Acting Semicircular Elements Induced by Orthogonal Projections on Von-Neumann-Algebras

Abstract: In this paper, we construct a free semicircular family induced by |Z|-many mutually-orthogonal projections, and construct Banach ∗-probability spaces containing the family, called the free filterizations. By acting a free filterization on fixed von Neumann algebras, we construct the corresponding Banach ∗-probability spaces, called affiliated free filterizations. We study free-probabilistic properties on such new structures, determined by both semicircularity and free-distributional data on von Neumann algebras. In particular, we study how the freeness on free filterizations, and embedded freeness conditions on fixed von Neumann algebras affect free-distributional data on affiliated free filterizations.


Introduction
There are different approaches to construct semicircular elements (e.g., [1][2][3]) in topological * -probability spaces (e.g., C * -probability spaces, or W * -probability spaces, or Banach * -probability spaces, etc.).In [4], we introduced how to construct semicircular elements in certain topological * -probability spaces.The construction of [4] is highly motivated by that of weighted-semicircular elements in a Banach * -probability space in the sense of [5,6].In this paper, we put our semicircular elements on a fixed W * -probability space, and then consider structure theorems of such Banach * -probabilistic structures under our actions, and study free-distributional data from the structures.

Motivation and Background
The main purpose of this paper are (i) to construct (weighted-)semicircular elements from orthogonal projections, (ii) to act them to von Neumann algebras, and (iii) to study free-distributional data determined both by these (weighted-)semicircular elements, and free distributions on von Neumann algebras.In particular, the construction of our (weighted-)semicircular elements are highly motivated by the constructions of [5,6].
In [7], the author and Gillespie studied free-probabilistic models of certain embedded sub-structures of Hecke algebras H(G p ) generated by the generalized linear groups G p = GL 2 (Q p ) over p-adic number fields Q p , for fixed primes p.In addition, such a free-probabilistic model is generalized in [8] fully on H(G p ). Motivated by [7,8], independently, the author mimicked the techniques and ideas to construct weighted-semicircular elements and corresponding semicircular elements induced by certain orthogonal projections on Q p in [6].In [5], as an application of the main results of [6], we studied free stochastic calculus for the weighted-semicircular laws in the sense of [6].
Our constructions of weighted-semicircular, and semicircular elements in this paper is understood as a pure operator-theoretic version of those of [5,6].

Overview
Here, we generalize the free probability on free filterizations (which are Banach * -probability spaces generated by the semicircular elements obtained in [4]).By using these free filterizations to arbitrarily fixed von Neumann algebras M, we consider M-affiliated free filterizations, and establish suitable free-probabilistic models on them.
In Section 2, we briefly mention background theories for our proceeding works.
In Section 3, we introduce fundamental free-probabilistic settings from given |Z|-many mutually orthogonal projections.
In Sections 4 and 5, we construct weighted-semicircular, and semicircular elements induced by given orthogonal projections.
In Section 5, from the ingredients of Sections 3, 4 and 5, we construct free filterizations as free product Banach * -probability spaces, and consider fundamental free-distributional data on them.
In Section 7, we act an arbitrarily fixed free filterization to a given von Neumann algebra, and construct the corresponding von-Neumann-algebra-affiliated free filterizations, and study how our semicircular elements work on such structures.
In Section 8, from the free-distributional data obtained in Section 7, we construct-and-study weighted-semicircular, and semicircular elements in affiliated free filterizations.By doing that, one can see how the freeness on our free filterizations affects the free probability on affiliated structures.
In Section 9, by considering (embedded, or full) freeness conditions on given von Neumann algebras, free-distributional data on affiliated free filterizations are studied.We show how the freeness conditions on von Neumann algebras affect the affiliated structures.
In Section 10, an example for the main results of Sections 7, 8 and 9 will be considered.In particular, we are interested in the case where a fixed von Neumann algebra is given to be a free group factor L(F n ) (e.g., [1]) generated by the free group F n with n-generators.

Preliminaries
Readers can check fundamental analytic-and-combinatorial free probability theory from [2,3,9] (and cited papers therein).Free probability is understood as the noncommutative operator-algebraic version of classical probability theory and statistics.The classical independence is replaced by the freeness, by replacing measures on sets to linear functionals on (noncommutative algebraic, or topological * -) algebras.It has various applications not only in pure mathematics (e.g., [1,2]), but also in related topics (e.g., [8] through [7]).In particular, we will use combinatorial free-probabilistic approach of Speicher (e.g., [9]).Free moments andfree cumulantsof operators (representing free-distributional data of operators), or free probability spaces, or free product of algebras will be considered without introducing detailed concepts.
An operator a of A is said to be a free random variable whenever it is regarded as an element of (A, ψ).As usual, we say a is self-adjoint (as an operator in A), if a * = a in A, where a * is the adjoint of a. Definition 1.A self-adjoint free random variable a is said to be weighted-semicircular in (A, ψ) with its weight t 0 ∈ C × = C \ {0}, (or, in short, t 0 -semicircular), if a satisfies the free-cumulant computation, for all n ∈ N, where k ψ n (...) is the free cumulant on A in terms of ψ, under the Möbius inversion of [9].If t 0 = 1 in (3.1), the 1-semicircular element a is simply said to be semicircular in (A, ψ), By definition, a free random variable a is semicircular in (A, ψ), if a satisfies for all n ∈ N.
By the Möbius inversion of [9], one can characterize the weighted-semicircularity (3.1) as follows: a is t 0 -semicircular in (A, ψ), if and only if where for all n ∈ N, where c k are the k-th Catalan numbers, 2) and (3.3), a free random variable a is semicircular in (A, ψ), if and only if for all n ∈ N, where ω n are in the sense of (3.3).Thus, one can use the t 0 -semicircularity (3.1) (respectively, the semicircularity (3.2)), and its characterization (3.3) (respectively, (3.4)) alternatively.
Recall that, if a free random variable x ∈ (A, ψ) is self-adjoint, then the sequences provide equivalent free distributions of x.Indeed, the Möbius inversion (of [9]) satisfies where NC(n) is the lattice of all noncrossing partitions over {1, ..., n}, and "V ∈ π" means "V is a block of π," and where µ is the Möbius functional in the incidence algebra over (see [9]).Now, let A be a given C * -algebra, and let q j ∈ A be a projection in the sense that: for all j ∈ Z, where Z is the set of all integers.Moreover, assume that the projections {q j } j∈Z are mutually orthogonal from each other in the sense that: q i q j = δ i,j q j in A, for all i, j ∈ Z, (3.5) where δ means the Kronecker delta.
Remark 1.Such mutually orthogonal |Z|-many projections {q j } j∈Z can be found in a C * -algebra A, naturally, or artificially.One can find such projections naturally as in [3,4].If {q j } N j=1 is a finite family of mutually orthogonal projections in a certain C * -algebra A 0 , for some N ∈ N, then one can construct a C * -algebra A, under product topology, and then we obtain the mutually orthogonal |Z|-many projections, Similarly, if N = ∞, and {q j } ∞ j=1 forms a family of mutually orthogonal projections in a certain C * -algebra A 0 , then one can construct a C * -algebra A, with a family of mutually orthogonal |Z|-many projections, Therefore, from below, we always assume a given C * -algebra A has a family {q j } j∈Z of mutually orthogonal |Z|-many projections.
Note that we are not interested in operator-algebraic structures or properties of A, but interested in induced weighted-semicircularity or semicircularity from projections in a C * -algebra A. Now, we fix a family {q j } j∈Z of mutually orthogonal projections of a fixed C * -algebra A, and we denote it by Q; Q = {q j : j ∈ Z} in A, (3.6) satisfying (3.5).
In addition, let Q be the C * -subalgebra of A generated by Q of (3.6), Then, it is easy to get the following structure theorem.

Proposition 1.
Let Q be a C * -subalgebra (3.7) of a given C * -algebra A, generated by the family Q of (3.6).Then, Proof.The structure theorem (3.8) is proven by the mutual-orthogonality (3.5) of the generator set Q of (3.6) in A.
Now, assume that we fix a bounded linear functional ψ on the C * -algebra A, creating the corresponding C * -probability space (A, ψ).From this fixed C * -probability space (A, ψ), define now linear functionals ψ j on Q by for all j ∈ Z, where ψ, on the right-hand side of (3.9), is the restricted linear functional of ψ on the C * -subalgebra Q of A. Remark that such linear functionals {ψ j } j∈Z of (3.9) are well-defined on Q by (3.8).Therefore, if q ∈ Q, then q = ∑ j∈Z t j q j (with t j ∈ C), and, hence, ψ j (q) = ψ j t j q j = t j ψ(q j ), by the definition (3.9) of ψ j , for all j ∈ Z.It shows that the system {ψ j } j∈Z of the linear functionals (3.9) filterize, or sectionize Q free-probabilistically.
Definition 2. The C * -probability spaces Q, ψ j are called the j-th C * -probability spaces of Q in (A, ψ), where Q is the C * -subalgebra (3.7) of A, and ψ j are in the sense of (3.9), for all j ∈ Z.
Now, let us define bounded linear transformations c and a "acting on the C * -algebra Q" by c q j = q j+1 , and a q j = q j−1 , (3.10) for all j ∈ Z.Then, c and a are indeed well-defined bounded linear operators "on Q," understood as elements of the operator space B(Q), consisting of all bounded linear transformations on Q (e.g., [10]).Without loss of generality, one can regard c and a of (3.10) as Banach-space operators on a Banach space Q.
Definition 3. We call these Banach-space operators c and a of (3.10), the creation, respectively, the annihilation on Q.
Define now a new Banach-space operator l in the operator space B(Q) by where c and a are the creation, respectively, the annihilation on Q.
Definition 4. We call the Banach-space operator l of (3.11), the radial operator on Q.
By the definition (3.11), one has l ∑ j∈Z t j q j = ∑ j∈Z t j q j+1 + q j−1 .
Now, define a Banach subspace of B(Q), generated by the radial operator l, equipped with the operator norm, where .Q is the C * -norm on Q, where X .mean the operator-norm closures of subsets X in B(Q).By the definition (3.12), it is not difficult to see that this Banach-subspace L forms a Banach algebra inside B(Q).
On the Banach algebra L of (3.12), define a unary operation ( * ) by where z are the conjugates of z in C.
Then, the operation (3.13) is a well-defined adjoint on L, and hence, all elements of L are adjointable (in the sense of [10]) in B(Q).Thus, the Banach algebra L of (3.12) forms a Banach * -algebra.Definition 5. We call the Banach * -algebra L of (3.12), the radial (Banach * -)algebra on Q.Now, let L be the radial algebra on Q. Construct now the tensor product Banach * -algebra, (3.14) Definition 6.We call the tensor product Banach * -algebra L Q of (3.14), the radial projection (Banach * -)algebra on Q.

Weighted-Semicircular Elements Induced by Q
Throughout this section, we fix the settings of Section 3, and construct weighted-semicircular elements induced by the family Q of mutually orthogonal |Z|-many projections in a fixed C * -probability space (A, ψ).Let (Q, ψ j ) be j-th C * -probability space of Q in (A, ψ), where ψ j are the linear functionals (3.9), for all j ∈ Z, and let L Q be the radial projection algebra (3.14) on Q.
Remark that, if u j = l ⊗ q j ∈ L Q , for all j ∈ Z, (4.1) then u n j = l ⊗ q j n = l n ⊗ q j , for all n ∈ N, since q n j = q j , for all n ∈ N, for j ∈ Z.Then, one can construct a linear functional ϕ j on the radial projection algebra L Q by a linear morphism satisfying that for all n ∈ N, for all i, j ∈ Z.Note that such linear functionals ϕ j of (4.2) are well-defined by (3.8) and (3.14).
Now, consider the elements l n (q i ) in Q, for all n ∈ N, i ∈ Z. Observe first that, if c and a are the creation, respectively, the annihilation on Q in the sense of (3.10), then where 1 Q is the identity operator on Q in the operator space B(Q), satisfying 1 Q (q) = q, for all q ∈ Q.
Indeed, for any q j ∈ Q in Q, ca q j = c a q j = c q j−1 = q j−1+1 = q j , and ac q j = a c q j = a q j+1 = q j+1−1 = q j , for all j ∈ Z.By (4.4), one can get that and Furthermore, since the radial algebra L, which is a tensor-factor of L Q , is generated by a single generator l, one has in L, for all n ∈ N, by (4.4) and (4.4) , where Note that, for any n ∈ N, by (4.5).Therefore, the formula (4.6) does not contain 1 Q -terms by (4.4) .Note also that, for any n ∈ N, one has by (4.5).
Theorem 1. Fix j ∈ Z, and let u j = l ⊗ q j be the corresponding generating operator of the j-th probability space (L Q , ϕ j ).Then, where ω n are in the sense of (3.3), and c n 2 are the ( n 2 )-th Catalan numbers, for all n ∈ N.
Motivated by the free-distributional data (4.12) of the generating operator u j = l ⊗ q j of the radial projection algebra L Q of (3.14), we define the following morphism ] means the minimal integer greater than or equal to n 2 , for example, The linear transformations E j,Q of (4.13) are well-defined linear transformations on L Q because of the construction (3.14) of L Q = L ⊗ C Q, and by the structure theorem (3.8) of the radial algebra L.
Define now a new linear functional τ j on L Q by where ϕ j are in the sense of (4.2).By the linearity of ϕ j and E j,Q , the above morphisms τ j are indeed well-defined linear functionals on L Q , for all j ∈ Z. Definition 8.The well-defined Banach * -probability spaces are called the j-th filtered (Banach- * -)probability spaces of the radial projection algebra L Q on Q, for all j ∈ Z.
On the j-th filtered probability space L Q (j) of ( 4.15), one can obtain that i.e., we can get that 16) for all n ∈ N, for j ∈ Z, by (4.12).Theorem 2. Let L Q (j) = (L Q , τ j ) be the j-th filtered probability space of the radial projection algebra L Q on Q, for an arbitrarily fixed j ∈ Z.Then, for all n ∈ N, for all i ∈ Z, where ω n are in the sense of (3.3).
Proof.If i = j in Z, then the free momental data (4.17)holds true by (4.16), for all n ∈ N.
If i = j in Z, then, by the very definition (4.13) of the j-th filterization E j,Q , and also by the definition (4.2) of ϕ j , τ j u n i = 0, for all n ∈ N.
Therefore, the above formula (4.17) holds, for all i ∈ Z.
The following corollary is a direct consequence of the above free distribution (4.17).
Corollary 1.Let L Q (j) be the j-th filtered probability space of L Q , for a fixed j ∈ Z, and let u j = l ⊗ q j be the j-th generating operator of L Q .Then, u j is ψ(q j ) 2 -semicircular in L Q (j).
Proof.First, remark that the j-th generating operator The ψ(q j ) 2 -semicircularity of u j is proven by the above self-adjointness, the free-moment computation (4.17), and the weighted-semicircularity characterization (3.3).
Readers can check that the j-th generating operator u j satisfies the free-cumulant formula for all n ∈ N, by the Möbius inversion of [9], where k j n (...) is the free cumulant on L Q in terms of τ j , for all j ∈ Z.Thus, by the definition (3.1), the free random variables u j are ψ(q j ) 2 -semicircular in the j-th filtered probability spaces L Q (j) = L Q , τ j , for all j ∈ Z.
Remark that, the k-th generating operators u k of the j-th filtered probability space L Q (j) have zero-free distributions, whenever k = j in Z, also, by (4.17).Therefore, in summary, we have the following theorem.Theorem 3. Let u k = l ⊗ q k be the generating operators of the j-th filtered probability space L Q (j), for all k ∈ Z, for a fixed j ∈ Z.Then, the j-th generating operator u j is ψ(q j ) 2 -semicircular in L Q (j), (4.19) the k-th generating operators u k have zero-free distributions, for all k = j in Z. (4.20) Proof.The proof of the statement (4.19) is done by (4.17) and (4.18).The statement (4.20) is also shown by (4.17).Indeed, if k = j in Z, then τ j u n k = 0, for all n ∈ N, by (4.17).Thus, the free distributions of these self-adjoint operators u k of L Q (j), where k = j in Z, are characterized by the following free-moment sequences: ∞ n=1 = (0, 0, 0, 0, ...) .Therefore, the free distributions of u k are the zero-free distribution in L Q (j), whenever k = j in Z.
The above two statements (4.19) and (4.20) fully characterize the free distributions of all generating operators u k of the j-th filtered probability spaces L Q (j), for all k, j ∈ Z.

Semicircular Elements Induced by Q
As in Section 4, we keep working on the j-th filtered probability spaces, The main results of Section 4 show that, for a fixed j ∈ Z, the j-th generating operator equivalently, for all n ∈ N.
Recall now that we assumed for convenience that Under our assumption, the generating operators u k of the projection-radial algebra 2) for all k, j ∈ Z. ψ(q k ) u k be free random variables (5.2) of the j-th filtered probability space L Q (j), for all k ∈ Z, for a fixed j ∈ Z. (5. 3) The operators U k have zero-free distributions in L Q (j), whenever k = j in Z. (5.4) Proof.Note first that the k-th generating operators u k have zero-free distributions in the j-th filtered probability space L Q (j), whenever k = j in Z, by (4.20).Since the corresponding operators U k of (5.2) are the scalar-multiplies of u k , if k = j in Z, then the operators U k also have zero-free distributions in L Q (j).It shows that the statement (5.4) holds.Assume now that U j is in the sense of (5.2) in L Q (j), for j ∈ Z, and suppose ψ(q j ) ∈ R × in C. Since ψ(q j ) ∈ R × , the corresponding operator U j is not only well-defined in L Q , but also self-adjoint in L Q (j) by the self-adjointness of u j .Therefore, this operator Under self-adjointness of U j , observe that by the ψ(q j ) 2 -semicircularity (5.1) of u j , for all n ∈ N. Therefore, by the semicircularity characterization (3.4), this operator U j is semicircular in L Q (j), whenever ψ(q j ) ∈ R × .Therefore, the statement (5.3) holds.
The above theorem shows that the operators U j of (5.2), generated by our ψ(q j ) 2 -semicircular elements u j , are semicircular in the j-th filtered probability spaces L Q (j), for all j ∈ Z, whenever ψ(q j ) ∈ R × .Assumption 5.1 (in short, A 5.1, from below) If there is no confusion, then we automatically assume ψ(q j ) ∈ R × in C, for all j ∈ Z, for all q j ∈ Q.
The above assumption, A 5.1, will guarantee that, if we have the ψ(q j ) 2 -semicircular elements u j in the j-th filtered probability space L Q (j), we also have the corresponding semicircular element U j = 1 ψ(q j ) u j in L Q (j) for all j ∈ Z.
6.The Free Product Banach * -Probability Space j∈Z L Q (j) A family {a n } n∈Λ in an arbitrary (topological or pure-algebraic) free probability space (B, ϕ) is said to be a free family, if all elements a n of the family are mutually free from each other in (B, ϕ), where Λ is a countable (finite or infinite) index set.For such a free family {a n } n∈Λ , if every element a n is weighted-semicircular (or semicircular), then we call the free family, free weighted-semicircular (respectively, semicircular) family in (B, ϕ).
Recall that, for a fixed C * -probability space (A, ψ), if there exists a mutually-orthogonal projections {q j } j∈Z , then one can construct ψ(q j ) 2 -semicircular elements u j = l ⊗ q j in the j-th filtered probability spaces L Q (j) = (L Q , τ j ), for all j ∈ Z, with where ω n are in the sense of (3.3), equivalently, for all n ∈ N, for all j ∈ Z, by (4.16) and (4.17).Moreover, one can construct corresponding semicircular elements for all j ∈ Z, by (5.3), under A 5.1.Now, we will construct the free product Banach * -probability space (L Q (Z), τ), by satisfying where ( ) means free product (over C) in the sense of [3,9].Note that the free product of [3,9] is different from a pure-algebraic free product.It is totally depending on given linear functionals.Definition 9.The free product Banach * -probability space (L Q (Z), τ) of (6.3) is called the free filterization of Q. Sometimes, the Banach * -algebra L Q (Z) is also said to be the free filterization of Q.
By the very construction (6.3) of the free filterization (L Q (Z), τ) of Q, we obtain the following proposition immediately.Proposition 3. Let (L Q (Z), τ) be the free filterization (6.3), and let u j and U j be in the sense of (6.1) and (6.2), respectively, for all j ∈ Z.
The family {u j ∈ L Q (j)} j∈Z is a free weighted-semicircular family in L Q (Z). (6.4) The family {U j ∈ L Q (j)} j∈Z is a free semicircular family in L Q (Z), under A 5.1.(6.5) Proof.By the very definition (6.3) of free filterizations, u j s are free from each other in (L Q (Z), τ), for all j ∈ Z. Indeed, each u j is taken from the free block L Q (j) of L Q (Z), τ .Therefore, the family for all n ∈ N, for all j ∈ Z.Thus, this family is a free weighted-semicircular family in L Q (Z).Therefore, statement (6.4) holds.
Similarly, one can conclude the family {U j ∈ L Q (j)} j∈Z is a free semicircular family in L Q (Z), showing that the statement (6.5) holds.

Weighted-Semicircularity on Affiliated Free Filterizations
Let (A, ψ) be a fixed C * -probability space, and let Q = {q j } j∈Z be a family in A, consisting of all mutually orthogonal projections.Let Q be the C * -subalgebra C * (Q) of A generated by Q, and L Q , the corresponding radial projection algebra on Q, inducing the corresponding j-th filtered probability spaces L Q (j) = L Q , τ j , for all j ∈ Z. Remember that, by A 5.1, be the free filterization (6.3), and let and be the free weighted-semicircular family (6.4), respectively, the free semicircular family (6.5).Now, we fix j ∈ Z, and focus on the free block L Q (j) = L Q , τ j of the free filterization L Q (Z).In addition, consider the compressed C * -subalgebra A j , A j de f = q j Aq j , for all j ∈ Z, be C * -subalgebras of A. Remark 3.There are no typical reasons why we take a unital tracial W * -probability space (M, tr).One may / can regard (M, tr) as a unital tracial C * -probability space.However, on the von Neumann algebra M, trace-depending operator theory, and operator algebra theory work well, and have been widely studied (as in I I 1 , I I ∞ , I I I λ -factor theories, etc.), and such structures have lots of interesting applications not only in operator theory but also in related science fields like quantum physics (under W * -topological settings).
One of the possible reasons would be from the main results of [1].We want to mimic the constructions, and apply the main results of [1] here, as applications of our results in Sections 4-7.In addition, we want to allow a variety of topological settings in our Banach * -probability structures, as generalizations of the results in previous sections.Now, for our j-th filtered probability space L Q (j) = (L Q , τ j ), a free block of the free filterization L Q (Z), for j ∈ Z, construct the tensor product Banach * -algebra, and define a linear functional τ M j on L M Q , by a linear morphism, for all n ∈ N 0 , for all m ∈ (M, tr), for all generators u j = l ⊗ q j of the radial projection algebra L Q .
Then, one has well-defined Banach * -probability spaces L M Q , τ M j , for all j ∈ Z.
Definition 10.The Banach * -algebra L M Q of (7.3) is called the M(-affiliated)-radial projection algebra.The Banach * -probability spaces L M Q , τ M j of the M-radial projection algebra L M Q , and the linear functionals τ M j of (7.4) are said to be the j-th M(-affiliated)-filtered probability spaces, for all j ∈ Z.For convenience, we denote our j-th M-filtered probability spaces , for all j ∈ Z.
) be our j-th M-filtered probability space (7.5), for all j ∈ Z. Construct the free-product Banach * -probability space Definition 11.The free-product Banach * -probability space It is not difficult to check that the elements , where 1 M is the identity element of M, and u j = l ⊗ q j ∈ L Q (j), for all j ∈ Z.Then, u o j are ψ(q j ) 2 -semicircular in the M-free filterization L M Q (Z). (7.7) , where U j = 1 ψ(q j ) u j ∈ L Q (j), under A 5.1, for all j ∈ Z.Then, by (7.5) and (7.6), for all n ∈ N. Therefore, by (7.9), (3.1) and (3.3), the free random variables u o j are ψ(q j ) 2 -semicircular in the M-free filterization L M Q (Z), for all j ∈ Z, i.e., the statement (7.7) holds.Similarly, since U o j is in the free block for all n ∈ N. Therefore, by (7.10), (3.2) and (3.4), the free random variables U o j are semicircular in L M Q (Z).
Corollary 2. Let L M Q (Z) be M-free filterization (7.6) of Q and (M, tr), and suppose L M Q (j) are the free blocks (7.5) of L M Q (Z), for all j ∈ Z.
The family The family Proof.By (7.7) and (7.8), the operators are ψ(q j ) 2 -semicircular in the M-free filterization L M Q (Z), and the operators are semicircular in L Q (Z), respectively.Moreover, since all elements u o j (or U o j ) are contained in the mutually-distinct free blocks L M Q (j) of L M Q (Z), for all j ∈ Z, the free random variables u o j (resp., U o j ) are mutually free from each other in L Q (Z).Therefore, the statements (7.11) and (7.12) hold.Now, we take and j∈Z in the M-free filterization L M Q (Z) of Q and (M, tr).By (7.11) and (7.12), the family Q (resp., X ) of (7.13) is a free weighted-semicircular (Respectively, semicircular) family in L M Q (Z).From the free families Q and X of (7.13), let us construct families, and where m j are elements of M, satisfying m j = 0 M , and m j = 1 M , where 0 M means the zero element of M, for all j ∈ Z.
Note that, by (7.6), the families (7.14) are free families in L M Q (Z).Now, consider certain type of free families (7.14).
Theorem 5. Let m ∈ (M, tr) be nonzero, and assume that (i) m is self-adjoint, and (ii) there exists t 0 ∈ C × , such that tr(m n ) = t n 0 , for all n ∈ N.
The family {m ⊗ U j ∈ L M Q (j)} j∈Z in the sense of (7.14) is a free t 2 0 -semicircular family in L M Q (Z).
(7.15)The family {m ⊗ u j ∈ L M Q (j)} j∈Z in the sense of (7.14) is a free weighted-semicircular family in L M Q (Z).In particular, each element m ⊗ u o j is t 0 ψ(q j ) 2 -semicircular in L M Q (Z), for all j ∈ Z. where Q and X are in the sense of (7.13), where m ∈ (M, tr) is given as above.
First of all, by the self-adjointness of x ∈ Q ∪ X , since m is assumed to be self-adjoint in M, all elements m ⊗ u j , m ⊗ U j ∈ mX ∪ mQ are self-adjoint in L M Q (Z).All elements m ⊗ U o j ∈ mX are contained in the mutually-distinct free blocks, for all j ∈ Z, these operators m ⊗ U j are mutually free from each other in L M Q (Z), for all j ∈ Z. Observe now that for all n ∈ N, by the assumption that tr(m n ) = t n 0 , for all n ∈ N, for some t 0 ∈ C × .It shows that the self-adjoint free random variables m ⊗ U j ∈ mX are t 2 0 -semicircular in the M-free filterization L M Q (Z), by (3.3).Therefore, the family mX is a free t 2 0 -semicircular family in L M Q (Z).
Similarly, since , for all j ∈ Z, the family mQ is a free family in L M Q (Z) because mX is.By A 5.1, and by the condition m is self-adjoint, all entries m ⊗ u j of mQ are self-adjoint in L Q (Z).
In addition, one has that = ω n t 0 ψ(q j ) n c n 2 , for all n ∈ N.
Therefore, by (3.4), each entry m ⊗ u j of the family is t 0 ψ(q j ) 2 -semicircular in L M Q (Z), for all j ∈ Z, and, hence, the family mQ is a free weighted-semicircular family in the M-free filterization L M Q (Z).
Let's denote the families for m j ∈ (M, tr), for all j ∈ Z.
Under A 5.1, every semicircular element U j (which is a tensor-factor of m j ⊗ U j ∈ X M ) is well-defined as the scalar-product 1 ψ(q j ) u j of the ψ(q j ) 2 -semicircular element u j in the free block , for all j ∈ Z.Thus, one can understand u j as ψ(q j )U j , and, hence, , for all j ∈ Z.
It means that the family Q M (or X M ) is generated by the family X M (resp., Q M ).Therefore, in the following, we concentrate on studying properties of the operators of L M Q (Z) induced by X M (covering the properties of those induced by Q M in the above senses).

Free Distributions on Affiliated Free Filterizations
In this section, we fix a M-free filterization L M Q (Z) = (L M Q (Z), τ M ) in the sense of (7.6) for a fixed unital tracial W * -probability space (M, tr), and study free-distributional data of certain free random variables of } j∈Z be a free semicircular family of (7.12) in L M Q (Z), where L M Q (j) are the free blocks of L M Q (Z), for all j ∈ Z. Now, we construct free random variables with m j ∈ (M, tr), and k j ∈ N, where the summands of (8.1) satisfy where 1 M ⊗ U j ∈ X , for all j ∈ Z.
For an operator T of (8.1), define the support of T, denoted by Supp(T), by Therefore the formula (8.3) holds.
Observe first that if T is in the sense of (8.1) in the M-free filterization L M Q (Z), then each summand m j ⊗ U k j j , such that m j = 0 M , equivalently, with j ∈ Supp(T), are contained in a free block , for all j ∈ Supp(T), in L M Q (Z).Therefore, one can conclude the following result.Proposition 6.Let T be in the sense of (8.1) induced by a fixed W * -probability space (M, tr), and the free semicircular family X of (7.12) in L M Q (Z).Then, all nonzero summands m j ⊗ U n j j of T are free from each other in the M-free filterization L M Q (Z), for all j ∈ Supp(T).Equivalently, this operator T is a free sum in L M Q (Z).
Proof.The proof is straightforward from the very construction (8.1) of the operator T in L M Q (Z), as we discussed in the very above paragraph.Now, we concentrate on studying the free distribution of a free sums T of (8.1).Consider first that where for all n ∈ N, where τ is the trace of (6.3) on the free filterization L Q (Z) of Q.
The above formula (8.5) shows that computing free moments of the free sum T of (8.1) is reduced to compute the joint free moments of semicircular elements } j∈Z be the free semicircular family (6.5) in the free filterization L Q (Z) of Q, and let where j 1 , ..., j n ∈ Z, for n ∈ N.
If j 1 = ... = j n = j in Z, then the operator U of (8.6) satisfies that If the sequence (j 1 , ..., j n ) is alternating in Z, in the sense that Proof.Let U be an operator (8.6) in the free filterization by the semicircularity (6.5) of U j in L Q (Z).
Similarly, by the self-adjointness of U j , one can get that and, hence, with N = ∑ n l=1 k l as above in N. Thus, the statement (8.7) holds.
Assume now that the sequence (j 1 , ..., j n ) is alternating in Z.Then, by the freeness (6.5) of the family X Q in L Q (Z), one obtains that by the semicircularity on X Q in L Q (Z).In addition, one can get that Therefore, the statement (8.8) holds.
The above results (8.7) and (8.8) in fact characterize the free distributions of the product operators of L Q (Z) in X Q because of the freeness on the free semicircular family X Q .Indeed, every product T in X Q has its unique form, where (j 1 , ..., j n ) is alternating in Z.The resulted unique forms under product are said to be the free reduced words of , where L Q (j) are the j-th filtered probability spaces, the free blocks of L Q (Z).In other words, this product operator X in the free family X Q is the free reduced word . Therefore, indeed, the above lemma characterizes the full free-distributional data obtained from the free semicircular family X Q of (6.5) in L Q (Z).
For example, in the very above product operator X, one can induce the corresponding integer-sequence, (−1, −1, 0, −1, 4, 4, 4), with its alternating partition, ((−1, − 1), (0), (−1), ( It is trivial that if an integer-sequence (j 1 , ..., j n ) is alternating in Z, then its alternating partition is ((j 1 ), (j 2 ), (j 3 ), ..., (j n )) .Now, let W = (j 1 , ..., j n ) be a finite integer sequence regarded as its unique alternating partition, where [j l 1 ], ..., [j l N ] are the blocks of the alternating partition of W, with N ≤ n in N, satisfying [j l s ] s = (j l s , j l s , ..., j l s ) in W, for all s = 1, ..., N, with We say that the cardinality N of blocks in W the (alternating-)partition size of W. One can define the following quantities |[j l s ]| for a fixed size-N alternating partition of the sequence W |[j l s ] s | = the cardinality of [j l s ] s in W for all l = 1, ..., N. We call these quantities |[j l s ] s | the block-sizes of W, for all s = 1, ..., N.
For example, if the product operator X is a free reduced word, is as above inducing the size-4 alternating partition of its integer-sequence, We can realize that the block-sizes are identical to the powers of free-factors of X.
Example 1.Let X Q = {U j ∈ L Q (j)} j∈Z be the free semicircular family (6.5) in the free filterization L Q (Z) of Q, and let be a product operator of L Q (Z) in X Q .Then, this operator X is identical to the free reduced word inducing the size-4 alternating partition of the corresponding integer-sequence, and Based on the above new concepts we discussed, let's refine the computations (8.7) and (8.8).
Lemma 3. Let U = n Π l=1 U j l be a product operator of L Q (Z) in the free semicircular family X Q of (6.5), for n ∈ N. Assume that U induces the size-N alternating-partition ([j l 1 ] 1 , ..., [j l N ] N ) of its integer-partition (j 1 , ..., j n ), with the block-sizes for some N ≤ n in N.Then, this product U is the free reduced word, satisfying Proof.Let U be given as above in the free filterization L Q (Z) of Q.Then, by the very above discussion, this product operator U in X Q is the free reduced word where N s are the block-sizes of the size-N alternating partition of (j 1 , ..., j n ) in Z.Thus, the product U is identified with the free reduced word of (8.9) in L Q (Z).
By the above three lemmas, we obtain the following free-distributional data of the free sum T in the sense of (8.1) in the M-free filterization L M Q (Z).
The following corollary is a direct consequence of the above theorem.satisfy (8.12), for all (j 1 , ..., j n ) ∈ Supp(T) n , for all n ∈ N.
by the self-adjointness of the semicircular elements U j in the free filterization L Q (Z) (under A 5.1), for all j ∈ Z.Therefore, similar to (8.11) and (8.12), the formula (8.13) holds.
The following result is immediately obtained by (8.12) and (8.13).

Corollary 4.
Let T be the free sum (8.1) in L M Q (Z).Assume that m j ∈ (M, tr) are self-adjoint in M, for all j ∈ Supp(T) in Z.Then, the free distribution of T is characterized by the free-moment sequence, τ M (T n ) ∞ n=1 , whose entries are determined by (8.12).
Proof.Under hypothesis, the free sum T is self-adjoint in L M Q (Z) in the sense that T * = T. Thus, by (8.12) and (8.13), this corollary is proven.Now, we generalize the free-distributional data (8.12) and (8.13).Let Z N = {(j k ) N k=1 : j k ∈ Z}, for all N ∈ N. Define a subset Alt Z N of Z N by for all N ∈ N.
and, hence, in the free block As we considered above, if W ∈ Z N , then there exists a unique W ∈ Alt Z N , such that as a free reduced word.It shows that, without loss of generality, one can reduce his interests in alternating sequences (instead of all sequences in Z N ) in Alt(Z N ), for N ∈ N.Moreover, if the fixed alternating N-tuple W = (j 1 , ..., j N ) satisfies Thus, the free-distributional data (8.17) are obtained.Assume now that a fixed alternating N-tuple W = (j 1 , ..., j N ) satisfies  for all n ∈ N, by (8.17).Therefore, the free-probabilistic information (8.18) is obtained.
As we have seen above, our main results of Section 8, the free-distributional data induced by the free semicircular family X , are affected by the freeness (6.3) on the free filterization L Q (Z) of Q in the M-free filterization L M Q (Z).In Section 9, let us consider freeness conditions and corresponding free-distributional information on L M Q (Z) affected by the freeness on (M, tr).

Certain Freeness Conditions on L
In this section, we consider freeness conditions on our M-free filterization affected by the freeness on a fixed unital tracial W * -probability space (M, tr).
Since L Q (Z) is defined to be the free product of j-th filtered probability spaces {L Q (j)} j∈Z , the freeness (6.3) on the free filterization L Q (Z) of Q affects the free-distributional information on L M Q (Z) canonically (see Section 8; e.g., (8.12), (8.13), (8.17) and (8.18)), and it affects the freeness on L M Q (Z) (see Section 7; e.g., (7.11), (7.12), (7.15) and (7.16)).Therefore, it is natural to ask how the freeness on the other tensor-factor (M, tr) affects the freeness on the M-free filterization L M Q (Z).Assume that a fixed W * -probability space (M, tr) satisfies Then, the free blocks where L Q (j) are the j-th filtered probability spaces, the free blocks of the free filterization because, as a Banach * -algebra, L Q (j) = L Q , the projection radial algebra (for each j ∈ Z), where L is the radial algebra (3.12), by (3.14) * -iso .
in the sense of (3.12) * -iso in the sense of (7.4), for all l = 1, 2, for all j ∈ Z.By (9.2) and (9.3), we obtain the following structure theorems.
Theorem 8. Let j ∈ Z be arbitrarily fixed, and let L M Q (j) be the corresponding free block of the M-free filterization L M Q (Z) of Q. Assume that a fixed unital tracial W * -probability space (M, tr) satisfies the freeness (9.1).Then, where Proof.The structure theorem (9.4) is proven by (9.2), where Q in the sense of (7.3) equipped with their linear functionals τ M l j of (9.3) as in (7.4).By (9.4), one can get the following corollary immediately.
Corollary 5. Suppose a given W * -probability space (M, tr) is the free product W * -probability space, (M, tr) = s∈Λ (M s , tr s ), (9.5) for an countable (finite or infinite) index set Λ (under suitable product topology).Then, the free blocks (e.g., [1,11]).For example, the free group factor L(F n ) is a group von Neumann algebra generated by F n , as a W * -subalgebra of the operator algebra B l 2 (F n ) , where l 2 (F n ) is the l 2 -Hilbert space generated by the group F n , satisfying the following factor-ness.A von Neumann algebra M (contained in the operator algebra B(H) of all operators on a Hilbert space H) is a factor, if Recall that every von Neumann algebra M is decomposed by factors of different types.For more about von Neumann algebras and factors, see [11].Note also that the free group factors L(F n ) are indeed well-determined factors (e.g., [1] because F n is an i.c.c.group), for all n ∈ N >1 ∞ .By construction, all elements m of L(F n ) are expressed by m = ∑ g∈F n t g g, with t g ∈ C, in L(F n ) (as finite sums or infinite sums under limit), with their adjoint, where g * is the adjoint of g (as an operator in L(F n )), and g −1 is the group-inverse of g (as a group-element of F n ).
The free group factors L(F n ) are equipped with their canonical traces tr n on them, defined by tr n ∑ g∈F n t g g de f = t e n , (10.1) where e n are the group-identities of F n , for all n ∈ N >1 ∞ .i.e., if m ∈ L(F n ), as a (possibly infinite) linear combination in F n , then tr n (m) is regarded as the process taking the coefficient t e n of m, for the group identity e n of F n .
Therefore, every free group factor L(F n ) is automatically understood as a W * -probability space (L(F n ), tr n ), where tr n is the canonical trace (10.1) on L(F n ), for all n ∈ N >1 ∞ .From below, if we write L(F n ), then it means either the free group factor, or the corresponding W * -probability space (L(F n ), tr n ).
It is not hard to check that L(F n ) forms a unital tracial W * -probability spaces, for all n ∈ N >1 ∞ .Thus, under our settings, one can establish the corresponding L(F n )-free filterization L L(F n ) q (Z) of Q. for all n ∈ N >1 ∞ , by regarding Z as the infinite cyclic abelian group (Z, +) (up to group-isomorphisms). Radulescu showed in [8] that either the statement (10.4)L Q (n l , Z), under (10.6).Therefore, the statement (10.9) holds.

Notation
In addition, the statement (10.10) holds as a special case of (10.9).
From below, let's fix n ∈ N >1 ∞ and the corresponding L(F n )-free filterization L Q (n, Z).Assume first that N = 1 in N, and hence T = g 1 ⊗ U n 1 j 1 ∈ L Q (n, Z).Then, for all k ∈ N, where e n is the group-identity of F n (and, hence, the identity element of L(F n )).
Suppose N > 1 in N, and assume that T is in the sense of (10.11) in L Q (n, Z), and the corresponding N-tuple (j 1 , ..., j N ) is alternating in Z. Proof.Let T = g 1 ⊗ U n 1 j 1 ∈ L Q (n, Z), for g 1 ∈ F n , U j 1 ∈ L Q (j 1 ).Then, this operator T is contained in the free block L L(F n ) Q (j 1 ) in the L(F n )-free filterization L Q (n, Z).Thus,

Theorem 4 .
Let U k = 1 which are interested.Motivated by the above remark, we now fix an arbitrary unital tracial W * -probability space (M, tr), consisting of the von Neumann algebra M, and a bounded linear functional tr on M; i.e., tr(1 M ) = 1, for the identity operator 1 M of M, and tr(m 1 m 2 ) = tr(m 2 m 1 ), for all m 1 , m 2 ∈ M.
For convenience, let us denote the two families of (7.15) and (7.16) by mX , respectively, mQ,

. 2 ) 5 .ω k j tr m j c k j 2 , ( 8 . 3 )m j ω k j c k j 2 .
PropositionLet T be a free random variable(8.1)  in the M-free filterization L M Q (Z).Then,where ω k are in the sense of (3.3), for all k ∈ N, and Supp(T) is the support (8.2) of T in Z.

Corollary 3 .
Let T be the free sum(8.1)  in the M-free filterization L M Q (Z).Then, τ M ((T * ) n ) = ∑ (j 1 ,...,j n )∈Supp(T) whose entries are from (M, tr), i.e., m j k ∈ (M, tr), for all k = 1, ..., N.In addition, for the N-tuple W, let η = (n j 1 , ..., n j N )be an N-tuple of natural numbers, n j k ∈ N, for all k = 1, ..., N k , where j 1 , ..., j N are the entries of W.For such N-tuples W, X and η, define an operator T the M-free filterization L M Q (Z), for all k = 1, ..., n.Now, let a fixed N-tuple W is taken from Alt Z N of (8.14), and let

Theorem 7 .
Consider free distributions of free random variables T X,η W of (8.16), for W ∈ Alt Z N .Let W ∈ Alt Z N , for some N ∈ N, and let T X,η W be the operator(8.16) in the M-free filterization L M Q (Z).Then, By the alternating-ness of W, the operator T X,,η W forms a free reduced word in L M Q (Z), by(7.6).Therefore, one can get that by the semicircularity of U j 's in L Q (Z).Similarly, Under this additional condition, one can realize that the nN-Alt Z nN , for all n ∈ N, i.e., W n are alternating in Z, for all n ∈ N. It guarantees that the operators T {l}] means the polynomial-algebra in l with M-coefficients, and where Y means the norm-topology-closure under the product topology for the W * -topology for M 1 M 2 (or that for M) and the Banach-topology for C[{l}] Denote the L(F n )-free filterizations L L(F n ) Q (Z) simply by L Q (n, Z), for all n ∈ N >1 ∞ .It is well-known that, if n ∈ N >1 ∞ , and if n 1 , n 2 ∈ N ∞ = N ∪ {∞}, such that n = n 1 + n 2 in N ∞ , then L(F n ) * -iso = L(F n 1 ) L(F n 2 )(10.2)(e.g., see[1,3]), where " * -iso = " means "being W * -algebra-isomorphic".More generally, ifn 1 + n 2 + ... + n k = n in N >1∞ , with n 1 , ..., n k ∈ N ∞ , for k ∈ N, then L(F n )

Corollary 9 .
Let T l = g l ⊗ U n l j l ∈ L Q (n, Z), where g l ∈ F n (and hence, they are generating operators of L(F n )), and U j l ∈ L Q (j l ) are semicircular elements in the free filterization L Q (Z), and n l ∈ N, for all l = 1, ..., N, for N ∈ N.Let T = N Π l=1 T l ∈ L Q (n, Z). (10.11)