*<sup>τ</sup>t*1<*t*2 *Qnp*,*<sup>j</sup>* − *t*0 → 0,

*where:*

$$t\_0 = \begin{cases} 0 & \text{if } j < -1 \\ \omega\_n c\_{\frac{\pi}{2}} & \text{if } j = -1, \end{cases} \tag{105}$$

*for all n* ∈ N.

> *Under the same hypothesis, if j* > −1 *in* Z, *then:*

$$\left| \left. \pi\_{t\_1 < t\_2} \left( Q\_{p,j}^n \right) \right| \right| \to \infty,\tag{106}$$

*for all n* ∈ N.

**Proof.** The estimations (105) and (106), for suitably big *t*1 ∈ R, are obtained by (103).

### **10. Semicircularity of Certain Free Sums in** LS*<sup>t</sup>*1<*t*2

As in Section 9, we will let LS be the Banach ∗-subalgebra (72) of the free Adelic filtration LS, and let LS0 be the semicircular filtration (LS, *τ*<sup>0</sup>) of (80).

Let (*<sup>A</sup>*, *ϕ*) be an arbitrary topological ∗-probability space and *a* ∈ (*<sup>A</sup>*, *ϕ*). We say a free random variable *a* is a *free sum* in (*<sup>A</sup>*, *ϕ*), if:

$$a = \sum\_{l=1}^{N} \mathbf{x}\_l,\text{ with } \mathbf{x}\_l \in (A, \,\,\phi)\_l$$

 and the summands *x*1, ..., *xN* of *a* are free from each other in (*<sup>A</sup>*, *ϕ*), for *N* ∈ N \ {1}.

Let *t*1 < *t*2 be suitable in R in the sense of **NA 9.2**, and let LS*<sup>t</sup>*1<*t*2 be the corresponding semicircular [*<sup>t</sup>*1, *<sup>t</sup>*2]-filtration. Now, we define free random variables *X* and *Y* of LS,

$$X = \sum\_{l=1}^{N} \mathbb{Q}\_{p\_l, \dot{\jmath}\_l}^{\eta\_l} and Y = \sum\_{l=1}^{N} \mathbb{G}\_{p\_l, \dot{\jmath}\_l}^{\eta\_l} \tag{107}$$

for *Qpl*,*jl* ∈ Q and <sup>Θ</sup>*pl*,*jl* ∈ Θ, for all *l* = 1, ..., *N*, for *N* ∈ N \ {1}.

Remark that, the operator *X* (or *Y*) of (107) is a free sum in LS, if and only if the summands *Qnl pl*,*jl* (resp., Θ*nl pl*,*jl*), which are the free reduced words with their lengths one, are free from each other in LS, if and only if *Qpl*,*jl* (resp., <sup>Θ</sup>*pl*,*jl*) are contained in the mutually-distinct free blocks <sup>C</sup>[{*Qpl*,*jl* }] of LS by (74), if and only if the pairs (*pl*, *jl*) are mutually distinct from each other in the Cartesian product P × Z, for all *l* = 1, ..., *N*. i.e., the given operators *X* and *Y* of (107) are free sums in LS, if and only if:

$$<\langle p\_{l\_1}, j\_{l\_1} \rangle \neq (p\_{l\_2}, j\_{l\_2}) \text{in} \mathcal{P} \times \mathbb{Z}\_\star \tag{108}$$

for all *l*1 = *l*2 in {1, ..., *<sup>N</sup>*}.

**Lemma 4.** *Let X and Y be in the sense of* (107) *in the semicircular filtration* LS0. *Assume that the pairs* (*pl*, *jl*) *are mutually distinct from each other in* P × Z, *for all l* = 1, *..., N*, *for N* ∈ N \ {1}. *Then:*

$$
\pi^0(X) = \sum\_{l=1}^N \left(\omega\_{n\_l} p\_l^{2(j\_l+1)} c\_{\frac{n\_l}{2}}\right),
$$

$$
\pi^0(Y) = \sum\_{l=1}^N \left(\omega\_{n\_l} c\_{\frac{n\_l}{2}}\right).
\tag{109}
$$

*and:*

\*\*Proof.\*\* Let  $\mathbb{X}$  and  $\mathbb{Y}$  be given as above in  $\mathbb{L}\mathbb{S}\_{0}$ . By the assumption that the pairs  $(p\_{\ell}, j\_{\ell})$  are mutually distinct from each other in  $\mathcal{P} \times \mathbb{Z}$ , these operators  $X$  and  $Y$  satisfy the condition (108); equivalently, they are free sums in  $\mathbb{L}\mathbb{S}\_{0}$ .

Therefore, one has that:

$$\begin{aligned} \tau^0(X) &= \sum\_{l=1}^N \tau^0 \left( Q^{n\_l}\_{p\_l, j\_l} \right) = \Sigma^N\_{l=1} \, \tau^0\_{p\_l, j\_l} \left( Q^{n\_l}\_{p\_l, j\_l} \right), \\ &= \Sigma^N\_{l=1} \left( \omega\_{n\_l} p\_l^{2(j\_l+1)} c\_{\frac{n\_l}{2}} \right). \end{aligned}$$

by the *p*<sup>2</sup>(*jl*+<sup>1</sup>) *l* -semicircularity of *Qpl*,*jl* ∈ Q, for all *l* = 1, ..., *N*.

Similarly, one can ge<sup>t</sup> that:

$$
\pi^0(\mathcal{Y}) = \Sigma\_{l=1}^N \pi\_{p\_l, j\_l}^0 \left( \Theta\_{p\_l, j\_l}^{n\_l} \right) = \Sigma\_{l=1}^N \left( \omega\_{n\_l} c\_{\frac{n\_l}{2}} \right),
$$

by the semicircularity of <sup>Θ</sup>*pl*,*jl* ∈ Θ, for all *l* = 1, ..., *N*.

Now, for the operators *X* and *Y* of (107), we consider how our truncation distorts the free-distributional data (109).

For a given closed interval [*<sup>t</sup>*1, *t*2] in R, where *t*1 < *t*2 are suitable in R, we define:

$$\mathcal{P}\_{[t\_1, t\_2]} = \{ p \in \mathcal{P} : t\_1 \le p \le t\_2 \} = \mathcal{P} \cap [t\_1, t\_2]\_{\mathcal{I}}$$

and:

$$\mathcal{P}\_{[t\_1, t\_2]}^{\varepsilon} = \mathcal{P} \backslash \mathcal{P}\_{[t\_1, t\_2]} \tag{110}$$

in P.

By (110), the family {P[*<sup>t</sup>*1,*t*2], <sup>P</sup>*<sup>c</sup>*[*<sup>t</sup>*1,*t*2]} forms a partition of the set P of all primes for the fixed interval [*<sup>t</sup>*1, *<sup>t</sup>*2]. Of course, if *t*1 < *t*2 are not suitable, then:

$$\mathcal{P}\_{[t\_1, t\_2]} = \mathcal{Q}\_{\prime} \text{ and hence, } \mathcal{P} = \mathcal{P}\_{[t\_1, t\_2]}^{\varepsilon}.$$

**Theorem 10.** *Let X and Y be the operators* (107)*, and assume they are free sums in the semicircular filtration* LS0; *and let* LS*<sup>t</sup>*1<*t*2*be the semicircular* [*<sup>t</sup>*1, *<sup>t</sup>*2]*-filtration for suitable t*1 < *t*2 *in* R*. Then:*

$$\pi\_{t\_1 < t\_2}(X) = \sum\_{p\_I \in \mathcal{P}\_{[t\_1, t\_2] : (p\_1, \dots, p\_N)}} \left( \omega\_{\mathcal{W}\_I} p\_I^{2(j\_l + 1)} \mathfrak{c}\_{\frac{n}{2}} \right),$$

*and:*

$$\pi\_{t\_1 < t\_2}(Y) = \sum\_{p\_l \in \mathcal{P}\_{[t\_1 < t\_2] : (p\_1, \dots, p\_N)}} \left( \omega\_{n\_l} c\_{\frac{n\_l}{2}} \right), \tag{111}$$

*where:*

$$\mathcal{P}\_{[t\_1, t\_2]:(p\_1, \dots, p\_N)} = \mathcal{P}\_{[t\_1, t\_2]} \cap \{p\_1, \dots, p\_N\} \text{ in } \mathcal{P}\_{\prime}$$

*where* <sup>P</sup>[*<sup>t</sup>*1,*t*2] *is in the sense of* (110) *in* P. *Clearly, if* <sup>P</sup>[*<sup>t</sup>*1,*t*2]:(*p*1,...,*pN*) *is empty in* P, *then the formulas in* (111) *vanish.*

**Proof.** The proof of (111) is done by (95), (96), (98), and (109). Indeed, if:

$$\mathcal{P}\_{[t\_1, t\_2]:(p\_1, \dots, p\_N)} = \mathcal{P}\_{[t\_1, t\_2]} \cap \{p\_1, \dots, p\_N\} \text{ in } \mathcal{P}\_{\prime}$$

where <sup>P</sup>[*<sup>t</sup>*1,*t*2] is in the sense of (110), and if:

$$\mathcal{P}\_{[t\_1, t\_2]:(p\_1, \dots, p\_N)} \neq \mathcal{Q}\_{\prime \prime}$$

then:

$$\pi\_{t\_1 < t\_2}(X) = \sum\_{p\_l \in \mathcal{P}\_{[t\_1, t\_2] \colon (p\_1, \dots, p\_N)}} \pi\_{p\_l, j\_l}^0 \left( Q\_{p\_l, j\_l}^{n\_l} \right),$$

by (98)

$$=\sum\_{p\_l \in \mathcal{P}\_{[\mathfrak{l}\_1, \mathfrak{l}\_2] : (\mathcal{P}\_1, \dots, \mathcal{P}\_N)}} \left(\omega\_{n\_l} p\_l^{2(j\_l+1)} c\_{\frac{n\_l}{2}}\right) \dots$$

by the *p*<sup>2</sup>(*j*+<sup>1</sup>)-semicircularity of *Qp*,*<sup>j</sup>* ∈ Q.

> Similarly, one can ge<sup>t</sup> that:

$$\pi\_{t\_1 < t\_2}(\mathcal{Y}) = \sum\_{p\_l \in \mathcal{P}\_{[t\_1, t\_2] : (\mathcal{P}\_1, \dots, \mathcal{P}N)}} \left( \omega\_{n\_l} c\_{\#} \right) \cdot \zeta$$

by the semicircularity of <sup>Θ</sup>*p*,*<sup>j</sup>* ∈ Θ. Therefore, the free-distributional data (111) holds, whenever:

$$\mathcal{P}\_{[t\_1, t\_2]:(p\_1, \dots, p\_N)} \neq \mathcal{Q} \text{ in } \mathcal{P}.$$

Definitely, if:

$$\mathcal{P}\_{[t\_1, t\_2]:(p\_1, \dots, p\_N)} = \mathcal{Q}\_{\prime}$$

then:

$$
\pi\_{t\_1 < t\_2}(X) = O(X) = 0 = O(Y) = \pi\_{t\_1 < t\_2}(Y).
$$

Therefore, the truncated free-distributional data (111) hold.

**Remark 4.** *Let us compare the free-distributional data* (109) *and* (111)*. One can check the differences between them dictated by the choices of* [*<sup>t</sup>*1, *t*2] *in* R. *Thus, the formula* (111) *also illustrates how our truncations on* P *distort the original free-probabilistic information on the semicircular filtration* LS0*.*

Let *q*0 be a fixed prime in P. Choose *t*0 < *s*0 ∈ R such that: (i) these quantities *t*0 and *s*0 satisfy:

$$t\_0 \le q\_0 \le s\_0 \text{ in } \mathbb{R}\_\prime$$

and (ii) *q*0 is the only prime in the closed interval [*<sup>t</sup>*0, *s*0] in R.

By the *Archimedean property* on R (or *the axiom of choice*), the existence of such interval [*<sup>t</sup>*0, *<sup>s</sup>*0], satisfying (i) and (ii) for the fixed prime *q*0, is guaranteed; however, the choices of the quantities *t*0 < *s*0 are of course not unique.

**Definition 18.** *Let q*0 ∈ P, *and let t* < *s* ∈ R *be the real numbers satisfying the conditions (i) and (ii) of the above paragraph. Then, the suitable closed interval* [*t*, *s*] *is called a q*0*-neighborhood.*

Depending on prime-neighborhoods, one can obtain the following semicircularity condition on our semicircular truncated-filtrations.

**Corollary 5.** *Let p* ∈ P, [*t*, *s*] *be a p-neighborhood in* R, *and* LS*<sup>t</sup>*<*s be the corresponding semicircular* [*t*, *s*]*-filtration. If X and Y are free sums formed by* (107) *in the semicircular filtration* LS0, *then:*

$$
\pi\_{l

$$
\pi\_{l$$
$$

*where δ is the Kronecker delta.*

*and:*

**Proof.** The free-distributional data (112) are a special case of (111), under the prime-neighborhood condition. Indeed, in this case,

$$\mathcal{P}\_{[t,s]:(p\_1,\ldots,p\_N)} = \{p\} \cap \{p\_1,\ldots,p\_N\} = \left\{\begin{array}{l} \{p\} \quad \text{or} \\ \mathcal{Q}\_{\prime} \end{array}\right\}$$

where <sup>P</sup>[*<sup>t</sup>*,*<sup>s</sup>*]:(*p*1,...,*pN*) is in the sense of (111).

More general to (112), we obtain the following result.

**Proposition 10.** *Let p* ∈ P *and* [*t*, *s*] *be a p-neighborhood in* R, *and let* LS*<sup>t</sup>*<*s be the corresponding semicircular* [*t*, *s*]*-filtration. Then, a free random variable T* ∈ LS*<sup>t</sup>*<*s has its non-zero free distribution, if and only if there exists a non-zero summand T*0 *of T*, *such that:*

$$T\_0 \in \mathbb{L} \mathbb{S}\_p \\ \|f\|\_{\ell^\infty(\mathbb{S}^\*)} \tag{113}$$

*where* LS*p* = *<sup>j</sup>*∈ZC[{<sup>Θ</sup>*p*,*j*}] *is a Banach* ∗*-subalgebra* (82) *of* LS. **Proof.** By (98), if *T* ∈ LS*<sup>t</sup>*<*s* has its non-zero free distribution, then there exists a non-zero summand *T*0 of *T* which can be a linear combination of free reduced words contained in - *<sup>t</sup>*≤*q*≤*s*in P LS*q*, and hence,

$$T0 \in \mathop{\mathbb{K}\_{t \le p}}\_{t \le q} \mathop{\star}\_{\le s} \mathbb{L} \mathcal{B}\_{q\_{\prime}} \tag{114}$$

where LS*q* are in the sense of (82), for *q* ∈ P.

> Since [*t*, *s*] is a *p*-neighborhood, the relation (114) is equivalent to:

$$T\_0 \in \mathbb{L} \mathbb{S}\_p. \tag{115}$$

Clearly, the converse holds true as well, by (98).

Therefore, a free random variable *T* ∈ LS*<sup>t</sup>*<*s* has its non-zero free distribution, if and only if *T* contains its non-zero summand *T*0 ∈ LS*p*, by (115); equivalently, the statement (113) holds true.

By (112) and (113), we obtain the following interesting result.

**Theorem 11.** *Let X*1 = ∑*Nl*=<sup>1</sup> *Qpl*,*jl and Y*1 = ∑*Nl*=<sup>1</sup> <sup>Θ</sup>*pl*,*jl be in the sense of* (107) *in the semicircular filtration* LS0, *and assume that* (*pl*, *jl*) *are mutually distinct in* P × Z, *for l* = 1, *..., N*, *for N* ∈ N \ {1}. *Suppose we fix:*

$$p\_{l\_0} \in \{p\_{1\prime}, \dots, p\_N\}\_{\prime}$$

*and take a pl*0*-neighborhood* [*<sup>t</sup>*0, *s*0] *in* R. *Then:*

$$X\_1 
is\_{l\_0}^{2(j\_{l\_0}+1)} - 
smatrix 
in \mathbb{L} \mathbb{S}\_{t\_0 < s\_{0\*}} \tag{116}$$

$$\mathcal{Y}\_1 \text{ is semicircular in } \mathbb{L}\mathbb{S}\_{t\_0 \times s\_0 \prime} \tag{117}$$

*where* LS*<sup>t</sup>*0<*s*0*is the semicircular* [*<sup>t</sup>*0, *<sup>s</sup>*0]*-filtration.*

**Proof.** Let *X*1 and *Y*1 be given as above in LS, and fix *pl*0 ∈ {*p*1, ..., *pN*}. Note that, by the assumption, these operators *X*1 and *Y*1 form free sums in the semicircular filtration LS0, having *N*-many summands. Note also that they are self-adjoint in LS by the self-adjointness of their summands.

By (113), if an operator *T* has its non-zero free distribution in the semicircular [*<sup>t</sup>*0, *<sup>s</sup>*0]-filtration LS*<sup>t</sup>*0<*s*0 , where [*<sup>t</sup>*0, *s*0] is a *pl*0-neighborhood in R, then it must have its non-zero summand *T*0,

$$T\_0 \in \mathbb{L} \mathbb{S}\_{p\_{l\_0}} \text{ in } \mathbb{L} \mathbb{S}\_{t\_0 < s\_0}$$

.

By the very construction of *X*1 and *Y*1, they contain their summands,

$$
\mathbb{Q}\_{p\_{l\_0}, j\_{l\_0}, \prime} \oplus\_{p\_{l\_0}, j\_{l\_0}} \in \mathbb{L} \mathbb{S}\_{p\_{l\_0}} \text{ in } \mathbb{L} \mathbb{S}\_{t\_0 \times s\_0}.
$$

Consider now that:

$$\begin{array}{lcl} X\_1^{\mathfrak{n}} &=& \sum\_{(\mathring{\imath}\_1,\ldots,\mathring{\imath}\_n)\in\{1,\ldots,N\}^{\mathfrak{n}}} \left( \prod\_{k=1}^N Q\_{p\_{\mathring{\imath}\_k,\mathring{\jmath}\_k}} \right) \\ &=& Q\_{p\_{\mathring{l}\_0,\mathring{l}\_0}}^{\mathfrak{n}} + \sum\_{(\mathring{\imath}\_1,\ldots,\mathring{\imath}\_n)\neq(\mathring{l}\_0,\ldots,\mathring{l}\_0)} \left( \prod\_{k=1}^N Q\_{p\_{\mathring{k}\_l},\mathring{\jmath}\_k} \right) \\ &=& Q\_{p\_{\mathring{l}\_0,\mathring{l}\_0}}^{\mathfrak{n}} + [\text{Rest Terms}]\_{\prime} \end{array}$$

and similarly,

$$Y\_1^n = \Theta\_{p\_{l\_0}, j\_{l\_0}}^n + [\text{RestTerms}]',\tag{118}$$

.

for all *n* ∈ N.

> It is not difficult to check that:

$$
\pi\_{t\_0 < s\_0} \left( [\text{Rest Terms}] \right) = 0 = \pi\_{t\_0 < s\_0} \left( [\text{RestTerms}]' \right) \,\text{},\tag{119}
$$

by (98) and (113), where [Rest Terms], and [Rest Terms] are from (118).

Therefore, one obtains that:

$$\begin{aligned} \tau\_{t\_0 < \varsigma\_0} \left( X^n\_1 \right) &= \tau\_{t\_0 < \varsigma\_0} \left( Q^n\_{p\_{l\_0, \dot{l}\_0}} \right) = \tau^0\_{p\_{l\_0}} \left( Q^n\_{p\_{l\_0, \dot{l}\_0}} \right) \\ &= \tau^0\_{p\_{l\_0, \dot{l}\_0}} \left( Q^n\_{p\_{l\_0, \dot{l}\_0}} \right) = \omega\_n p\_{l\_0}^{2(\dot{j}\_{l\_0} + 1)} c\_{\frac{n}{2}} \end{aligned}$$

and, similarly,

$$
\pi\_{l\_0 < \varepsilon\_0} \left( Y\_1^n \right) = \pi\_{p\_{l\_0, \hat{l}\_0}}^0 \left( \Theta\_{p\_{l\_0, \hat{l}\_0}}^n \right) = \omega\_n \varepsilon\_{\frac{n}{2}}.\tag{120}
$$

for all *n* ∈ N, by (119).

Therefore, the free sum *X*1 ∈ LS is *p*<sup>2</sup>(*jl*0+<sup>1</sup>) *l*0 -semicircular in LS*<sup>t</sup>*0<*s*0 ; and the free sum *Y*1 ∈ LS is semicircular in LS*<sup>t</sup>*0<*s*0 , by (120). Therefore, the statements (116) and (117) hold true.

The above theorem shows that, if there is a free sum *T* in the semicircular filtration LS0 and if we "nicely" truncate the linear functional *τ*0 on LS, then one can focus on the non-zero summand *T*0 of *T*, whose the free distribution not only determines the truncated free distribution of *T*, but also follows the (weighted-)semicircular law.

### **11. Applications of Prime-Neighborhoods**

In Section 9, we considered the semicircular truncated-filtrations LS*<sup>t</sup>*<*s* for *t* < *s* ∈ R and studied how [*<sup>t</sup>*,*<sup>s</sup>*]-truncations on P affect, or distort, the original free-distributional data on the semicircular filtration LS0 = -LS, *<sup>τ</sup>*<sup>0</sup>. As a special case, in Section 10, we introduced *p*-neighborhoods for primes *p* and considered corresponding truncated free distributions on LS.

In this section, by using prime-neighborhoods, we provide a completely "new" model of truncated free probability on LS and study how the original free-distributional data on LS0 are distorted under this new truncation model.

Let us now regard the set P of all primes as a *totally ordered set* (a TOset),

$$\mathcal{P} = \{q\_1 < q\_2 < q\_3 < q\_4 < q\_5 < \dots\} \tag{121}$$

under the usual inequality (≤) on P, i.e.,

$$q\_1 = 2, q\_2 = 3, q\_3 = 5, q\_4 = 7, q\_5 = 11, \text{etc.}$$

For each *qk* ∈ P of (121), determine a *qk*-neighborhood *Bk*

$$B\_k \stackrel{donde}{=} [t\_k, s\_k] \\in \mathbb{R}\_\prime \tag{122}$$

for all *k* ∈ N.

Let *<sup>τ</sup>Bk*be our truncated linear functionals *<sup>τ</sup>tk*<sup>&</sup>lt;*sk*of (92) on the Banach ∗-algebra LS, i.e.,

$$
\pi\_{\mathbb{R}\_k} = \pi\_{l\_k \le s\_{k'}} \text{ for all } k \in \mathbb{N}. \tag{123}
$$

Then, by the truncated linear functionals of (123), one can have the corresponding semicircular *Bk*-filtrations,

$$\mathbb{LES}\_{\mathbb{B}\_k} = \mathbb{LES}\_{\mathbb{H}\_k \le s\_k} = \left(\mathbb{LES}, \,\,\tau \mathfrak{g}\_k\right), \tag{124}$$

for all *k* ∈ N.

> We now focus on the system:

$$\mathbf{T} = \{\tau \mathbf{z}\_k\}\_{k=1}^{\infty} \tag{125}$$

of *qk*-neighborhood-truncated linear functionals (123) for all *k* ∈ N.

Let *F* be a "finite" subset of the TOset P of (121), and for such a set *F*, define a new linear functional *τF* on LS induced by the system **T** of (125), by:

$$\tau\_{\mathcal{F}} = \sum\_{\emptyset \neq \mathcal{F}} \tau\_{B\_k} on \mathbb{L} \mathbb{S}. \tag{126}$$

Before proceeding, let us consider the following result obtained from (113).

**Lemma 5.** *Let p* ∈ P *and* [*t*, *s*] *be a p-neighborhood in* R, *and let* LS*<sup>t</sup>*<*s be the semicircular* [*t*, *s*]*-filtration. Let τ*0*p* = *<sup>j</sup>*∈Z*τ*0*p*,*<sup>j</sup> be the linear functional* (81) *on the Banach* ∗*-subalgebra* LS*p of* (82) *in the semicircular filtration* LS0. *Define a linear functional τp on the Banach* ∗*-algebra* LS *by:*

$$
\pi^p(T) \stackrel{def}{=} \begin{cases}
\pi\_p^0(T) & \text{if } T \in \mathbb{L}\mathbb{S}\_p \text{ in } \mathbb{L}\mathbb{S}, \\
O(T) = 0 & \text{otherwise},
\end{cases}
$$

*for all T* ∈ LS. *Then, the Banach* ∗*-probability space* (LS, *τ<sup>p</sup>*) *is free-isomorphic to* LS*<sup>t</sup>*<*s*, *i.e.,*

$$[t, s] \text{ is a } p-m \text{-} neighborhood \implies \mathbb{L} \mathbb{S}\_{t \le s} \text{ and } (\mathbb{L} \mathbb{S}, \ \mathbf{r}^p) \text{ are free}-isomorphic. \tag{127}$$

**Proof.** Under the hypothesis, it is not hard to check:

$$
\pi\_{t < s} = \tau^p \text{ on } \mathbb{L}\mathbb{S}.
$$

Therefore, the identity map on LS becomes a free-isomorphism from LS*<sup>t</sup>*<*s* onto (LS, *<sup>τ</sup><sup>p</sup>*).

If a finite subset *F* is a singleton subset of P, then the free probability on LS determined by the corresponding linear functional *τF* of (126) is already considered in Section 10 and in (127). Therefore, we now restrict our interests to the cases where finite subsets *F* have more than one element in P.

**Lemma 6.** *Let F be a finite subset of the TOset* P *of* (121)*, and let τF be the corresponding linear functional* (126) *on* LS. *Then:*

$$\pi\_{\mathcal{F}} = \sum\_{\emptyset \neq \mathcal{F}} \pi^{\emptyset \mathbb{k}} on \mathbb{L} \mathbb{S}, \tag{128}$$

*where τqk are in the sense of* (127)*.*

**Proof.** The proof of (128) is done by (126) and (127) because:

$$(\mathbb{L}\mathbb{S}, \tau^{q\_k}) \text{ and } \mathbb{L}\mathbb{S}\_{B\_{k}}$$

are free-isomorphic for all *qk* ∈ *F*. Therefore, the linear functional *τF* of (126) satisfies that:

$$\mathsf{TF} = \sum\_{q\_k \in F} \mathsf{\tau}\_{\mathsf{B}\_k} = \sum\_{q\_k \in F} \mathsf{\tau}^{q\_k} \text{ on } \mathsf{L}\mathbb{S}.$$

By (113), (116), (117), and (128), one obtains the following result.

**Theorem 12.** *Let T* = *N* Π *<sup>l</sup>*=<sup>1</sup>*Qnl pl*,*jl or S* = *N*Π*<sup>l</sup>*=1Θ*nl pl*,*jl be a free reduced word of* LS *with its length-N*, *for N* ∈ N*. If: F* ∩ {*p*1, *..., pN*} = ∅ *in* P,

*then:*

$$
\tau\_{\mathbb{F}}(T) = 0 = \tau\_{\mathbb{F}}(S). \tag{129}
$$

*While, if F* ∩ {*p*1, ..., *pN*} = ∅ *in* P, *then:*

> *<sup>τ</sup>F*(*T*) = ∑ *q*∈*<sup>F</sup>*∩{*p*1,...,*pN*} *ωnq*<sup>2</sup>(*j*+<sup>1</sup>)*c n*2 ,

*respectively,*

$$\pi\_{\mathcal{F}}(S) = |F \cap \{p\_{1\prime}, \dots, p\_N\}| \left(\omega\_n \mathfrak{c}\_{\frac{n}{2}}^n\right) \,. \tag{130}$$

*where* |*X*| *mean the cardinalities of sets X*.

**Proof.** Let *T* and *S* be given free reduced words with their length-*N* in LS, for *N* ∈ N. If:

$$F \cap \{p\_{1\prime}, \dots, p\_N\} = \mathcal{Q} \text{ in } \mathcal{P}\_{\prime}$$

then we obtain the formula (129) by (127) and (128). Indeed,

$$\tau\_F(T) = \sum\_{q \in F} \tau^q(T) = 0 = \sum\_{q \in F} \tau^q(S) = \tau\_F(S).$$

Now, assume that:

$$F \cap \{p\_{1\prime}, \dots, p\_N\} = \{p\_{i\_1 \prime}, \dots, p\_{i\_k}\} \text{ in } \mathcal{P}\_{\prime \prime}$$

for some *k* ∈ N, such that 1 ≤ *k* ≤ *N*. Then:

$$\pi\_F(T) = \left(\sum\_{l=1}^k \pi\_{p\_{i\_l}}^0\right)(T) = \sum\_{l=1}^k \pi\_{p\_{i\_l}}^0(T).$$

by (126) and (128)

$$\begin{split} \boldsymbol{\tau} &= \boldsymbol{\Sigma}\_{l=1}^{k} \boldsymbol{\tau}\_{p\_{\dot{\boldsymbol{\eta}}}}^{0} \left( \boldsymbol{Q}\_{p\_{\dot{\boldsymbol{\eta}}\_{l}}, \dot{\boldsymbol{\eta}}\_{l}}^{n\_{\boldsymbol{\eta}\_{l}}} \right) = \boldsymbol{\Sigma}\_{l=1}^{k} \, \boldsymbol{\tau}\_{p\_{\dot{\boldsymbol{\eta}}\_{l}}, \dot{\boldsymbol{\eta}}\_{l}}^{0} \left( \boldsymbol{Q}\_{p\_{\dot{\boldsymbol{\eta}}\_{l}}, \dot{\boldsymbol{\eta}}\_{l}}^{n\_{\boldsymbol{\eta}\_{l}}} \right) \\ &= \boldsymbol{\Sigma}\_{l=1}^{k} \left( \boldsymbol{\omega}\_{\boldsymbol{\eta}\_{l}} p\_{\boldsymbol{\eta}\_{l}}^{2(j\_{l}+1)} c\_{\frac{\boldsymbol{\eta}\_{l}}{2}} \right) . \end{split} \tag{131}$$

Similarly,

$$\pi\_{\mathcal{F}}(S) = \sum\_{l=1}^{k} \left( \omega\_{n\_l} c\_{\frac{n\_l}{2}} \right) = k \cdot \left( \omega\_{n\_l} c\_{\frac{n\_l}{2}} \right).$$

Therefore, the free-distributional data (130) hold.

The above free-distributional data (129) and (130) characterize the free-probabilistic information on Banach ∗-probability spaces

> (LS, *<sup>τ</sup>F*),

for any finite subsets *F* of P.

**Author Contributions:** The authors contributted equally in this article.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.
