FreeBoolean independence with amalgamation
Abstract.
In this paper, we develop the notion of freeBoolean independence in an amalgamation setting. We construct freeBoolean cumulants and show that the vanishing of mixed freeBoolean cumulants is equivalent to our freeBoolean independence with amalgamation. We also provide a characterization of freeBoolean independence by conditions in terms of mixed moments. In addition, we study freeBoolean independence over a algebra and prove a positivity property.
1. Introduction
Free probability theory is a probability theory that studies noncommutative random variables with highest noncommutativity. This theory, due to Voiculescu, is based on the notion of free independence which is an analogue of the classical independence. In [18], Voiculescu generalized his notion of free independence to free independence with amalgamation over an arbitrary algebra in details. To be specific, moments of random variables are no longer scalar numbers but elements from a given algebra. On the other aspect, Voiculescu started to study pairs of random variables simultaneously thereby generalized the notion of free independence to a notion of bifree independence [19]. Further more, the notion of bifree independence with amalgamation, defined by Voiculescu [19], was fully developed in [2]. there are exactly two unital universal independence relations, namely Voiculescu’s free independence relation, the classical independence relation [15]. It was mentioned that we would obtain more independence relations by decreasing the number of axioms for universal products [8]. For instance, people introduced Boolean independence [17], monotone independence [10], conditionally independence [1] in various contexts. Their operatorvalued generalization were studied as well [9, 12, 13]. Recently, their corresponding independence relations for pairs of random variables, analog of Voiculescu’s bifree theory, were introduced and studied [4, 3, 6]. Furthermore, the conditionally bifree independence with amalgamation is studied in [5].
In [8], the firstnamed author introduced a notion of mixed independence relations for pairs of random variables, where random variables in different faces exhibit different kinds of noncommutative independence. In particular, the combinatorics of freeBoolean independence relation was fully developed. In this paper, we generalize the notion of freeBoolean independence to an amalgamation setting. Relevant combinatorial tools are extended to study this new independence. Beyond the corresponding combinatorial results, we address the positivity of freeBoolean independence with amalgamation. Therefore, it is possible to study the relation in topological probability spaces but not only algebraic probability spaces. For instance, we can study our freeBoolean independence with amalgamation over a algebra, which is a suitable framework to address some probabilistic questions. We plan to study probabilistic results such as operatorvalued infinitely divisible laws in a forthcoming paper.
The paper is organized as follows: Besides this introduction, in Section 2, we give the definition of freeBoolean independence with amalgamation over an algebra. In Section 3, we review some relevant combinatorial tools. In Section 4, we demonstrate that freeBoolean independence can be characterized by the property of the vanishing of mixed freeBoolean cumulants. In Section 5, we prove an operatorvalued version of freeBoolean central limit law. In Section 6, we provide an equivalent characterization of freeBoolean independence by certain momentsconditions. In Section 7, we study the positivity property for freeBoolean independence relation.
2. Preliminaries and Notations
In this section, we give the motivation and the definition for freeBoolean independence relation with amalgamation over an algebra.
Definition 2.1.
A bimodule with a specified projection is a triple , where is a direct sum of bimodules , and is the projection
Denote by the algebra of linear operators with respect to the bimodule structure. The expectation from onto is the linear map defined by
We now recall the definition of the reduced free product of bimodules with specified projections [16, 18]. Let be a family of bimodules with specified projections. The reduce free product of with amalgamation over is defined to be the bimodule with a specified projection , where and is the bimodule defined by
For each , we denote by
and let be the natural isomorphism of bimodules
For each , is a unital homomorphism defined by
and is a linear map defined by
where is the natural projection onto and vanishes on the other direct summands.
Proposition 2.2.
For any , we have .
Proof.
Notice that the reduced free product of bimodules with specified projections can be decomposed as
The space is invariant under for any . We can check directly that the space is also invariant under for any by the definition of . Hence the result follows. ∎
The preceding result implies the next corollary.
Corollary 2.3.
The map is a homomorphism.
Definition 2.4.
A valued probability space is a pair consisting of an algebra over and an bimodule map , i.e. a linear map such that
for all and .
Definition 2.5.
Let be a valued probability space. A family of faces of is a family of (not necessarily unital) subalgebras of such that are bimodules for each . The family of faces is said to be freeBoolean with amalgamation over if

are unital algebras

there are bimodules with specified projections such that there are unital homomorphisms , (not necessarily unital) homomorphisms ,

Let be the reduce free product of , so that the joint distribution of the family in is equal to the joint distribution of the family of operators in the probability space . That is,
where
3. Intervalnoncrossing partitions
In this section, we review some combinatorial tools which will be used
to define operatorvalued freeBoolean cumulants. We give a characterization of freeBoolean independence with amalgamation thereby generalizes results in [8] to the operatorvalued framework. In noncommutative probability theory, noncrossing partitions are used in the combinatorics of free probability and the interval partitions are used in the combinatorics of Boolean independence. It turns out the partitions used in the combinatorics of freeBoolean independence are socalled intervalnoncrossing partitions introduced in [8]. All results without proof in this section are taken from [8].
3.1. Intervalnoncrossing partitions
Throughout this section, we let , and , for some fixed index set . We will denote by the set for .
Definition 3.1.
Let be a linearly ordered set. A partition of the set consists of a collection disjoint, nonempty sets whose union is . The sets are called the blocks of . Given , we write if the two elements are in the same block.

A partition is called noncrossing if there is no quadruple such that , , and , where are two disjoint blocks of . The set of all noncrossing partitions of will be denoted by .

A block of is called interval if for any and , we have . A partition is called an interval partition if all blocks are interval blocks.

A block of a partition is said to be inner if there is another block and such that for all . A block is outer if it is not inner.

Let . We denote by the partition whose blocks are the sets .

Given two partitions and , we say if each block of is contained in a block of . This relation is called the reversed refinement order.

We denote by the partition of consists of blocks and by the partition of consists of exactly one block.
Definition 3.2.
Let . A partition of is said to be intervalnoncrossing with respect to if is noncrossing, and are in the same block whenever , and . We denote by the set of all intervalnoncrossing partitions of the set with respect to .
Remark 3.3.
The set does not depend on the value of at and . In particular, when , we have .
For example, let such that . Given two noncrossing and
of the set , then and . In pictures below, we use ""to denote elements in
and ""to denote elements in
.
Assume now
and set . We denote by the
interval .
For each , we denote by the restriction of to the interval
.
We also denote by the
restriction of to the interval
and the restriction of
to the interval . Note that each can be any noncrossing partition of the set ,
since there is no such that .
Proposition 3.4.
Let be defined by
and be defined by
Then and are isomorphisms of partial ordered sets. The set is a lattice with respect to the reverse refinement order on partitions.
We provide pictures below to illustrate the preceding proposition. Let , and which is an intervalnoncrossing of the set with respect to as shown in the following diagram.
In the above diagram, ,
, . Therefore,
, , , and are illustrated
in the following diagrams:
Proposition 3.5.
Let and let such that , i.e. each block of is contained in a block of . Then if and only if for all .
Proposition 3.6.
Let . Denote by the set of all such that . Then
3.2. Möbius functions on intervalnoncrossing partitions
One can define the convolution for functions on the lattice following the standard procedure for partially ordered sets (see [14]). Once the map is fixed, the lattice structure of caputred from the lattice of the product of noncrossing partitions according to the natural isomorphism described in Proposition 3.4.
Let . Given two complexvalued functions defined on the set . The convolution of and is given by
The delta function defined as follows:
We then define the zeta function by
and the Möbius function is the inverse of the zeta function in the following sense:
We will use the following product formula in [7, Section 6].
Proposition 3.7.
Let , and such that . Suppose that and , then
where , , and is the restriction of to the set .
Corollary 3.8.
Let and . Denote by , and . Then,
In particular, we have
for and .
4. Vanishing cumulants condition for freeBoolean independence
In this section, we introduce the notion of operatorvalued freeBoolean cumulants for pairs of random variables and give an alternative characterization of freeBoolean independence by using the freeBoolean cumulants.
4.1. FreeBoolean cumulants
Let be a valued probability space . Let be the Blinear map from to defined as
Then, for each noncrossing partition , we can write , where is an interval block of and . We define an Blinear map recursively as follows:
(1) 
For example, let be a noncrossing partition of . Then,
Definition 4.1.
Given any , and a tuple of elements in , the freeBoolean cumulant is an linear map defined as follows:
We start to show that the operatorvalued freeBoolean cumulants have the following multiplicative property.
Theorem 4.2.
Let and be noncommutative random variables in a valued probability space . Suppose that is an interval block of , then
where
Proof.
The preceding theorem shows that is completely determined by cumulant functionals of the form for and .
Definition 4.3.
Let be a family of pairs of faces of in a valued probability space . We say that the family is combinatorially freeBoolean independent with amalgamation over if
whenever , , and is not a constant.
Proposition 4.4.
Let be a family of pairs of faces in a valued probability space . Then has the following additivity property:
whenever , , , and .
Proof.
By a direct calculation, we have
Since are combinatorially freeBoolean independent, by Definition 4.3, we have
if for some . The result follows. ∎
Proposition 4.5.
Let be a combinatorially freeBoolean independent family of pairs of faces in a valued probability space . Assume that . Then
whenever , , and is not a constant on a block of .
Proof.
We prove the statement by induction on the number of blocks of .
4.2. FreeBoolean independence is equivalent to combinatorially freeBoolean independence
In this subsection, we will prove that freeBoolean independence defined in Definition 2.5 is equivalent to the combinatorially freeBoolean independence given in Definition 4.3. We will show that mixed moments are uniquely determined by lower order mixed moments in the same way for both freeBoolean independence and combinatorially freeBoolean independence.
The proof for following result is essentially the same as the proof of in [11, Proposition 10.6] in free probability context and we thus leave the details to the reader. Applying Theorem 4.2, we have the following result.
Lemma 4.6.
Let and be noncommutative random variables in a valued probability space . Then
For combinatorially freeBoolean independent random variables, we have the following result.
Lemma 4.7.
Let be a family of combinatorially freeBoolean independent pairs of faces in a valued probability space . Assume that , where , . Let . Then,
() 
Proof.
By Lemma 4.6, we have
For each , write its blocks as . Since are combinatorially freeBoolean independent, by Lemma 4.5, we have
if is not a constant on some block of . In other words, only if is a constant on for all , which implies that is contained in a block of for all , i.e., . Therefore, we have
This finishes the proof. ∎
We now turn to consider the case that the family is freeBoolean independent in in the sense of Definition 2.5. In what follows, we assume that , where , . Let , the kernel of . Recall that . Let (or ) be the restriction of (or ) to respectively. Let (or ) be the restriction of (or ) to the interval