arXiv:0704.0040v3  [math.OA]  5 Sep 2008
MULTILINEAR FUNCTION SERIES IN CONDITIONALLY
FREE PROBABILITY WITH AMALGAMATION
MIHAI POPA
Abstract. As in the cases of freeness and monotonic independence, the
notion of conditional freeness is meaningful when complex-valued states are
replaced by positive conditional expectations. In this framework, the paper
presents several positivity results, a version of the central limit theorem and
an analogue of the conditionally free R-transform constructed by means of
multilinear function series.
1. Introduction
The paper addresses a topic related to conditionally free (or, shortly, using the
term from [2], c-free) probability. This notion was developed in the ’90’s (see [1],
[2]) as an extension of freeness within the framework of ∗-algebras endowed with
not one, but two states. Namely, given a family of unital algebras {A}i∈I, each Ai
endowed with two expectations ϕi, ψi : Ai −→ C, their c-free product is the triple
(A, ϕ, ψ), where:
(i) A = ∗i∈IAi is the free product of the algebras Ai.
(ii) ψ = ∗i∈Iψi and ϕ = ∗(ψi)i∈Iϕi are expectations given by the relations
(a) ψ(a1 · · · an) = 0
(b) ϕ(a1 · · · an) = ϕε(1)(a1) · · ·ϕε(n)(an)
for all aj ∈ Aε(j), j = 1, . . . , n such that ψε(j)(aj) = 0 and ε(1) 6= · · · 6= ε(n).
A key result is that if the Ai are ∗-algebras and ϕi, ψi are positive functionals,
then ϕ and ψ are also positive.
In [6], the positivity of the free product maps ϕ, ψ is proved for the case
when ϕ1, ϕ2 are positive conditional expectations in a common C∗-subalgebra,
but ψ1, ψ2 remain positive C-valued maps. A more general situation is indeed
discussed (see Theorem 3, Section 6, from [6]), but the question if ϕ, ψ re positive
for ϕ1,2, ψ1,2 arbitrary positive conditional expectations is left unanswered.
A first answer was given in [8], where we showed that for A a ∗-algebra, the
analogous construction with both ϕ and ψ valued in a C∗-subalgebra B of A still
retains the positivity. The present paper further develops this result (see Theorem
2.3) and also demonstrates the use of multilinear function series in c-free setting.
2000 Mathematics Subject Classification. Primary 45L53; Secondary 46L08.
Key words and phrases. conditional freeness, conditional expectation, R-transform, multi-
linear function series.
http://arxiv.org/abs/0704.0040v3
2 MIHAI POPA
In [2] is constructed a c-free version of Voiculescu’s R-transform, which we will
call the cR-transform, with the property that cRX+Y =
cRX +
cRY if X and Y
are c-free elements from the algebra A relative to ϕ and ψ (i.e. the relations (a)
and (b) from the definition of the c-free product hold true for the subalgebras
generated by X and Y .)
The apparatus of multilinear function series is used in recent work of K. Dykema
([3] and [4]) to construct suitable analogues for the R and S-transforms in the
framework of freeness with amalgamation. We will show that this construction
is also appropriate for the cR-transform mentioned above. The techniques used
differ from the ones of [3], the Fock space type construction being substituted by
combinatorial techniques similar to [2] and [7]. Particularly, Theorems 3.3 and 3.6
contain new (shorter) proves of the results 6.1–6.13 from [3].
The paper is structured in four sections. In Section 2 are stated the basic
definitions and are proved the main positivity results. Section 3 describes the
construction and the basic property of the multilinear function series cR-transform
and Section 4 treats the central limit theorem and a related positivity property.
2. Definitions and positivity results
Definition 2.1. Let {Ai}i∈I be a family of algebras, all containing the subalgebra
B. Suppose D is a subalgebra of B and Ψi : Ai −→ D and Φi : Ai −→ B are con-
ditional expectations, i ∈ I. We say that the triple (A,Φ,Ψ) = ∗i∈I(Ai,Φi,Ψi)
is the conditionally free product with amalgamation over (B,D), or shortly, the
c-free product, of the triples (Ai,Φi,Ψi)i∈I if
(1) A is the free product with amalgamation over B of the family (Ai)i∈I
(2) Ψ = ∗i∈IΨi and Φ = ∗(Ψi)i∈IΦi are determined by the relations
Ψ(a1a2 . . . an) = 0
Φ(a1a2 . . . an) = Φ(a1)Φ(a2) . . .Φ(an),
for all ai ∈ Aε(i), ε(i) ∈ I such that ε(1) 6= ε(2) 6= · · · 6= ε(n) and Ψε(i)(ai) = 0.
When D = C, this definition reduces to the one given in [6]. When both B and
D are equal to C, this definition was given in [2].
When discussing positivity, we need a ∗-structure on our algebras. We will
demand that B and D be C∗-algebras, while Ai and A are only required to be
∗-algebras.
The following results are slightly modified versions of Lemma 6.4 and Theorem
6.5 from [8].
Lemma 2.2. Let B be a C∗-algebra and A1, A2 be two ∗-algebras containing B
as a ∗-subalgebra, endowed with positive conditional expectations Φj : Aj −→ B,
j = 1, 2. If a1, . . . , an ∈ A1, an+1, . . . , an+m ∈ A2 and A = (Ai,j)i,j ∈ Mn+m(B)
is the matrix with the entries
Ai,j =
i aj) if i, j ≤ n
i )Φ2(aj) if i ≤ n, j > n
i )Φ1(aj) if i > n, j ≤ n
i aj) if i, j > n
MULTILINEAR FUNCTION SERIES, C-FREE PROBABILITY WITH AMALGAMATION 3
then A is positive.
Proof. The vector space E = B ⊕ ker(Φ1) ⊕ ker(Φ2) has a B-bimodule structure
given by the algebraic operations on A1 and A2. Consider the B-sesquilinear
pairing
〈·, ·〉 : E× E −→ B
determined by the relations:
〈b1, b2〉 = b∗1b2, for b1, b2 ∈ B
〈uj, vj〉 = Φj(u∗jvj), for uj , vj ∈ ker(Φj), j = 1, 2
〈u1, u2〉 = 〈u2, u1〉 = 0 for u1 ∈ ker(Φ1), and u2 ∈ ker(Φ2).
〈b, uj〉 = 〈uj , b〉 = 0 for all b ∈ B, uj ∈ Aj
With this notation, we have that Ai,j = 〈ai, aj〉, hence it suffices to show that
〈a, a〉 ≥ 0 for all a ∈ E.
Indeed, for an element a = b + u1 + u2 with b ∈ B, uj ∈ ker(Φj), j = 1, 2, we
have:
〈a, a〉 = 〈b + u1 + u2, b+ u1 + u2〉
= 〈b, b〉+ 〈u1, u1〉+ 〈u2, u2〉
= b∗b+Φ1(u
1u1) + Φ2(u
Theorem 2.3. Let B be a C∗-algebra and D a C∗-subalgebra of B. Suppose that
A1, A2 are ∗-algebras containing B, each endowed with two positive conditional
expectations Φj : Aj −→ B, and Ψj : Aj −→ D, j = 1, 2 and consider the c-free
product (A,Φ,Ψ) = ∗i=1,2(Ai,Φi,Ψi).
Then the maps Φ and Ψ are positive.
Proof. The positivity of Ψ is by now a classical result in the theory of free proba-
bility with amalgamation over a C∗-algebra (for example, see [9], Theorem 3.5.6).
For the positivity of Φ we have to show that Φ(a∗a) ≥ 0 for any a ∈ A.
Any element of A can be written as
s1,k . . . sn(k),k,
where sj,k ∈ Aε(j,k) ε(1, k) 6= ε(2, k) 6= · · · 6= ε(n(k), k).
Writing
s(j,k) = s(j,k) −Ψ(s(j,k)) +Ψ(s(j,k))
and expanding the product, we can consider a of the form
a = d+
a1,k . . . an(k),k
with d ∈ D ⊂ B and aj,k ∈ Aε(j,k) such that Ψε(j,k)(aj,k) = 0 and ε(1, k) 6=
ε(2, k) 6= · · · 6= ε(n(k), k).
4 MIHAI POPA
Therefore
Φ(a∗a) = Φ
d+ d∗
a1,k . . . an(k),k
a1,k . . . an(k),k
a1,k . . . an(k),k
]∗[ N∑
a1,k . . . an(k),k
Since Φ is a conditional expectation and d ∈ D ⊂ B, the above equality becomes
Φ(a∗a) = d∗d+
d∗Φ(a1,k . . . an(k),k) +
Φ(a∗n(k),k . . . a
1,k)d
k,l=1
a∗n(k),k . . . a
1,ka1,l . . . an(l),l
Using the definition of the conditionally free product with amalgamation over B
and that Ψε(j,k)(aj,k) = 0 for all j, k, one further has
Φ(a∗a) = d∗d+
Φ(d∗a1,k)Φ(a2,k) . . .Φ(an(k),k)
Φ(an(k),k)
. . .Φ(a∗2,k)Φ(a
1,kd)
k,l=1
Φ(an(k),k)
∗ . . .Φ(a∗2,k)
Φ(a∗1,ka1,l)Φ(a2,l) . . .Φ(an(l),l)
that is
Φ(a∗a) = d∗d+
Φ(d∗a1,k)
Φ(a2,k) . . .Φ(an(k),k)
Φ(a2,k) . . .Φ(an(k),k)
Φ(a∗1,kd)
k,l=1
Φ(a2,k) . . .Φ(an(k),k)
Φ(a∗1,ka1,l)
Φ(a2,l) . . .Φ(an(l),l)
From Lemma 2.2, the matrix S =
Φ(a∗1,ia1,j)
i,j=1
is positive in MN+1(B),
therefore
S = T ∗T, for some T ∈MN+1(B).
MULTILINEAR FUNCTION SERIES, C-FREE PROBABILITY WITH AMALGAMATION 5
Denote now a1,N+1 = d and vk = Φ(a2,k) . . .Φ(an(k),k).The identity for Φ(a
becomes:
Φ(a∗a) = (v1, . . . , vN , 1)
∗T ∗T (v1, . . . , vN , 1)
≥ 0, as claimed.
Theorem 2.4. Assume that I =
j∈J Ij is a partition of I. Then:
∗j∈J (∗i∈Ij (Ai,Φi,Ψi)) = ∗i∈I(Ai,Φi,Ψi)
Proof. The proof is identical to the proofs of similar results in [6] and [2]. Consider
ai ∈ Aε(i), 1 ≤ i ≤ m such that ε(1) 6= ε(2 6= · · · 6= ε(m) and Ψε(i)(ai) = 0. Let
1 = i0 < i1 < · · · < ik = m and Jl = {ε(i), il−1 ≤ i < il}.
Since
(∗j∈JlΨj) ((ail−1 · · · ail)) = 0,
it suffices to show that
Φ(a1 · · · am) =
(∗(Ψj),j∈JlΦj)(ail−1 · · · ail)] .
Φ(a1 · · · am) = Φε(1)(a1) · · ·Φε(m)(am)
while, since Ψε(i)(ai) = 0,
(∗(Ψj),j∈JlΦj)(ail−1 · · · ail) = Φil−1(ail−1) · · ·Φil(ail)
and the conclusion follows. �
Definition 2.5. Let A be an algebra (respectively a ∗-algebra),B a subalgebra (∗-
subalgebra) of A and D a subalgebra (∗-subalgebra) of B. Suppose A is endowed
with the conditional expectations Ψ : A −→ D and Φ : A −→ D.
(i) The subalgebras (∗-subalgebras) (Ai)i∈I of A are said to be c-free with
respect to (Φ,Ψ) if:
(a) (Ai)i∈I are free with respect to Ψ.
(b) if ai ∈ Aε(i), 1 ≤ i ≤ m, are such that ε(1) 6= · · · 6= ε(m) and
Ψ(ai) = 0, then Φ(a1 · · ·am) = Φ(a1) · · ·Φ(am).
(ii) The elements (Xi)i∈I of A are said to be c-free with respect to (Φ,Ψ) if
the subalgebras (∗-subalgebras) generated by B and Xi are c-free with
respect to (Φ,Ψ).
We will denote by B〈ξ〉 the non-commutative algebra of polynomials in the
symbol ξ and with coefficients from B (the coefficients do not commute with
the symbol ξ). If I is a family of indices, B〈{ξi}i∈I〉 will denote the algebra of
polynomials in the non-commuting variables {ξ}i∈I and with coefficients from B.
We will identify B〈{ξi}i∈I〉 with the free product with amalgamation over B of
the family {B〈ξi〉}i∈I .
6 MIHAI POPA
If A is a ∗-algebra and B is with the C∗-algebra, B〈ξ〉 will also be considered
with a ∗-algebra structure, by taking ξ∗ = ξ. If X is a selfadjoint element from A,
we define the conditional expectations
ΦX ,ΨX : B〈ξ〉 −→ B
given by ΦX(f(ξ)) = Φ(f(X)) and ΨX(f(ξ)) = Ψ(f(X)) , for any f(ξ) ∈ B〈ξ〉.
Corollary 2.6. Suppose that A is a ∗-algebra and X and Y are c-free selfadjoint
elements of A such that the maps ΦX ,ΨX and ΦY ,ΨY are positive. Then the
maps ΦX+Y and ΨX+Y are also positive.
Proof. The positivity of ΨX+Y is an immediate consequence of the fact that X
and Y are free with amalgamation over B with respect to Ψ. It remains to prove
the positivity of ΦX+Y .
Since the maps ΦX : B〈ξ1〉 −→ B and ΦY : B〈ξ2〉 −→ B are positive, from
Theorem 2.3 so is
Φx ∗(ΨX ,ΨY ) ΦY : B〈ξ1〉 ∗B B〈ξ2〉 = B〈ξ1, ξ2〉 −→ B
Remark also that
iZ : B〈ξ〉 ∋ f(ξ) 7→ f(X + Y ) ∈ B〈ξ1〉 ∗B B〈ξ2〉
is a positive B-functional.
The conclusion follows from the fact that the c-freeness of X and Y is equivalent
ΦX+Y = (ΦX ∗(ΨX ,ΨY ) ΦY ) ◦ iX+Y .
3. Multilinear function series and the cR-transform
Let A be a ∗-algebra containing the C∗-algebra B, endowed with a conditional
expectation Ψ : A −→ B. If X is a selfadjoint element of A, then by the moment
of order n of X we will understand the map
X : B× · · · ×B︸ ︷︷ ︸
n−1 times
X (b1, . . . , bn−1) = Ψ(Xb1X . . .Xbn−1X)
If B = C, then the moment-generating series of X
mX(z) =
Ψ(Xn)zn
encodes all the information about the moments of X . For B 6= C, the straightfor-
ward generalization
mX(z) =
Ψ(Xn)zn
generally fails to keep track of all the possible moments of X . A solution to
this inconvenience was proposed in [3], namely the moment-generating multilin-
ear function series of X . Before defining this notion, we will briefly recall the
construction and several results on multilinear function series.
MULTILINEAR FUNCTION SERIES, C-FREE PROBABILITY WITH AMALGAMATION 7
Let B be an algebra over a field K. We set B̃ equal to B if B is unital and
to the unitization of B otherwise. For n ≥ 1, we denote by Ln(B) the set of all
K-multilinear maps
ωn : B× · · · ×B︸ ︷︷ ︸
n times
A formal multilinear function series over B is a sequence ω = (ω0, ω1, . . . ),
where ω0 ∈ B̃ and ωn ∈ Ln(B) for n ≥ 1. According to [3], the set of all
multilinear function series over B will de denoted by Mul[[B]].
For α, β ∈Mul[[B]], the formal sum α+ β and the formal product αβ are the
elements from Mul[[B]] defined by:
(α+ β)n(b1, . . . , bn) = αn(b1, . . . , bn) + βn(b1, . . . , bn)
(αβ)n(b1, . . . , bn) =
αk(b1, . . . , bk)βn−k(bk+1, . . . , bn)
for any b1, . . . , bn ∈ B.
If β0 = 0, then the formal composition α ◦ β ∈Mul[[B]] is defined by
(α ◦ β)0 = α0
and, for n ≥ 1, by
(α ◦ β)n(b1, . . . , bn) =
βp1(b1, . . . , bp1), . . . , βpk(bqk+1, . . . , bqk+pk)
where the second summation is done over all k-tuples p1, . . . , pk ≥ 1 such that
p1 + · · ·+ pk = n and qj = p1 + · · ·+ pj−1.
One can work with elements ofMul[[B]] as if they were formal power series. The
relevant properties are described in [3], Proposition 2.3 and Proposition 2.6. As in
[3], we use 1, respectively I, to denote the identity elements of Mul[[B]] relative
to multiplication, respectively composition. In other words, 1 = (1, 0, 0, . . . ) and
I = (0, idB, 0, 0, . . . ). We will also use the fact that an element α ∈Mul[[B]] has
an inverse with respect to formal composition, denoted α〈−1〉, if and only if α has
the form (0, α1, α2, . . . ) with α1 an invertible element of L1(B).
Definition 3.1. With the above notation, the moment-generating multilinear func-
tion series MX of X is the element of Mul[[B]] such that:
MX,0 = Ψ(X)
MX,n(b1, . . . , bn) = Ψ(Xb1X · · ·XbnX).
Given an element α ∈ Mul[[B]], the multilinear function series Rα is defined
by the following equation (see [3], Def 6.1):
(1 + αI)
◦ (I + IαI)〈−1〉. (3.1)
A key property of R is that for any X,Y ∈ A free over B, we have
RMX+Y = RMX +RMY . (3.2)
8 MIHAI POPA
These relations were proved earlier in the particular case B = C. One can also
describe Rα by combinatorial means, via the recurrence relation
αn(b1, . . . , bn) =
[b1αp(1)(b3, . . . , bi1−2)bi1−1], . . .
. . . , [bi(k−1)αp(k)(bi(k−1)+1, . . . , bi(k)−2)bi(k)−1]
bi(k)αn−ik(bik+1 , . . . , bn)
where the second summation is done over all 1 = i(0) < i(1) < · · · < i(k) ≤ n and
p(k) = i(k)− i(k − 1)− 2.
Following an idea from [2], the above equation can be graphically illustrated by
the picture:
In the case of scalar c-free probability, an analogue of the Voiculescu’s R-
transform is developed in [2]. In order to avoid confusions, we will denote it
by cR.
The cR-transform has the property that it linearizes the c-free convolution of
pairs of compactly supported measures. In particular, if X and Y are c-free
elements from some algebra A, then
cRX+Y =
cRX +
cRY .
If the ∗-algebraA is endowed with the C-valued states ϕ, ψ andX is a selfadjoint
element of A, then (see [2]), the coefficients {cRm}m ≥ 0 of cRX are defined by
the recurrence:
ϕ(Xn) =
l(1),...,l(k)≥0
l(1)+···+l(k)=n−k
cRk · ψ(X l(1)) · · ·ψ(X l(k−1))ϕ(X l(k))
equation that can be graphically illustrated by the picture, were the dark boxes
stand for the application of ϕ and the light ones for the application of ψ:
The above considerations lead to the following definition:
Definition 3.2. Let β, γ ∈Mul[[B]]. The multilinear function series cRβ,γ is the
element of Mul[[B]] defined by the recurrence relation
βn(b1, . . . , bn) =
cRβ,γ,k
[b1γp(1)(b3, . . . , bi1−2)bi1−1], . . .
. . . , [bi(k−1)γp(k)(bi(k−1)+1, . . . , bi(k)−2)bi(k)−1]
bi(k)βn−ik(bik+1 , . . . , bn)
where the second summation is done over all 1 = i(0) < i(1) < · · · < i(k) ≤ n and
p(k) = i(k)− i(k − 1)− 2.
MULTILINEAR FUNCTION SERIES, C-FREE PROBABILITY WITH AMALGAMATION 9
The following analytical description of cRβ,γ also shows that it is unique and
well-defined:
Theorem 3.3. For any β, γ ∈Mul[[B]],
Rβ,γ =
β(1 + Iβ)−1
◦ (I + IγI)〈−1〉 (3.3)
Before proving the theorem, remark that the right-hand side of (3.3) is well-
defined and unique, since 1 + Iγ is invertible with respect to the formal multipli-
cation, I+ IβI is invertible with respect to formal composition and its inverse has
0 as first component (see [3]). We will need the following auxiliary result:
Lemma 3.4. Let β be an element of Mul[[B]] and I the identity element with
respect to formal composition, I = (0, idB, 0, 0 . . . ).
(i) the multilinear function series Iβ is given by:
(Iβ)0 = 0
(Iβ)n(b1, . . . , bn) = b1βn−1(b2, . . . , bn)
(ii) the multilinear function series IβI is given by
(IβI)0 = 0
(IβI)1(b1) = 0
(IβI)n(b1, . . . , bn) = b1βn−2(b2, . . . , bn−1)bn
Proof. Since I = (0, idB, 0, . . . ), one has:
(Iβ)0 = I0β0 = 0.
If n ≥ 1,
(Iβ)n(b1, . . . , bn) =
Ik(b1, . . . , bk)βn−k(bk+1, . . . , bn)
= I1(b1)βn−1(bk+1, . . . , bn)
= b1βn−1(bk+1, . . . , bn).
For IβI, the same computations give:
(IβI)0 = (Iβ)0I0 = 0
(IβI)1 = (Iβ)0I1(b1) + (Iβ)1(b1)I0
If n ≥ 2, one has:
(IβI)n(b1, . . . , bn) =
(Iβ)k(b1, . . . , bk)In−k(bk+1, . . . , bn)
= (Iβ)n−1(b1, . . . , bk)I1(b1)
= b1βn−2(b2, . . . , bn−1)bn
10 MIHAI POPA
Proof of the Theorem 3.3: Set σ = I + IβI. Then
(cRβ,γ ◦ σ)n (b1, . . . , bn) =
p1,...,pk≥1
p1+···+pk=n
Rβ,γ,k
σp1 (b1, . . . , bp1), . . . ,
σpk(bqk+1, . . . , bqk+pk)
where qi = p1 + · · ·+ pi−1.
From Lemma (3.4)(ii), we have that
σn(b1, . . . , bn) = (I + IβI)n(b1, . . . , bn)
therefore Definition 3.2 is equivalent to
βn(b1, . . . , bn) =
(cRβ,γ ◦ (I + IβI)k(b1, . . . , bk)) bk+1βn−k−2(bk+2, . . . , bn)
Considering now Lemma 3.4(i), the above relation becomes
βn(b1, . . . , bn) =
(cRβ,γ ◦ (I + IβI)k(b1, . . . , bk)) (I + Iβ)n−k(bk+1, . . . , bn)
therefore
β = [cRβ,γ ◦ (I + IγI)] (1 + Iβ)
which is equivalent to (3.3). �
Remark 3.5. Up to a shift in the coefficients, equation (3.3) is similar to the result
in the case B = C from [2], Theorem 5.1.
Let X be a selfadjoint element of A. If A is endowed with two B-valued condi-
tional expectations Φ,Ψ, the element X will have two moment-generating multi-
linear function series, one with respect to Ψ, that we will denote by MX , and one
with respect to Φ, denoted MX . For brevity, we will use the notation
cRX for the
multilinear function series cRMX ,MX .
Theorem 3.6. Let X and Y be two elements of A that are c-free with respect to
the pair of conditional expectations (Φ,Ψ). Then
cRX+Y =
cRX +
Proof. Let A be an algebra containing B as a subalgebra and endowed with the
conditional expectations Φ,Ψ : A −→ B. Consider the set A0 = A \ B (set
difference). For n ≥ 1 define the maps
cr : A0 × · · · × A0︸ ︷︷ ︸
n times
given by the recurrence formula:
Φ(a1 · · · an) =
l(1)<···<l(k)
1<l(1),l(k)≤n
rk(a1[Ψ(a2 · · · al(1)−1)], . . . ,
. . . , al(k−1)[Ψ(al(k−1)+1 · · · al(k)1)], al(k)[Φ(al(k)+1 · · · an)])
MULTILINEAR FUNCTION SERIES, C-FREE PROBABILITY WITH AMALGAMATION 11
Note that crn is well defined, and that, for any b1, . . . , bn ∈ B,
crn+1(X, b1X, . . . , bnX) =
cRX,n(b1, . . . , bn). (3.4)
As in Section 2, consider B〈ξi〉, the noncommutative algebras of polynomi-
als in the symbols ξi, i = 1, 2 and with coefficients from B and the conditional
expectations
ΦX ,ΨX : B〈ξ1〉 −→ B
given by
ΦX(f(ξ1)) = Φ(f(X))
ΨX(f(ξ1)) = Ψ(f(X))
and their analogues ΦY ,ΨY for B〈ξ2〉.
OnB〈ξ1, ξ2〉, identified toB〈ξ1〉∗BB〈ξ2〉, consider the conditional expectations
Ψ0,Φ0, ϕ given by:
Ψ0 = ΨX ∗ΨY
Φ0(f(ξ1, ξ2)) = Φ(f(X,Y ))
ϕ(a1a2 . . . an) =
l(1)<···<l(k)
1<l(1),l(k)≤n
ρk(a1[Ψ0(a2 · · · al(1)−1)], . . . ,
. . . , al(k−1)[Ψ0(al(k−1)+1 · · ·al(k)1 )], al(k)[ϕ(al(k)+1 · · · an)])
where a1, . . . , an are elements of the set B〈ξ1, ξ2〉0 = B〈ξ1〉 ∪B〈ξ2〉 \B, and the
ρn : B〈ξ1, ξ2〉0 × . . .B〈ξ1, ξ2〉0︸ ︷︷ ︸
n times
are given by:
ρn(a1, . . . , an) =
cr(a1, . . . , an) if all a1, . . . , an ∈ B〈ξ1〉
cr(a1, . . . , an) if all a1, . . . , an ∈ B〈ξ2〉
0 otherwise
We will show that ϕ = Φ0, in particular ϕ is also well-defined. Consider the
element a ∈ B〈ξ1, ξ2〉 of the form a = a1 · · · an with aj ∈ B〈ξε(j)〉, such that
ε(1) 6= ε(2) 6= · · · 6= ε(n) and Ψ0(aj) = 0. The computation of ϕ(a1 · · · an) is done
via the recurrence relation above. Because of the definition of ρ and the fact that
Ψ0 = ΨX ∗ΨY , only the term with k = 1 contribute at the sum, i.e.
ϕ(a1 · · · an) = ϕ(a1ϕ(a2 · · · an))
= ϕε(1)(a1ϕ(a2 · · · an)
= ϕε(1)(a1)ϕ(a2 · · · an)
and the identity between ϕ and Φ0 follows by induction over n.
Since ϕ = Φ0, the maps ρn and
crn are satisfying the same recurrence relation,
hence
ρn(a1, . . . , an) =
cr(a1, . . . , an).
12 MIHAI POPA
In particular
RX+Y,n(b1, . . . , bn) =
rn+1((X + Y )b1(X + Y ) . . . (X + Y )bn(X + Y ))
= ρn+1((X + Y )b1(X + Y ) . . . (X + Y )bn(X + Y ))
= ρn+1((X)b1(X) . . . (X)bn(X)) + ρn+1((Y )b1(Y ) . . . (Y )bn(Y ))
= cRX,n(b1, . . . , bn) +
cRY,n(b1, . . . , bn).
4. Central limit theorem
Consider the ordered set 〈n〉 = {1, 2, . . . , n} and π a partition of 〈n〉 with blocks
B1, . . . , Bm:
〈n〉 = B1 ⊔B2 ⊔ · · · ⊔Bm.
The blocks Bp and Bq of π are said to be crossing if there exist i < j < k < l
in 〈n〉 such that i, k ∈ Bp and j, l ∈ Bq.
The partition π is said to be non-crossing if all pairs of distinct blocks of π
are not crossing. We will denote by NC2(n) the set of all non-crossing partitions
of 〈n〉 whose blocks contain exactly 2 elements and by NC≤s(n) the set of all
non-crossing partitions of 〈n〉 whose blocks contain at most s elements.
Let now γ be a non-crossing partition of 〈n〉 and B and C be two blocks of
π. We say that B is interior to C if there exist two indices i < j in 〈n〉 such
that i, j ∈ C and B ⊂ {i + 1, . . . , j − 1}. The block B is said to be outer if it is
not interior to any other block of γ. In a non-crossing partition of 〈n〉, the block
containing 1 is always outer.
Consider now an element X of A. Let π be a partition from NC2(n+ 1) (n =
odd) and B1 = (1, k) be the block of π containing 1. We define, by recurrence,
the following expressions:
Vπ(X, b1, . . . , bn) = Ψ(Xb1Vπ|{2,...,j−1}(X, b2, . . . , bk−2)bk−1X)bk
Vπ|{k+1,...,n+1}(X, bk+1, . . . , bn)
Wπ(X, b1, . . . , bn) = Φ(Xb1Vπ|{2,...,j−1}(X, b2, . . . , bk−2)bk−1X)bk
Wπ|{k+1,...,n+1}(X, bk+1, . . . , bn)
Theorem 4.1. (Central Limit Theorem) Let (Xn)n≥1 be a sequence of c-free
elements of A such that:
(1) all Xn have the same moment-generating multilinear function series, M
with respect to Φ and M with respect to Ψ.
(2) Ψ(Xn) = Φ(Xn) = 0.
X1 + · · ·+XN√
Then:
(i) lim
RSN = (0,M1(·), 0, . . . )
MULTILINEAR FUNCTION SERIES, C-FREE PROBABILITY WITH AMALGAMATION 13
(ii) lim
RSN = (0,M1(·), 0, . . . )
(iii) there exist two conditional expectations ν : B〈ξ〉 −→ B, depending only
on M1(·), and µ : B〈ξ〉 −→ B, depending only on M1(·) and M1(·), such
ΨSN = ν
ΦSN = µ
in the weak sense; in particular,
ν(ξb1ξ . . . bnξ) =
π∈NC2(n)
Vπ(X1, b1, . . . , bn)
µ(ξb1ξ . . . bnξ) =
π∈NC2(n)
Wπ(X1, b1, . . . , bn).
Proof. Let X be an element of A with the same moment generating series as Xj,
j ≥ 1. As shown in [3],
RSN =
R Xk√
= NR X√
Also, from Theorem 2.4 and Theorem 3.6, it follows that
cRSN =
cR Xk√
= N cR X√
Since R and cR are multilinear and M0 = M0 = 0, we have that
cRSN ,n = lim
cRX,n
0 if n 6= 1
M1(·) if n = 1
and the similar relations for RSN ,n, hence (i) and (ii) are proved.
For (iii) it suffices to check the relations for ν(ξb1ξ . . . bnξ) and µ(ξb1ξ . . . bnξ),
which are a trivial corollary of (i), (ii), and the recurrence formulas that define R
and cR. �
Remark 4.2. For B = C, the theorem is a weaker version of Theorem 4.3 from [2].
If Ψ is C-valued, then the result is similar to Corollary 5.1 from [6]. Also, under
the assumptions that for some a, b ∈ B we have that:
NΨ(X1 · · ·XN ) = a
NΨ(X1 · · ·XN ) = b
the same techniques lead to a Poisson-type limit Theorem, similar to Corollary 2,
Section 5 of [6].
14 MIHAI POPA
In the following remaining pages we will describe the positivity of the limit
functionals µ and ν in terms of Φ and Ψ. The central result is Corollary 4.4.
For simplicity, suppose that B is a unital ∗-algebra (otherwise, we can replace
B by its unitisation). Consider the symbol ξ, the ∗-algebra B〈ξ〉 of polynomials
in ξ with coefficients from B, as defined before, and consider also the linear space
BξB generated by the set {b1ξb2; b1, b2 ∈ B} with the B-bimodule structure
given by
a1b1ξb2a2 = (a1b1)ξ(b2a2)
for all a1, a2, b1, b2 ∈ B.
Lemma 4.3. For any positive B-sesquilinear pairing 〈·, ·〉 on BξB there exists a
positive conditional expectation ϕ : B〈ξ〉 −→ B such that for any b1, b2 ∈ B one
has that
ϕ(ξb∗1b2ξ) = 〈b1ξ, b2ξ〉
Proof. Without loss of generality, we can suppose that B is unital (otherwise we
can replace B by its unitization).
Consider the Full Fock bimodule over BξB
F〈ξ〉 = B⊕
BξB⊗B · · · ⊗B BξB︸ ︷︷ ︸
n times
with the pairing given by
〈a, b〉 = a∗b
〈a1ξ ⊗ · · · ⊗ anξ, b1ξ ⊗ · · · ⊗ bmξ〉 = δm,n〈anξ, 〈. . . , 〈a1ξ, b1ξ〉b2ξ〉, . . . bnξ〉.
(a, aj , b, bj ∈ B, j = 1, . . . , n)
Note that the B-linear operators A1, A2 : F〈ξ〉 −→ F〈ξ〉 described by the
relations
A1b = ξb
A1(a1ξ ⊗ · · · ⊗ anξb) = ξ ⊗ a1ξ ⊗ · · · ⊗ anξb
A2b = 0
A2(a1ξ ⊗ · · · ⊗ anξb) = 〈ξ, a1ξ〉a2ξ ⊗ · · · ⊗ anξb
are self-adjoint to each other, in the sense that
〈A1ζ̃1, ζ̃2〉 = 〈ζ̃1, A2ζ̃2〉
for any ζ̃1, ζ̃2 ∈ F〈ξ〉, therefore S = A1 +A2 is selfadjoint.
Moreover, for any a, b ∈ B,
〈1, Sa∗bS1〉 = 〈aS1, bS1〉
= 〈a(A1 +A2)1, b(A1 +A2)1〉
= 〈aξ, bξ〉
and the conclusion follows by setting ϕ(p(ξ)) = 〈1, p(S)1〉 for all p ∈ B〈ξ〉. �
Corollary 4.4. The maps µ and ν from Theorem 4.1 are positive if and only if
for any b ∈ B one has that Φ(Xb∗bX) ≥ 0 and Ψ(Xb∗bX) ≥ 0.
MULTILINEAR FUNCTION SERIES, C-FREE PROBABILITY WITH AMALGAMATION 15
Proof. One implication is trivial, since, if ν and µ are positive, then
Ψ(Xb∗bX) = ν(Xb∗bX) = ν((bX)∗bX) ≥ 0
Φ(Xb∗bX) = µ(Xb∗bX) = µ((bX)∗bX) ≥ 0.
Suppose now that Φ(Xb∗bX) ≥ 0 and Ψ(Xb∗bX) ≥ 0 for all b ∈ B. We will
use the same argument as in [9] and [8].
Consider the set of selfadjoint symbols {ξi}i≥1. On each B-bimodule BξiB we
have the positive B-sesquilinear pairings 〈·, ·〉Φ and 〈·, ·〉Ψ determined by
〈aξi, bξi〉Φ = Φ(Xa∗bX)
〈aξi, bξi〉Ψ = Ψ(Xa∗bX).
As shown in Lemma 4.3, the above B-sesquilinear pairings determine positive
conditional expectations ϕ1, ψi : Ai −→ B, where Ai = B〈ξi〉 be the ∗-algebras of
polynomials in ξ with coefficients from B, i ≥ 1.
For τ : B〈ξ〉 −→ B a conditional expectation, and λ ≥ 0, note with Dλτ the
dilation with λ of τ , i.e.
Dλτ(ξb1ξ · · · bnξ) = λn+1τ(ξb1ξ · · · bnξ)
Remark that if τ is positive, then Dλτ is also positive.
With the notations above, consider, as in Definition 2.1, the conditionally free
product (A,Φ,Ψ) = ∗i∈I(Ai,Φi,Ψi). The elements {ξi}i≥1 are conditionally free
in A, so Theorem 4.1 implies that:
µ = lim
Φ ξ1+···+ξN√
= D 1√
Φξ1+···+ξN
ν = lim
Ψ ξ1+···+ξN√
= D 1√
Ψξ1+···+ξN
= D 1√
(∗Ni=1Ψξi
We have that ∗Ni=1Ψξi ≥ 0 since it is the free product of states (see, for example
[9]), hence the positivity of ν.
Also, Theorem 2.4 and Corollary 2.6 imply that Φξ1+···+ξN ≥ 0, therefore µ ≥ 0.
Acknowledgment. This research was partially supported by the Grant 2-CEx06-
11-34 of the Romanian Government. I am thankful to Marek Bożejko for present-
ing me the basics of c-freeness and bringing to my attention the references [2] and
[6]. I thank also Hari Bercovici for his constant support and his many advices
during the work on this paper.
16 MIHAI POPA
References
1. Bożejko, M. and Speicher, R. : ψ-independent and symmetrized white noises., in: Quantum
Probability and Related Topics, VI (1991), 219-236, World Scientific, Singapore.
2. Bożejko, M., Leinert, M. and Speicher, R.: Convolution and Limit Theorems for Condi-
tionally free Random Variables., in: Pac. J. Math. 175 (1996), 357-388.
3. Dykema, K.: Multilinear function series and transforms in Free Probability theory, in: Ad-
vances in Mathematics 208 (2007) 351407.
4. Dykema, K.: On the S-transform over a Banach algebra, in: Journal of Functional Analysis
231 (2006) 90110
5. Lance, E. C.: Hilbert C*-modules. A toolkit for operator algebraists, London Mathematical
Society Lecture Note Series 210, Cambridge University Press 1990.
6. M lotkowski, W.: Operator-valued version of conditionally free product, in: Studia Mathe-
matics 153 (1) (2002)
7. Nica, A. and Speicher, R.: Lectures on the Combinatorics of the Free Probability, London
mathematical Society Lecture Note Series 335, Cambridge University Press 2006
8. Popa, M.: A combinatorial approach to monotonic independence over a C*-algebra Preprint,
arXiv: math.OA/0612570, 01/2007
9. Speicher, R.: Combinatorial Theory of the Free Product with amalgamation and Operator-
Valued Free Probability Theory, in: Mem. AMS, Vol 132, No 627 (1998)
Mihai Popa: Department of Mathematics, Indiana University at Bloomington,
Rawles Hall, 931 E 3rd St, Bloomington, IN 47405
E-mail address: mipopa@indiana.edu

  As in the cases of freeness and monotonic independence, the notion of
conditional freeness is meaningful when complex-valued states are replaced by
positive conditional expectations. In this framework, the paper presents
several positivity results, a version of the central limit theorem and an
analogue of the conditionally free R-transform constructed by means of
multilinear function series.

Introduction
The paper addresses a topic related to conditionally free (or, shortly, using the
term from [2], c-free) probability. This notion was developed in the ’90’s (see [1],
[2]) as an extension of freeness within the framework of ∗-algebras endowed with
not one, but two states. Namely, given a family of unital algebras {A}i∈I, each Ai
endowed with two expectations ϕi, ψi : Ai −→ C, their c-free product is the triple
(A, ϕ, ψ), where:
(i) A = ∗i∈IAi is the free product of the algebras Ai.
(ii) ψ = ∗i∈Iψi and ϕ = ∗(ψi)i∈Iϕi are expectations given by the relations
(a) ψ(a1 · · · an) = 0
(b) ϕ(a1 · · · an) = ϕε(1)(a1) · · ·ϕε(n)(an)
for all aj ∈ Aε(j), j = 1, . . . , n such that ψε(j)(aj) = 0 and ε(1) 6= · · · 6= ε(n).
A key result is that if the Ai are ∗-algebras and ϕi, ψi are positive functionals,
then ϕ and ψ are also positive.
In [6], the positivity of the free product maps ϕ, ψ is proved for the case
when ϕ1, ϕ2 are positive conditional expectations in a common C∗-subalgebra,
but ψ1, ψ2 remain positive C-valued maps. A more general situation is indeed
discussed (see Theorem 3, Section 6, from [6]), but the question if ϕ, ψ re positive
for ϕ1,2, ψ1,2 arbitrary positive conditional expectations is left unanswered.
A first answer was given in [8], where we showed that for A a ∗-algebra, the
analogous construction with both ϕ and ψ valued in a C∗-subalgebra B of A still
retains the positivity. The present paper further develops this result (see Theorem
2.3) and also demonstrates the use of multilinear function series in c-free setting.
2000 Mathematics Subject Classification. Primary 45L53; Secondary 46L08.
Key words and phrases. conditional freeness, conditional expectation, R-transform, multi-
linear function series.
http://arxiv.org/abs/0704.0040v3
2 MIHAI POPA
In [2] is constructed a c-free version of Voiculescu’s R-transform, which we will
call the cR-transform, with the property that cRX+Y =
cRX +
cRY if X and Y
are c-free elements from the algebra A relative to ϕ and ψ (i.e. the relations (a)
and (b) from the definition of the c-free product hold true for the subalgebras
generated by X and Y .)
The apparatus of multilinear function series is used in recent work of K. Dykema
([3] and [4]) to construct suitable analogues for the R and S-transforms in the
framework of freeness with amalgamation. We will show that this construction
is also appropriate for the cR-transform mentioned above. The techniques used
differ from the ones of [3], the Fock space type construction being substituted by
combinatorial techniques similar to [2] and [7]. Particularly, Theorems 3.3 and 3.6
contain new (shorter) proves of the results 6.1–6.13 from [3].
The paper is structured in four sections. In Section 2 are stated the basic
definitions and are proved the main positivity results. Section 3 describes the
construction and the basic property of the multilinear function series cR-transform
and Section 4 treats the central limit theorem and a related positivity property.
2. Definitions and positivity results
Definition 2.1. Let {Ai}i∈I be a family of algebras, all containing the subalgebra
B. Suppose D is a subalgebra of B and Ψi : Ai −→ D and Φi : Ai −→ B are con-
ditional expectations, i ∈ I. We say that the triple (A,Φ,Ψ) = ∗i∈I(Ai,Φi,Ψi)
is the conditionally free product with amalgamation over (B,D), or shortly, the
c-free product, of the triples (Ai,Φi,Ψi)i∈I if
(1) A is the free product with amalgamation over B of the family (Ai)i∈I
(2) Ψ = ∗i∈IΨi and Φ = ∗(Ψi)i∈IΦi are determined by the relations
Ψ(a1a2 . . . an) = 0
Φ(a1a2 . . . an) = Φ(a1)Φ(a2) . . .Φ(an),
for all ai ∈ Aε(i), ε(i) ∈ I such that ε(1) 6= ε(2) 6= · · · 6= ε(n) and Ψε(i)(ai) = 0.
When D = C, this definition reduces to the one given in [6]. When both B and
D are equal to C, this definition was given in [2].
When discussing positivity, we need a ∗-structure on our algebras. We will
demand that B and D be C∗-algebras, while Ai and A are only required to be
∗-algebras.
The following results are slightly modified versions of Lemma 6.4 and Theorem
6.5 from [8].
Lemma 2.2. Let B be a C∗-algebra and A1, A2 be two ∗-algebras containing B
as a ∗-subalgebra, endowed with positive conditional expectations Φj : Aj −→ B,
j = 1, 2. If a1, . . . , an ∈ A1, an+1, . . . , an+m ∈ A2 and A = (Ai,j)i,j ∈ Mn+m(B)
is the matrix with the entries
Ai,j =
i aj) if i, j ≤ n
i )Φ2(aj) if i ≤ n, j > n
i )Φ1(aj) if i > n, j ≤ n
i aj) if i, j > n
MULTILINEAR FUNCTION SERIES, C-FREE PROBABILITY WITH AMALGAMATION 3
then A is positive.
Proof. The vector space E = B ⊕ ker(Φ1) ⊕ ker(Φ2) has a B-bimodule structure
given by the algebraic operations on A1 and A2. Consider the B-sesquilinear
pairing
〈·, ·〉 : E× E −→ B
determined by the relations:
〈b1, b2〉 = b∗1b2, for b1, b2 ∈ B
〈uj, vj〉 = Φj(u∗jvj), for uj , vj ∈ ker(Φj), j = 1, 2
〈u1, u2〉 = 〈u2, u1〉 = 0 for u1 ∈ ker(Φ1), and u2 ∈ ker(Φ2).
〈b, uj〉 = 〈uj , b〉 = 0 for all b ∈ B, uj ∈ Aj
With this notation, we have that Ai,j = 〈ai, aj〉, hence it suffices to show that
〈a, a〉 ≥ 0 for all a ∈ E.
Indeed, for an element a = b + u1 + u2 with b ∈ B, uj ∈ ker(Φj), j = 1, 2, we
have:
〈a, a〉 = 〈b + u1 + u2, b+ u1 + u2〉
= 〈b, b〉+ 〈u1, u1〉+ 〈u2, u2〉
= b∗b+Φ1(u
1u1) + Φ2(u
Theorem 2.3. Let B be a C∗-algebra and D a C∗-subalgebra of B. Suppose that
A1, A2 are ∗-algebras containing B, each endowed with two positive conditional
expectations Φj : Aj −→ B, and Ψj : Aj −→ D, j = 1, 2 and consider the c-free
product (A,Φ,Ψ) = ∗i=1,2(Ai,Φi,Ψi).
Then the maps Φ and Ψ are positive.
Proof. The positivity of Ψ is by now a classical result in the theory of free proba-
bility with amalgamation over a C∗-algebra (for example, see [9], Theorem 3.5.6).
For the positivity of Φ we have to show that Φ(a∗a) ≥ 0 for any a ∈ A.
Any element of A can be written as
s1,k . . . sn(k),k,
where sj,k ∈ Aε(j,k) ε(1, k) 6= ε(2, k) 6= · · · 6= ε(n(k), k).
Writing
s(j,k) = s(j,k) −Ψ(s(j,k)) +Ψ(s(j,k))
and expanding the product, we can consider a of the form
a = d+
a1,k . . . an(k),k
with d ∈ D ⊂ B and aj,k ∈ Aε(j,k) such that Ψε(j,k)(aj,k) = 0 and ε(1, k) 6=
ε(2, k) 6= · · · 6= ε(n(k), k).
4 MIHAI POPA
Therefore
Φ(a∗a) = Φ
d+ d∗
a1,k . . . an(k),k
a1,k . . . an(k),k
a1,k . . . an(k),k
]∗[ N∑
a1,k . . . an(k),k
Since Φ is a conditional expectation and d ∈ D ⊂ B, the above equality becomes
Φ(a∗a) = d∗d+
d∗Φ(a1,k . . . an(k),k) +
Φ(a∗n(k),k . . . a
1,k)d
k,l=1
a∗n(k),k . . . a
1,ka1,l . . . an(l),l
Using the definition of the conditionally free product with amalgamation over B
and that Ψε(j,k)(aj,k) = 0 for all j, k, one further has
Φ(a∗a) = d∗d+
Φ(d∗a1,k)Φ(a2,k) . . .Φ(an(k),k)
Φ(an(k),k)
. . .Φ(a∗2,k)Φ(a
1,kd)
k,l=1
Φ(an(k),k)
∗ . . .Φ(a∗2,k)
Φ(a∗1,ka1,l)Φ(a2,l) . . .Φ(an(l),l)
that is
Φ(a∗a) = d∗d+
Φ(d∗a1,k)
Φ(a2,k) . . .Φ(an(k),k)
Φ(a2,k) . . .Φ(an(k),k)
Φ(a∗1,kd)
k,l=1
Φ(a2,k) . . .Φ(an(k),k)
Φ(a∗1,ka1,l)
Φ(a2,l) . . .Φ(an(l),l)
From Lemma 2.2, the matrix S =
Φ(a∗1,ia1,j)
i,j=1
is positive in MN+1(B),
therefore
S = T ∗T, for some T ∈MN+1(B).
MULTILINEAR FUNCTION SERIES, C-FREE PROBABILITY WITH AMALGAMATION 5
Denote now a1,N+1 = d and vk = Φ(a2,k) . . .Φ(an(k),k).The identity for Φ(a
becomes:
Φ(a∗a) = (v1, . . . , vN , 1)
∗T ∗T (v1, . . . , vN , 1)
≥ 0, as claimed.
Theorem 2.4. Assume that I =
j∈J Ij is a partition of I. Then:
∗j∈J (∗i∈Ij (Ai,Φi,Ψi)) = ∗i∈I(Ai,Φi,Ψi)
Proof. The proof is identical to the proofs of similar results in [6] and [2]. Consider
ai ∈ Aε(i), 1 ≤ i ≤ m such that ε(1) 6= ε(2 6= · · · 6= ε(m) and Ψε(i)(ai) = 0. Let
1 = i0 < i1 < · · · < ik = m and Jl = {ε(i), il−1 ≤ i < il}.
Since
(∗j∈JlΨj) ((ail−1 · · · ail)) = 0,
it suffices to show that
Φ(a1 · · · am) =
(∗(Ψj),j∈JlΦj)(ail−1 · · · ail)] .
Φ(a1 · · · am) = Φε(1)(a1) · · ·Φε(m)(am)
while, since Ψε(i)(ai) = 0,
(∗(Ψj),j∈JlΦj)(ail−1 · · · ail) = Φil−1(ail−1) · · ·Φil(ail)
and the conclusion follows. �
Definition 2.5. Let A be an algebra (respectively a ∗-algebra),B a subalgebra (∗-
subalgebra) of A and D a subalgebra (∗-subalgebra) of B. Suppose A is endowed
with the conditional expectations Ψ : A −→ D and Φ : A −→ D.
(i) The subalgebras (∗-subalgebras) (Ai)i∈I of A are said to be c-free with
respect to (Φ,Ψ) if:
(a) (Ai)i∈I are free with respect to Ψ.
(b) if ai ∈ Aε(i), 1 ≤ i ≤ m, are such that ε(1) 6= · · · 6= ε(m) and
Ψ(ai) = 0, then Φ(a1 · · ·am) = Φ(a1) · · ·Φ(am).
(ii) The elements (Xi)i∈I of A are said to be c-free with respect to (Φ,Ψ) if
the subalgebras (∗-subalgebras) generated by B and Xi are c-free with
respect to (Φ,Ψ).
We will denote by B〈ξ〉 the non-commutative algebra of polynomials in the
symbol ξ and with coefficients from B (the coefficients do not commute with
the symbol ξ). If I is a family of indices, B〈{ξi}i∈I〉 will denote the algebra of
polynomials in the non-commuting variables {ξ}i∈I and with coefficients from B.
We will identify B〈{ξi}i∈I〉 with the free product with amalgamation over B of
the family {B〈ξi〉}i∈I .
6 MIHAI POPA
If A is a ∗-algebra and B is with the C∗-algebra, B〈ξ〉 will also be considered
with a ∗-algebra structure, by taking ξ∗ = ξ. If X is a selfadjoint element from A,
we define the conditional expectations
ΦX ,ΨX : B〈ξ〉 −→ B
given by ΦX(f(ξ)) = Φ(f(X)) and ΨX(f(ξ)) = Ψ(f(X)) , for any f(ξ) ∈ B〈ξ〉.
Corollary 2.6. Suppose that A is a ∗-algebra and X and Y are c-free selfadjoint
elements of A such that the maps ΦX ,ΨX and ΦY ,ΨY are positive. Then the
maps ΦX+Y and ΨX+Y are also positive.
Proof. The positivity of ΨX+Y is an immediate consequence of the fact that X
and Y are free with amalgamation over B with respect to Ψ. It remains to prove
the positivity of ΦX+Y .
Since the maps ΦX : B〈ξ1〉 −→ B and ΦY : B〈ξ2〉 −→ B are positive, from
Theorem 2.3 so is
Φx ∗(ΨX ,ΨY ) ΦY : B〈ξ1〉 ∗B B〈ξ2〉 = B〈ξ1, ξ2〉 −→ B
Remark also that
iZ : B〈ξ〉 ∋ f(ξ) 7→ f(X + Y ) ∈ B〈ξ1〉 ∗B B〈ξ2〉
is a positive B-functional.
The conclusion follows from the fact that the c-freeness of X and Y is equivalent
ΦX+Y = (ΦX ∗(ΨX ,ΨY ) ΦY ) ◦ iX+Y .
3. Multilinear function series and the cR-transform
Let A be a ∗-algebra containing the C∗-algebra B, endowed with a conditional
expectation Ψ : A −→ B. If X is a selfadjoint element of A, then by the moment
of order n of X we will understand the map
X : B× · · · ×B︸ ︷︷ ︸
n−1 times
X (b1, . . . , bn−1) = Ψ(Xb1X . . .Xbn−1X)
If B = C, then the moment-generating series of X
mX(z) =
Ψ(Xn)zn
encodes all the information about the moments of X . For B 6= C, the straightfor-
ward generalization
mX(z) =
Ψ(Xn)zn
generally fails to keep track of all the possible moments of X . A solution to
this inconvenience was proposed in [3], namely the moment-generating multilin-
ear function series of X . Before defining this notion, we will briefly recall the
construction and several results on multilinear function series.
MULTILINEAR FUNCTION SERIES, C-FREE PROBABILITY WITH AMALGAMATION 7
Let B be an algebra over a field K. We set B̃ equal to B if B is unital and
to the unitization of B otherwise. For n ≥ 1, we denote by Ln(B) the set of all
K-multilinear maps
ωn : B× · · · ×B︸ ︷︷ ︸
n times
A formal multilinear function series over B is a sequence ω = (ω0, ω1, . . . ),
where ω0 ∈ B̃ and ωn ∈ Ln(B) for n ≥ 1. According to [3], the set of all
multilinear function series over B will de denoted by Mul[[B]].
For α, β ∈Mul[[B]], the formal sum α+ β and the formal product αβ are the
elements from Mul[[B]] defined by:
(α+ β)n(b1, . . . , bn) = αn(b1, . . . , bn) + βn(b1, . . . , bn)
(αβ)n(b1, . . . , bn) =
αk(b1, . . . , bk)βn−k(bk+1, . . . , bn)
for any b1, . . . , bn ∈ B.
If β0 = 0, then the formal composition α ◦ β ∈Mul[[B]] is defined by
(α ◦ β)0 = α0
and, for n ≥ 1, by
(α ◦ β)n(b1, . . . , bn) =
βp1(b1, . . . , bp1), . . . , βpk(bqk+1, . . . , bqk+pk)
where the second summation is done over all k-tuples p1, . . . , pk ≥ 1 such that
p1 + · · ·+ pk = n and qj = p1 + · · ·+ pj−1.
One can work with elements ofMul[[B]] as if they were formal power series. The
relevant properties are described in [3], Proposition 2.3 and Proposition 2.6. As in
[3], we use 1, respectively I, to denote the identity elements of Mul[[B]] relative
to multiplication, respectively composition. In other words, 1 = (1, 0, 0, . . . ) and
I = (0, idB, 0, 0, . . . ). We will also use the fact that an element α ∈Mul[[B]] has
an inverse with respect to formal composition, denoted α〈−1〉, if and only if α has
the form (0, α1, α2, . . . ) with α1 an invertible element of L1(B).
Definition 3.1. With the above notation, the moment-generating multilinear func-
tion series MX of X is the element of Mul[[B]] such that:
MX,0 = Ψ(X)
MX,n(b1, . . . , bn) = Ψ(Xb1X · · ·XbnX).
Given an element α ∈ Mul[[B]], the multilinear function series Rα is defined
by the following equation (see [3], Def 6.1):
(1 + αI)
◦ (I + IαI)〈−1〉. (3.1)
A key property of R is that for any X,Y ∈ A free over B, we have
RMX+Y = RMX +RMY . (3.2)
8 MIHAI POPA
These relations were proved earlier in the particular case B = C. One can also
describe Rα by combinatorial means, via the recurrence relation
αn(b1, . . . , bn) =
[b1αp(1)(b3, . . . , bi1−2)bi1−1], . . .
. . . , [bi(k−1)αp(k)(bi(k−1)+1, . . . , bi(k)−2)bi(k)−1]
bi(k)αn−ik(bik+1 , . . . , bn)
where the second summation is done over all 1 = i(0) < i(1) < · · · < i(k) ≤ n and
p(k) = i(k)− i(k − 1)− 2.
Following an idea from [2], the above equation can be graphically illustrated by
the picture:
In the case of scalar c-free probability, an analogue of the Voiculescu’s R-
transform is developed in [2]. In order to avoid confusions, we will denote it
by cR.
The cR-transform has the property that it linearizes the c-free convolution of
pairs of compactly supported measures. In particular, if X and Y are c-free
elements from some algebra A, then
cRX+Y =
cRX +
cRY .
If the ∗-algebraA is endowed with the C-valued states ϕ, ψ andX is a selfadjoint
element of A, then (see [2]), the coefficients {cRm}m ≥ 0 of cRX are defined by
the recurrence:
ϕ(Xn) =
l(1),...,l(k)≥0
l(1)+···+l(k)=n−k
cRk · ψ(X l(1)) · · ·ψ(X l(k−1))ϕ(X l(k))
equation that can be graphically illustrated by the picture, were the dark boxes
stand for the application of ϕ and the light ones for the application of ψ:
The above considerations lead to the following definition:
Definition 3.2. Let β, γ ∈Mul[[B]]. The multilinear function series cRβ,γ is the
element of Mul[[B]] defined by the recurrence relation
βn(b1, . . . , bn) =
cRβ,γ,k
[b1γp(1)(b3, . . . , bi1−2)bi1−1], . . .
. . . , [bi(k−1)γp(k)(bi(k−1)+1, . . . , bi(k)−2)bi(k)−1]
bi(k)βn−ik(bik+1 , . . . , bn)
where the second summation is done over all 1 = i(0) < i(1) < · · · < i(k) ≤ n and
p(k) = i(k)− i(k − 1)− 2.
MULTILINEAR FUNCTION SERIES, C-FREE PROBABILITY WITH AMALGAMATION 9
The following analytical description of cRβ,γ also shows that it is unique and
well-defined:
Theorem 3.3. For any β, γ ∈Mul[[B]],
Rβ,γ =
β(1 + Iβ)−1
◦ (I + IγI)〈−1〉 (3.3)
Before proving the theorem, remark that the right-hand side of (3.3) is well-
defined and unique, since 1 + Iγ is invertible with respect to the formal multipli-
cation, I+ IβI is invertible with respect to formal composition and its inverse has
0 as first component (see [3]). We will need the following auxiliary result:
Lemma 3.4. Let β be an element of Mul[[B]] and I the identity element with
respect to formal composition, I = (0, idB, 0, 0 . . . ).
(i) the multilinear function series Iβ is given by:
(Iβ)0 = 0
(Iβ)n(b1, . . . , bn) = b1βn−1(b2, . . . , bn)
(ii) the multilinear function series IβI is given by
(IβI)0 = 0
(IβI)1(b1) = 0
(IβI)n(b1, . . . , bn) = b1βn−2(b2, . . . , bn−1)bn
Proof. Since I = (0, idB, 0, . . . ), one has:
(Iβ)0 = I0β0 = 0.
If n ≥ 1,
(Iβ)n(b1, . . . , bn) =
Ik(b1, . . . , bk)βn−k(bk+1, . . . , bn)
= I1(b1)βn−1(bk+1, . . . , bn)
= b1βn−1(bk+1, . . . , bn).
For IβI, the same computations give:
(IβI)0 = (Iβ)0I0 = 0
(IβI)1 = (Iβ)0I1(b1) + (Iβ)1(b1)I0
If n ≥ 2, one has:
(IβI)n(b1, . . . , bn) =
(Iβ)k(b1, . . . , bk)In−k(bk+1, . . . , bn)
= (Iβ)n−1(b1, . . . , bk)I1(b1)
= b1βn−2(b2, . . . , bn−1)bn
10 MIHAI POPA
Proof of the Theorem 3.3: Set σ = I + IβI. Then
(cRβ,γ ◦ σ)n (b1, . . . , bn) =
p1,...,pk≥1
p1+···+pk=n
Rβ,γ,k
σp1 (b1, . . . , bp1), . . . ,
σpk(bqk+1, . . . , bqk+pk)
where qi = p1 + · · ·+ pi−1.
From Lemma (3.4)(ii), we have that
σn(b1, . . . , bn) = (I + IβI)n(b1, . . . , bn)
therefore Definition 3.2 is equivalent to
βn(b1, . . . , bn) =
(cRβ,γ ◦ (I + IβI)k(b1, . . . , bk)) bk+1βn−k−2(bk+2, . . . , bn)
Considering now Lemma 3.4(i), the above relation becomes
βn(b1, . . . , bn) =
(cRβ,γ ◦ (I + IβI)k(b1, . . . , bk)) (I + Iβ)n−k(bk+1, . . . , bn)
therefore
β = [cRβ,γ ◦ (I + IγI)] (1 + Iβ)
which is equivalent to (3.3). �
Remark 3.5. Up to a shift in the coefficients, equation (3.3) is similar to the result
in the case B = C from [2], Theorem 5.1.
Let X be a selfadjoint element of A. If A is endowed with two B-valued condi-
tional expectations Φ,Ψ, the element X will have two moment-generating multi-
linear function series, one with respect to Ψ, that we will denote by MX , and one
with respect to Φ, denoted MX . For brevity, we will use the notation
cRX for the
multilinear function series cRMX ,MX .
Theorem 3.6. Let X and Y be two elements of A that are c-free with respect to
the pair of conditional expectations (Φ,Ψ). Then
cRX+Y =
cRX +
Proof. Let A be an algebra containing B as a subalgebra and endowed with the
conditional expectations Φ,Ψ : A −→ B. Consider the set A0 = A \ B (set
difference). For n ≥ 1 define the maps
cr : A0 × · · · × A0︸ ︷︷ ︸
n times
given by the recurrence formula:
Φ(a1 · · · an) =
l(1)<···<l(k)
1<l(1),l(k)≤n
rk(a1[Ψ(a2 · · · al(1)−1)], . . . ,
. . . , al(k−1)[Ψ(al(k−1)+1 · · · al(k)1)], al(k)[Φ(al(k)+1 · · · an)])
MULTILINEAR FUNCTION SERIES, C-FREE PROBABILITY WITH AMALGAMATION 11
Note that crn is well defined, and that, for any b1, . . . , bn ∈ B,
crn+1(X, b1X, . . . , bnX) =
cRX,n(b1, . . . , bn). (3.4)
As in Section 2, consider B〈ξi〉, the noncommutative algebras of polynomi-
als in the symbols ξi, i = 1, 2 and with coefficients from B and the conditional
expectations
ΦX ,ΨX : B〈ξ1〉 −→ B
given by
ΦX(f(ξ1)) = Φ(f(X))
ΨX(f(ξ1)) = Ψ(f(X))
and their analogues ΦY ,ΨY for B〈ξ2〉.
OnB〈ξ1, ξ2〉, identified toB〈ξ1〉∗BB〈ξ2〉, consider the conditional expectations
Ψ0,Φ0, ϕ given by:
Ψ0 = ΨX ∗ΨY
Φ0(f(ξ1, ξ2)) = Φ(f(X,Y ))
ϕ(a1a2 . . . an) =
l(1)<···<l(k)
1<l(1),l(k)≤n
ρk(a1[Ψ0(a2 · · · al(1)−1)], . . . ,
. . . , al(k−1)[Ψ0(al(k−1)+1 · · ·al(k)1 )], al(k)[ϕ(al(k)+1 · · · an)])
where a1, . . . , an are elements of the set B〈ξ1, ξ2〉0 = B〈ξ1〉 ∪B〈ξ2〉 \B, and the
ρn : B〈ξ1, ξ2〉0 × . . .B〈ξ1, ξ2〉0︸ ︷︷ ︸
n times
are given by:
ρn(a1, . . . , an) =
cr(a1, . . . , an) if all a1, . . . , an ∈ B〈ξ1〉
cr(a1, . . . , an) if all a1, . . . , an ∈ B〈ξ2〉
0 otherwise
We will show that ϕ = Φ0, in particular ϕ is also well-defined. Consider the
element a ∈ B〈ξ1, ξ2〉 of the form a = a1 · · · an with aj ∈ B〈ξε(j)〉, such that
ε(1) 6= ε(2) 6= · · · 6= ε(n) and Ψ0(aj) = 0. The computation of ϕ(a1 · · · an) is done
via the recurrence relation above. Because of the definition of ρ and the fact that
Ψ0 = ΨX ∗ΨY , only the term with k = 1 contribute at the sum, i.e.
ϕ(a1 · · · an) = ϕ(a1ϕ(a2 · · · an))
= ϕε(1)(a1ϕ(a2 · · · an)
= ϕε(1)(a1)ϕ(a2 · · · an)
and the identity between ϕ and Φ0 follows by induction over n.
Since ϕ = Φ0, the maps ρn and
crn are satisfying the same recurrence relation,
hence
ρn(a1, . . . , an) =
cr(a1, . . . , an).
12 MIHAI POPA
In particular
RX+Y,n(b1, . . . , bn) =
rn+1((X + Y )b1(X + Y ) . . . (X + Y )bn(X + Y ))
= ρn+1((X + Y )b1(X + Y ) . . . (X + Y )bn(X + Y ))
= ρn+1((X)b1(X) . . . (X)bn(X)) + ρn+1((Y )b1(Y ) . . . (Y )bn(Y ))
= cRX,n(b1, . . . , bn) +
cRY,n(b1, . . . , bn).
4. Central limit theorem
Consider the ordered set 〈n〉 = {1, 2, . . . , n} and π a partition of 〈n〉 with blocks
B1, . . . , Bm:
〈n〉 = B1 ⊔B2 ⊔ · · · ⊔Bm.
The blocks Bp and Bq of π are said to be crossing if there exist i < j < k < l
in 〈n〉 such that i, k ∈ Bp and j, l ∈ Bq.
The partition π is said to be non-crossing if all pairs of distinct blocks of π
are not crossing. We will denote by NC2(n) the set of all non-crossing partitions
of 〈n〉 whose blocks contain exactly 2 elements and by NC≤s(n) the set of all
non-crossing partitions of 〈n〉 whose blocks contain at most s elements.
Let now γ be a non-crossing partition of 〈n〉 and B and C be two blocks of
π. We say that B is interior to C if there exist two indices i < j in 〈n〉 such
that i, j ∈ C and B ⊂ {i + 1, . . . , j − 1}. The block B is said to be outer if it is
not interior to any other block of γ. In a non-crossing partition of 〈n〉, the block
containing 1 is always outer.
Consider now an element X of A. Let π be a partition from NC2(n+ 1) (n =
odd) and B1 = (1, k) be the block of π containing 1. We define, by recurrence,
the following expressions:
Vπ(X, b1, . . . , bn) = Ψ(Xb1Vπ|{2,...,j−1}(X, b2, . . . , bk−2)bk−1X)bk
Vπ|{k+1,...,n+1}(X, bk+1, . . . , bn)
Wπ(X, b1, . . . , bn) = Φ(Xb1Vπ|{2,...,j−1}(X, b2, . . . , bk−2)bk−1X)bk
Wπ|{k+1,...,n+1}(X, bk+1, . . . , bn)
Theorem 4.1. (Central Limit Theorem) Let (Xn)n≥1 be a sequence of c-free
elements of A such that:
(1) all Xn have the same moment-generating multilinear function series, M
with respect to Φ and M with respect to Ψ.
(2) Ψ(Xn) = Φ(Xn) = 0.
X1 + · · ·+XN√
Then:
(i) lim
RSN = (0,M1(·), 0, . . . )
MULTILINEAR FUNCTION SERIES, C-FREE PROBABILITY WITH AMALGAMATION 13
(ii) lim
RSN = (0,M1(·), 0, . . . )
(iii) there exist two conditional expectations ν : B〈ξ〉 −→ B, depending only
on M1(·), and µ : B〈ξ〉 −→ B, depending only on M1(·) and M1(·), such
ΨSN = ν
ΦSN = µ
in the weak sense; in particular,
ν(ξb1ξ . . . bnξ) =
π∈NC2(n)
Vπ(X1, b1, . . . , bn)
µ(ξb1ξ . . . bnξ) =
π∈NC2(n)
Wπ(X1, b1, . . . , bn).
Proof. Let X be an element of A with the same moment generating series as Xj,
j ≥ 1. As shown in [3],
RSN =
R Xk√
= NR X√
Also, from Theorem 2.4 and Theorem 3.6, it follows that
cRSN =
cR Xk√
= N cR X√
Since R and cR are multilinear and M0 = M0 = 0, we have that
cRSN ,n = lim
cRX,n
0 if n 6= 1
M1(·) if n = 1
and the similar relations for RSN ,n, hence (i) and (ii) are proved.
For (iii) it suffices to check the relations for ν(ξb1ξ . . . bnξ) and µ(ξb1ξ . . . bnξ),
which are a trivial corollary of (i), (ii), and the recurrence formulas that define R
and cR. �
Remark 4.2. For B = C, the theorem is a weaker version of Theorem 4.3 from [2].
If Ψ is C-valued, then the result is similar to Corollary 5.1 from [6]. Also, under
the assumptions that for some a, b ∈ B we have that:
NΨ(X1 · · ·XN ) = a
NΨ(X1 · · ·XN ) = b
the same techniques lead to a Poisson-type limit Theorem, similar to Corollary 2,
Section 5 of [6].
14 MIHAI POPA
In the following remaining pages we will describe the positivity of the limit
functionals µ and ν in terms of Φ and Ψ. The central result is Corollary 4.4.
For simplicity, suppose that B is a unital ∗-algebra (otherwise, we can replace
B by its unitisation). Consider the symbol ξ, the ∗-algebra B〈ξ〉 of polynomials
in ξ with coefficients from B, as defined before, and consider also the linear space
BξB generated by the set {b1ξb2; b1, b2 ∈ B} with the B-bimodule structure
given by
a1b1ξb2a2 = (a1b1)ξ(b2a2)
for all a1, a2, b1, b2 ∈ B.
Lemma 4.3. For any positive B-sesquilinear pairing 〈·, ·〉 on BξB there exists a
positive conditional expectation ϕ : B〈ξ〉 −→ B such that for any b1, b2 ∈ B one
has that
ϕ(ξb∗1b2ξ) = 〈b1ξ, b2ξ〉
Proof. Without loss of generality, we can suppose that B is unital (otherwise we
can replace B by its unitization).
Consider the Full Fock bimodule over BξB
F〈ξ〉 = B⊕
BξB⊗B · · · ⊗B BξB︸ ︷︷ ︸
n times
with the pairing given by
〈a, b〉 = a∗b
〈a1ξ ⊗ · · · ⊗ anξ, b1ξ ⊗ · · · ⊗ bmξ〉 = δm,n〈anξ, 〈. . . , 〈a1ξ, b1ξ〉b2ξ〉, . . . bnξ〉.
(a, aj , b, bj ∈ B, j = 1, . . . , n)
Note that the B-linear operators A1, A2 : F〈ξ〉 −→ F〈ξ〉 described by the
relations
A1b = ξb
A1(a1ξ ⊗ · · · ⊗ anξb) = ξ ⊗ a1ξ ⊗ · · · ⊗ anξb
A2b = 0
A2(a1ξ ⊗ · · · ⊗ anξb) = 〈ξ, a1ξ〉a2ξ ⊗ · · · ⊗ anξb
are self-adjoint to each other, in the sense that
〈A1ζ̃1, ζ̃2〉 = 〈ζ̃1, A2ζ̃2〉
for any ζ̃1, ζ̃2 ∈ F〈ξ〉, therefore S = A1 +A2 is selfadjoint.
Moreover, for any a, b ∈ B,
〈1, Sa∗bS1〉 = 〈aS1, bS1〉
= 〈a(A1 +A2)1, b(A1 +A2)1〉
= 〈aξ, bξ〉
and the conclusion follows by setting ϕ(p(ξ)) = 〈1, p(S)1〉 for all p ∈ B〈ξ〉. �
Corollary 4.4. The maps µ and ν from Theorem 4.1 are positive if and only if
for any b ∈ B one has that Φ(Xb∗bX) ≥ 0 and Ψ(Xb∗bX) ≥ 0.
MULTILINEAR FUNCTION SERIES, C-FREE PROBABILITY WITH AMALGAMATION 15
Proof. One implication is trivial, since, if ν and µ are positive, then
Ψ(Xb∗bX) = ν(Xb∗bX) = ν((bX)∗bX) ≥ 0
Φ(Xb∗bX) = µ(Xb∗bX) = µ((bX)∗bX) ≥ 0.
Suppose now that Φ(Xb∗bX) ≥ 0 and Ψ(Xb∗bX) ≥ 0 for all b ∈ B. We will
use the same argument as in [9] and [8].
Consider the set of selfadjoint symbols {ξi}i≥1. On each B-bimodule BξiB we
have the positive B-sesquilinear pairings 〈·, ·〉Φ and 〈·, ·〉Ψ determined by
〈aξi, bξi〉Φ = Φ(Xa∗bX)
〈aξi, bξi〉Ψ = Ψ(Xa∗bX).
As shown in Lemma 4.3, the above B-sesquilinear pairings determine positive
conditional expectations ϕ1, ψi : Ai −→ B, where Ai = B〈ξi〉 be the ∗-algebras of
polynomials in ξ with coefficients from B, i ≥ 1.
For τ : B〈ξ〉 −→ B a conditional expectation, and λ ≥ 0, note with Dλτ the
dilation with λ of τ , i.e.
Dλτ(ξb1ξ · · · bnξ) = λn+1τ(ξb1ξ · · · bnξ)
Remark that if τ is positive, then Dλτ is also positive.
With the notations above, consider, as in Definition 2.1, the conditionally free
product (A,Φ,Ψ) = ∗i∈I(Ai,Φi,Ψi). The elements {ξi}i≥1 are conditionally free
in A, so Theorem 4.1 implies that:
µ = lim
Φ ξ1+···+ξN√
= D 1√
Φξ1+···+ξN
ν = lim
Ψ ξ1+···+ξN√
= D 1√
Ψξ1+···+ξN
= D 1√
(∗Ni=1Ψξi
We have that ∗Ni=1Ψξi ≥ 0 since it is the free product of states (see, for example
[9]), hence the positivity of ν.
Also, Theorem 2.4 and Corollary 2.6 imply that Φξ1+···+ξN ≥ 0, therefore µ ≥ 0.
Acknowledgment. This research was partially supported by the Grant 2-CEx06-
11-34 of the Romanian Government. I am thankful to Marek Bożejko for present-
ing me the basics of c-freeness and bringing to my attention the references [2] and
[6]. I thank also Hari Bercovici for his constant support and his many advices
during the work on this paper.
16 MIHAI POPA
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Mihai Popa: Department of Mathematics, Indiana University at Bloomington,
Rawles Hall, 931 E 3rd St, Bloomington, IN 47405
E-mail address: mipopa@indiana.edu
