FREE DENDRIFORM ALGEBRAS. PART I. A PARENTHESIS SETTING

In Section 2, we propose a reformulation of the free dendriform algebra over the generator via a parenthesis setting. We propose at the same time both a brief survey on trees and new results proved from the parenthesis setting. In Section 3, we present a bijection between planar rooted binary trees and noncrossing partitions. This allows the introduction of the concept of NCP-operads, whose axioms look like regular operads ones. We then show, in Section 4, how to use the free dendriform algebra on one generator to reformulate some results obtained by Speicher in free probability. In this paper, K is a characteristic zero field, N is the semiring of integers, and Nb stands for the set {v := (v1, . . . ,vn)∈Nn; for all 1≤ i≤ n, 0 < vi ≤ i}, in bijection with Sn, the symmetric group over n elements. If S is a finite set, then card(S) denotes its cardinal, KS, the K-vector space spanned by S, and 〈S〉, the free associative semigroup generated by S. Rooted planar binary trees will be called binary trees for short. For all n > 0, we mean by Yn the set of planar rooted binary trees with n+ 1 leaves. If k ∈ K , v ∈ Kn, then (k+ v) is the vector v whose coordinates have been shifted by k.


Introduction
In Section 2, we propose a reformulation of the free dendriform algebra over the generator via a parenthesis setting.We propose at the same time both a brief survey on trees and new results proved from the parenthesis setting.In Section 3, we present a bijection between planar rooted binary trees and noncrossing partitions.This allows the introduction of the concept of NCP-operads, whose axioms look like regular operads ones.We then show, in Section 4, how to use the free dendriform algebra on one generator to reformulate some results obtained by Speicher in free probability.
In this paper, K is a characteristic zero field, N is the semiring of integers, and N n b stands for the set {v := (v 1 ,...,v n ) ∈ N n ; for all 1 ≤ i ≤ n, 0 < v i ≤ i}, in bijection with S n , the symmetric group over n elements.If S is a finite set, then card(S) denotes its cardinal, KS, the K-vector space spanned by S, and S , the free associative semigroup generated by S. Rooted planar binary trees will be called binary trees for short.For all n > 0, we mean by Y n the set of planar rooted binary trees with n + 1 leaves.If k ∈ K, v ∈ K n , then (k + v) is the vector v whose coordinates have been shifted by k.

Arithmetics on trees from operads
For details about operads, the reader should consult the literature, for instance, [1,4,10].Dendriform algebras have been introduced by Loday [5] as dual, in the operadic sense, to associative dialgebras, themselves motivated by K-theory.The free dendriform algebra on 2 Free dendriform algebras.Part I.A parenthesis setting one generator is then closely related to binary trees.Major developments [7] have been put forward by using the Hopf algebra structure on the regular representations of the permutation groups founded by Malvenuto and Reutenauer [9] and connections between permutations and binary trees.Since then, an arithmetic on trees has been introduced by Loday [6].The aim of this section is to present another way to handle the free dendriform algebra on one generator.Instead of starting with coding binary trees via permutations, we focus on the parenthesizing the meaning of binary trees.

Binary trees versus vectors.
Contrary to permutations used in [6,8], we propose in this section another way to encode binary trees compatible with the Tamari order which will play a key role to explicit operations on trees.For that, we associate with a planar binary tree of Y n a unique vector of N n in the following way.To any binary tree τ corresponds a unique parenthesizsing, and therefore a unique monomial in x 1 ,...,x n+1 ,(,) and thus a unique monomial in x 1 ,...,x n+1 ,( obtained by forgetting all right parentheses.Proceeding this way, we obtain an injection: Exp : Y n → x 1 ,...,x n+1 ,( .In the sequel, to ease notation, the unique parenthesizing associated with the binary tree τ will be also represented by Exp(τ) as in the following example: ( Exp (x 1 (x 2 ((x 3 x 4 )x 5 ))) Exp (x 1 (x 2 ((x 3 x 4 )x 5 ))) ( Encode the parentheses of Exp(τ) of the binary tree τ in a vector v := (v 1 ,v 2 ,...,v n ) of N n by declaring that for all 1 ≤ i ≤ n, v i := i if and only if there exists a left parenthesizing at the left-hand side of x i , that is, ...( p x i ..., with p > 0, occurs in the monomial Exp(τ).Otherwise, there exists a unique rightmost parenthesis at the right-hand side of x i which closes a unique left parenthesis, say, open at x j .In this case, v i := j.Observe that this framework works since binary trees, via their leaves, model all parentheses one can obtain from a binary operation.We then obtain an injective map; name : Y n → N n , which maps any tree τ into a vector, name(τ), also denoted by τ for short, called the name of τ.We warn the reader that we use the word "name" with a different meaning that in [6].
In the sequel, name(Y n ) will be denoted by A n (A, like Appellation), and by complete expression, we mean a monomial of x 1 ,...,x n+1 ,(,) in one-to-one correspondence with a rooted planar binary tree, that is, every parenthesis ( is closed by a unique parenthesis ).
There is a unique monomial ( q1 x 1 ( q2 x 2 ••• ( qn x n x n+1 from x 1 ,...,x n+1 ,(,) associated with v, where q i is the number of i's appearing in v.Such an algorithm gives a surjective map Tree : , where q j is the number of j's appearing in v. Observe that q j = n and 1 ≤ q j ≤ n − j + 1 since j may appear only Philippe Leroux 3 from x j .Take the highest j, with q j = 0, that is, consider ( qj x j x j+1 ••• x n .As parentheses model a binary operation, there is a unique way to set right parentheses, namely, Proposition 2.1 is in fact an error-correcting code.Let us apply it to (1,2,1,2).This gives ((x 1 ((x 2 x 3 x 4 x 5 , that is, ((x 1 ((x 2 x 3 )x 4 ))x 5 ), that is, the tree named by (1,2,2,1).

Corollary 2.2 (reconstruction criterion). A vector v ∈ N n
b is the name of a binary tree if and only if name(Tree(v)) = v.
The grafting operation can be extended by bilinearity to KA ∞ := n≥0 KA n .In the sequel, we set KA ∞ * := n≥1 KA n , A • := n≥0 A n , and Coding the over and under operations.Before going on, recall that an associative L-algebra is a K-vector space A equipped with two binary operations , : A ⊗2 → A and obeying three constraints.The two operations are associative and verify (x y) z := x (y z).L-monoids are straightforward to define.From a (co)algebraic point of view, Lcoalgebras have been introduced on graphs in [2,3].In [8], Loday and Ronco introduced the operations over and under on trees, denoted, respectively, by , : Y n × Y m → Y n+m , for all n,m = 0, where π τ is the tree τ with its leftmost leaf identified with the root of π and where π τ is the tree π with its rightmost leaf identified with the root of τ.These two operations have a common unit which is .To define the analogue of these two operations on vectors, consider the map : where k +v i := k + v i , for v i = 1 and k +1 := 1 (otherwise stated, 1 is a right annihilator for the operation +).
where the presence of a strict inequality on the left-hand side induces a strict one in the right-hand side.Moreover, v w ≤ v w holds.
Proof.The proof is complete by using Propositions 2.3 and 2.4.
it is straightforward to prove that the free L-algebra over one generator x is isomorphic to (KA ∞ * , , ) by mapping x to the generator (1) (see also [11]).By anticipating the ideas of Loday (explained in detail below), one can convert operations , into set operations called L-additions denoted by + ,+ : A n × A m → A n+m where v + u := v u and v + u := v u.These additions are associative and noncommutative.Similarly, there is a notion of L-multiplication.As (KA ∞ * , , ) is the free L-algebra on the generator (1), one can uniquely write any name of binary trees via only the operations and and (1).Such a formula, for a vector v, is called its universal expression and is denoted by v ((1)), obtained by the following induction v (( 1) where m is a prime number, will be prime for the L-arithmetics.Consider now the K-vector space K[X] L spanned by {X v , v ∈ (A • * ,+ ,+ , )}.This is the free L-algebra over the generator X (1) where, as expected, operations are defined by X u X v := X u+ v , X u X v := X u+ v , and (X u ) v := X u v , imitating the usual ploynomial algebra on one variable endowed with the usual arithmetics over N.There is also a dendriform involution †, described in Section 2.2.2.We summarize our investigation by the following theorem.
6 Free dendriform algebras.Part I.A parenthesis setting Theorem 2.6.The set A • * equipped with the L-additions, + , and + , and with the dendriform involution † is an involutive-graded L-monoid.The L-multiplication, , is left distributive associative though noncommutative.For any names of trees, u, v, } is the free involutive associative L-algebra over the generator X (1) .Proposition 2.7.Fix n,m = 0 and x, y ∈ (A n ,<) and a,b ∈ (A m ,<).With regards to the trivial partial order, the map x : A m → A nm is a lattice morphism, that is, x a < x b ⇔ a < b and the map a : A n → A nm is also a lattice morphism, that is, x a < y a ⇔ x < y.
Proof.Keep notation of Proposition 2.7 For the first claim, proceed by induction.It is true for x = (1), for x = (1,1) and for x = (1,2).By Corollary 2.5, The proof is complete by induction.Concerning the second claim, if x < y, then there exist say k Tamari moves between the trees associated with x and y.Suppose k = 1.Then, in the definitions of x, y, this means the existence of three vectors say v 1 , v 2 , v 3 such that we have The operations , have a common unit which is (0) ≡ .However, the link axiom of L-algebras is not compatible with this unit since it forces = .Using the trivial partial order, we will exhibit an associative operation, sum of two nonassociative operations obeying three axioms.This operation has first been introduced by Loday and Ronco by using techniques in permutation groups [8].One of the main advantages of our coding is to give easier proofs to these results.
Proposition 2.8.The following binary operation: and is associative.The last claim is obvious since (0) is by definition a unit for the operations and .
The sum in the definition of the associative product can be split into two parts corresponding to two operations.Proposition 2.9.Let v and w be names of some trees.Then, the set I := {u; v w ≤ u ≤ v w} splits into two disjoint subsets: Proof.First of all, observe that (v, v w) = (v, v w l , 1, v + 1 + w l + w r ).Therefore, we have only to compare (v w l , 1) and (v + w l , 1) in I 2 .Similarly, concerning I 1 , observe that (v, . Therefore, we have to compare the vector (v l + 1 + v r w l , v l + 2) with (v + w l , v + 1).As v l represents a complete expression, jumps of coordinates situated after v l cannot take values below v l .From this remark, one obtains that I 1 and I 2 are disjoint and I 1 ∪ I 2 = I.
We recover from a vectorial framework the dendriform algebra introduced in [5].Recall that a K-vector space E is a dendriform algebra [5] if it is equipped with 2 binary operations ≺ and satisfying the following axioms for all x, y ∈ E: where, by definition, x y := x ≺ y + x y, for all x, y ∈ E, where turns out to be associative.We now adapt the following theorem appearing in [5,8].

Arithmetree on planar binary trees.
After the reformulations of constructions developed in [5,8], let us recall a deep notion introduced by Loday.We follow [6].A grove is simply a nonempty subset of Y n , that is, a disjoint union of binary trees with same degree such that each tree appears only once.The set of groves over Y n is denoted by Y n and is of cardinal 2 cn − 1.For instance, in low degrees, (2.7) 8 Free dendriform algebras.Part I.A parenthesis setting Similarly, we define A n in the same way.Instead of binary trees, we work with the set A n , which are the names of groves of Y n .Hence, A 0 := {(0)}, A 1 := {(1)}, A 2 := {(1, 1),(1,2), (1,1) ∪ (1,2)}, and continue to call grove such a union of vectors.The idea is now to convert the associative operation in Proposition 2.8 into an addition with values in groves.

The dendriform addition.
Definition 2.12 (dendriform addition [6]).The dendriform addition of two vectors v and w associated with some planar binary trees is defined by .This is extended to groves by distributivity of both sides, that is, i v i j w j := i j (v i w j ).Theorem 2.14 below proves that this definition has a meaning.This fundamental result has been found by Loday by using the following lemma whose proof can be simplified by using our vector codes.Lemma 2.13.Let w ∈ A .Then, there exists unique u ∈ A n and v ∈ A m such that (2.9) Proof (cf., to [6], Proposition 2.3, and Corollary 2.4).Recall that for u ∈ A n and v ∈ A m , we get u v = (u,u v) and u v = (u,u + v).Take the first n coordinates of w ∈ A n+m .This gives a unique vector u ∈ A n according to Proposition 2.1.Consider the vector v 1 defined by v 1 := (w n+1 ,...,w n+m ).Make the translation of −n := −u to obtain v 1 − u = (w n+1 − u,...,w n+m − u).The vector v ∈ A m we are looking for is obtained by replacing all negative or null coordinates by 1. Observe that Theorem 2.14 (Loday [6]).The dendriform addition of two groves is still a grove, that is,

(left and right cancellations). Let u,v
Proof.Using Proposition 2.4, recall that u v = (u;u + v) and u w = (u;u + w).The equality u v = u w entails that u v = u w; hence v = w.The second equality is straightforward.

The dendriform involution.
There is an involution on A • := n≥0 A n denoted by † and defined by (v ∨ w) ) is an involutive graded monoid.Observe that (1) † = (1) and by convention, we set (0) † := (0).We now state some properties of the involution on trees.
Proof.Observe that v = v † if and only if there exists a unique w such that v = w ∨ w † .
[Trick to name v † ].Fix v ∈ A n .There exists a very simple way to name v † .Associate with v, its complete expression in x 1 ,...,x n ,x n+1 ,(,) .Relabel x n+1 by x 1 , x n by x 2 and so on.Read therefore from left to right such a monomial.The vector v † ∈ A n is obtained from the following construction.The coordinate v † i := i, for 1 ≤ i ≤ n, if and only if there is a ) at the right-hand side of x i and v † i := j if the leftmost parenthesis ( at the left-hand side of x i closes a ) open in x j .This works since the involution on binary trees is a symmetry with regards to the root axis, which can also be viewed as a symmetry with regards to an axis perpendicular to it-the Mirror axis-giving then the mirror image of the tree and thus its involution: Root axis ((x 4 (x 3 x 2 )) x 1 ) Name: (122) Mirror axis ((x 1 (x 2 x 3 )) x 4 ) Name: (121) (2.10) Proposition 2.17 (lattice anti-automorphism).Let v,w ∈ A n .Then, the dendriform involution is a lattice anti-automorphism, that is, v < w ⇔ w † < v † .Consequently, M(v,w) = M(w † ,v † ), for any names of trees.
Proof.Fix v,w ∈ A n with v < w.We will check the case when both v i = i = w i .In this case v † n+1−i = j and w † n+1−i = j with j ≤ j < N + 1 − i.Indeed, suppose the existence of a ), the most external parenthesis standing at the right-hand side of x j and closing one ( open in x i in the complete expression associated with w.As v < w, we get v j ≤ w j = i.If v j = k < i, then this means that the most external parenthesis ) standing at the righthand side of x j in the complete expression associated with v closes one ( open in x k .This implies that v † n+1−i ≤ w † n+1−i .Checking every possibility leads to the conclusion that The proof is complete since the dendriform involution is an involution.For the last claim, recall that the dual lattice of (A n ,<) with order ≤ * is defined such that v ≤ * w ⇔ w ≤ v. Therefore, v ≤ * w ⇔ v † ≤ w † , for any vectors of A n .The last claim holds since M * (v,w) = M(w,v) (see [13]).

The dendriform multiplication.
The following idea developed by Loday consists in replacing the polynomial ring K[X] (basis (X n ) n∈N ) and well-known equations X n X m := X n+m and (X n ) m := X nm related to the usual arithmetic on N by planar binary trees.Instead of writing K[X], one could have chosen K [N] to denote this polynomial ring.Consider the K-vector space ] has a natural dendriform algebraic structure given by: X u ≺ X v := X u v and X u X v := X u v , with the convention X u∪v := X u + X v .As expected, X u X v := X u v , where is the associative product, sum of ≺ and .This nonunital associative algebra, another presentation of the free dendriform algebra on one generator, here X (1) , can be augmented by adding the unit 1 := X (0) so that, . By convention, we set X ∅ = 0.As usual, the operations ≺ and can be partially extended to K[A • ] by declaring that 1 X v := X v =: X v ≺ 1, for v = (0) and vanish otherwise, explaining the presence of the empty set.For instance, 1 ≺ X v := X (0) v := X ∅ := 0, as expected.The notation K[N] := K[X] stands for the usual polynomial algebra on one variable say X.As X n X m := X n+m and N is invariant by addition, one can use also the notation K[N] without any ambiguity.However, K[A • ] is not invariant by the dendriform addition, that is why we choose the notation Definition 2.18 (dendriform multiplication [6]).The dendriform multiplication : A n × A m → A nm is given by u v := ω u (v), for all u and v are names of binary trees and extended to groves via distributivity on the left with respect to the disjoint union, that is, . The dendriform multiplication is associative, not commutative, distributive on the left with regards to that the dendriform addition has the neutral element (1) and is compatible with the involution †, (u v) † = u † v † .Moreover, the neutral element for , that is, (0), is by convention a left annihilator for , that is, (0) u = (0).A vector w ∈ A n is said to be prime if there exists no vector v ∈ A m and v ∈ A m , with n = mm such that w = v v .In general, the dendriform product of two vectors gives a grove.However, observe there are two unique ways to obtain a vector.The first one is to consider (1) (1) v, with v := v 1 ∨ (0) and the second one is to consider (1) (1) v, with v := (0) ∨ v 1 .In the first case, we obtain (1) (1) v = v 1 ∨ (v 1 ∨ (0)) and in the second case, (1) (1) v = ((0) ∨ v 1 ) ∨ v 1 .We summarize our discussion by the following proposition.Proposition 2.19.Any vector of A 2n+1 is prime for the arithmetree just described.Whereas there exist 2c n nonprime vectors in A 2n+2 .They are of the forms ((0

(right and left cancellations). Let
Proof.The first claim is obtained by observing that the first operation appearing in ω v ((1)) is either or .Therefore, in both cases, the vectors composing the groves v u and v w will start with (u,...), respectively, with (w,...).The same remark applies also for the second claim.To complete the proof, observe that the dendriform multiplication acting on the right-hand side is the unique dendriform automorphism which maps the generator X (1) to The construction of the inverse map is left to the reader.We therefore obtain a bijection between rooted binary trees and noncrossing partitions.

Free probability
Free probability has been introduced by Voiculescu.Later, a complementary point of view was given by Speicher [15], inspired by previous works of Rota [12].We will focus on [15].

4.1.
Action of arithmetree on Ꮾ − Ꮾ-bimodule and operads.In the sequel, Ꮾ denotes a unital associative algebra (most of the time a unital C * -algebra for applications) and ᏹ is a Ꮾ − Ꮾ-bimodule.We denote by ᏹ ⊗Ꮾn the space ᏹ ⊗ Ꮾ ᏹ ⊗ Ꮾ ••• ⊗ Ꮾ ᏹ, n times and by convention ᏹ ⊗ Ꮾ 0 := B. By abuse of language and sometimes to ease notation, we will use equivalently trees and/or their names.One of the aims of this part is to describe the action of the space K[Y ∞ ]-or equivalently K[A • ]-equipped with its arithmetree onto the bimodule ᏹ.
Definition 4.1.Let Ꮾ be an associative K-algebra.A NCP(Ꮾ)-operad P (without unit) over a Ꮾ − Ꮾ-bimodule ᏹ is the data of a family of finite dimensional K-vector spaces (P(n)) n>0 , whose basis elements μ are Ꮾ − Ꮾ-bimodule n-ary operations with values in Ꮾ, that is, μ : ᏹ ⊗ Ꮾ n → Ꮾ, and equipped with a family of composition maps ((• i ) i>0 ) verifying the following relations.
(1) For all μ ∈ P(m) and ν ∈ P(n) and 1 (2) For all λ ∈ P(l), μ ∈ P(m), and ν ∈ P(n), Philippe Leroux 13 Remark 4.2.This concept is inspired from [15] but is different, though similar, from the definition of a regular K-linear operad over a vector space ᏹ.Indeed, take Ꮾ as the ground field.In the usual operad theory, the basis elements μ are n-ary operations with values in ᏹ, that is, μ : ᏹ ⊗ Ꮾ n → ᏹ, and not with values in the ground field.This entails a slight modification of the usual axioms of an operad over a vector space ᏹ presented just above.
A noncrossing partition μ ∈ NC(n) is said to be decorated by a set Col if a unique color of Col is associated with each interval composing it.Observe that decorated noncrossing partitions give special decorated binary trees, that is, binary trees whose all vertices of a SW-NE-branch have the same color.We now give an example of such NCP(Ꮾ)-operad by mixing results in Section 2 on the free dendriform algebra on one generator and ideas developed from noncommutative probability.* ] induces a dendriform algebra structure on the following K-vector space: Dend Ꮾ (ᏹ) : for any tensor κ ∈ ᏹ ⊗ Ꮾ n and κ ∈ ᏹ ⊗ Ꮾ n .Inspired by [15], we define a family of n-ary operations, , for all n > 0 and b,b ∈ Ꮾ and a 1 ,...,a n ∈ ᏹ.With each family ( f (n) ) n≥1 , we associate the following operator valued function f , acting on the whole Dend Ꮾ (ᏹ): and defined via the following recursive prescription.With any monomial X v , a unique noncrossing partition Pr(v) is associated, constructed from the algorithm described in the previous section.Identify this partition to the tensor a Localize the most nested block of length p ≤ n and apply the p-ary operations, giving thus an operator in Ꮾ.Then, reapply this procedure.In the sequel, we will write to denote that action of the noncrossing partition Pr(v).The following examples will be better than a fastidious description.Here are three examples (recall that a ⊗ Ꮾ ba = ab ⊗ Ꮾ a , for b ∈ Ꮾ). ( 14 Free dendriform algebras.Part I.A parenthesis setting (3) Let Pr(v) = (1,9)(2,6,7)(3,4)(5)(8)(10) be the noncrossing partition represented in Section 3 and get and obtain f (1) (a 8 )a 9 ) f (1)  (a 10 ).
Remark 4.3.Proceeding that way, observe that f Remark 4.4.We can slightly reformulate this framework using the concept of NCP(Ꮾ)operad.Let Col := { f (n) , n > 0} be the color set made out of the n-ary operations f (n) .
Observe that with each noncrossing partition, a unique decorated noncrossing partition can be associated.Introduce the object P[ f ] made out of a family of the K-vector spaces (P[ f ](n)) n>0 and the family of composition (• i ) i>0 defined by induction as follows.The K-vector space P[ f ](1) is spanned by f (1) and P[ f ](p) by the elements f (p) and the μ for all a 1 ,...,a n ∈ ᏹ and not in Ꮾ.From a noncrossing partition, one can easily write its action on tensor elements in terms of composition maps.The following example will fix ideas.
Example 4.5.Consider again Pr(v) := (1,9)(2,6,7)(3,4)(5)(8) (10), the noncrossing partition represented in Section 3. Read the partition from left to right.Take the first encountered interval, say, with p elements (here {1, 9} and p := 2) and (thus) starting with 1.Take the second encountered interval, starting with say n, and with say q elements (here {2, 6,7} and q := 3) and write f (p) • n f (q) ••• .Reapply the algorithm.We obtain The following theorem summarizes the previous discussion.Philippe Leroux 15 4.2.Cumulants and moments in free probability.We recall some properties of free probability [15].Fix B, a unital associative algebra.Let (ᏹ,φ) be a noncommutative probability space, that is, a B − B-bimodule ᏹ endowed with a unital associative algebra structure equipped with a B − B-bimodule map φ : ᏹ → B such that φ(1) = 1, that is, φ(b) = b, for any b ∈ B. Let ᏹ 1 ,...,ᏹ n be n unital B − B-subalgebras of ᏹ.It is said that ᏹ 1 ,...,ᏹ n are stochastically free if φ(a 1 ••• a n ) = 0 under the following conditions.For all 1 ≤ i ≤ n, φ(a i ) = 0 and for a 1 ∈ ᏹ 1 ,...,a n ∈ ᏹ n , 1 = 2 , 2 = 3 ,..., n−1 = n .It has been shown by Speicher, that this definition can be reformulated in terms of noncrossing partitions equipped with the refinement order.For that, he introduced in [15], the set n>1 NC(n) × ᏹ ⊗ Ꮾ n and a family of functions φ = (φ (n) ) n>1 : n>1 NC(n) × ᏹ ⊗ Ꮾ n → B and defined φ(π)(a 1 ⊗ Ꮾ ••• ⊗ Ꮾ a n ), where π is a noncrossing partition, as explained in the previous section.The idea is to replace the object n>1 NC(n) × ᏹ ⊗ Ꮾ n by a much more structured one, that is, Dend Ꮾ (ᏹ) equipped with the NCP(Ꮾ)-operad P[ φ].We are now able to reformulate the major result of Speicher [15].Recall that a moment function [15] φ is defined by φ (1) (1) = 1 and by (n > 1), (4.8) In this case, one can choose In our framework, Speicher showed also that the cumulant function C obtained by convolution of φ with the Zeta function associated with the refinement order of the noncrossing partitions is still a map from Dend Ꮾ (ᏹ) to B. We now reformulate the result of Speicher [15].
Theorem 4.7 [15].Fix B, a unital associative algebra.Let (ᏹ,φ) be a noncommutative probability space and φ a moment function.Let ᏹ 1 ,...,ᏹ n be n unital B − B-subalgebras of ᏹ.Consider the set I := {X n ⊗ a 1 ⊗ Ꮾ ••• ⊗ Ꮾ a n ∈ ᏹ; for all n > 1; such that ∃ i, ja i ∈ ᏹ i , a j ∈ ᏹ j , and i = j }.Then, ᏹ 1 ,...,ᏹ n are stochastically free if and only if I ⊆ ker C, where C : Dend Ꮾ (ᏹ) → B is the cumulant function associated with φ via the convolution with the Zeta function with respect to the refinement order.
. The second claim holds for k = 1, and thus for all k.