Differential independence via an associative product of
infinitely many linear functionals
Takahiro Hasebe
Graduate School of Science, Kyoto University
Kyoto 606-8502, Japan
E-mail: hsb@kurims.kyoto-u.ac.jp
Abstract
We generalize the infinitesimal independence appearing in free probability of type
B to two directions: higher order derivatives and the other natural independences, i.e.,
tensor, monotone and Boolean ones. Such generalized infinitesimal independences can beformulated as associative products of infinitely many linear functionals, and therefore theassociated cumulants can be defined. These products can be seen as the usual natural
products of linear maps with values in formal power series.

1

Introduction

Free probability theory [24] was initiated by Voiculescu to solve problems in operator algebras.
Many concepts in probability theory have analogues in free probability by replacing the concept
of independence with free independence for random variables. For instance, cumulants whichappear in probability theory are replaced by free cumulants in free probability. There are otherexamples such as the central limit theorem and infinitely divisible distributions [23].
In probability theory, a standard model of independent random variables is constructedby taking the direct product of probability spaces. In the algebraic description, the usual
independence is realized on the tensor product of algebras equipped with the tensor product ofstates. In analogy to this, free independent random variables can be realized in the free productof algebras equipped with free product of states [24]. An important point is that the tensor
and free products of states are associative; this property enables us to define unique cumulants
associated to free independence.
There are other associative products of states on the free product of algebras (with orwithout a unit), i.e., Boolean [4, 22], monotone and anti-monotone [16, 18] products. Under
natural conditions, Schürmann, Speicher, Ben Ghorbal and Muraki [2, 18, 17, 19, 21] provedthat an associative product of states is necessarily one of the tensor, free, Boolean, monotone
2010 Mathematics Subject Classification: Primary 46L53; Secondary 46L54.
Key words and phrases: free probability of type B, tensor independence, monotone independence, Boolean
independence.

1and anti-monotone products. Among them, only monotone and anti-monotone products are notsymmetric, that is, independence of random variables X and Y does not imply independence ofY and X. This makes it impossible to define cumulants in the usual sense (a unified treatmentof cumulants in the “usual sense” can be found in [15]). In papers [14] and [13], however, it
is proved that there exist uniquely defined cumulants in a generalized sense. We will use thisgeneralized framework later.
There are also several attempts to construct an associative product of more than one state.
A conditionally (or c- for simplicity) free product were introduced by Bożejko, Leinert andSpeicher in [6, 5] as a product of two states. This product is important since it interpolates
free and Boolean products, preserving the associative laws. Moreover, in [10] Franz foundthat the c-free product also unifies monotone and anti-monotone products, not preserving theassociative laws (the reader is referred to [12] for details). Then a c-monotone product of
two states was introduced to unify monotone and Boolean products in [11]. Furthermore, anordered free product and an indented product were defined in [12]; the former one is defined fortwo states and unifies free and monotone products, and the latter one is defined for three states
unifying free, Boolean, monotone and anti-monotone products (and also c-free and c-monotone
products).
In this paper, we construct an associative product of infinitely many linear functionals for agiven natural independence. The idea of a product of infinitely many states or linear functionals
was proposed by Cabanal-Duvillard and Ionescu in [7]. They constructed a product of infinitely
many states as an extension of the c-free product. This product is, however, not associative.
The product defined in this paper extends the infinitesimal aspect of free probability oftype B. Free probability of type B was originally defined by Biane, Goodman and Nica [3] tofind a free probability associated to type B non-crossing partitions. Later, in [1] Belinschi and
Shlyakhtenko found that free probability of type B appears as an infinitesimal property of freeprobability. More precisely, let {ϕt }0≤t<ε be a family of states on an algebra A for ε > 0. If Aj
are free independent w.r.t. ϕt for all t, then free probability of type B can be realized w.r.t. thepair (ϕ0 , ϕ00 ), where ϕ00 is defined by the linear functional X 7→ dtd ϕt (X)|t=0 . Février and Nica
investigated combinatorial properties of infinitesimal free probability in [9].
We extend free probability of type B from two viewpoints: higher derivatives and natural
independence. This is the contents of Section 2. This extension can be understood as an
associative product of infinitely many linear functionals. By using associativity, we define
cumulants in Section 3. These cumulants are obtained by formally differentiating the usual
cumulants for natural independence. In Section 4, we study two examples: one is formal
multi-variate Lévy processes, and the other is dual derivation systems introduced in [9]. Many
contents of Sections 2–4 can easily be extended to more general settings. We will mention suchgeneralizations in Section 5.
Note. The author has learned that Février also extended free probability of type B tohigher order derivatives, independently of this paper. He also clarified combinatorics of higher
order derivatives; see [8].

2Differential independence

In this section we extend the infinitesimal realization of free probability of type B to higher
derivatives, and also extend free independence to the other natural independences, i.e., tensor,
2monotone and Boolean independences.
Throughout this paper, symbols such as A and Ai always denote algebras over C. We
denote by A∗ the set of the linear functionals from A to C. We also denote by A1 ∗ A2 the freeproduct of A1 and A2 without identification of units. Let ? be any one of the natural productsof linear functionals, i.e., a product? : A∗1 × A∗2 → (A1 ∗ A2 )∗ ,defined for arbitrary algebras A1 and A2 , satisfying some natural conditions in terms of category
theory. Natural products were classified into five ones: tensor, free, Boolean, monotone andanti-monotone ones. It is known that natural products of linear functionals preserve positivity,
and therefore a natural product of states is again a state.
Let CJtK be the unital ring of formal power series. We introduce CJtK-valued linear maps
to treat all the objects algebraically. First we consider a natural product of CJtK-valued linear
maps ϕt : A → CJtK which is defined essentially in the same way as the C-valued case. For
instance, the concept of joint moments is almost the same as those for C-valued independence.
In this paper we do not discuss anti-monotone independence since what we prove for monotone independence can easily be translated into anti-monotone independence. This is only
within the scope of this paper and we do not insist that every property of anti-monotone
independence is always translated into that of monotone independence.
Now we define CJtK-valued natural products of linear maps on the free product of algebras.
In the following definitions, (λ1 , λ2 , λ3 , · · · , λn ) is any alternating sequence of 1 and 2 of arbitrary length n, that is, (λ1 , λ2 , λ3 , · · · ) = (1, 2, 1, · · · ) or (2, 1, 2, · · · ). Moreover, Xk is any
element in Aλk for 1 ≤ k ≤ n.
Definition 2.1. (Tensor product) The tensor product ϕt1 ⊗ ϕt2 on A1 ∗ A2 is defined by
ϕt1 ⊗ ϕt2 (X1 · · · Xn ) = ϕt1³−−Y−−→ ´ ³−−Y−−→ ´
Xi ϕt2Xi .
i:Xi ∈A1−→
Q

j∈V Xj denotes the ordered product Xj1 · · · Xjk for V

i:Xi ∈A2= {j1 , · · · , jk }, j1 < · · · < jk .

Definition 2.2. (Free product) We assume that Ai is unital for i = 1, 2 and CJtK is containedin the center of Ai , and moreover that the unit of Ai is the same as the unit of CJtK. We assume
that ϕt (1) = 1. Let A1 ∗CJtK A2 be the amalgamated free product of algebras over CJtK. The
free product ϕt1 ∗ ϕt2 on A1 ∗CJtK A2 is defined by the following rule:ϕt1 ∗ ϕt2 (X1 · · · Xn ) = 0
holds whenever ϕtλi (Xi ) = 0 for all i.
Definition 2.3. (Boolean product) The Boolean product ϕt1 ¦ ϕt2 on A1 ∗ A2 is defined by
ϕt1 ¦ ϕt2 (X1 · · · Xn ) = ϕtλ1 (X1 )ϕtλ2 (X2 ) · · · ϕtλn (Xn ).
Definition 2.4. (Monotone product) The monotone product ϕt1 . ϕt2 on A1 ∗ A2 is defined by
ϕt1 . ϕt2 (X1 · · · Xn ) = ϕt1³−−´−→ ´³ YYtXi
ϕ2 (Xi ) .
Xi ∈A1

3

Xi ∈A2

The tensor product can be defined on both A1 ∗ A2 and A1 ∗CJtK A2 . By contrast, theBoolean and monotone products cannot be defined on A1 ∗CJtK A2 . We defined the free productof CJtK-valued linear maps on A1 ∗CJtK A2 under special assumptions on the algebras. It is
however also possible to define the free product on A1 ∗ A2 for arbitrary algebras Ai . To do so,
let Aei be the unitization of Ai defined by Aei = C ⊕ Ai . We consider Aei -valued formal power
series CJt; Aei K. Then CJtK is embedded into the center of CJt; Aei K. CJtK-valued linear maps on
Ai can be naturally extended to unit-preserving ones on CJt; Aei K. Then we can define the freeproduct by Definition 2.2, and restrict it to A1 ∗ A2 , to obtain the free product on A1 ∗ A2 .
We can define universal calculation rules [18]1 for tensor, free, Boolean and monotone products of CJtK-valued linear maps. Then we can formulate independence easily as follows.
Definition 2.5. We fix an arbitrary natural independence. Let (A, ϕt ) be a pair of an algebra
and a CJtK-valued linear map on A. Subalgebras (Ai )i≥1 of A are called independent if for any
elements Xk ∈ Aik and indices i1 6= · · · =6 in , the universal calculation rule holds.
An important property of a natural product is the associative law. A product ? : A∗1 ×A∗2 →(A1 ∗ A2 )∗ defined for any A1 , A2 is said to be associative if (ϕ1 ? ϕ2 ) ? ϕ3 = ϕ1 ? (ϕ2 ? ϕ3 ) under
the natural identification (A1 ∗ A2 ) ∗ A3 ∼= A1 ∗ (A2 ∗ A3 ). Associativity can be defined forany family of linear functionals as follows. Let (A∗ )Λ be the set of functions from a set Λ withvalues in A∗ . In most cases we take Λ to be N = {0, 1, 2, · · · }. We consider a pair (A, (ϕ(λ) )λ∈Λ )³´Λ
of an algebra and a function in (A∗ )Λ . A product ? : (A∗1 )Λ × (A∗2 )Λ → (A1 ∗ A2 )∗ , defined
¡ (λ)
¡ (λ)
(λ) ¢
(λ)
(λ)
(λ) ¢
for any A1 , A2 , is said to be associative if (ϕ1 ) ? (ϕ2 ) ? (ϕ3 ) = (ϕ1 ) ? (ϕ2 ) ? (ϕ3 )under the natural isomorphism (A1 ∗ A2 ) ∗ A3 ∼= A1 ∗ (A2 ∗ A3 ).
Associativity can also be defined for linear maps with values in a common algebra. In
particular, we often consider associative products defined for linear maps with values in CJtK.
We have defined products of algebraic probability spaces (A, ϕt ), where ϕt are CJtK-valued
linear maps. By the way, ϕt can be identified with infinitely many linear functionals (ϕ(n) )n≥0
byt2
ϕt (X) = ϕ(0) (X) + ϕ(1) (X)t + ϕ(2) (X) + · · · .
2!
nFormally, ϕ(n) (X) = dtd n ϕt (X)|t=0 . Let LiMap(A, CJtK) be the set of all linear maps from A
to CJtK. Then the map F : ϕt 7→ (ϕ(n) )n≥0 is bijective from LiMap(A, CJtK) to (A∗ )N . Let ?
be any one of the natural products of CJtK-valued linear maps. We can define an associative
product(n)(n)(A1 , (ϕ1 )n≥0 ) ?D (A2 , (ϕ2 )n≥0 ) = (A1 ∗ A2 , (ϕ(n) )n≥0 )
³
´
(n)
(n)
by (ϕ(n) )n≥0 := F F −1 ((ϕ1 )n≥0 ) ? F −1 ((ϕ2 )n≥0 ) . By definition this product is associative.
In the case of the tensor and free products, we can replace A1 ∗ A2 by A1 ∗CJtK A2 . The
sequence (ϕ(n) ) contains information on infinitesimal properties of ϕt . Therefore, the following
terminology is reasonable.
Definition 2.6. We call ?D a differential product associated to a natural product ?. Moreconcretely, if ? = ⊗, ∗, ¦, ., we respectively call the products ⊗D , ∗D , ¦D , .D a differentiallytensor product, a differentially free product, a differentially Boolean product and a differentiallymonotone product.
1

Muraki called these rules quasi-universal calculation rules.

4

We can also define a product up to the n-th derivative. That is, let CJtKn be the unital ringof power series with relation tn+1 = 0. This ring can be realized by upper triangular matricesof the forms
a0 a1 a2 . . . an
 0 a0 a1 . . . an−1  .. .. . . . .
.
.
Gn := {A ∈ Mn+1 (C); A =  . .
}.
.
.
.


 0 0 . . . a 0 a1 
0 0 ... 0a0
G1 is used in free probability of type B (see [3]). Let LiMap(A, CJtKn ) be the set of linear maps
from A to CJtKn . ϕt ∈ LiMap(A, CJtKn ) is of the formϕt (X) =nXϕ(k) (X)

k=0

tk
.
k!

We can define natural products of CJtKn -valued linear maps analogously to Definitions 2.1–2.4.
Let Fn : LiMap(A, CJtKn ) → (A∗ )n be the map defined by Fn (ϕt ) = (ϕ(k) )nk=0 . Then we define
a product(k)(k)(A1 , (ϕ1 )nk=0 ) ?D (A2 , (ϕ2 )nk=0 ) := (A1 ∗ A2 , (ϕ(k) )nk=0 )
³
´
(k) n
(k) n
(k) n
−1
−1
by (ϕ )k=0 := Fn Fn ((ϕ1 )k=0 ) ? Fn ((ϕ2 )k=0 ) . We call this product an n-differential
product. All results in this paper can easily be proved for n-differential product; therefore, weonly focus on infinitely many linear functionals.
Now we show how to calculate joint moments. By definition,
we can calculate the joint
P∞ (n)
n(n)tmoments ϕ (X1 · · · Xn ) by derivatives of ϕ (X1 · · · Xn ) = n=0 ϕ (X1 · · · Xn ) tn! .
(n)Theorem 2.7. Let (A, (ϕ(n) )n≥0 ) := ?Dk≥1 (Ak , (ϕk )n≥0 ) be a differential product. For any
elements Xk ∈ Aik and indices i1 6= · · · =6 in , we have the following.
(1) Let ? be the tensor product. Then for m ≥ 0
ϕ

(m)

∞
³−−Y
−−→ ´
Y
m!
(ki )
(X1 · · · Xn ) =ϕXj .
k1 !k2 ! · · · i=1
j:X ∈Ak +k +···=m;k ≥0

X

1

2

j

i

i

(2) Let ? be the monotone product. If j satisfies ij−1 < ij > ij+1 thenϕ(m) (X1 · · · Xn ) =mXm!
ϕ(k) (Xj )ϕ(m−k) (X1 · · · Xj−1 Xj+1 · · · Xn ).
k!(m
−
k)!
k=0

(3) Let ? be the Boolean product. Then
ϕ

(m)

n
Y
m!
(X1 · · · Xn ) =ϕ(ki ) (Xi ).
k1 ! · · · kn ! i=1
k +···+k =m;k ≥0

X

1

n

i

Thus we obtained the calculation rules of higher order derivatives for tensor, monotone andBoolean cases. The 1-differential free product is an important aspect of free probability of typeB, and a calculation rule for it was clarified in [3]. However, higher order cases are complicatedand we do not calculate them in this paper. Anyway, if we use a universal calculation rule [21]5for free independent algebras, we can write down the calculations of moments for higher orderderivatives in an abstract way.
The above calculation rules for differentially tensor, free, monotone and Boolean products
are called differential universal calculation rules in this paper. Then we can formulate differential independence of subalgebras.
Definition 2.8. We fix an arbitrary natural independence. Let (A, (ϕ(n) )n≥0 ) be a pair of analgebra and a sequence of linear functionals. We say that subalgebras (Ai )i≥1 are differentiallyindependent if for any elements ak ∈ Aik and subscripts i1 6= · · · =6 in , the differential universal
calculation rule holds.
Example 2.9. (1) Let {X, X 0 } and {Y, Y 0 } be differentially tensor independent. Thenϕ(0) (XY X 0 Y 0 ) = ϕ(0) (XX 0 )ϕ(0) (Y Y 0 ),ϕ(1) (XY X 0 Y 0 ) = ϕ(1) (XX 0 )ϕ(0) (Y Y 0 ) + ϕ(0) (XX 0 )ϕ(1) (Y Y 0 ),ϕ(2) (XY X 0 Y 0 ) = ϕ(2) (XX 0 )ϕ(0) (Y Y 0 ) + 2ϕ(1) (XX 0 )ϕ(1) (Y Y 0 ) + ϕ(0) (XX 0 )ϕ(2) (Y Y 0 ).
(2) Let {X, X 0 } and Y be differentially free independent. Thenϕ(0) (XY X 0 ) = ϕ(0) (XX 0 )ϕ(0) (Y ),ϕ(1) (XY X 0 ) = ϕ(1) (XX 0 )ϕ(0) (Y ) + ϕ(0) (XX 0 )ϕ(1) (Y ),ϕ(2) (XY X 0 ) = ϕ(2) (XX 0 )ϕ(0) (Y ) + 2ϕ(1) (XX 0 )ϕ(1) (Y ) + ϕ(0) (XX 0 )ϕ(2) (Y ).
(3) Let {X, X 0 } and {Y, Y 0 } be differentially monotone independent. Thenϕ(0) (XY X 0 Y 0 ) = ϕ(0) (XX 0 )ϕ(0) (Y )ϕ(0) (Y 0 ),ϕ(1) (XY X 0 Y 0 ) = ϕ(1) (XX 0 )ϕ(0) (Y )ϕ(0) (Y 0 ) + ϕ(0) (XX 0 )ϕ(1) (Y )ϕ(0) (Y 0 )+ ϕ(0) (XX 0 )ϕ(0) (Y )ϕ(1) (Y 0 ),ϕ(2) (XY X 0 Y 0 ) = ϕ(2) (XX 0 )ϕ(0) (Y )ϕ(0) (Y 0 ) + ϕ(0) (XX 0 )ϕ(2) (Y )ϕ(0) (Y 0 )+ ϕ(0) (XX 0 )ϕ(0) (Y )ϕ(2) (Y 0 ) + 2ϕ(1) (XX 0 )ϕ(1) (Y )ϕ(0) (Y 0 )+ 2ϕ(1) (XX 0 )ϕ(0) (Y )ϕ(1) (Y 0 ) + 2ϕ(0) (XX 0 )ϕ(1) (Y )ϕ(1) (Y 0 ).
Remark 2.10. Let us consider only the zeroth and first derivatives. In this section, we haveconstructed an associative product
³
´
(0)
(1)
(0)
(1)
(A1 , ϕ1 , ϕ1 ), (A2 , ϕ2 , ϕ2 ) 7→ (A1 ∗ A2 , ϕ(0) , ϕ(1) ).
The product for the second components is the usual natural product. The marginal distribution(1)of ϕ(1) on Ai is ϕi . Therefore the situation is similar to c-free and c-monotone products.
However, the calculation rule of second moments is different from the c-free and c-monotone
(1)
(0)
(0)
(1)
cases since ϕ(1) (ab) = ϕ(1) (ba) = ϕ1 (a)ϕ2 (b) + ϕ1 (a)ϕ2 (b) for a ∈ A1 and b ∈ A2 .
By definition, we have the following.
Proposition 2.11. Let (A, (ϕ(n) )n≥0 ) be a pair of an algebra and a sequence of linear functionals. For each concept of natural independence, subalgebras (Ai )i≥1 of A are differentiallyindependent if and only if they are independent (in the sense of Definition 2.5) in the algebraicPkprobability space (A, ϕt ), where ϕt is defined by ϕt (X) = nk=0 ϕ(k) (X) tk! .
6

3

Cumulants associated with differential independence

Cumulants can be defined for a natural independence along the line of [13]. Now we areconsidering infinitely many linear functionals, but the idea for a single linear functional can beextended easily. We note that Lehner’s approach [15] is applicable to all natural independences
except for monotone independence. We outline how to define cumulants. Proofs are the sameas in [13] if we use Proposition 2.11.
Definition 3.1. We consider any differential independence among tensor, free, monotone andBoolean ones. Let (A, (ϕ(n) )n≥0 ) be a pair of an algebra and a sequence of linear functionals.
e (ϕWe take copies {X (j) }j≥1 of every X ∈ A in an extended pair (A,e(n) )n≥0 ) such thate(1) For each j ≥ 1, the map X 7→ X (j) is a homomorphism from A to A;
(j)

(j)

(j)

(2) ϕ
e(k) (X1 X2 · · · Xn ) = ϕ(k) (X1 X2 · · · Xn ) for any Xi ∈ A, j, n ≥ 1, k ≥ 0;(3) the subalgebras A(j) := {X (j) : X ∈ A}, j ≥ 1 are differentially independent.
We define a dot operation N.X to be the sum of i.i.d. random variables:
N.X = X (1) + · · · + X (N )for X ∈ A and N ∈ N. 0.X is defined to be 0.
Lemma 3.2. The dot operation is associative:¡¢¡¢ϕ(k) N.(M.X1 ) · · · N.(M.Xn ) = ϕ(k) (M N ).X1 · · · (M N ).Xn

(3.1)

for any Xi ∈ A, n ≥ 1, k ≥ 0.
Lemma 3.3. ϕ(k) (N.X1 · · · N.Xn ) is a polynomial of N and ϕ(m) (Xi1 · · · Xip ) for i1 < · · · < ip ,1 ≤ p ≤ n, m ≤ k, and there appears no constant term with respect to N .
(k)Definition 3.4. For each natural independence, a (k; n)-differential cumulant Kn (X1 , · · · , Xn )is defined to be the coefficient of N in ϕ(k) (N.X1 · · · N.Xn ).
Proposition 3.5. We have the following properties.
(k)(1) (Multilinearity) Kn are multilinear.
(k)(2) (Polynomiality) There exist polynomials Pnsuch that¡¢
Kn(k) (X1 , · · · , Xn ) = ϕ(k) (X1 · · · Xn ) + Pn(k) {ϕ(m) (Xi1 · · · Xip )}1≤p≤n−1;m≤k; .
i1 <···<ip(3) (Extensivity)
Kn(k) (N.X1 , · · · , N.Xn ) = N Kn(k) (X1 , · · · , Xn ).
Moreover, differential cumulants satisfying the above three properties are unique.

7

Cumulants can be calculated from the CJtK-valued usual cumulants because of Proposition
2.11. We know the CJtK-valued moment-cumulant formula for each natural independence since
the proof needs no changes. Let TKnt (X1 , · · · , Xn ), FKnt (X1 , · · · , Xn ), MKnt (X1 , · · · , Xn ) andB t
Kn (X1 , · · · , Xn ) be the CJtK-valued tensor, free, monotone and Boolean cumulants, respectively. We prepare notation: for p-multilinear functional fp , we define fp (XV ) := fn (Xi1 , · · · , Xip )for any subset V = {i1 , · · · , ip } ⊂ {1, · · · , n}, i1 < · · · < ip . Then
X Y
T t
K|V | (XV ),(3.2)
ϕt (X1 · · · Xn ) =π∈P(n) V ∈πϕt (X1 · · · Xn ) =

X

Y

F

t
K|V
| (XV ),

(3.3)

π∈N C(n) V ∈π

X Y 1
M t
K|V | (XV ),|π|!
π∈M(n) V ∈π̄X Y
B t
ϕt (X1 · · · Xn ) =K|V | (XV ),ϕt (X1 · · · Xn ) =

(3.4)
(3.5)

π∈I(n) V ∈πwhere P(n), N C(n), M(n) and I(n) are the sets of partitions, non-crossing partitions, monotone partitions and interval partitions, respectively. π̄ denotes the projection from M(n) onto
N C(n). (3.3) was proved in [20], (3.4) in [13] and (3.5) in [15]. The reader is referred to [13, 15]
for the definitions of the above partitions.
Differential cumulants can be calculated from the above formulae as follows.
Pn(n)(X) tn! be a CJtK-valued linear map on an algebra A.
Proposition 3.6. Let ϕt (X) = ∞
n=0 ϕ
(k)
Then the (k; n)-differential cumulant Kn associated to a natural independence is calculated as
the k-th derivative of CJtK-valued n-th cumulant Knt :Kn(k) (X1 , · · · , Xn ) =dk t
K (X1 , · · · , Xn )|t=0 .
dtk n

Proof. By Proposition 2.11, the dot operation for differential independence coincides with thatk(k)
for CJtK-valued independence. We define Ln (X1 , · · · , Xn ) := dtd k Knt (X1 , · · · , Xn )|t=0 . We
notice that Knt (N.X1 , · · · , N.Xn ) = N Knt (X1 , · · · , Xn ) by definition. Therefore, the extensivity(k)(k)
Ln (N.X1 , · · · , N.Xn ) = N Ln (X1 , · · · , Xn ) holds. Multilinearity and polynomiality are easy(k)(k)
to prove. By the uniqueness of differential cumulants, Ln coincides with Kn .
We can prove the vanishing property of cumulants.
Theorem 3.7. We consider any one of tensor, free and Boolean independences. Let (A, (ϕ(n) )n≥0 )be a pair of an algebra and a sequence of linear functionals. Subalgebras Ai (i ≥ 1) of A are
differentially independent if and only if the vanishing property of cumulants holds:(m)(Vanishing property) For any i1 , · · · , in and Xk ∈ Aik for 1 ≤ k ≤ n, Kn (X1 , · · · , Xn ) =0 holds for all m ≥ 0 unless all ik ’s are the same numbers.
Proof. The proof of the first implication is the same as in Proposition 3.6 of [13]. The converse
implication follows from the formulae (3.2), (3.3) or (3.5). Indeed, these moment-cumulant
formulae enable us to calculate the joint moments ϕ(m) (X1 · · · Xn ) for any n ≥ 1, i1 , · · · , in ≥ 1
and any Xk ∈ Aik only by sums and products of ϕ(Xj1 · · · Xjp ), where all Xjk belong to thesame Ar .
8

4

Examples

4.1

Formal multi-variate convolution semigroups

We introduced the dot operation N.X in Section 2. Let CJX1 , · · · , Xn K be the unital ring
generated by n non-commuting indeterminates. Let ϕ be a linear functional on CJX1 , · · · , Xn K
with ϕ(1) = 1. ϕ((N.Xj1 ) · · · (N.Xjp )) is a polynomial of N , and hence we can define a newlinear functional ϕt by ϕt (Xj1 · · · Xjp ) := ϕ((t.Xj1 ) · · · (t.Xjp )) for t ∈ R, p ≥ 1 and 1 ≤j1 , · · · , jp ≤ n. Formally, this corresponds to a multi-variate convolution semigroup for eachconcept of natural independence. In particular, if n = 1 and the probability distribution µ of
X1 is infinitely divisible, then ϕt (X1p ) becomes the p-th moment of the associated Lévy process
at time t ≥ 0.
F (1)
Kn was calculated in [9]. We can also calculate the other differential cumulants associatedto each natural independence.
Proposition 4.1. Let P(n)(m) , N C(n)(m) , M(n)(m) and I(n)(m) be respectively the set ofthe partitions, non-crossing partitions, monotone partitions and interval partitions of the set{1, · · · , n} with m blocks. If the considered independence is tensor, free, monotone and Boolean,
we respectively obtainX Y
T
ϕ(m) (Xj1 · · · Xjp ) = m!
K|V | (Xj(V ) ),(4.1)
π∈P(p)(m) V ∈πϕ(m) (Xj1 · · · Xjp ) = m!

X

Y

F

K|V | (Xj(V ) ),(4.2)

π∈N C(p)(m) V ∈π

X

Y 1
M
K|V | (Xj(V ) ),|π|!
π∈M(p)(m) V ∈π̄X Y
Bϕ(m) (Xj1 · · · Xjp ) = m!
K|V | (Xj(V ) )ϕ(m) (Xj1 · · · Xjp ) = m!

(4.3)
(4.4)

π∈I(p)(m) V ∈π

for 0 ≤ m ≤ p. K|V | (Xj(V ) ) is defined by K|V | (Xji1 , · · · , Xjir ) for V = {i1 , · · · , ir }, i1 < · · · <ir . ϕ(m) (Xj1 · · · Xjp ) = 0 for m > p.
The proof is easy from the extensivity of cumulants.

4.2

Dual derivation systems

Février and Nica defined dual derivation systems in [9] which give us many examples of differential cumulants and independence. We basically follow the notations and definitions in [9],except for the range of an operator dn . Let M be the set of sequences (fn )n≥1 , where fn is ann-multilinear functional from An to C for each n ≥ 1. For a partition π = {V1 , · · · , Vp } ∈ P(n)and f = (fn ) ∈ M, we define an n-multilinear functional Jπ (f ) byY
Jπ (f )(X1 , · · · , Xn ) =f|V | (XV ).
V ∈πJπ (f ) only depends on f|V1 | , · · · , f|V|π| | . Therefore, we may denote Jπ (f ) by Jπ (f|V1 | , · · · , f|V|π| | ).
Let Multn be the multiplication map from An to A defined by Multn (X1 , · · · , Xn ) = X1 · · · Xn .
9Definition 4.2. Let Mn the set of all n-multilinear functionals from An to C. A family of
linear maps (dn : Dn → Dn )n≥1 , where Dn is a subspace of Mn , is called a dual derivation
system if it satisfies the following conditions.
(i) Let D be the set of sequences (fn )n≥1 satisfying fn ∈ Dn . Thendn (Jπ (f|V1 | , · · · , f|V|π| | )) =|π|XJπ (f|V1 | , · · · , d|Vk | f|Vk | , · · · , f|V|π| | )k=1

for all f ∈ D and π ∈ P(n).
(ii) For every f ∈ D1 and every n ≥ 1, f ◦ Multn ∈ Dn anddn (f ◦ Multn ) = (d1 f ) ◦ Multn .
If a linear functional ϕ ∈ D1 and a dual derivation system (dn : Dn → Dn )n≥1 are given,
we can construct a CJtK-valued linear map ϕt byt

ϕ = exp(td1 )ϕ =

∞
X
tn
n=0

n!

dn1 ϕ.

In other words, ϕ(k) is defined by ϕ(k) = dk1 ϕ for k ≥ 0. In this setting, we can calculate thedifferential cumulants as follows.
Proposition 4.3. We fix a natural independence. Let Kn be the cumulants associated to the(k)fixed independence. Then the differential cumulants Kn associated to the sequence (dn1 ϕ)n≥0(k)
are given by Kn = dkn Kn . Equivalently, the CJtK-valued cumulants Knt associated to ϕt :=exp(td1 )ϕ are given by Knt = exp(tdn )Kn .
Proof. We prove the latter equivalent claim. Moreover, we only prove the claim for tensor
independence; the other cases are proved in the same way.
First we notice that exp(tdn ) satisfies thatexp(tdn )Jπ ((fk )k≥1 ) = Jπ ((exp(tdk )fk )k≥1 )for every π ∈ P(n) and every (fk )k≥1 ∈ D.
The cumulants TKn for ϕ are defined byϕ(X1 · · · Xn ) =X Y

TK|V | (XV ),π∈P(n) V ∈πwhich is equivalent toϕ ◦ Multn =X

Jπ (TK|V1 | , · · · , TK|V|π| | ).

π=(V1 ,··· ,V|π| )∈P(n)If we operate exp(tdn ) on the above equality, thenX
(exp(td1 )ϕ) ◦ Multn =Jπ (exp(td|V1 | )TK|V1 | , · · · , exp(td|V|π| | )TK|V|π| | ),π=(V1 ,··· ,V|π| )∈P(n)which completes the proof.
10

Let ϕ be a linear functional on A. If D : A → A is a derivation, then we can define
dn : Mn → Mn by
dn fn (X1 , · · · , Xn ) =nX

fn (X1 , · · · , DXk , · · · , Xn ).

k=1

It is easy to check that (dn : Mn → Mn )n≥1 is a dual derivation system. Let {αt }t∈R be theformal automorphism group αt = exp(tD). Then ϕt := ϕ ◦ αt is the CJtK-valued linear map
associated to D.
Example 4.4. Let A = CJxK be the unital ring generated by one indeterminate and let D be
the derivation Dxn = nxn−1 . For Ra probability measure µ with finite moments of all orders, wedenote by ϕµ the state ϕµ (xn ) = R xn µ(dx). Then, ϕt = ϕ ◦ etD has the moments of x + t:
ϕt (xn ) = ϕµ ((x + t)n )for all n ≥ 0. Let Kn (x) := Kn (x, · · · , x) be the n-th cumulant of x for any one of tensor,
free and Boolean independences. Then (etD Kn )(x) = Kn (x + t, · · · , x + t) = Kn (x) + tδ1n . We
excluded monotone independence since the above calculation does not hold.

5

Further generalizations

It is easy to generalize many results in this paper to a more general setting. Let R be acommutative algebra over C. We take a basis {eλ }λ∈Λ of the vector space
P R over C. We
would like to consider infinite sums by identifying the formal expression λ∈Λ cλ eλ (cλ ∈ C)
with the map λP7→ cλ which
P is denoted by (cλ )λ∈Λ . For this purpose, we assume that themultiplication ( λ∈Λ cλ eλ )( λ∈Λ dλ eλ ) is well-defined for any cλ , dλ ∈ C. That is, we assume
that the coefficient of each eλ , appearing in the result of the multiplication, is a finite sum. For
instance, this is true for a graded algebra with finite dimensional homogeneous components.
An R-valued linear map ϕ on an algebra A can be written asXϕ(λ) (X)eλϕ(X) =λ∈Λfor some linear functionals ϕ(λ) from A to C. By this correspondence, we identify ϕ with
(ϕ(λ) )λ∈Λ . Along the same line of Section 2, we can define an associative product of linear
functionals indexed by Λ for each natural product. We show some examples.
(1) Let CJt1 , · · · , tn K be the unital and commutative algebra over C generated by t1 , · · · , tn .
We introduce the set of multi-indices An consisting of α = (α1 , · · · , αn ) satisfying αk ∈Z, αk ≥ 0 for each k. We define |α| := α1 + · · · + αn . A CJt1 , · · · , tn K-valued linear|α|
map ϕt1 ,··· ,tn can be identified with (ϕ(α) )α∈An by ϕ(α) (X) = ∂∂tα ϕt1 ,··· ,tn (X)|(t1 ,··· ,tn )=(0,··· ,0) =∂ |α|ϕt1 ,··· ,tn (X)|(t1 ,··· ,tn )=(0,··· ,0) . Using this identification, we can define an associative prodα∂t 1 ···∂tαn
1n

uct for multi-indexed linear functionals (ϕ(α) )α∈An .
(2) Let G be a finite abelian group and CG be its group algebra. It is known that a
finite abelian group is the direct product of cyclic groups. CG consists of every element of
the form (cg )g∈G , cg ∈ C. We note that the example (1) corresponds to the semigroup algebraof Nn , where N = {0, 1, · · · }. More concretely, let G be consisting of the unit e and g with11

the relation g 2 = e. A CG-valued linear map ϕ can be identified with a pair (ϕ(e) , ϕ(g) ) byϕ(X) = ϕ(e) (X)e + ϕ(g) (X)g. If {X, X 0 } and {Y, Y 0 } are free independent under the CG-valued
linear map ϕ, we recall that
ϕ(XY ) = ϕ(X)ϕ(Y ),ϕ(XY X 0 ) = ϕ(XX 0 )ϕ(Y ),ϕ(XY X 0 Y 0 ) = ϕ(XX 0 )ϕ(Y )ϕ(Y 0 ) + ϕ(X)ϕ(X 0 )ϕ(Y Y 0 )− ϕ(X)ϕ(X 0 )ϕ(Y )ϕ(Y 0 ).
Therefore, we obtain the following calculations.
ϕ(e) (XY ) = ϕ(e) (X)ϕ(e) (Y ) + ϕ(g) (X)ϕ(g) (Y ),ϕ(g) (XY ) = ϕ(e) (X)ϕ(g) (Y ) + ϕ(g) (X)ϕ(e) (Y ),ϕ(e) (XY X 0 ) = ϕ(e) (XX 0 )ϕ(e) (Y ) + ϕ(g) (XX 0 )ϕ(g) (Y ),ϕ(g) (XY X 0 ) = ϕ(e) (XX 0 )ϕ(g) (Y ) + ϕ(g) (XX 0 )ϕ(e) (Y ),³´Xϕ(e) (XY X 0 Y 0 ) =ϕ(a) (XX 0 )ϕ(b) (Y )ϕ(c) (Y 0 ) + ϕ(a) (X)ϕ(b) (X 0 )ϕ(c) (Y Y 0 )(a,b,c)=(e,e,e),(g,g,e),(g,e,g),(e,g,g)X−

ϕ(a) (X)ϕ(b) (X 0 )ϕ(c) (Y )ϕ(d) (Y 0 ),(a,b,c,d)=(e,e,e,e),(g,g,e,e),(g,e,g,e),(g,e,e,g),(e,g,g,e),(e,g,e,g),(e,e,g,g),(g,g,g,g)³

X

ϕ(g) (XY X 0 Y 0 ) =´
ϕ(a) (XX 0 )ϕ(b) (Y )ϕ(c) (Y 0 ) + ϕ(a) (X)ϕ(b) (X 0 )ϕ(c) (Y Y 0 )(a,b,c)=(g,e,e),(e,g,e),(e,e,g),(g,g,g)−X

ϕ(a) (X)ϕ(b) (X 0 )ϕ(c) (Y )ϕ(d) (Y 0 ).

(a,b,c,d)=(g,e,e,e),(e,g,e,e),(e,e,g,e),(e,e,e,g),(g,g,g,e),(g,g,e,g),(g,e,g,g),(e,g,g,g)

So far we have discussed in the setting of linear functionals without positivity. A natural
question is: when does the associative product preserves the positivity of states? We, however,
do not treat this problem in this paper.
One may expect that the c-free and c-monotone products can be understood as commutative
algebra-valued free and monotone products, respectively. This is, however, impossible for thefollowing reason. For instance, let us consider random variables X, X 0 , Y such that {X, X 0 }and Y are free independent for a linear map ϕ with values in a two-dimensional commutative
algebra. Let ϕ be identified with (ϕ(0) , ϕ(1) ). Then the calculation rules for ϕ(1) (XY ) andϕ(1) (XY X 0 ) must be the same. If we try to realize the c-free independence with respect tothe two linear functionals (ϕ(0) , ϕ(1) ), then we need to choose the commutative algebra andits basis such that ϕ(1) (XY ) coincides with ϕ(1) (X)ϕ(1) (Y ). This automatically implies thatϕ(1) (XY X 0 ) = ϕ(1) (XX 0 )ϕ(1) (Y ) by definition of free independence. Therefore, we cannot
realize the c-free independence.
There are however connections to multi-state cases in a different context. There are natural
products for more than one state: a c-free product [6], a c-monotone product [11], an ordered
free product and an indented product [12]. Differential products can be defined for theseproducts of states. For instance, let ∗ be the c-free product: (ϕ, ψ) = (ϕ1 , ψ1 ) ∗ (ϕ2 , ψ2 ). We
can define the c-free product of pairs of CJtK-valued linear maps, similarly to the free case. A
pair of CJtK-valued linear maps (ϕt , ψ t ) can be identified with a sequence of pairs (ϕ(n) , ψ (n) )n≥0 .
Then we can introduce an associative product for such sequences of pairs.
12Acknowledgements
The author would like to thank Dr. Hayato Saigo for discussing and improving the proof of
associative laws of differential products. He is grateful to Professor Izumi Ojima for discussions
about independence and interpretation of the multi-state setting. He is also grateful to Professor
Marek Bożejko for suggesting the author to generalize a field C to other fields; this generalizationis related to the contents of the final section. He thanks Professor Uwe Franz for indicating
independence with respect to infinitely many states. This work was supported by Grant-in-Aid
for JSPS Research Fellows (KAKENHI 21-5106).

References[1] S.T. Belinschi and D. Shlyakhtenko, Free probability of type B: analytic interpretation
and applications, arXiv:0903.2721v1.
[2] A. Ben Ghorbal and M. Schürmann, Non-commutative notions of stochastic independence,
Math. Proc. Comb. Phil. Soc. 133 (2002), 531–561.
[3] Ph. Biane, F. Goodman and A. Nica, Non-crossing cumulants of type B, Trans. Amer.
Math. Soc. 335, No. 6 (2003), 2263–2303.
[4] M. Bożejko, Positive definite functions on the free group and the noncommutative Riesz
product, Bull. Un. Mat. Ital. (6) 5-A (1986), 13–21.
[5] M. Bożejko, M. Leinert and R. Speicher, Convolution and limit theorems for conditionally
free random variables, Pac. J. Math. 175 (1996), No. 2, 357–388.
[6] M. Bożejko and R. Speicher, ψ-independent and symmetrized white noises, Quantum
Probability and Related Topics (L. Accardi, ed.), World Scientific, Singapore VI (1991),
219–236.
[7] T. Cabanal-Duvillard and V. Ionescu, Un théorème central limite pour des variablesaléatoires non-commutatives, C.R. Acad. Sci. Paris. 325, Série I (1997), 1117–1120.
[8] M. Février, Higher order infinitesimal freeness, arXiv:1009.2389v1.
[9] M. Février and A. Nica, Infinitesimal non-crossing cumulants and free probability of typeB, arXiv:0906.2017v1.
[10] U. Franz, Multiplicative monotone convolutions, Banach Center Publ., 73 (2006), 153–
166.
[11] T. Hasebe, Conditionally monotone independence I: Independence, additive convolutions
and related convolutions, accepted for publicatoin in Infin. Dimens. Anal. Quantum
Probab. Relat. Topics, arXiv:0907.5473v4.

13

[12] T. Hasebe, New associative product of three states generalizing free, monotone,
anti-monotone, Boolean, conditionally free and conditionally monotone products,
arXiv:1009.1505v1.
[13] T. Hasebe and H. Saigo, Joint cumulants for natural independence, arXiv:1005.3900v2.
[14] T. Hasebe and H. Saigo, The monotone cumulants, to appear in Ann. Inst. Henri PoincaréProbab. Stat. arXiv:0907.4896v3.
[15] F. Lehner, Cumulants in noncommutative probability theory I, Math. Z. 248 (2004), 67–100.
[16] N. Muraki, Monotonic convolution and monotonic Lévy-Hinčin formula, preprint, 2000.
[17] N. Muraki, The five independences as natural products, Infin. Dimens. Anal. Quantum
Probab. Relat. Topics 6, No. 3 (2003), 337–371.
[18] N. Muraki, The five independences as quasi-universal products, Infin. Dimens. Anal.
Quantum Probab. Relat. Topics 5, No. 1 (2002), 113–134.
[19] M. Schürmann, Direct sums of tensor products and noncommutative independence, J.
Funct. Anal. 133 (1995), 1–9.
[20] R. Speicher, Multiplicative functions on the lattice of non-crossing partitions and freeconvolution, Math. Ann. 298 (1994), 611–628.
[21] R. Speicher, On universal products, in Free Probability Theory, papers from a Workshop
on Random Matrices and Operator Algebra Free Products, Toronto, Canada 1995, ed.
D. Voiculescu, Fields Inst. Commun. 12 (Amer. Math. Soc., 1997), 257–266.
[22] R. Speicher and R. Woroudi, Boolean convolution, in Free Probability Theory, papers
from a Workshop on Random Matrices and Operator Algebra Free Products, Toronto,
Canada 1995, ed. D. Voiculescu, Fields Inst. Commun. 12 (Amer. Math. Soc., 1997),
267–280.
[23] D. Voiculescu, Addition of certain non-commutative random variables, J. Funct. Anal. 66(1986), 323–346.
[24] D. Voiculescu, Symmetries of some reduced free product algebras, Operator algebras andtheir connections with topology and ergodic theory, Lect. Notes in Math. 1132, Springer(1985), 556–588.

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