J , all consecutive pairs ( i l , i l +1 ) and ( j l , j l +1 ) have the order. Accord-ing to the definition, we have i l ′ − < i l ′ = i l ′ +1 · · · = i l ′ > j l ′ +1 If follows that j l ′ − < j l ′ = j l ′ +1 · · · = j l ′ > j l ′ +1 , thus ( j l ′ , ..., j l ′ ) is a crest of J . (cid:3) Now, we will introduce some ∼ m preserving operations on index sequences. The firstoperation is to remove a crest from a sequence. Let ( i l ′ , ..., i l ′ ) be an interval of I =( i , ..., i L ), we denote by I \ ( i l ′ , ..., i l ′ ) the new sequence ( i , ..., i l ′ − , i l ′ +1 , ..., i L ). Wedenote by the empty set ∅ = I \ I and we assume ∅ ∼ m ∅ . Lemma 5.11.
Let I = ( i , ...., i l ) , J = ( j , ..., j L ) ∈ Z L such that I 6∼ m J . If ( i l ′ , ..., i l ′ ) is a crest of I and ( j l ′ , ..., j l ′ ) is a crest of J . Then, I \ ( i l ′ , ..., i l ′ ) m J \ ( j l ′ , ..., j l ′ ) Proof. If I \ ( i l ′ , ..., i l ′ ) is empty, then J \ ( j l ′ , ..., j l ′ ) must be empty because the lengthsof I , J are the same. If I \ ( i l ′ , ..., i l ′ ) is non empty, then I can be written as( i , ..., i l ′ , ..., i l ′ , ..., i L )and I \ ( i l ′ , ..., i l ′ ) = ( i , ..., i l ′ − , i l ′ +1 , ..., i L ) = ( i ′ , ..., i ′ l ′ − , i ′ l ′ , ..., i ′ L − l ′ + l ′ − )and J \ ( j l ′ , ..., j l ′ ) = ( j , ..., j l ′ − , j l ′ +1 , ..., j L ) = ( j ′ , ..., j ′ l ′ − , j ′ l ′ , ..., j ′ L − l ′ + l ′ − )For any indices 1 ≤ l < l < L − l ′ + l ′ − i l > max { i ′ l , i ′ l } :If l , l ≤ l ′ − l , l ≥ l ′ , then ( i ′ l , ..., i ′ l ) is an interval of I . Since I ∼ m J , ( i ′ l , i ′ l )and ( j ′ l , j ′ l ) have the same order .If l < l ′ ≤ l , then i ′ l = i l + l ′ − l ′ +1 . We have i l > i l ′ − ≥ max { i ′ l , i ′ l } for all l ′ ≤ l ≤ l ′ . It follows that i l > max { i l , i l } for all l < l < l + l ′ − l ′ + 1. It follows that ( i l , i l + l ′ − l ′ +1 ) and ( j l , j l + l ′ − l ′ +1 ) havethe same order. Thus, ( i ′ l , i ′ l ) and ( j ′ l , j ′ l ) have the same order.The proof is complete. (cid:3) The same as the previous proof, by checking the definition of ∼ m , we have Lemma 5.12.
Let I = ( i , ...., i L ) ∈ Z L and ( i l ′ , ..., i l ′ ) is a crest of I , then we have I = ( i , ...i L ) ∼ m ( i , ..., i l ′ − , i l ′ + K, ..., i l ′ + K, i l ′ +1 , ..., i l ) for any integer K such that i l ′ + K > max { i l ′ − , i l ′ +1 } . The following proposition shows a deep relation between the set of standard generatorsof M ( n, k ) and ∼ m : Proposition 5.13.
Given two sequences I = { i , ..., i L } ∈ [ k ] L , J = { j , ..., j L } ∈ [ n ] L ,let { u ( m ) i,j } i =1 ,...,n ; j =1 ,...,k be the set of standard generators of M ( n, k ) , then we have X ( q ,...,q L ) ∼ m J u ( m ) q ,i · · · u ( m ) q L ,i L P = (cid:26) P if J ∼ m I otherwiseProof. We will prove the proposition by induction.When L = 1, the statement is apparently true.Suppose the statement is true for all L ≤ L ′ . Let us consider the case L = L ′ + 1. Let( i l ′ , ..., i l ′ ) be a crest of I : Case 1:
If ( j l ′ , ..., j l ′ ) is not a crest of J , then I 6∼ m J and one of the following caseshappens:1. There exists an index j l ′ of J such that j l ′ = j l ′ +1 for some l ′ ≤ l ′ < l ′ .2. j l ′ ≤ j l ′ − .3. j l ′ ≤ j l ′ +1 .But, for all Q = ( q , ..., q L ) ∼ m J , we have: ONCOMMUTATIVE SPREADABILITY 25
1. ( q l ′ , q l ′ − ) and ( j l ′ , j l ′ − ) have the same order.2. ( q l ′ , q l ′ − ) and ( j l ′ , j l ′ − ) have the same order.3. ( q l ′ , q l ′ +1 ) and ( j l ′ , j l ′ +1 ) have the same order.Therefore, we have:1. q l ′ = q l ′ − and i l ′ = i l ′ − for some l ′ ≤ l ′ < l ′ .2. q l ′ ≤ q l ′ − and i l ′ > i l ′ − .3. q l ′ ≤ q l ′ +1 and i l ′ > i l ′ +1 .According to the definition of M i ( n, k ), we have one of the following equations:1. u ( m ) q l ′ ,i l ′ u ( m ) q l ′ ,i l ′ = 0 for some l ′ ≤ l ′ < l ′ .2. u ( m ) q l ′ − ,i l ′ − u ( m ) q l ′ ,i l ′ = 0.3. u ( m ) q l ′ ,i l ′ u ( m ) q l ′ ,i l ′ = 0.In this case, we have X ( q ,...,q L ) ∼ m J u ( m ) q ,i · · · u ( m ) q L ,i L P = 0 . Case 2:
If ( j l ′ , ..., j l ′ ) is a crest of J , then ( q l ′ , ..., q l ′ ) is a crest of Q . Therefore, u ( m ) q l ′ ,i l ′ · · · u ( m ) q l ′ ,i l ′ = u ( m ) q l ′ ,i l ′ . By Lemma 5.12, if we fix the indices of
Q \ ( q l ′ , ..., q l ′ ), then q l ′ , ..., q l ′ can be anyintegers such that q l ′ = ... = q l ′ and max { q l ′ − , q l ′ +1 ) } < q l ′ ≤ n . Therefore, we have P max { q l ′ − ,q l ′ ) } Q \ ( q l ′ , ..., q l ′ ) ∼ m J \ ( j l ′ , ..., j l ′ ). If we denote by( i ′ , ..., i ′ L ′′ ) the sequence I \ ( i l ′ , ..., i l ′ ), then we have P ( q ,...,q L ) ∼ m J u ( m ) q ,i · · · u ( m ) q L ,i L P = P ( q ′ ,...,q ′ L ′′ ) ∼ m J \ ( j l ′ ,...,j l ′ ) u ( m ) q ′ ,i ′ · · · u ( m ) q ′ L ′′ ,i ′ L ′′ P = (cid:26) P if J \ ( j l ′ , ..., j l ′ ) ∼ m I \ ( i l ′ , ..., i l ′ )0 otherwiseThe last equality comes from the assumption of our induction. By Lemma5.10 andLemma5.11, J \ ( j l ′ , ..., j l ′ ) ∼ m I \ ( i l ′ , ..., i l ′ ) iff J ∼ m I .The proof is complete. (cid:3) Operator valued monotone sequences are monotonically spreadable. Inthis subsection, we will show that operator valued monotone finite sequences of randomvariables are monotonically spreadable. To achieve it, we need to consider the positionsof the smallest elements of indices sequences. Definition 5.14. Let I = ( i , ..., i L ) be a sequence of ordered indices and a = min { i , ..., i L } .We call the set § ( I ) = { l | i l = a } the positions of the smallest elements of I . An intervalof ( i l ′ , ..., i l ′ ) is called a hill of I if i l ′ − = i l ′ +1 = a and i ′ l = a for all l ′ ≤ l ′ ≤ l ′ , herewe assume i = i L +1 = a for convenience. Example: (1 , , , , , , 1) has two hills (2 , , 4) and (2). (1 , , , , , ) has two hills(2) and (3 , , , , , 1) has no hill. Lemma 5.15. Given two sequences I = { i , ..., i L } , J = { j , ..., j L } ∈ [ n ] L such that I ∼ m J , then § ( I ) = § ( J ) . Let ( i l ′ , ..., i l ′ ) be a hill of I , then ( i l ′ , ..., i l ′ ) ∼ m ( j l ′ , ..., j l ′ ) . Proof. Let us check the values of J one by one. Suppose § ( I ) = { l ′′ < · · · < l ′′ k ′ } , where k ′ is the number of elements of § ( I ). Let b = min { j , ...j L } , we want to show that j l ′′ = · · · = j l ′′ k ′ = b and j l > b for all l 6∈ § ( I ).Given an integer 1 ≤ p < k ′ , we have i l > a = i l ′′ p = i l ′′ p +1 for all l ′′ p < l < l ′′ p +1 . According to the definition of ∼ m and Lemma5, we have j l ′′ p = j l ′′ p +1 and j l > max { j l ′′ p , j l ′′ p +1 } for all l ′′ p < l < l ” p +1 . The left is to check the elements j l with l < l ′′ or l > l ′′ k ′ . If thereexists and l < l ′′ such that j l ≤ j l ′′ , we chose the greatest such l . Then, we have j l ′ > max { j l , j l ′′ } for all l < l ′ < l ′′ . Therefore, we have i l ≤ i l ′′ which is a contradiction. It implies that j l > j l ′′ for all l < l ′′ . the same we have j l > j l ′′ for all l > l ′′ k . Therefore, j l ′′ = · · · = j l ′′ k ′ = min { j , ..., j L } . The last statement is obviousfrom the definition of ∼ m . The proof is complete. (cid:3) Given I = { i , ..., i L } ∈ Z L , we will denote by x I = x i x i · · · x j L for short . Then, wehave ONCOMMUTATIVE SPREADABILITY 27 Proposition 5.16. Let ( A , B , E ) be an operator valued probability space, and ( x i ) i =1 ,...,n be a sequence of random variables in A . If ( x i ) i =1 ,...,n are identically distributed andmonotonically independent. Then, for indices sequences I = { i , ..., i L } , J = { j , ..., j L } ∈ [ n ] L such that I ∼ m J , L ∈ N , we have E [ x I ] = E [ x J ] . Proof. When L = 1, the statement is true since the sequence is identically distributed.Suppose the statement is true for all L ≤ L ′ ≥ 1. Let us consider the case L = L ′ + 1.If I has no hill, then i = · · · = i L which implies j = · · · = j L . The statement is truefor this case, because the sequence is identically distributed. Suppose I has hills I , ..., I l and a = min { i , ..., i L } . Then, x I can be written as x n a x I x n a x I · · · x n l a x I l x n l +1 a , where n , ..., n l ∈ Z + and n , n l +1 ∈ Z ∪ { } . Since ( x i ) i =1 ,...,n are monotonically indepen-dent, we have E [ x I ] = E [ x n a E [ x I ] x n a E [ x I ] · · · x n l a E [ x I l ] x n l +1 a ] . Let b = min { j , ..., j L } , by Lemma 5.15, J has hills J , ..., J l whose positions of elementscorrespond to the positions of elements of I , ..., I l and J l ′ ∼ m J l ′ for all 1 ≤ l ′ ≤ k ′ .Therefore, we have E [ x J ] = E [ x n b E [ x J ] x n b E [ x J ] · · · x n l b E [ x J l ] x n l +1 b ]= E [ x n b E [ x I ] x n b E [ x I ] · · · x n l b E [ x I l ] x n l +1 b ]= E [ x n a E [ x I ] x n a E [ x I ] · · · x n l a E [ x I l ] x n l +1 a ]= E [ x I ] , where the second equality follows the induction and the third equality holds because x a and x b are identically distributed. The proof is complete. (cid:3) Proposition 5.17. Let ( A , B , E ) be an operator valued probability space, and ( x i ) i =1 ,...,n be a sequence of random variables in A . If ( x i ) i =1 ,...,n are identically distributed andmonotonically independent with respect to E . Let φ be a state on A such that φ ( · ) = φ ( E [ · ]) . Then, ( x i ) i =1 ,...,n is monotonically spreadable with respect to φ .Proof. For fixed natural numbers n, k ∈ N , let ( u i,j ) i =1 ,...,n ; j =1 ,...,k be standard generatorsof M i ( n, k ). Let J = ( j , ..., j L ) ∈ [ k ] L and denote x j · · · x j L by x J . We denote the equiv-alent class of [ n ] L associated with ∼ m by [ n L ]. For each I ∈ [ n ] L , we denote u i ,j · · · u i L ,j L by u I , J . Then, by proposition 5.13, we have P I∈ [ n ] L φ ( x I ) P u I , J P = P I∈ [ n ] L φ ( E [ x I ]) P u I , J P = P ¯ Q ∈ [ n ] L P I∈ ¯ Q φ ( E [ x I ]) P u I , J P = P J 6∈ ¯ Q ∈ [ n ] L P I∈ ¯ Q φ ( E [ x I ]) P u I , J P + P J ∈ ¯ Q ∈ [ n ] L P I∈ ¯ Q φ ( E [ x I ]) P u I , J P = P J 6∈ ¯ Q ∈ [ n ] L P I∈ ¯ Q φ ( E [ x Q ]) P u I , J P + P I∼ m J φ ( E [ x J ]) P u I , J P = 0 + φ ( E [ x J ]) P = φ ( x J ) P Since n, k are arbitrary, the proof is complete. (cid:3) Tail algebras In the previous work on distributional symmetries, infinite sequences of objects areindexed by natural numbers. For this kind of infinite sequences of random variables, theconditional expectations in de Finetti type theorems are defined via the limit of unilateralshifts. It is shown in [14] that unilateral shift is an isometry frome A to itself if ( A , φ )is a W ∗ -probability space generated by a spreadable sequence of random variables and φ is faithful. Therefore, WOT continuous conditional expectations defined via the limit ofunilateral shift exist in a very weak situation, i.e. the sequence of random variables justneed to be spreadable. However, our works are in a more general situation that the state φ is not necessarily faithful. In our framework, we will provide an example in which thesequence of random variables is monotone spreadable but the unilateral shift is not anisometry. Therefore, we can not get an extended de Finetti type theorem for monotoneindependence in the usual way. The key change in this paper is that we will considerbilateral sequences of random variables. We begin with an interesting example :6.1. Unbounded spreadable sequences. Unlike the situation in probability spaceswith faithful states, an infinite spreadable sequence of random variables indexed by naturalnumbers needs not to be bounded. Even more, there exists an infinite monotonicallyspreadable unbounded sequence of bounded random variables in a non-degenerated W ∗ -probability space. Example: Let H be the standard 2-dimensional Hilbert space with orthonormal basis { v = (cid:18) (cid:19) , w = (cid:18) (cid:19) } . Let p, A, x ∈ B ( H ) be operators on H with the following matrix forms: p = (cid:18) (cid:19) , A = (cid:18) (cid:19) , x = (cid:18) (cid:19) . ONCOMMUTATIVE SPREADABILITY 29 Let H = ∞ N n =1 H the infinite tensor product of H . Let { x i } ∞ i =1 be a sequence of selfadjointoperators in B ( H ) defined as follows: x i = i − O n =1 A ⊗ x ⊗ ∞ O m =1 p Let φ be the vector state h· v, v i on H and Φ = ∞ N n =1 φ be a state on B ( H ). It is obviousthat Φ( x ni ) = φ ( x n ) for for i . Therefore, the sequence ( x i ) i ∈ N is identically distributed.For any x, y ∈ B ( H ), an elementary computation shows φ ( xpy ) = φ ( x ) φ ( y ) . For convenience, we will denote A ⊗ i − = i − N n =1 A and P ⊗∞ = ∞ N n =1 P . Also, we denote x i · · · x i L = x I for I = ( i , ..., i L ) ∈ N L . We will show that the sequence { x i } i ∈ N is M i ( n, k )-spreadable with respect to Φ. Lemma 6.1. For indices sequences I = ( i , ..., i L ) , J = ( j , ..., j L ) ∈ [ n ] L such that I ∼ m J and L ∈ Z + , we have Φ( x I ) = Φ( x J ) Proof. When L = 1, the statement is true since the sequence is identically distributed.Suppose the statement is true for all L ≤ L ′ . Let us consider the case L = L ′ + 1. If I has no hill, then i = · · · = i L which implies j = · · · = j L . The statement is true for thiscase, because the sequence is identically distributed. Also, we denote by x ( n ) i the n -thecomponent of x i . Then, x ( n ) i = a if n < ix if n = ip if n > i and x ( n ) I = x ( n ) i x ( n ) i · · · x ( n ) i L .According to the definition of Φ, we have thatΦ( x i x i · · · x j L ) = ∞ Y n =1 φ ( L Y l =1 x ( n ) i ) . Notice that all the terms φ ( L Q l =1 x ( n ) i ) are 1 except finite terms. Suppose I has hills I , ..., I l and a = min { i , ..., i L } , then x I can be written as x n a x I x n a x I · · · x n l a x I l x n l +1 a . Therefore, φ ( L Y l =1 x ( n ) i ) = n < aφ ( x n A |I | x n A |I | · · · x n l A |I l | x n l +1 ) if n = aφ ( px ( n ) I px ( n ) I p · · · px ( n ) I l p ) if n > a It follows that φ ( L Y l =1 x ( n ) i ) = ∞ Y n ≥ min {I} φ ( L Y l =1 x ( n ) i ) . Because φ ( px ( n ) I px ( n ) I p · · · px ( n ) I l p ) = φ ( x ( n ) I ) φ ( x ( n ) I ) · · · φ ( x ( n ) I l ) , we have Φ( x i x i · · · x j L )= φ ( x n A |I | x n A |I | · · · x n l A |I l | x n l +1 ) ∞ Q n>a φ ( px ( n ) I px ( n ) I p · · · px ( n ) I l p ) . = φ ( x n A |I | x n A |I | · · · x n l A |I l | x n l +1 ) ∞ Q n>a φ ( x ( n ) I ) φ ( x ( n ) I ) · · · φ ( x ( n ) I l )= φ ( x n A |I | x n A |I | · · · x n l A |I l | x n l +1 )Φ( x I )Φ( x I ) · · · Φ( x I l )Let b = min { j , ..., j L } , by Lemma5.15, J has hills J , ..., J l whose positions of elementscorrespond to the positions of elements of I , ..., I l and J l ′ ∼ m J l ′ for all 1 ≤ l ′ ≤ k ′ .Therefore, we haveΦ( x J ) = Φ( x i x i · · · x i L )= φ ( x n A |J | x n A |J | · · · x n l A |J l | x n l +1 )Φ( x J )Φ( x J ) · · · Φ( x J l )= φ ( x n A |I | x n A |I | · · · x n l A |I l | x n l +1 )Φ( x I )Φ( x I ) · · · Φ( x I l )= Φ( x I )where the second equality follows the induction and the true that J k ∼ m I k and |J k | = |I k | for all 1 ≤ k ≤ l . The proof is complete. (cid:3) Proposition 6.2. The joint distribution of ( x i ) i ∈ N with respect to Φ is monotonicallyspreadable.Proof. Fixed n > k ∈ N , let { u ( m ) i,j } i =1 ,...,n ; j =1 ,...,k be the set of standard generators of M ( n, k ). For all I = ( i , ..., i L ) ∈ [ k ] L , we denote by [ n ] L the ∼ m equivalence class of ONCOMMUTATIVE SPREADABILITY 31 [ n ] L , then we have P µ x ,...,x n ⊗ ( id M ( n,k ) )( α ( m ) n,k ( X I )) P = P J ∈ [ n ] L µ x ,...,x n ( X J ) P u ( m ) J , I P = P ¯ Q∈ [ n ] L P J ∈ ¯ Q µ x ,...,x n ( X J ) P u ( m ) J , I P = P I6∈ ¯ Q∈ [ n ] L P J ∈ ¯ Q µ x ,...,x n ( X J ) P u ( m ) J , I P + P J ∼ m I µ x ,...,x n ( X J ) P u ( m ) J , I P = P I6∈ ¯ Q∈ [ n ] L P J ∈ ¯ Q µ x ,...,x n ( X Q ) P u ( m ) J , I P + P J ∼ m I µ x ,...,x n ( X I ) P u ( m ) J , I P = P I6∈ ¯ Q∈ [ n ] L µ x ,...,x n ( X Q ) P J ∈ ¯ Q P u ( m ) J , I P + P J ∼ m I µ x ,...,x n ( X I ) P u ( m ) J , I P = P I6∈ ¯ Q∈ [ n ] L µ x ,...,x n ( X Q ) · P J ∼ m I µ x ,...,x n ( X I ) P u ( m ) J , I P = P J ∼ m I µ x ,...,x n ( X I ) P u ( m ) J , I P = Φ( x I ) P The proof is complete. (cid:3) By direct computations, we have n Y i =1 x n +1 − i v ⊗∞ = w ⊗ n ⊗ v ⊗∞ and(4) x n +1 w ⊗ n ⊗ v ⊗∞ = 2 n w ⊗ n +1 ⊗ v ⊗∞ Let ( H ′ , π ′ , ξ ′ ) be the GNS representation of the von Neumann algebra generated by( x i ) i =1 ,..., ∞ associated with Φ. We have k π ′ ( x n +1 ) k ≤ k x n +1 k = 2 n , but equation 4 shows that k π ′ ( x n +1 ) k ≥ n . Therefore, k π ′ ( x n +1 ) k = 2 n .Therefore, there is no bounded endomorphism α on A such that α ( x i ) = x i +1 .6.2. Tail algebras of bilateral sequences of random variables. In the last subsec-tion, we showed that, in a W ∗ -probability space with a non-degenerated normal state,the unilateral shift of a spreadable unilateral sequence of random variables may not beextended to be a bounded endomorphism. Therefore, in general, we can not define anormal condition expectation by taking the limit of unilateral shifts of variables. Themain reason here is that the spreadability of variables does not give enough restrictionsto control the norms of the variables in our probability space. In ( A , φ ), a W ∗ -probabilityspace with a faithful state, the norm of a selfadjoint random variable x ∈ A is controlledby the moments of X , i.e. k x k = lim n →∞ φ ( | x | n ) n . But, in our non-degenerated W ∗ -probability spaces, the norm of a random variable de-pends on all mixed moments which involve it. To make the conditional expectation exist,we will consider spreadable sequences of random variables indexed by Z but not N . Inthis case, the sequence ( x i ) i ∈ Z is bilateral. As a consequence, we will have two choices totake limits on defining normal conditional expectations and tail algebras. Before studyingproperties of tail algebras of bilateral sequences, we introduce some necessary notationsand assumptions first.Let ( A , φ ) is a W ∗ − probability space generated by a spreadable bilateral sequence ofbounded random variables ( x i ) i ∈ Z and φ is a non-degenerated normal state. We assumethat the unit of A is contained in the WOT-closure of the non-unital algebra generatedby ( x i ) i ∈ Z . Let ( H , π, ξ ) be the GNS representation of A associated with φ . Then, { π ( P ( x i | i ∈ Z )) ξ | P ∈ C h X i | i ∈ Z i} is dense in H . For convenience, we will denote π ( y ) ξ by ˆ y for all y ∈ A . When there is no confusion, we will write y short for π ( y ). Wedenote by A k + the non-unital algebra generated by ( x i ) i ≥ k and A k − the non-unital algebragenerated by ( x i ) i ≤ k . Let A + k and A − k be the WOT-closure of A k + and A k − , respectively. Definition 6.3. Let ( A , φ ) be a no-degenerated noncommutative W ∗ -probability space,( x i ) i ∈ Z be a bilateral sequence of bounded random variables in A such that A is the WOTclosure of the non-unital algebra generated by ( x i ) i ∈ Z . The positive tail algebra A + tail of( x i ) i ∈ Z is defined as following: A + tail = \ k> A + k . In the opposite direction, we define the negative tail algebra A − tail of ( x i ) i ∈ Z as following: A − tail = \ k< A − k . Remark 6.4. In general, the positive tail algebra and the negative tail algebra are differ-ent. Even though our framework looks quit different from the framework in [14], we canshow that there exists a normal bounded shift of the sequence in a similar way. Forcompleteness, we provide the details here. Lemma 6.5. There exists a unitary map U : H → H such that U ( P ( x i | i ∈ Z )) ξ = P ( x i +1 | i ∈ Z ) ξ Proof. Since ( x i ) i ∈ Z is spreadable, we have φ (( P ( x i | i ∈ Z )) ∗ P ( x i | i ∈ Z )) = φ (( P ( x i +1 | i ∈ Z )) ∗ P ( x i +1 | i ∈ Z )) . It implies that U ( P ( x i | i ∈ Z ) ξ ) = P ( x i +1 | i ∈ Z ) ξ is a well defined isometry on { π ( P ( x i | i ∈ Z )) ξ | P ∈ C h X i | i ∈ Z i} . Since { π ( P ( x i | i ∈ Z )) ξ | P ∈ C h X i | i ∈ Z i} is dense in H , U can be extended to the whole space H . Itis obvious that { π ( P ( x i | i ∈ Z )) ξ | P ∈ C h X i | i ∈ Z i} is contained in the range of U .Therefore, the extension of U is a unitary map on H . (cid:3) ONCOMMUTATIVE SPREADABILITY 33 Now, we can define an automorphism α on A by the following formula: α ( y ) = U yU − . Lemma 6.6. α is the bilateral shift of ( x i ) i ∈ Z , i.e. α ( x k ) = x k +1 for all k ∈ Z .Proof. For all y = P ( x i | i ∈ Z ) ξ , we have α ( x k ) y = U x k U − P ( x i | i ∈ Z ) ξ = U x k P ( x i − | i ∈ Z ) ξ = x k +1 P ( x i | i ∈ Z ) ξ. By the density of { π ( P ( x i | i ∈ Z )) ξ | P ∈ C h X i | i ∈ Z i} , we have α ( x k ) = x k +1 . The proofis complete. (cid:3) Since α is a normal automorphism of A , we have Corollary 6.7. For all k ∈ Z , we have α ( A + k ) = A + k +1 . Lemma 6.8. Fix n ∈ Z . Let y , y ∈ A n − . Then, we have h α l ( a )ˆ y , ˆ y i = h a ˆ y , ˆ y i , where l ∈ N and a ∈ A + n +1 .Proof. It is sufficient to prove the statement under the assumption that l = 1. Since a ∈ A + n +1 , by Kaplansky’s theorem, there exists a sequence ( a m ) m ∈ N ⊂ A ( n +1)+ such that k a m k ≤ k a k for all m and a m converges to a in WOT. Then, by the spreadability of( x i ) i ∈ Z , we have h α ( a )ˆ y , ˆ y i = lim m →∞ h α ( a m )ˆ y , ˆ y i = lim m →∞ φ ( y ∗ a m ˆ y ) = h a ˆ y , ˆ y i (cid:3) In the following context, we fix k ∈ Z . Lemma 6.9. For all a ∈ A + k , we have that E + [ a ] = W OT − lim l →∞ α l ( a ) exists. Moreover, E + [ a ] ∈ A + tail Proof. For all y , y ∈ { π ( P ( x i | i ∈ Z )) ξ | P ∈ C h X i | i ∈ Z i} , there exits n ∈ Z such that y , y ∈ A n − . For all l > n − k , we have α l ( a ) ∈ A ( n +1)+ . By Lemma 6.8, we have h α n +1 − k ( a ) y , y i = h α n +2 − k ( a ) y , y i = · · · . Therefore, lim l →∞ h α l ( a ) y , y i = h α n +1 − k ( a ) y , y i .α l ( a ) converges pointwisely to an element E + [ a ]. Since for all n > 0, we have α l ( a ) ∈ A + n for all l > n − k + 1. It follows that W OT − lim l →∞ α l ( a ) ∈ A + n for all n . Hence, E + [ a ] ∈A + tail . (cid:3) Proposition 6.10. E + is normal on A + k for all k ∈ Z . Proof. Let ( a m ) m ∈ N ⊂ A + k be a bounded sequence which converges to 0 in WOT. For all y , y ∈ { π ( P ( x i | i ∈ Z )) ξ | P ∈ C h X i | i ∈ Z i} , there exits n ∈ Z such that y , y ∈ A n − .Then, we have lim m →∞ h E + [ a m ] y , y i = lim m →∞ h α n +1 − k ( a m ) y , y i = 0 . The last equality holds because α l is normal for all l ∈ N . The proof is complete. (cid:3) Remark 6.11. E + is defined on S k ∈ Z A + k but not on A . In general, we can not extend E + to the whole algebra A . Lemma 6.12. E + [ a ] = a for all a ∈ A + tail .Proof. For all ˆ y , ˆ y ∈ { π ( P ( x i | i ∈ Z )) ξ | P ∈ C h X i | i ∈ Z i} , there exits n ∈ Z such that y , y ∈ A n − . Since a ∈ A + tail ⊂ A + n +1 , by Kaplansky’s theorem, there exists a sequence of( a m ) m ∈ N ⊂ A ( n +1)+ such that a m → a in WOT and k a m k ≤ k a k for all m . Then we have. h a ˆ y , ˆ y i = lim m →∞ h a ˆ y , ˆ y i = lim m →∞ h α ( a m )ˆ y , ˆ y i = h α ( a )ˆ y , ˆ y i . Since y , y are arbitrary, we have a = α ( a ). (cid:3) Remark 6.13. One should be careful that A + tail could be a proper subset of the fixed pointsset of α . Lemma 6.14. E + [ a ba ] = a E + [ b ] a for all b ∈ A + k , a , a ∈ A + tail .Proof. By Lemma6.12, we have E + [ a ba ] = lim l →∞ α l ( a ba ) = lim l →∞ α l ( a ) α l ( b ) α l ( a ) = lim l →∞ a α l ( b ) a = a E + [ b ] a (cid:3) Conditional expectations of bilateral monotonically spreadablesequence In this section, we assume that the joint distribution of ( x i ) i ∈ Z is monotonically spread-able. Lemma 7.1. Fix n > k ∈ N , let ( u i,j ) i =1 ,...,n ; j =1 ,...,k be the standard generators of M i ( n, k ) .Then, we have φ ( a x l i b x l i b · · · b m − x l m i m a ) P = n X j ,...,j m =1 φ ( a x l j b x l j b · · · b m − x l m j m a ) P u j ,i · · · u j m ,i m P , where ≤ i , ...i m ≤ k , b , ..., b m − ∈ A ( n +1)+ and a , a ∈ A − .Proof. Without loss of generality, we assume that there exist n , n ∈ N such that a , a ∈ A [ − n +1 , and b , ..., b m − ∈ A [ n +1 ,n + k ] . ONCOMMUTATIVE SPREADABILITY 35 Since the map is linear, we just need to consider the case that a , a and b , ..., b m − areproducts of ( x i ) i ∈ Z . Let a = x s , · · · x s ,t and a = x s , · · · x s ,t for some t , t ∈ N and − n + 1 ≤ s c,d ≤ 0. Let b i = x r i, · · · x r i,t ′ i for t ′ , ..., t ′ m − ∈ N ∪ { } and n + 1 ≤ r c,d ≤ k + n . Then, ( x − n +1 , ..., x n + n ) is asequence of length n + n + n , we denote it by ( y , ..., y n + n + n ). Let n ′ = n + n + n and k ′ = k + n + n . By our assumption, a x l i b x l i b · · · b m − x l m i m a is in the algebra generatedby ( y , ...., y k ′ ). Let ( u ′ i,j ) i =1 ,...,n ′ ; j =1 ,...,k ′ be the standard generators of M i ( n ′ , k ′ ) and P ′ be the invariant projection. Let π be the C ∗ -homomorphism in Lemma 3.13 and id bethe identity may on C h X , ...., X n ′ i . Since 1 ≤ s c,d + n ≤ n , we have id ⊗ π ( α ( m ) n ′ ,k ′ ( X s i, + n · · · X s i,t + n )) = X s i, + n · · · X s i,t + n ⊗ P . Since n + n + 1 ≤ r c,d + n ≤ n + n + k , we have id ⊗ π ( α ( m ) n ′ ,k ′ ( X r i, + n · · · X r ,t ′ + n )) = X r i, + n + n − k · · · X r i,t ′ i + n + n − k ⊗ I, where I is the identity of M i ( n, k ). According to our assumption, we have 1 ≤ i t ≤ k for t = 1 , ...., m . Then id ⊗ π ( α ( m ) n ′ ,k ′ ( X l t i t + n ) = n X j t =1 X l t j t + n ⊗ u j t ,i t . According to the monotone spreadability of ( y , ..., y n ′ ) and Lemma3.13, we have φ ( a x l i b x l i b · · · b m − x l m i m a ) P = µ y ,...,y k ′ ( X s , + n · · · X s ,t + n X l i + n · · · X l m i m + n X s , + n · · · X s ,t + n ) π ( P ′ )= P µ y ,...,y n ′ ⊗ π ( α ( m ) n ′ ,k ′ ( X s , + n · · · X s ,t + n X l i + n · · · X l m i m + n X s , + n · · · X s ,t + n )) P = n P j ,...,j m =1 µ y ,...,y n ′ ( X s , + n · · · X s ,t + n X l j + n X r , + n + n − k · · · X r m − ,t ′ m − n + n − k X l m + n j m X s , + n · · · X s ,t ) P u j ,i · · · u j m ,i m P Notice that ( y , ..., y n ′ ) is spreadable and n + 1 ≤ r , , the above equation becomes φ ( a x l i b x l i b · · · b m − x l m i m a ) P = n P j ,...,j m =1 µ y ,...,y n ′ ( X s , + n · · · X s ,t + n X l j + n X r , + n · · · X r m − ,t ′ m − n X l m j m + n X s , + n · · · X s ,t ) P u j ,i · · · u j m ,i m P = n P j ,...,j m =1 φ ( x s , · · · x s ,t x l j x r , · · · x r m − ,t ′ m − x l m j m x s , · · · x s ,t ) P u j ,i · · · u j m ,i m P = n P j ,...,j m =1 φ ( a x l j b x l j b · · · b m − x l m j m a ) P u j ,i · · · u j m ,i m P The proof is complete. (cid:3) Lemma 7.2. Fix n > k ∈ N , let ( u i,j ) i =1 ,...,n ; j =1 ,...,k be the standard generators of M i ( n, k ) .Then, we have E + [ x l i b x l i b · · · b m − x l m i m ] ⊗ P = n X j ,...,j m =1 E + [ x l j b x l j b · · · b m − x l m j m ] ⊗ P u j ,i · · · u j m ,i m P , where ≤ i , ...i m ≤ k , b , ..., b m − ∈ A ( n +1)+ .Proof. It is necessary to check the two sides of the equation equal to each other pointwisely,i.e.(5) φ ( a E + [ x l i b x l i b · · · b m − x l m i m ] a ) P = n X j ,...,j m =1 φ ( a E + [ x l j b x l j b · · · b m − x l m j m ] a ) P u j ,i · · · u j m ,i m P for all a , a ∈ A [ −∞ , ∞ ] . Given a , a ∈ A [ −∞ , ∞ ] , then there exists M ∈ N such that a , a ∈ A M − . Then, α − m ( a ) , α − m ( a ) ∈ A − for all m > M . By Lemma 7.1, we have φ ( α − m ( a ) x l i b x l i b · · · b m − x l m i m α − m ( a )) P = n P j ,...,j m =1 φ ( α − m ( a ) x l j b x l j b · · · b m − x l m j m α − m ( a )) P u j ,i · · · u j m ,i m P . Therefore, for all m > M ,we have φ ( a α m ( x l i b x l i b · · · b m − x l m i m ) a ) P = n P j ,...,j m =1 φ ( a α m ( x l j b x l j b · · · b m − x l m j m ) a ) P u j ,i · · · u j m ,i m P . Let m go to + ∞ , we get equation 5.The proof is complete since a , a are arbitrary. (cid:3) Proposition 7.3. Let ( A , φ ) be a W ∗ -probability space, ( x i ) i ∈ Z a sequence of selfadjointrandom variables in A , E + be the conditional expectation onto the positive tail algebra A + tail . Assume that the joint distribution of ( x i ) i ∈ Z is monotonically spreadable, then thesame is true for the joint distribution with respect to E + , i.e. for fixed n > k ∈ N and ( u i,j ) i =1 ,...,n ; j =1 ,...,k the standard generators of M i ( n, k ) , we have that E + [ x l i b x l i b · · · b m − x l m i m ] ⊗ P = n X j ,...,j m =1 E + [ x l i b x l i b · · · b m − x l m i m ] ⊗ P u j ,i · · · u j m ,i m P , ≤ i , ..., i m ≤ k , l , ..., l m ∈ N and b , ..., b n ∈ A + tail .Proof. Since b , ..., b m − ∈ A + tail ∈ A + n , by Kaplansky’s theorem, there exists sequences { b s,t } s =1 ,...m − t ∈ N ⊂ A n + such that k b s,t k ≤ k b s k and lim n →∞ b s,t = b s in SOT for each s = 1 , ..., m − 1. Therefore, SOT − lim t →∞ x l i b ,t x l i b ,t · · · b m − ,t m x l m i m = x l i b x l i b ,t · · · b m − ,t m x l m i m . ONCOMMUTATIVE SPREADABILITY 37 By Lemma 7.2, we have E + [ x l i b ,t x l i b ,t · · · b m − ,t m x l m i m ] ⊗ P = n X j ,...,j m =1 E + [ x l j b ,t x l j b ,t · · · b m − ,t m − x l m j m ] ⊗ P u j ,i · · · u j m ,i m P Let t go to + ∞ , by normality of E + , we have E + [ x l i b x l i b ,t · · · b m − ,t m x l m i m ] ⊗ P = n X j ,...,j m =1 E + [ x l j b x l j b ,t · · · b m − ,t m − x l m j m ] ⊗ P u j ,i · · · u j m ,i m P Again, take t , ..., t m − to + ∞ , we have(6) E + [ x l i b x l i b · · · b m − x l m i m ] ⊗ P = n X j ,...,j m =1 E + [ x j b x j b · · · b m − x j m ] ⊗ P u j ,i · · · u j m ,i m P (cid:3) According to the universal conditions of M i ( n, k ), if i s = i s +1 for some s , then the termson the right hand side are not vanished only if j s = j s +1 . Therefore we can shorten theproduct on the right hand side of 6 if i s = i s +1 for some s . We have Proposition 7.4. Let ( A , φ ) be a W ∗ -probability space, ( x i ) i ∈ Z a sequence of selfadjointrandom variables in A , E + be the conditional expectation onto the positive tail algebra A + tail . Assume that the joint distribution of ( x i ) i ∈ Z is monotonically spreadable, for fixed n > k ∈ N and ( u i,j ) i =1 ,...,n ; j =1 ,...,k the standard generators of M i ( n, k ) , we have that E + [ p ( x i ) · · · p m ( x i m )] ⊗ P = n X j ,...,j m =1 E + [ p ( x j ) · · · p m ( x j m )] ⊗ P u j ,i · · · u j m ,i m P , whenever ≤ i , ..., i m ≤ k , i = · · · 6 = i m and p , ..., p m ∈ A + tail h X i . Lemma 7.5. Let ( A , φ ) be a W ∗ -probability space, ( x i ) i ∈ Z a sequence of selfadjoint ran-dom variables in A , E + be the conditional expectation onto the positive tail algebra A + tail .Assume that the joint distribution of ( x i ) i ∈ Z is monotonically spreadable, then E + [ p ( x i ) · · · p s ( x i s ) · · · p m ( x i m )] = E + [ p ( x i ) · · · E + [ p s ( x i s )] · · · p m ( x i m )] whenever i s > i t for all t = s , i = · · · 6 = i m and p , ..., p m ∈ A + tail h X i .Proof. Since ( x i ) i ∈ Z is spreadable, by Lemma 6.9, we have that α ( p t ( x i t )) = p t ( α ( x i t ))and E + [ α k ′ ( a )] = E + [ a ]for all a ∈ S n ′ ∈ Z A + n ′ and k ′ ∈ Z .Therefore, it is sufficient to prove the statement under the assumption that i , ..., i m > Let i s = k , ( u i,j ) i =1 ,...,n +1; j =1 ,...,k the standard generators of M i ( n + k, k ). By proposition7.4, we have E + [ p ( x i ) · · · p m ( x i m )] ⊗ P = n + k X j ,...,j m =1 E + [ p ( x j ) · · · p m ( x j m )] ⊗ P u j ,i · · · u j m ,i m P . Now, apply proposition 3.12 by letting l = · · · = l k − = 1 and l k = n + 1, then we have E + [ p ( x i ) · · · p s ( x i s ) · · · p m ( x i m )] ⊗ P = 1 n + 1 n + k X j s = k E + [ p ( x i ) · · · p s ( x j s ) · · · p m ( x i m )] ⊗ P . Since n is arbitrary, and E + is normal on A +0 , we have E + [ p ( x i ) · · · p s ( x i s ) · · · p m ( x i m )]= n +1 n + k P j s = k E + [ p ( x i ) · · · p s ( x j s ) · · · p m ( x i m )]= WOT − lim n →∞ E + [ p ( x i ) · · · ( n +1 n + k P j s = k p s ( x j s )) · · · p m ( x i m )]= WOT − lim n →∞ E + [ p ( x i ) · · · ( n +1 n P t =0 α t ( p s ( x i s )) · · · p m ( x i m )]= WOT − lim n →∞ E + [ p ( x i ) · · · E + [ p s ( x i s )] · · · p m ( x i m )] . The proof is complete. (cid:3) Now, we turn to consider the case that the maximal index is not unique. Proposition 7.6. Let ( A , φ ) be a W ∗ -probability space, ( x i ) i ∈ Z a sequence of selfadjointrandom variables in A , E + be the conditional expectation onto the positive tail algebra A + tail . Assume that the joint distribution of ( x i ) i ∈ Z is monotonically spreadable, then E + [ p ( x i ) · · · p s ( x i s ) · · · p m ( x i m )] = E + [ p ( x i ) · · · E + [ p s ( x i s )] · · · p m ( x i m )] whenever i s = max { i , ..., i n } for all t = s , i = · · · 6 = i m and p , ..., p m ∈ A + tail h X i .Proof. Again, we can assume that i , ..., i t > { i , ..., i m } = k . Suppose thenumber k appears t times in the sequence, which are { i l j } j = 1 , ..., t such that i l j = k and l < l < · · · < l t . Fix n, k and consider M i ( n + k, k ), by proposition 7.4 and proposition3.12, we have E + [ p ( x i ) · · · p l ( x i l ) · · · p l ( x i l ) · · · p m ( x i m )] ⊗ P = k + n P j l ,j l ,...j lt = k E + [ p ( x i ) · · · p l ( x j l ) · · · p l ( x j l ) · · · p m ( x i m )] ⊗ P P j l ,k P P j l ,k P · · · u j lt ,k P = n +1) t k + n P j l ,j l ,...j lt = k E + [ p ( x i ) · · · p l ( x j l ) · · · p l ( x j l ) · · · p m ( x i m )]] ⊗ P = n +1) t ( N P j ls = j lr if s = r E + [ p ( x i ) · · · p l ( x j l ) · · · p l ( x j l ) · · · p m ( x i m )] ⊗ P + N P j ls = j lt for some s = t E + [ p ( x i ) · · · p l ( x j l ) · · · p l ( x j l ) · · · p m ( x i m )] ⊗ P ) ONCOMMUTATIVE SPREADABILITY 39 In the first part of the sum, apply proposition 7.5 on indices j l , ...j l lt recursively, it followsthat E + [ p ( x i ) · · · p s ( x j l ) · · · p s ( x j l ) · · · p m ( x i m )] = E + [ p ( x i ) · · · E [ p l ( x j l )] · · · E [ p l ( x j l )] · · · p m ( x i m )] . Since E [ p s ( x j l )] = E [ p s ( x k )], for all j l , ..., j l t , E + [ p ( x i ) · · · p l ( x j l ) · · · p l ( x j l ) · · · p m ( x i m )] = E + [ p ( x i ) · · · E [ p l ( x k )] · · · E [ p l ( x k )] · · · p m ( x i m )] . Then, we have n +1) t ( N P j ls = j lr if s = r E + [ p ( x i ) · · · p l ( x j l ) · · · p l ( x j l ) · · · p m ( x i m )] ⊗ P = t − Q s =0 ( n +1 − s ) n +1 t E + [ p ( x i ) · · · E [ p l ( x k )] · · · E [ p l ( x k )] · · · p m ( x i m )] ⊗ P , which converges to E [ p s ( x k )] · · · E [ p s ( x k )] · · · p m ( x i m )] ⊗ P in norm as n goes to + ∞ .To the second part of the sum, we have k E + [ p ( x i ) · · · p l ( x j l ) · · · p l ( x j l ) · · · p m ( x i m )] k≤ k p ( x i ) · · · p l ( x j l ) · · · p l ( x j l ) · · · p m ( x i m ) k≤ k p ( x i ) k · · · k p l ( x j l ) k · · · k p l ( x j l ) k · · · k p m ( x i m ) k≤ k p ( x ) k · · · k p l ( x ) k · · · k p l ( x ) k · · · k p m ( x ) k which is finite. Therefore, | N P j ls = j lt for some s = t E + [ p ( x i ) · · · p l ( x j l ) · · · p l ( x j l ) · · · p m ( x i m )] k≤ (1 − t − Q s =0 ( n +1 − s )( n +1) t ) k p ( x ) k · · · k p l ( x ) k · · · k p l ( x ) k · · · k p m ( x ) k goes to 0 as n goes to + ∞ .Therefore, we have E + [ p ( x i ) · · · p l ( x i l ) · · · p l ( x i l ) · · · p m ( x i m )] = E + [ p ( x i ) · · · E [ p l ( x k )] · · · E [ p l ( x k )] · · · p m ( x i m )]The same we can show that E + [ p ( x i ) · · · p l ( x k ) · · · E + [ p s ( x i s )] · · · p l ( x k ) · · · p m ( x i m )]= E + [ p ( x i ) · · · E [ p l ( x k )] · · · E [ p l ( x k )] · · · p m ( x i m )]which implies E + [ p ( x i ) · · · p s ( x i s ) · · · p m ( x i m )] = E + [ p ( x i ) · · · E + [ p s ( x i s )] · · · p m ( x i m )] (cid:3) de Finetti type theorem for monotone spreadability Proof of main theorem 1. Now, we turn to prove our main theorem for monotoneindependence: Theorem 8.1. Let ( A , φ ) be a non degenerated W ∗ -probability space and ( x i ) i ∈ Z be abilateral infinite sequence of selfadjoint random variables which generate A . Let A + k bethe WOT closure of the non-unital algebra generated by { x i | i ≥ k } . Then the followingare equivalent: a) The joint distribution of ( x i ) i ∈ Z is monotonically spreadable. b) For all k ∈ Z , there exits a φ preserving conditional expectation E k : A + k → A + tail such that the sequence ( x i ) i ≥ k is identically distributed and monotone with respect E k . Moreover, E k | A k ′ = E k ′ when k ≥ k ′ .Proof. “ b ) ⇒ a ) ”follows corollary 5.17We will prove “ a ) ⇒ b ) ”by induction. Since the sequence is spreadable, it is sufficesto prove a ) ⇒ b ) for k = 1:By the results in the previous two sections, there exists a conditional expectation E k : A + k → A + tail such that the sequence ( x i ) i ≥ k is identically distributed with respect to E k and E k | A k ′ = E k ′ when k ≥ k ′ . Actually, E k is the restriction of E + on A + k . Since thesequence is spreadable, we just need to show that the sequence ( x i ) i ∈ N is monotonicallyindependent with respect to E , i.e.(7) E + [ p ( x i ) · · · p s ( x i s ) · · · p m ( x i m )] = E + [ p ( x i ) · · · E + [ p s ( x i s )] · · · p m ( x i m )] i s − < i s > i s +1 , i = · · · 6 = i m , i , ..., i m ∈ N and p , ..., p m ∈ A + tail h X i .Now, we prove this equality by induction on the maximal index of { i , ..., i m } :When max { i , ..., i m } = 1, then equality is true because i s = 1 and the length of thesequence ( i , ..., i m ) can only be 1.Suppose the equality holds for max { i , ..., i m } = n . When max { i , ..., i m } = n + 1, wehave two cases: Case 1: i s = n + 1 . In this case the equality follows proposition 7.6. Case 2: i s ≤ n . Suppose the number n + 1 appears t times in the sequence, whichare { i l j } j = 1 , ..., t such that i l j = k and l < l < · · · < l t . Since i s − < i s > i s +1 , i s − , i s , i s +1 = n + 1. By proposition 7.6, we have: E + [ p ( x i ) · · · p l ( x i l ) · · · p s − ( x i s − ) p s ( x i s ) p s +1 ( x i s +1 ) · · · p l t ( x i lt ) · · · p m ( x i m )]= E + [ p ( x i ) · · · E + [ p l ( x i l )] · · · p s − ( x i s − ) p s ( x i s ) p s +1 ( x i s +1 ) · · · E + [ p l t ( x i lt )] · · · p m ( x i m ))]Notice that p ( x i ) · · · E + [ p l ( x i l )] · · · p s − ( x i s − ) p s ( x i s ) p s +1 ( x i s +1 ) · · · E + [ p l t ( x i lt )] · · · p m ( x i m ) ∈ A + tail h X , ..., X n i by induction, we have E + [ p ( x i ) · · · E + [ p l ( x i l )] · · · p s − ( x i s − ) p s ( x i s ) p s +1 ( x i s +1 ) · · · E + [ p l t ( x i lt ) · · · p m ( x i m )]= E + [ p ( x i ) · · · E + [ p l ( x i l )] · · · p s − ( x i s − ) E + [ p s ( x i s )] p s +1 ( x i s +1 ) · · · E + [ p l t ( x i lt )] · · · p m ( x i m )]= E + [ p ( x i ) · · · p l ( x i l ) · · · p s − ( x i s − ) E + [ p s ( x i s )] p s +1 ( x i s +1 ) · · · p l t ( x i lt ) · · · p m ( x i m )] ONCOMMUTATIVE SPREADABILITY 41 The last equality follows proposition 7.6. This our desired conclusion. (cid:3) Conditional expectation E − . We do not know whether we can extend E + to thewhole space A . But, the conditional expectation E − can be extended to the whole algebra A if the bilateral sequence ( x i ) i ∈ Z is monotonically spreadable. Given a, b, c ∈ A [ −∞ , ∞ ] ,then there exists L ∈ N such that a, b, c ∈ A [ − L,L ] . Therefore, α − L ( c ) ∈ A [ − L, − L ] . Since( x − L , x − L +1 , ... ) is monotonically with respect to E + , we have φ ( aE − [ b ] c )= lim n →∞ φ ( aα − n ( b ) c )= lim n →∞ ,n> L φ ( aα − n ( b ) c )= lim n →∞ ,n> L φ ( E + [ aα − n ( b ) c ])= lim n →∞ ,n> L φ ( E + [ E + [ a ] α − n ( b ) E + [ c ]])= lim n →∞ φ ( E + [ a ] α − n ( b ) E + [ c ])= lim n →∞ φ ( E + [ a ] E − [ b ] E + [ c ])Since A is generated by countablely many operators, by Kaplansky’s density theorem,for all y ∈ A , there exists a sequence { y n } n ∈ N ⊂ A [ −∞ , ∞ ] such that k y n k ≤ k y k for all n and y n converges to y in WOT. Then, for all a, c ∈ A [ −∞ , ∞ ] we havelim n →∞ φ ( aE − [ y n ] c ) = lim n →∞ φ ( E + [ a ] y n E + [ c ]) = φ ( E + [ a ] yE + [ c ])Therefore, E − [ y n ] converges to an element y ′ pointwisely. Moreover, y ′ depends only on y . If we define E − [ y ] = y ′ , then we have Proposition 8.2. Let ( A , φ ) be a non-degenerated W ∗ -probability space and ( x i ) i ∈ Z be abilateral infinite sequence of selfadjoint random variables which generate A . If ( x i ) i ∈ Z ismonotonically spreadable, then the negative conditional expectation E − can be extend tothe whole algebra A such that φ ( aE − [ y ] b ) = φ ( E + [ a ] yE + [ c ]) for all y ∈ A and a, c ∈ A [ −∞ , ∞ ] . Moreover, the extension is normal. de Finetti type theorem for boolean spreadability In this section, we assume that ( A , φ ) is a W ∗ -probability space with a non-degeneratednormal state and A is generated by a bilateral sequence of random variables ( x i ) i ∈ Z and( x i ) i ∈ Z are boolean spreadable. Lemma 9.1. Let y i = x − i for all i ∈ Z , then ( y i ) i ∈ Z is also boolean spreadable.Proof. By proposition 3.22, it suffices to show that ( y i ) i =1 ,...,n is boolean spreadable forall n ∈ N . Given a natural number k < n , assume the standard generators of B i ( n, k ) are { u i,j } i =1 ,...,n ; j =1 ,...,k and invariant projection P . Consider the matrix { u ′ i,j } i =1 ,...,n ; j =1 ,...,k such that u ′ i,j = u n +1 − i,k +1 − j . . it is obvious thatthe entries of the matrix are are orthogonal projections and n X i =1 u ′ i,j P = n X i =1 u i,k +1 − j P = P . Given j, j ′ , i, i ′ ∈ N such that 1 ≤ j < j ′ ≤ k and 1 ≤ i ≤ i ′ ≤ n . Then, we have n + 1 − i ≤ n + 1 − i ′ and k + 1 − j < k + 1 − j ′ . Therefore, u ′ i,j u ′ i ′ ,j ′ = u n +1 − i,k +1 − j u n +1 − i ′ ,k +1 − j ′ = 0 . It implies that { u ′ i,j } i =1 ,...,n ; j =1 ,...,k and P satisfy the universal conditions of B i ( n, k ). Itfollows that there exists a unital C ∗ -homomorphism Φ : B i ( n, k ) → B i ( n, k ) such that:Φ( u i,i ) = u ′ i,j , and Φ( P ) = P . Let z i = x i − n − for i = 1 , ..., n . Since ( x i ) i ∈ Z are boolean spreadable, ( z i ) i =1 ,...,n is booleanspreadable. Therefore, for i , ..., i L ∈ [ k ], we have φ ( y i · · · y i L ) P = φ ( y n − k + i · · · y n − k + i L ) P = φ ( x − n + k − i · · · x n − k − i L ) P = Φ( φ ( z k +1 − i · · · z k +1 − i L ) P )= Φ( n P j ,...,j L =1 φ ( z j · · · z j L ) P u j ,k +1 − i · · · u j L ,k +1 − i L P )= n P j ,...,j L =1 φ ( z j · · · z j L ) P u n +1 − j ,i · · · u n +1 − j L ,i L P = n P j ,...,j L =1 φ ( x j − n − · · · x j L − n − ) P u n +1 − j ,i · · · u n +1 − j L ,i L P = n P j ,...,j L =1 φ ( y n +1 − j · · · y n +1 − j L ) P u n +1 − j ,i · · · u n +1 − j L ,i L P = n P j ,...,j L =1 φ ( y j · · · y j L ) P u j ,i · · · u j L ,i L P which completes the proof. (cid:3) Proposition 9.2. ( A , φ ) is a W ∗ -probability space with a non-degenerated normal stateand A is generated by a bilateral sequence of random variables ( x i ) i ∈ Z and ( x i ) i ∈ Z areboolean spreadable. Then, E − and E + can be extend to the whole algebra A . Moreover, E − = E + Proof. Since ( x i ) i ∈ Z is boolean spreadable, ( x i ) i ∈ Z is monotonically spreadable. By propo-sition 8.2 E − can be extended to the whole algebra. By Lemma 9.1, ( x − i ) i ∈ Z is also booleanspreadable and its negative-conditional expectation is exactly the positive conditional ex-pectation of ( x i ) i ∈ Z . Therefore, E + can also be extended the whole algebra A normally.Give a, b, c ∈ A [ −∞ , ∞ ] , by Lemma 8.2, we have ONCOMMUTATIVE SPREADABILITY 43 φ ( aE − [ b ] c ) = φ ( E + [ a ] bE + [ c ])= φ ( E + [ E + [ a ] bE + [ c ]])= φ ( E + [ a ] E + [ b ] E + [ c ])= lim n →∞ φ ( α n ( a ) E + [ b ] E + [ c ])= lim n →∞ lim m →∞ φ ( α n ( a ) E + [ b ] α m ( c ))Notice that, for fixed n, m , φ ( α n ( a ) E + [ b ] α m ( c )) = φ ( α n ( a ) α L ( b ) α m ( c ))for L ∈ N which is large enough. Since ( x − i ) i ∈ Z is monotonically spreadable, by theorem1.1, ( x − i ) i ∈ Z is monotonically independent with respect to E − . Therefore, we have φ ( α n ( a ) E + [ b ] α m ( c ))= φ ( α n ( a ) α L ( b ) α m ( c ))= φ ( E − [ α n ( a ) α L ( b ) α m ( c )])= φ ( E − [ α n ( a )] E − [ α L ( b )] E − [ α m ( c )])= φ ( E − [ a ] E − [ b ] E − [ c ])= φ ( E − [ E − [ a ] bE − [ c ]])= φ ( E − [ a ] bE − [ c ])= φ ( aE + [ b ] c ) φ ( aE − [ b ] c ) = φ ( E + [ a ] bE + [ c ])= lim n →∞ lim m →∞ φ ( α n ( a ) E + [ b ] α m ( c ))= lim n →∞ lim m →∞ φ ( aE + [ b ] c )= φ ( aE + [ b ] c )It implies that E + [ b ] = E − [ b ] for all b ∈ A [ −∞ , ∞ ] . Since A is the WOT closure of A [ −∞ , ∞ ] ,the proof is complete. (cid:3) Corollary 9.3. ( A , φ ) is a W ∗ -probability space with a non-degenerated normal state and A is generated by a bilateral sequence of random variables ( x i ) i ∈ Z and ( x i ) i ∈ Z are booleanspreadable. Then, the positive tail algebra and the negative tail algebra of ( x i ) i ∈ Z are thesame. Now, we are ready to prove theorem 1.3 Theorem 9.4. Let ( A , φ ) be a non degenerated W ∗ -probability space and ( x i ) i ∈ Z be a bilat-eral infinite sequence of selfadjoint random variables which generate A as a von Neumannalgebra. Then the following are equivalent: a) The joint distribution of ( x i ) i ∈ N is boolean spreadable. b) The sequence ( x i ) i ∈ Z is identically distributed and boolean independent with respectto the φ − preserving conditional expectation E + onto the non unital positive tailalgebra of the ( x i ) i ∈ Z Proof. “ b ) ⇒ a )”. If the sequence ( x i ) i ∈ Z is identically distributed and boolean indepen-dent with respect to a φ − preserving conditional expectation E , then sequence ( x i ) i ∈ Z isboolean exchangeable by theorem 7.1 in [16]. According the diagram in section 4, ( x i ) i ∈ Z is boolean spreadable.“ a ) ⇒ b )”. By Lemma9.2, ( x i ) i ∈ Z is monotone with respect to E + , ( x − i ) i ∈ Z is monotonewith respect to E − and E + = E − . 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