**2. Preliminaries**

In this section, we briefly introduce the backgrounds of our proceeding works.

### *2.1. Free Probability*

Readers can review *free probability theory* from [14,15] (and the cited papers therein). *Free probability* is understood as the noncommutative operator-algebraic version of classical *measure theory* and *statistics*. The classical *independence* is replaced by the *freeness*, by replacing *measures* on sets with *linear functionals* on noncommutative (∗-)algebras. It has various applications not only in pure mathematics (e.g., [16–20]), but also in related topics (e.g., see [2,8–11]). Here, we will use the *combinatorial free probability theory* of *Speicher* (e.g., see [14]).

In the text, without introducing detailed definitions and combinatorial backgrounds, *free moments* and *free cumulants* of operators will be computed. Furthermore, the *free product of* ∗*-probability spaces in the sense of* [14,15] is considered without detailed introduction.

Note now that one of our main objects, the ∗-algebra M*p* of Section 3, are commutative, and hence, (traditional, or usual "noncommutative") free probability theory is not needed for studying *functional analysis* or *operator algebra theory* on M*p*, because the freeness on this commutative structure is trivial. However, we are not interested in the free-probability-depending operator-algebraic structures of commutative algebras, but in statistical data of certain elements to establish (weighted-)semicircular elements. Such data are well explained by the free-probability-theoretic terminology and language. Therefore, as in [2], we use "free-probabilistic models" on M*p* to construct and study our (weighted-)semicircularity by using concepts, tools, and techniques from free probability theory "non-traditionally." Note also that, in Section 8, we construct "traditional" free-probabilistic structures, as in [1], from our "non-traditional" free-probabilistic structures of Sections 3–7 (like the *free group factors*; see, e.g., [15,19]).

### *2.2. Analysis of* Q*p*

For more about *p*-adic and Adelic analysis, see [7]. Let *p* ∈ P, and let Q*p* be the *p*-*adic number field*. Under the *p*-adic addition and the *p*-adic multiplication of [7], the set Q*p* forms a *field* algebraically. It is equipped with the *non-Archimedean norm* |.|*p* , which is the inherited *p*-*norm* on the set Q of all *rational numbers* defined by:

$$|x|\_p = \left| p^k \frac{q}{b} \right|\_p = \frac{1}{p^k}.$$

whenever *x* = *pk ab* in Q, where *k*, *a* ∈ Z, and *b* ∈ Z \ {0}. For instance,

$$\left|\frac{8}{3}\right|\_2 = \left|2^3 \times \frac{1}{3}\right|\_2 = \frac{1}{2^3} = \frac{1}{8'}$$

and:

$$\left|\frac{8}{3}\right|\_{3} = \left|3^{-1} \times 8\right|\_{3} = \frac{1}{3^{-1}} = 3\sqrt{2}$$

and:

 $\left|\frac{8}{3}\right|\_q = \frac{1}{q^5} = 1$ , whereever  $q \in \mathcal{P}$  \\
(2, 3).

The *p*-*adic number field* Q*p* is the maximal *p*-norm closure in Q. Therefore, under norm topology, it forms a *Banach space* (e.g., [7]).

Let us understand the *Banach field* Q*p* as a *measure space*,

$$
\mathbb{Q}\_p = \left( \mathbb{Q}\_{p\_{\prime}} \,\, \sigma(\mathbb{Q}\_p) \,\, \mu\_P \right) \,\,.
$$

where *<sup>σ</sup>*(Q*p*) is the *<sup>σ</sup>*-*algebra of* Q*p* consisting of all *<sup>μ</sup>p*-*measurable subsets*, where *μp* is a left-and-right additive invariant *Haar measure* on Q*p* satisfying:

$$
\mu\_p(\mathbb{Z}\_p) = 1,
$$

where Z*p* is the *unit disk of* Q*<sup>p</sup>*, consisting of all *p*-*adic integers x* satisfying |*x*|*p* ≤ 1. Moreover, if we define:

$$\mathcal{U}\_k = p^k \mathbb{Z}\_p = \{ p^k \mathbf{x} \in r \mathbb{Q}\_p : \mathbf{x} \in \mathbb{Z}\_p \},\tag{1}$$

for all *k* ∈ Z (with *U*0 = <sup>Z</sup>*p*), then these *<sup>μ</sup>p*-measurable subsets *Uk*'s of (1) satisfy:

$$\mathbb{Q}\_p = \underset{k \in \mathbb{Z}}{\cup} \mathcal{U}\_{k'}$$

and:

$$
\mu\_p\left(\mathcal{U}\_k\right) = \frac{1}{p^k} = \mu\_p\left(\mathbf{x} + \mathcal{U}\_k\right), \forall \mathbf{x} \in \mathbb{Q}\_{p^\prime} \tag{2}
$$

and:

$$\cdots \cdot \subset \mathcal{U}\_{\mathcal{U}} \subset \mathcal{U}\_{1} \subset \mathcal{U}\_{0} = \mathbb{Z}\_{p} \subset \mathcal{U}\_{-1} \subset \mathcal{U}\_{-2} \subset \cdots \dots$$

In fact, the family {*Uk*}*<sup>k</sup>*∈<sup>Z</sup> forms a *basis* of the Banach topology for Q*p* (e.g., [7]). Define now subsets *∂k* of Q*p* by:

$$
\partial\_k = \mathcal{U}\_k \nmid \mathcal{U}\_{k+1}, for \; allk \in \mathbb{Z}. \tag{3}
$$

We call such *<sup>μ</sup>p*-measurable subsets *∂k* the *k*th boundaries of *Uk* in Q*<sup>p</sup>*, for all *k* ∈ Z. By (2) and (3), one obtains that:

$$\mathbb{Q}\_p = \sqcup\_{k \in \mathbb{Z}} \partial\_{k'} $$

and:

$$
\mu\_p \left( \partial\_k \right) = \mu\_p \left( \mathcal{U}\_k \right) - \mu\_p \left( \mathcal{U}\_{k+1} \right) = \frac{1}{p^k} - \frac{1}{p^{k+1}} \tag{4}
$$

and:

$$
\partial\_{k\_1} \cap \partial\_{k\_2} = \begin{cases}
\partial\_{k\_1} & \text{if } k\_1 = k\_2 \\
\mathcal{Q} & \text{otherwise}
\end{cases}
$$

for all *k*, *k*1, *k*2 ∈ Z, where is the *disjoint union* and ∅ is the *empty set*.

Now, let M*p* be the algebra,

$$\mathcal{M}\_p = \mathbb{C}\left[ \{ \chi\_S : S \in \sigma(\mathbb{Q}\_p) \} \right],\tag{5}$$

where *χS* are the usual *characteristic functions* of *S* ∈ *σ* -Q*p* .

Then the algebra M*p* of (5) forms a well-defined <sup>∗</sup>-*algebra over* C, with its *adjoint*,

$$\left(\sum\_{S \in \sigma(\mathcal{G}\_p)} \mathsf{f}\_S \mathsf{X}\_S\right)^\* \stackrel{def}{=} \sum\_{S \in \sigma(\mathcal{G}\_p)} \overline{\mathsf{f}\_S} \,\mathsf{X}\_S \mathsf{A}\_S$$

where *tS* ∈ C, having their *conjugates tS* in C.

> Let ∑ *<sup>S</sup>*∈*σ*(*Gp*) *tSχS* ∈ M*p*. Then, one can define the *p*-*adic integral* by:

$$\int\_{\mathbb{Q}\_p} \left( \sum\_{S \in \sigma(\mathbb{Q}\_p)} t\_S \chi\_S \right) d\mu\_p = \sum\_{S \in \sigma(\mathbb{Q}\_p)} t\_S \mu\_p(S). \tag{6}$$

Note that, by (4), if *S* ∈ *<sup>σ</sup>*(Q*p*), then there exists a subset Λ*S* of Z, such that:

$$\Lambda\_{\mathbb{S}} = \{ j \in \mathbb{Z} : \mathbb{S} \cap \partial\_{j} \neq \mathcal{Q} \}, \tag{7}$$

satisfying:

$$\int\_{\mathbb{Q}\_p} \chi\_S \, d\mu\_p = \int\_{\mathbb{Q}\_p} \sum\_{j \in \Lambda\_S} \chi\_{S \cap \partial\_j} \, d\mu\_p$$

$$= \sum\_{j \in \Lambda\_S} \mu\_p \left( S \cap \partial\_j \right)$$

by (6)

$$\leq \sum\_{j \in \Lambda\_S} \mu\_p \left( \partial\_j \right) = \sum\_{j \in \Lambda\_S} \left( \frac{1}{p^j} - \frac{1}{p^{j+1}} \right) \, \, \, \tag{8}$$

by (4), for all *S* ∈ *<sup>σ</sup>*(Q*p*), where Λ*S* is in the sense of (7).

**Proposition 1.** *Let S* ∈ *<sup>σ</sup>*(Q*p*), *and let χS* ∈ M*p*. *Then, there exist rj* ∈ R, *such that:*

$$0 \le r\_{\dot{\jmath}} \le 1 \\ \text{in} \mathbb{R}, \text{for all} \dot{\jmath} \in \Lambda\_{\heartsuit} \tag{9}$$

.

*and:*

$$\int\_{\mathbb{Q}\_p} \chi\_S \, d\mu\_p = \sum\_{j \in \Lambda\_S} r\_j \left( \frac{1}{p^j} - \frac{1}{p^{j+1}} \right).$$

**Proof.** The existence of *rj* = *<sup>μ</sup>p*(*<sup>S</sup>*∩*∂j*) *<sup>μ</sup>p*(*<sup>∂</sup>j*) , for all *j* ∈ Z, is guaranteed by (7) and (8). The *p*-adic integral in (9) is obtained by (8).

### **3. Free-Probabilistic Model on** M*p*

Throughout this section, fix a prime *p* ∈ P, and let Q*p* be the corresponding *p*-adic number field and M*p* be the ∗-algebra (5) consisting of *<sup>μ</sup>p*-measurable functions on Q*<sup>p</sup>*. Here, we establish a suitable (non-traditional) free-probabilistic model on M*p* implying *p*-adic analytic data.

> Let *Uk* be the basis elements (1) of the topology for Q*p* with their boundaries *∂k* of (3), i.e.,

$$\mathcal{U}\_k = p^k \mathbb{Z}\_{p\_\prime} for all \, k \in \mathbb{Z}\_{\prime} \tag{10}$$

and:

$$
\partial\_k = \mathcal{U}\_k \text{ \\
\u{\urcorner}}\\ 
\mathcal{U}\_{k+1\prime} \text{ for all } k \in \mathbb{Z}.
$$

Define a linear functional *ϕp* : M*p* → C by the *p*-adic integration (6),

$$\mathfrak{op}\_p(f) = \int\_{\mathbb{Q}\_p} f d\mu\_{p\prime} for all f \in \mathcal{M}\_p. \tag{11}$$

Then, by (9) and (11), one obtains:

$$\varphi\_p\left(\chi\_{\mathcal{U}\_{\vec{\jmath}}}\right) = \frac{1}{p^{\vec{\jmath}}} \text{ and } \varphi\_p\left(\chi\_{\partial\_{\vec{\jmath}}}\right) = \frac{1}{p^{\vec{\jmath}}} - \frac{1}{p^{\vec{\jmath}+1}},$$

for all *j* ∈ Z.

**Definition 1.** *We call the pair* -M*p*, *<sup>ϕ</sup>p the p-adic (non-traditional) free probability space for p* ∈ P, *where ϕp is the linear functional* (11) *on* M*p*.

**Remark 1.** *As we discussed in Section 2.1, we study the measure-theoretic structure* -M*p*, *<sup>ϕ</sup>p as a free-probabilistic model on* M*p for our purposes. Therefore, without loss of generality, we regard* -M*p*, *<sup>ϕ</sup>p as a non-traditional free-probabilistic structure. In this sense, we call* -M*p*, *<sup>ϕ</sup>p the p-adic free probability space for p*. *The readers can understand* -M*p*, *<sup>ϕ</sup>p as the pair of a commutative* ∗*-algebra* M*p and a linear functional ϕp*, *having as its name the p-adic free probability space.*

Let *∂k* be the *k*th boundary *Uk* \ *Uk*+<sup>1</sup> of *Uk* in Q*<sup>p</sup>*, for all *k* ∈ Z. Then, for *k*1, *k*2 ∈ Z, one obtains that:

$$
\chi\_{\mathfrak{d}\_{k\_1}} \chi\_{\mathfrak{d}\_{k\_2}} = \chi\_{\mathfrak{d}\_{k\_1} \cap \mathfrak{d}\_{k\_2}} = \delta\_{k\_1 k\_2} \chi\_{\mathfrak{d}\_{k\_1} \mathcal{I}}
$$

by (4), and hence,

$$\begin{split} \left(\boldsymbol{\varrho}\_{p}\left(\chi\_{\boldsymbol{\eth}\_{k\_{1}}}\chi\_{\boldsymbol{\eth}\_{k\_{2}}}\right)\right) &= \boldsymbol{\delta}\_{k\_{1},k\_{2}}\boldsymbol{\varrho}\_{p}\left(\chi\_{\boldsymbol{\eth}\_{k\_{1}}}\right) \\ &= \boldsymbol{\delta}\_{k\_{1},k\_{2}}\left(\frac{1}{p^{k\_{1}}}-\frac{1}{p^{k\_{1}+1}}\right), \end{split} \tag{12}$$

where *δ* is the *Kronecker delta*.

**Proposition 2.** *Let* (*j*1, *..., jN*) ∈ <sup>Z</sup>*N*, *for N* ∈ N. *Then:*

$$\prod\_{l=1}^{N} \mathsf{X}\_{\mathsf{\eth}\_{\hat{l}\_l}} = \delta\_{(\hat{j}\_1, \dots, \hat{j}\_N)} \mathsf{X}\_{\mathsf{\eth}\_{\hat{j}\_1}} \text{ in } \mathsf{\mathcal{M}}\_{p^\*}$$

*and hence,*

$$\mathfrak{sp}\_p \left( \prod\_{l=1}^N \chi\_{\partial\_{j\_l}} \right) = \delta\_{(j\_1, \dots, j\_N)} \left( \frac{1}{p^{j\_1}} - \frac{1}{p^{j\_1+1}} \right),\tag{13}$$

*where:*

$$
\delta\_{(j\_1,\dots,j\_N)} = \left(\prod\_{l=1}^{N-1} \delta\_{\hat{j}l,\hat{j}l+1}\right) \left(\delta\_{\hat{j}N,\hat{j}1}\right).
$$

**Proof.** The proof of (13) is done by induction on (12).

Thus, one can ge<sup>t</sup> that, for any *S* ∈ *σ* -Q*p* ,

$$\mathfrak{q}\_p\left(\chi\_S\right) = \mathfrak{q}\_p\left(\sum\_{j \in \Lambda\_S} \chi\_{S \cap \mathfrak{d}\_j}\right) \tag{14}$$

where Λ*S* is in the sense of (7).

$$\begin{aligned} \mathcal{I} &= \sum\_{j \in \Lambda\_S} \wp\_p \left( \chi\_{S \cap \mathfrak{d}\_j} \right) = \sum\_{j \in \Lambda\_S} \mu\_p \left( \mathcal{S} \cap \partial\_j \right) \\ &= \sum\_{j \in \Lambda\_S} r\_j \left( \frac{1}{p^j} - \frac{1}{p^{j+1}} \right) \end{aligned} \tag{15}$$

by (13), where 0 ≤ *rj* ≤ 1 are in the sense of (9) for all *j* ∈ Λ*S*. Furthermore, if *S*1, *S*2∈ *σ* -Q*p* , then:

$$\begin{aligned} \chi\_{\mathbf{S}\_1 \mathbf{X} \mathbf{S}\_2} &= \left( \sum\_{k \in \Lambda\_{S\_1}} \chi\_{S\_1 \cap \mathfrak{d}\_k} \right) \left( \sum\_{j \in \Lambda\_{S\_2}} \chi\_{S\_2 \cap \mathfrak{d}\_j} \right) \\ &= \sum\_{(k,j) \in \Lambda\_{S\_1} \times \Lambda\_{S\_2}} \delta\_{k,j} \chi\_{(S\_1 \cap S\_2) \cap \mathfrak{d}\_j} \\ &= \sum\_{j \in \Lambda\_{S\_1} \times \mathfrak{d}\_2} \chi\_{(S\_1 \cap S\_2) \cap \mathfrak{d}\_{j'}} \end{aligned} \tag{16}$$

where:

$$
\Lambda\_{\mathcal{S}\_1, \mathcal{S}\_2} = \Lambda\_{\mathcal{S}\_1} \cap \Lambda\_{\mathcal{S}\_2}
$$

.

**Proposition 3.** *Let Sl* ∈ *<sup>σ</sup>*(Q*p*), *and let χSl* ∈ -M*p*, *<sup>ϕ</sup>p* , *for l* = 1, *..., N*, *for N* ∈ N. *Let:*

$$
\Lambda\_{\mathcal{S}\_1,\dots,\mathcal{S}\_N} = \bigcap\_{l=1}^N \Lambda\_{\mathcal{S}\_l} \text{ in } \mathbb{Z}\_r
$$

*where* <sup>Λ</sup>*Slare in the sense of* (7)*, for l* = 1, *..., N*. *Then, there exist rj* ∈ R, *such that:*

$$0 \le r\_{\rangle} \le 1 \text{ in } \mathbb{R}, \text{ for } j \in \Lambda\_{\mathbb{S}\_1, \dots, \mathbb{S}\_{N'}}$$

*and:* 
$$\varphi\_p \left( \prod\_{l=1}^N \chi\_{S\_l} \right) = \sum\_{j \in \Lambda\_{S\_1, \dots, S\_N}} r\_j \left( \frac{1}{p^j} - \frac{1}{p^{j+1}} \right) \,. \tag{17}$$

**Proof.** The proof of (17) is done by induction on (16) with the help of (15).

### **4. Representations of** -M*p*, *<sup>ϕ</sup>p*

Fix a prime *p* in P, and let -M*p*, *<sup>ϕ</sup>p* be the *p*-adic free probability space. By understanding Q*p* as a measure space, construct the *<sup>L</sup>*2-*space Hp*,

$$H\_p \stackrel{def}{=} L^2\left(\mathbb{Q}\_{p\prime} \,\,\sigma(\mathbb{Q}\_p), \,\,\mu\_p\right) = L^2\left(\mathbb{Q}\_p\right),\tag{18}$$

over C. Then, this *L*2-space *Hp* of (18) is a well-defined *Hilbert space* equipped with its *inner product* <, >2,

$$\langle h\_1, h\_2 \rangle\_2 \stackrel{def}{=} \int\_{\mathbb{Q}\_p} h\_1 h\_2^\* d\mu\_{p\prime} \tag{19}$$

for all *h*1, *h*2 ∈ *Hp*.

**Definition 2.** *We call the Hilbert space Hp of* (18)*, the p-adic Hilbert space.*

By the definition (18) of the *p*-adic Hilbert space *Hp*, our ∗-algebra M*p* acts on *Hp*, via an *algebra-action <sup>α</sup>p*,

$$a^p(f)\left(h\right) = fh\_\prime for allh \in H\_{p\prime} \tag{20}$$

for all *f* ∈ M*p*.

**Notation:** Denote *α<sup>p</sup>*(*f*) of (20) by *αpf* , for all *f* ∈ M*p*. Furthermore, for convenience, denote *<sup>α</sup>pχS* simply by *αpS*, for all *S* ∈ *σ* -Q*p* . -

By (20), the linear morphism *αp* is indeed a well-determined ∗-algebra-action of M*p* acting on *Hp* (equivalently, every *αpf* is a ∗-homomorphism from M*p* into the operator algebra *B*(*Hp*) of all bounded operators on *Hp*, for all *f* ∈ M*p*), since:

$$\begin{aligned} \mathfrak{a}\_{f\_1 f\_2}^p(h) &= f\_1 f\_2 h = f\_1 \,(f\_2 h) \\ &= f\_1 \left( \mathfrak{a}\_{f\_2}^p(h) \right) = \mathfrak{a}\_{f\_1}^p \mathfrak{a}\_{f\_2}^p(h), \end{aligned}$$

for all *h* ∈ *Hp*, implying that:

*αpf*1 *f*2 = *<sup>α</sup>pf*1*<sup>α</sup>pf*2 , (21)

for all *f*1, *f*2 ∈ M*p*; and:

$$\begin{aligned} \left< \mathfrak{a}\_f^p(h\_1), h\_2 \right>\_{\mathfrak{b}} &= \left< fh\_1, h\_2 \right>\_{\mathfrak{b}} = \int\_{\mathbb{Q}\_p} fh\_1 h\_2^\* d\mu\_p \\ &= \int\_{\mathbb{Q}\_p} h\_1 f h\_2^\* d\mu\_p = \int\_{\mathbb{Q}\_p} h\_1 \left( h\_2 f^\* \right)^\* d\mu\_p \\ &= \int\_{\mathbb{Q}\_p} h\_1 \left( f^\* h\_2 \right)^\* d\mu\_p = \left< h\_1, \mathfrak{a}\_{f^\*}^p(h\_2) \right>\_{\mathfrak{b}}. \end{aligned}$$

for all *h*1, *h*2 ∈ *Hp*, for all *f* ∈ M*p*, implying that:

$$\left(a\_f^p\right)^\* = a\_{f^\*}, for all \, f \in \mathcal{M}\_{p\_\prime} \tag{22}$$

where <, >2 is the inner product (19) on *Hp*.

**Proposition 4.** *The linear morphism αp of* (20) *is a well-defined* ∗*-algebra-action of* M*p acting on Hp*. *Equivalently, the pair* (*Hp*, *α<sup>p</sup>*) *is a Hilbert-space representation of* M*p*.

**Proof.** The proof is done by (21) and (22).

**Definition 3.** *The Hilbert-space representation* -*Hp*, *α<sup>p</sup> is said to be the p-adic representation of* M*p*.

Depending on the *p*-adic representation (*Hp*, *α<sup>p</sup>*) of M*p*, one can construct the *C*<sup>∗</sup>-*subalgebra Mp* of the operator algebra *<sup>B</sup>*(*Hp*).

**Definition 4.** *Define the C*∗*-subalgebra Mp of the operator algebra B*(*Hp*) *by:*

$$\mathcal{M}\_p \stackrel{def}{=} \overline{\mathfrak{a}^p \left( \mathcal{M}\_p \right)} = \overline{\mathbb{C} \left[ \mathfrak{a}\_f^p \;:\; f \in \mathcal{M}\_p \right]}\;\tag{23}$$

*where X mean the operator-norm closures of subsets X of <sup>B</sup>*(*Hp*). *Then, this C*∗*-algebra Mp is called the p-adic C*∗*-algebra of the p-adic free probability space* -M*p*, *<sup>ϕ</sup>p* .

### **5. Free-Probabilistic Models on** *Mp*

Throughout this section, let us fix a prime *p* ∈ P, and let -M*p*, *<sup>ϕ</sup>p* be the corresponding *p*-adic free probability space. Let -*Hp*, *α<sup>p</sup>* be the *p*-adic representation of M*p*, and let *Mp* be the *p*-adic *C*∗-algebra (23) of -M*p*, *<sup>ϕ</sup>p* .

We here construct suitable free-probabilistic models on *Mp*. In particular, we are interested in a system {*ϕpj* }*j*∈<sup>Z</sup> of *linear functionals on Mp*, determined by the *j*th boundaries {*∂j*}*j*∈<sup>Z</sup> of Q*<sup>p</sup>*.

Define a linear functional *ϕpj*: *Mp* → C by a linear morphism,

$$\left. \mathcal{P}\_{\dot{j}}^{p} \left( a \right) \stackrel{\text{def}}{=} \left\langle a \left( \chi\_{\mathfrak{d}\_{\dot{j}}} \right) \,, \left. \chi\_{\mathfrak{d}\_{\dot{j}}} \right\rangle\_{2} \right\rangle \right. \tag{24}$$

for all *a* ∈ *Mp*, for all *j* ∈ Z, where <, >2 is the inner product (19) on the *p*-adic Hilbert space *Hp* of (18). Remark that if *a* ∈ *Mp*, then:

$$a = \sum\_{S \in \sigma \left( \mathbb{Q}\_p \right)} t\_S \ a\_{S'}^p \text{ in } M\_{\mathbb{P}}$$

(with *tS* ∈ C), where ∑ is a finite or infinite (i.e., limit of finite) sum(s) under the *C*<sup>∗</sup>-topology for *Mp*. Thus, the linear functionals *ϕpj* of (24) are well defined on *Mp*, for all *j* ∈ Z, i.e., for any fixed *j* ∈ Z, one has that:

$$\begin{aligned} \left| \boldsymbol{\varphi}\_{j}^{p}(\boldsymbol{a}) \right| &= \left| \sum\_{S \in \boldsymbol{r}\mathcal{I}\left(\mathcal{Q}\_{\mathcal{P}}\right)} t\_{S} \left< \boldsymbol{\chi}\_{S \cap \partial\_{j} \boldsymbol{\epsilon}}, \mathcal{X} \boldsymbol{\mathfrak{a}}\_{j} \right>\_{2} \right| \\ &= \left| \sum\_{S \in \boldsymbol{r}\mathcal{I}\left(\mathcal{Q}\_{\mathcal{P}}\right)} t\_{S} \mu\_{p} \left( \boldsymbol{\chi}\_{S \cap \partial\_{j}} \right) \right| \\ &\leq \mu\_{p} \left( \partial\_{j} \right) \left| \sum\_{S \in \boldsymbol{r}\mathcal{I}\left(\mathcal{Q}\_{\mathcal{P}}\right)} t\_{S} \right| \leq \left( \frac{1}{p^{j}} - \frac{1}{p^{j+1}} \right) \left\| \boldsymbol{a} \right\|, \end{aligned} \tag{25}$$

where:

$$||a|| = \sup\left\{ ||a(h)||\_2 : h \in H\_p \text{ with } ||h||\_2 = 1 \right\}$$

is the *C*<sup>∗</sup>-norm on *Mp* (inherited by the operator norm on the operator algebra *B*(*Hp*)), and .<sup>2</sup> is the Hilbert-space norm,

$$\|f\|\_2 = \sqrt{\langle f, f \rangle\_{2'}} \,\forall f \in H\_{p'}$$

induced by the inner product <, >2 of (19). Therefore, for any fixed integer *j* ∈ Z, the corresponding linear functional *ϕpj*of (24) is bounded on *Mp*.

**Definition 5.** *Let j* ∈ Z, *and let ϕpj be the linear functional* (24) *on the p-adic C*∗*-algebra Mp*. *Then, the pair Mp*, *ϕpj is said to be the jth p-adic (non-traditional) C*<sup>∗</sup>*-probability space.*

**Remark 2.** *As in Section 4, the readers can understand the pairs Mp*, *ϕpj simply as structures consisting of a commutative C*∗*-algebra Mp and linear functionals ϕpj on Mp*, *whose names are jth p-adic C*<sup>∗</sup>*-probability spaces for all j* ∈ Z, *for p* ∈ P.

Fix *j* ∈ Z, and take the corresponding *j*th *p*-adic *C*<sup>∗</sup>-probability space *Mp*, *ϕpj* . For *S* ∈ *σ* -Q*p* and a generating operator *αpS*of *Mp*, one has that:

$$\begin{array}{rcl} \mathcal{O}\_{\boldsymbol{\upbeta}}^{p}(\boldsymbol{a}\_{\mathcal{S}}^{p}) &= \left< \boldsymbol{a}\_{\mathcal{S}}^{p}(\boldsymbol{\upchi}\_{\boldsymbol{\upbeta}\_{\boldsymbol{\upbeta}}}), \, \boldsymbol{\upchi}\_{\boldsymbol{\upbeta}\_{\boldsymbol{\upbeta}}} \right>\_{2} = \left< \boldsymbol{\upchi}\_{\boldsymbol{\upbeta} \cap \boldsymbol{\upbeta}\_{\boldsymbol{\upbeta}}} \, \boldsymbol{\upchi}\_{\boldsymbol{\upbeta}\_{\boldsymbol{\upbeta}}} \right>\_{2} \\ &= \int\_{\mathbb{Q}\_{p}} \boldsymbol{\upchi}\_{\boldsymbol{S} \cap \boldsymbol{\upbeta}\_{\boldsymbol{\upbeta}}} \boldsymbol{\upchi}\_{\boldsymbol{\upbeta}\_{\boldsymbol{\upbeta}}}^{\*} d\boldsymbol{\upmu}\_{p} = \int\_{\mathbb{Q}\_{p}} \boldsymbol{\upchi}\_{\boldsymbol{S} \cap \boldsymbol{\upbeta}\_{\boldsymbol{\upbeta}}} \boldsymbol{\upchi}\_{\boldsymbol{\upbeta}\_{\boldsymbol{\upbeta}}} d\boldsymbol{\upmu}\_{p} \end{array} \tag{26}$$

by (19)

$$\begin{array}{l} = \int\_{\mathbb{Q}\_p} \chi\_{\mathcal{S}\cap\partial\_{\hat{j}}} d\mu\_p = \mu\_p \left( \mathcal{S}\cap\partial\_{\hat{j}} \right) \\ = r\_{\mathcal{S}} \left( \frac{1}{p^{\prime}} - \frac{1}{p^{j+1}} \right), \end{array} \tag{27}$$

for some 0 ≤ *rS* ≤ 1 in R, for *S* ∈ *σ* -Q*p* .

**Proposition 5.** *Let S* ∈ *σ* -Q*p and αpS* = *<sup>α</sup>pχS* ∈ *Mp*, *ϕpj* , *for a fixed j* ∈ Z. *Then, there exists rS* ∈ R, *such that:*

$$0 \le r\_{\mathcal{S}} \le 1 \text{ in } \mathbb{R},$$

*and:*

$$\sigma\_j^p\left(\left(a\_S^p\right)^n\right) = r\_S \left(\frac{1}{p^j} - \frac{1}{p^{j+1}}\right), for all \ n \in \mathbb{N}.\tag{28}$$

**Proof.** Remark that the generating operator *αpS*is a projection in *Mp*, in the sense that:

$$\left(\boldsymbol{\alpha}\_{\boldsymbol{S}}^{p}\right)^{\*} = \boldsymbol{\alpha}\_{\boldsymbol{S}}^{p} = \left(\boldsymbol{\alpha}\_{\boldsymbol{S}}^{p}\right)^{2}, \text{ in } M\_{p}\boldsymbol{\omega}$$

so,

$$\left(\boldsymbol{\alpha}\_{\boldsymbol{S}}^{p}\right)^{n} = \boldsymbol{a}\_{\boldsymbol{S}'}^{p} \text{ for all } n \in \mathbb{N}.$$

Thus, for any *n* ∈ N, we have:

$$\rho\_{\dot{j}}^{p}\left(\left(a\_{S}^{p}\right)^{n}\right) = \rho\_{\dot{j}}^{p}(a\_{S}^{p}) = r\_{S}\left(\frac{1}{p^{\prime}} - \frac{1}{p^{\prime + 1}}\right),$$

for some 0 ≤ *rS* ≤ 1 in R, by (27).

> As a corollary of (28), one obtains the following corollary.

**Corollary 1.** *Let ∂k be the kth boundaries* (10) *of* Q*<sup>p</sup>*, *for all k* ∈ Z. *Then:*

$$\sigma\_{\dot{\gamma}}^{p}\left(\left(\mathbf{a}\_{\partial\_{k}}^{p}\right)^{\mathrm{tr}}\right) = \delta\_{\dot{\gamma},k}\left(\frac{1}{p^{j}} - \frac{1}{p^{j+1}}\right) \tag{29}$$

*for all n* ∈ N, *for all j* ∈ Z.

**Proof.** The formula (29) is shown by (28).

**6. Semigroup** *C*∗**-Subalgebras** <sup>S</sup>*p* **of** *Mp*

> Let *Mp* be the *p*-adic *C*∗-algebra (23) for an arbitrarily-fixed *p* ∈ P. Take operators:

$$P\_{p,j} = \mathfrak{a}\_{\mathfrak{d}\_j}^p \in \mathcal{M}\_{p,} \tag{30}$$

where *∂j* are the *j*th boundaries (10) of Q*<sup>p</sup>*, for the fixed prime *p*, for all *j* ∈ Z.

Then, these operators *Pp*,*<sup>j</sup>* of (30) are *projections* on the *p*-adic Hilbert space *Hp* in *Mp*, i.e.,

$$P\_{p\_{\vec{\sigma}}j}^{\*\*} = P\_{p\_{\vec{\sigma}}j} = P\_{p\_{\vec{\sigma}}j''}^2$$

for all *j* ∈ Z. We now restrict our interest to these projections *Pp*,*<sup>j</sup>* of (30).

**Definition 6.** *Fix p* ∈ P. *Let* <sup>S</sup>*p be the C*∗*-subalgebra:*

$$\mathfrak{S}\_{\mathcal{P}} = \mathbb{C}^\* \left( \{ P\_{p,j} \}\_{j \in \mathbb{Z}} \right) = \overline{\mathbb{C} \left[ \{ P\_{p,j} \}\_{j \in \mathbb{Z}} \right]} \operatorname{of} \mathcal{M}\_{\mathcal{P}} \tag{31}$$

*where Pp*,*<sup>j</sup> are projections* (30)*, for all j* ∈ Z. *We call this C*∗*-subalgebra* <sup>S</sup>*p the p-adic boundary (C*∗*-)subalgebra of Mp.*

The *p*-adic boundary subalgebra <sup>S</sup>*p* of the *p*-adic *C*∗-algebra *Mp* satisfies the following structure theorem.

**Proposition 6.** *Let* <sup>S</sup>*p be the p-adic boundary subalgebra* (31) *of the p-adic C*∗*-algebra Mp*. *Then:*

$$\mathfrak{S}\_p \stackrel{\*}{=} \stackrel{\*}{\underset{j \in \mathbb{Z}}{\rightleftarrows}} \left( \mathbb{C} \cdot P\_{p,j} \right) \stackrel{\*}{=} \mathbb{C}^{\oplus \left| \mathbb{Z} \right|} \text{.}\tag{32}$$

*in Mp*. **Proof.** The proof of (32) is done by the mutual orthogonality of the projections {*Pp*,*j*}*j*∈<sup>Z</sup> in *Mp*. Indeed, one has:

$$P\_{p,j\_1}P\_{p,j\_2} = \mathfrak{a}^p\_{\partial\_{j\_1}}\mathfrak{a}^p\_{\partial\_{j\_2}} = \mathfrak{a}^p\_{\partial\_{j\_1}\cap\partial\_{j\_2}} = \delta\_{j\_1,j\_2}P\_{p,j\_1}\mathfrak{a}$$

in <sup>S</sup>*p*, for all *j*1, *j*2 ∈ Z.

Define now linear functionals *ϕpj*(for a fixed prime *p*) by:

$$\left|\varphi\_{j}^{(p)}\right\rangle = \varphi\_{j}^{p}\left|\_{\mathfrak{S}\_{p}} \, on \mathfrak{S}\_{p}\right.\tag{33}$$

where *ϕpj*in the right-hand side of (33) are the linear functionals (24) on *Mp*, for all *j* ∈ Z.

### **7. Weighted-Semicircular Elements**

Let *Mp* be the *p*-adic *C*∗-algebra, and let <sup>S</sup>*p* be the *p*-adic boundary subalgebra (31) of *Mp*, satisfying the structure theorem (32). Fix *p* ∈ P. Recall that the generating projections *Pp*,*<sup>j</sup>* of <sup>S</sup>*p* satisfy:

$$\left(\boldsymbol{\varrho}\_{j}^{(p)}\right)\left(\boldsymbol{P}\_{p,j}\right) = \frac{1}{p^{j}} - \frac{1}{p^{j+1}}, \forall j \in \mathbb{Z},\tag{34}$$

by (33) (also see (28) and (29)).

> Now, let *φ* be the *Euler totient function*, an *arithmetic function:*

$$
\phi: \mathbb{N} \to \mathbb{C}\_\*
$$

defined by:

$$\phi(n) = \left| \{ k \in \mathbb{N} : k \le n, \gcd(n, k) = 1 \} \right|,\tag{35}$$

for all *n* ∈ N, where |*X*| mean the cardinalities of sets *X* and gcd is the *greatest common divisor*.

It is well known that:

$$\phi(n) = n \left( \prod\_{\emptyset \in \mathcal{P}\_I \neq \emptyset} \left( 1 - \frac{1}{\eta} \right) \right),$$

for all *n* ∈ N, where "*q* | *n*" means "*q* divides *<sup>n</sup>*." For instance,

$$
\phi(p) = p - 1 = p \left( 1 - \frac{1}{p} \right), \forall p \in \mathcal{P}. \tag{36}
$$

Thus:

$$\begin{array}{l} \, \rho\_{\hat{j}}^{(p)} \left( P\_{p,\hat{j}} \right) & = \left( \frac{1}{p^{\circ}} - \frac{1}{p^{\circ + 1}} \right) = \frac{1}{p^{\circ}} \left( 1 - \frac{1}{p} \right) \\\\ & = \frac{p}{p^{\circ + 1}} \left( 1 - \frac{1}{p} \right) = \frac{\rho(p)}{p^{\circ + 1}} \end{array}$$

by (34), (35), and (36), for all *Pp*,*<sup>j</sup>* ∈ <sup>S</sup>*p*. More generally,

$$\log\_{\hat{\jmath}}^{(p)}\left(P\_{p,k}\right) = \delta\_{\hat{\jmath},k}\left(\frac{\phi(p)}{p^{j+1}}\right), \forall p \in \mathcal{P}, k \in \mathbb{Z}.\tag{37}$$

Now, for a fixed prime *p*, define new linear functionals *τpj* on <sup>S</sup>*p*, by linear morphisms satisfying that:

$$
\pi\_j^p = \frac{1}{\phi(p)} \varphi\_j^{(p)} \, , \, on \mathfrak{S}\_{p\prime} \tag{38}
$$

for all *j* ∈ Z, where *ϕpj* are in the sense of (33). Then, one obtains new (non-traditional) *C*<sup>∗</sup>-probabilistic structures,

$$\{\mathfrak{S}\_{\mathcal{P}}(j) = \left(\mathfrak{S}\_{\mathcal{P}}, \,\,\tau\_{j}^{p}\right) : p \in \mathcal{P}, j \in \mathbb{Z}\},\tag{39}$$

where *τpj* are in the sense of (38).

**Proposition 7.** *Let* <sup>S</sup>*p*(*j*)=(<sup>S</sup>*p*, *τpj* ) *be in the sense of* (39)*, and let Pp*,*<sup>k</sup> be generating operators of* <sup>S</sup>*p*(*j*), *for p* ∈ P, *j* ∈ Z. *Then:*

$$\pi\_j^p \left( P\_{p,k}^{\mathfrak{u}} \right) = \frac{\delta\_{j,k}}{p^{j+1}}, for all n \in \mathbb{N}. \tag{40}$$

**Proof.** The formula (40) is proven by (37) and (38). Indeed, since *Pp*,*<sup>k</sup>* are projections in <sup>S</sup>*p*(*j*),

$$\pi\_{\!\!\!\!\!\!}^{p}\left(P\_{p,k}^{n}\right) = \pi\_{\!\!\!\!\!\!\/}^{p}\left(P\_{p,k}\right) = \delta\_{\!\!\!\!\/}k\left(\frac{1}{p^{\!\!\!\!\!\!\/+\!\!\!\!\/}\right),$$

for all *n* ∈ N, for all *p* ∈ P, and *j*, *k* ∈ Z.

### *7.1. Semicircular and Weighted-Semicircular Elements*

Let (*<sup>A</sup>*, *ϕ*) be an arbitrary (traditional or non-traditional) *topological* <sup>∗</sup>-*probability space* (*C*<sup>∗</sup>-probability space, or *<sup>W</sup>*<sup>∗</sup>-probability space, or Banach ∗-probability space, etc.), consisting of a (noncommutative, resp., commutative) topological ∗-algebra *A* (*C*∗-algebra, resp., *W*∗-algebra, resp., Banach ∗-algebra, etc.), and a (bounded or unbounded) linear functional *ϕ* on *A*.

**Definition 7.** *Let a be a self-adjoint element in* (*<sup>A</sup>*, *ϕ*). *It is said to be even in* (*<sup>A</sup>*, *ϕ*)*, if all odd free moments of a vanish, i.e.,*

$$\varrho\left(a^{2n-1}\right) = 0, for all n \in \mathbb{N}.\tag{41}$$

*Let a be a "self-adjoint," and "even" element of* (*<sup>A</sup>*, *ϕ*) *satisfying* (41)*. Then, it is said to be semicircular in* (*<sup>A</sup>*, *ϕ*), *if:*

$$
\varphi(a^{2n}) = \mathfrak{c}\_{n,\prime} \, for all n \in \mathbb{N},\tag{42}
$$

*where ck are the kth Catalan number,*

$$c\_k = \frac{1}{k+1} \left( \begin{array}{c} 2k \\ k \end{array} \right) = \frac{1}{k+1} \frac{(2k)!}{(k!)^2} = \frac{(2k)!}{k!(k+1)!}$$

*for all k* ∈ N0 = N ∪ {0}.

It is well known that, if *kn*(...) is the *free cumulant on A in terms of a linear functional ϕ* (in the sense of [14]), then a self-adjoint element *a* is *semicircular* in (*<sup>A</sup>*, *ϕ*), if and only if:

$$k\_n \left( \underbrace{a\_{\text{\textquotedblleft}a\_{\text{\textquotedblleft}b\_{\text{\textquotedblleft}c\_{\text{\textquotedblleft}b\_{\text{\textquotedblright}c\_{\text{\textquotedblright}}a}}}} \right)} \right) = \left\{ \begin{array}{cc} 1 & \text{if } n = 2 \\ 0 & \text{otherwise} \end{array} \right. \tag{43}$$

for all *n* ∈ N (e.g., see [14]). The above equivalent free-distributional data (43) of the semicircularity (42) are obtained by the *Möbius inversion of* [14].

Motivated by (43), one can define the *weighted-semicircularity*.

**Definition 8.** *Let a* ∈ (*<sup>A</sup>*, *ϕ*) *be a self-adjoint element. It is said to be weighted-semicircular in* (*<sup>A</sup>*, *ϕ*) *with its weight t*0 *(in short, t*0*-semicircular), if there exists t*0 ∈ C× = C \ {0}, *such that:*

$$k\_n \left( \underbrace{a\_\prime \, a\_\prime \ldots a\_\prime}\_{n\text{-times}} a \right) = \begin{cases} \, \, t\_0 & \text{if } n = 2\\ 0 & \, \, otherwise, \end{cases} \tag{44}$$

*for all n* ∈ N, *where kn*(...) *is the free cumulant on A in terms of ϕ.*

By the definition (44) and by the Möbius inversion of [14], one obtains the following free-moment characterization (45) of the weighted-semicircularity (44): A self-adjoint element *a* is *t*0-*semicircular* in (*<sup>A</sup>*, *ϕ*), if and only if there exists *t*0 ∈ C<sup>×</sup>, such that:

$$
\varphi(a^n) = \omega\_n \, t\_0^{\frac{\mu}{2}} \varepsilon\_{\frac{\mu}{2}}.
$$

$$
\omega\_n = \begin{cases}
1 & \text{if } n \text{ is even} \\
0 & \text{if } n \text{ is odd}
\end{cases}
\tag{45}
$$

where:

for all *n* ∈ N, where *cm* are the *m*th Catalan numbers for all *m* ∈ N0.

Thus, from below, we use the weighted-semicircularity (44) and its characterization (45) alternatively.

### *7.2. Tensor Product Banach* ∗*-Algebra* LS*p*

Let <sup>S</sup>*p*(*k*) = <sup>S</sup>*p*, *τ<sup>p</sup> k* be a (non-traditional) *C*<sup>∗</sup>-probability space (39), for *p* ∈ P, *k* ∈ Z. Define *bounded linear transformations* **c***p* and **<sup>a</sup>***p* "acting on the *p*-adic boundary subalgebra <sup>S</sup>*p* of *Mp*," by linear morphisms satisfying,

**c***p* -*Pp*,*<sup>j</sup>* = *Pp*,*j*+1, **<sup>a</sup>***p* - *Pp*,*<sup>j</sup>* = *Pp*,*j*−1, (46)

and:

on <sup>S</sup>*p*, for all *j* ∈ Z.

By (46), these linear transformations **c***p* and **<sup>a</sup>***p* are bounded under the operator-norm induced by the *C*<sup>∗</sup>-norm on <sup>S</sup>*p*. Therefore, the linear transformations **c***p* and **<sup>a</sup>***p* are regarded as Banach-space operators "acting on <sup>S</sup>*p*," by regarding <sup>S</sup>*p* as a Banach space (under its *C*∗-norm). i.e., **c***p* and **<sup>a</sup>***p* are elements of the *operator space B* - <sup>S</sup>*p* consisting of all bounded operators on the Banach space <sup>S</sup>*p*.

**Definition 9.** *The Banach-space operators* **c***p and* **<sup>a</sup>***p of* (46) *are called the p-creation, respectively, the p-annihilation on* <sup>S</sup>*p*, *for p* ∈ P. *Define a new Banach-space operator lp* ∈ *B* - <sup>S</sup>*p by:*

$$l\_p = \mathbf{c}\_p + \mathbf{a}\_p on \mathfrak{S}\_p. \tag{47}$$

*We call it the p-radial operator on* <sup>S</sup>*p*.

Let *lp* be the *p*-radial operator **c***p* + **<sup>a</sup>***p* of (47) on <sup>S</sup>*p*. Construct a *closed subspace* L*p* of *<sup>B</sup>*(<sup>S</sup>*p*) by:

$$\mathfrak{L}\_p = \overleftarrow{\mathbb{C}[l\_p]} \subset B(\mathfrak{S}\_p),\tag{48}$$

where *Y* means the operator-norm-topology closure of every subset *Y* of the operator space *<sup>B</sup>*(<sup>S</sup>*p*).

By the definition (48), L*p* is not only a closed subspace, but also a well-defined Banach algebra embedded in the vector space *<sup>B</sup>*(<sup>S</sup>*p*). On this Banach algebra <sup>L</sup>*p*, define the adjoint (∗) by:

$$\sum\_{k=0}^{\infty} s\_k l\_p^k \in \mathfrak{L}\_p \longmapsto \sum\_{k=0}^{\infty} \overline{s\_k} l\_p^k \in \mathfrak{L}\_{p'} \tag{49}$$

where *sk* ∈ C with their conjugates *sk* ∈ C.

Then, equipped with the adjoint (49), this Banach algebra L*p* of (48) forms a *Banach* ∗*-algebra* inside *<sup>B</sup>*(<sup>S</sup>*p*).

**Definition 10.** *Let* L*p be a Banach* ∗*-algebra* (48) *in the operator space <sup>B</sup>*(<sup>S</sup>*p*) *for p* ∈ P. *We call it the p-radial (Banach-*∗*-)algebra on* <sup>S</sup>*p*.

Let L*p* be the *p*-radial algebra (48) on <sup>S</sup>*p*. Construct now the tensor product Banach ∗-algebra LS*p* by:

$$\mathfrak{VC}\_p = \mathfrak{C}\_p \otimes\_{\mathbb{C}} \mathfrak{S}\_{p\prime} \tag{50}$$

where ⊗C is the *tensor product of Banach* ∗*-algebras* (Remark that <sup>S</sup>*p* is a *C*∗-algebra and L*p* is a Banach ∗-algebra; and hence, the tensor product Banach ∗-algebra LS*p* of (50) is well-defined.).

Take now a generating element *lkp* ⊗ *Pp*,*j*, for some *k* ∈ N0 = N ∪ {0}, and *j* ∈ Z, where *Pp*,*<sup>j</sup>* is in the sense of (30) in <sup>S</sup>*p*, with axiomatization:

$$I\_p^0 = \mathbf{1}\_{\mathfrak{S}\_{p'}} \text{ the identity operator on } \mathfrak{S}\_{p'}$$

in *<sup>B</sup>*(<sup>S</sup>*p*), satisfying:

$$1\_{\mathfrak{S}\_p} \left( P\_{p,j} \right) = P\_{p,j}, \text{ for all } P\_{p,j} \in \mathfrak{S}\_{p,j}$$

for all *j* ∈ Z.

> By (50) and (32), the elements *lkp* ⊗ *Pp*,*<sup>j</sup>* indeed generate LS*p* under linearity, because:

$$\left(l\_p \otimes P\_{p,j}\right)^k = l\_p^k \otimes P\_{p,j,r}$$

for all *k* ∈ N0, and *j* ∈ Z, for *p* ∈ P, and their self-adjointness. We now focus on such generating operators of LS*p*.

Define a linear morphism:

$$E\_P: \mathfrak{LSet}\_P \to \mathfrak{S}\_P$$

by a linear transformation satisfying that:

$$E\_p\left(l\_p^k \otimes P\_{p,j}\right) = \frac{\left(p^{j+1}\right)^{k+1}}{\left[\frac{k}{2}\right] + 1} l\_p^k (P\_{p,j})\_\prime \tag{51}$$

for all *k* ∈ N0, *j* ∈ Z, where *k*2 is the *minimal integer greater than or equal to k*2 , for all *k* ∈ N0; for example,

$$\left[\frac{3}{2}\right] = 2 = \left[\frac{4}{2}\right].$$

.

By the cyclicity (48) of the tensor factor L*p* of LS*p*, and by the structure theorem (32) of the other tensor factor <sup>S</sup>*p* of LS*p*, the above morphism *Ep* of (51) is a well-defined bounded surjective linear transformation.

Now, consider how our *p*-radial operator *lp* of (47) works on <sup>S</sup>*p*. Observe first that: if **c***p* and **<sup>a</sup>***p* are the *p*-creation, respectively, the *p*-annihilation on <sup>S</sup>*p*, then:

$$\mathbf{c}\_{\mathcal{P}} \mathbf{a}\_{\mathcal{P}} \left( P\_{p,j} \right) = P\_{p,j} = \mathbf{a}\_{\mathcal{P}} \mathbf{c}\_{\mathcal{P}} \left( P\_{p,j} \right) \, \prime$$

for all *j* ∈ Z, *p* ∈ P, and hence:

$$\mathbf{c}\_{p}\mathbf{a}\_{p} = \mathbf{1}\_{\mathfrak{S}\_{p}} = \mathbf{a}\_{p}\mathbf{c}\_{p}om\mathfrak{S}\_{p}.\tag{52}$$

**Lemma 1.** *Let* **<sup>c</sup>***p*, **<sup>a</sup>***p be the p-creation, respectively, the p-annihilation on* <sup>S</sup>*p*. *Then:*

$$\mathbf{c}\_{\mathcal{P}}^{\boldsymbol{n}} \mathbf{a}\_{\mathcal{P}}^{\boldsymbol{n}} = \left(\mathbf{c}\_{\mathcal{P}} \mathbf{a}\_{\mathcal{P}}\right)^{\boldsymbol{n}} = \mathbf{1}\_{\mathbb{S}\_{\mathcal{P}}} = \left(\mathbf{a}\_{\mathcal{P}} \mathbf{c}\_{\mathcal{P}}\right)^{\boldsymbol{n}} = \mathbf{a}\_{\mathcal{P}} \mathbf{c}\_{\mathcal{P}}.$$

*and:*

$$\mathbf{c}\_{p}^{n\_1}\mathbf{a}\_{p}^{n\_2} = \mathbf{a}\_{p}^{n\_2}\mathbf{c}\_{p}^{n\_1} on \mathfrak{S}\_{p, \prime} \tag{53}$$

*for all n*, *n*1, *n*2 ∈ N0.

**Proof.** The formula (53) holds by (52).

> By (53), one can ge<sup>t</sup> that:

$$d\_p^n = \left(\mathbf{c}\_p + \mathbf{a}\_p\right)^n = \sum\_{k=0}^n \binom{n}{k} \mathbf{c}\_p^k \mathbf{a}\_p^{n-k} \text{ on } \mathfrak{S}\_{p^n}$$

with identities;

$$\mathbf{c}\_{p}^{0} = \mathbf{1}\_{\mathfrak{S}\_{p}} = \mathbf{a}\_{p}^{0} \tag{54}$$

for all *n* ∈ N, where:

$$
\binom{n}{k} = \frac{n!}{k!(n-k)!}, \forall k \le n \in \mathbb{N}\_0.
$$

Thus, one obtains the following proposition.

**Proposition 8.** *Let lp* ∈ L*p be the p-radial operator on* <sup>S</sup>*p*. *Then:*

$$d\_p^{2m-1} \text{ does not contain } \mathbf{1}\_{\mathfrak{S}\_p} - term, \text{ and} \tag{55}$$

$$\mathbf{1}\_p^{2m} \text{ contains its } \mathbf{1}\_{\mathfrak{S}\_p} - term\_\prime \left(\begin{array}{c} 2m \\ m \end{array}\right) \cdot \mathbf{1}\_{\mathfrak{S}\_{p'}} \tag{56}$$

*for all m* ∈ N.

**Proof.** The proofs of (55) and (56) are done by straightforward computations by (53) and (54). See [1] for more details.

### *7.3. Weighted-Semicircular Elements Qp*,*<sup>j</sup> in* LS*p*

Fix *p* ∈ P, and let LS*p* = L*p* ⊗C <sup>S</sup>*p* be the tensor product Banach ∗-algebra (50) and *Ep* be the linear transformation (51) from LS*p* onto <sup>S</sup>*p*. Throughout this section, fix a generating element:

$$Q\_{p,j} = l\_p \circledast P\_{p,j} \text{of } \mathfrak{S} \mathfrak{S}\_{p,j} \tag{57}$$

for *j* ∈ Z, where *Pp*,*<sup>j</sup>* is a projection (30) generating <sup>S</sup>*p*. Observe that:

$$Q\_{p,j}^{n} = \left(l\_p \odot P\_{p,j}\right)^{n} = l\_p^n \odot P\_{p,j}.\tag{58}$$

for all *n* ∈ N, for all *j* ∈ Z.

> If *Qp*,*<sup>j</sup>* ∈ LS*p* is in the sense of (57) for *j* ∈ Z, then:

$$E\_p\left(Q\_{p,j}^n\right) = E\_p\left(l\_p^n \odot P\_{p,j}\right) = \frac{\left(p^{j+1}\right)^{n+1}}{\left[\frac{n}{2}\right] + 1} l\_p^n \left(P\_{p,j}\right),\tag{59}$$

by (51) and (58), for all *n* ∈ N.

> Now, for a fixed *j* ∈ Z, define a linear functional *τ*0*p*,*j* on LS*p* by:

$$
\pi^0\_{p,j} = \pi^p\_j \circ E\_p \\
on \mathfrak{S}\_{p,j} \tag{60}
$$

where *τpj* = 1 *φ*(*p*) *ϕ*(*p*) *j* is the linear functional (38) on <sup>S</sup>*p*.

By the bounded-linearity of both *τpj* and *Ep*, the morphism *τ*0*p*,*j* of (60) is a bounded linear functional on LS*p*.

By (59) and (60), if *Qp*,*<sup>j</sup>* is in the sense of (57), then:

$$
\pi\_{p,j}^{0}\left(Q\_{p,j}^{n}\right)\_{-}=\frac{\left(p^{j+1}\right)^{n+1}}{\left[\frac{n}{2}\right]+1}\pi\_{j}^{p}\left(l\_{p}^{n}(P\_{p,j})\right),\tag{61}
$$

for all *n* ∈ N.

**Theorem 1.** *Let Qp*,*<sup>j</sup>* = *lp* ⊗ *Pp*,*<sup>j</sup>* ∈ LS*p*, *τ*0*p*,*j*, *for a fixed j* ∈ Z. *Then, Qp*,*<sup>j</sup> is p*<sup>2</sup>(*j*+<sup>1</sup>)*-semicircular in* LS*p*, *τ*0*p*,*j* . *More precisely,*

$$\pi\_{p,j}^0 \left( Q\_{p,j}^{\mathbb{N}} \right) = \omega\_{\mathbb{R}} \left( p^{2(j+1)} \right)^{\frac{n}{2}} c\_{\frac{n}{2}\prime} \tag{62}$$

*for all n* ∈ N*, where ωn are in the sense of* (45)*. Equivalently, if <sup>k</sup>*0,*p*,*<sup>j</sup> n* (...) *is the free cumulant on* LS*p in terms of the linear functional τ*0*p*,*j of* (61) *on* LS*p, then:*

$$k\_n^{0,p,j} \left( \underbrace{\mathcal{Q}\_{p,j\prime} \, \mathcal{Q}\_{p,j\prime} \dots \mathcal{Q}\_{p,j}}\_{n\text{-times}} \right) = \begin{cases} \, \, p^{2(j+1)} & \text{if } n = 2\\ 0 & \text{otherwise} \end{cases} \tag{63}$$

*for all n* ∈ N.

**Proof.** The free-moment formula (62) is obtained by (55), (56) and (61). The free-cumulant formula (63) is obtained by (62) under the Möbius inversion of [14]. See [1] for details.

### **8. Semicircularity on** LS

For all *p* ∈ P, *j* ∈ Z, let:

$$\mathfrak{LSE}\_p(j) = \left(\mathfrak{LSE}\_{p\prime}, \tau^0\_{p,j}\right) \tag{64}$$

be a Banach ∗-probabilistic model of the Banach ∗-algebra LS*p* of (50), where *τ*0*p*,*j* is the linear functional (60).

**Definition 11.** *We call the pairs* LS*p*(*j*) *of* (64) *the jtextth p*-adic filters*, for all p* ∈ P, *j* ∈ Z.

Let *Qp*,*<sup>k</sup>* = *lp* ⊗ *Pp*,*<sup>k</sup>* be the *k*th generating elements of the *j*th *p*-adic filter LS*p*(*j*) of (64), for all *k* ∈ Z, for fixed *p* ∈ P, *j* ∈ Z. Then, the generating elements {*Qp*,*<sup>k</sup>*}*<sup>k</sup>*∈<sup>Z</sup> of the *j*th *p*-adic filter LS*p*(*j*) satisfy that:

$$k\_n^{0,p,j} \left( \mathbb{Q}\_{p,k'} \dots \mathbb{Q}\_{p,k} \right) = \begin{cases} \delta\_{j,k} \ p^{2(j+1)} & \text{if } n = 2\\ 0 & \text{otherwise} \end{cases}$$

and:

$$\pi\_{p,j}^{0}\left(Q\_{p,k}^{\mathfrak{n}}\right) = \delta\_{j,k}\left(\omega\_{\mathfrak{n}}\left(p^{2(j+1)}\right)^{\frac{\mathfrak{n}}{2}}c\_{\frac{\mathfrak{n}}{2}}\right),\tag{65}$$

for all *p* ∈ P, *j* ∈ Z, for all *n* ∈ N, by (62) and (63), where:

$$
\omega\_{\mathbb{H}} = \begin{cases} 1 & \text{if } n \text{ is even} \\\ 0 & \text{if } n \text{ is odd} \end{cases}
$$

for all *n* ∈ N.

> For the family:

$$\left\{ \mathfrak{L}\mathfrak{S}\_p(j) = \left( \mathfrak{L}\mathfrak{S}\_{p\prime} \, \, \pi^0\_{p,j} \right) : p \in \mathcal{P}\_{\prime} \, \, j \in \mathbb{Z} \right\}$$

of *j*th *p*-adic filters of (64), one can define the *free product Banach* ∗*-probability space*,

$$\mathfrak{L}\mathfrak{S}\stackrel{\text{denot}}{=}\left(\mathfrak{L}\mathfrak{S},\,\,\tau^{0}\right)\stackrel{\text{def}}{=}\limits\_{p\in\mathcal{P}\_{\*},j\in\mathbb{Z}}\mathfrak{L}\mathfrak{S}\_{p}(j).\tag{66}$$

as in [14,15], with:

$$\mathfrak{GE} = \underset{p \in \mathcal{P}\_{\prime}}{\star}\_{\text{p} \in \mathbb{Z}} \mathfrak{GE}\_{p\prime} \text{ and } \mathfrak{r}^0 = \underset{p \in \mathcal{P}\_{\prime}}{\star}\_{\text{p} \in \mathbb{Z}} \mathfrak{r}^0\_{p\_{\prime}j}.$$

Note that the pair LS = -LS, *τ*<sup>0</sup> of (66) is a well-defined "traditional or noncommutative" Banach ∗-probability space. For more about the (free-probabilistic) *free product* of free probability spaces, see [14,15].

**Definition 12.** *The Banach* ∗*-probability space* LS = -LS, *τ*<sup>0</sup> *of* (66) *is called the free Adelic filtration.*

Let LS be the free Adelic filtration (66). Then, by (65), one can take a subset:

$$\mathcal{Q} = \left\{ Q\_{p,j} = l\_p \otimes P\_{p,j} \in \mathfrak{L}\mathfrak{S}\_p(j) \right\}\_{p \in \mathcal{P}, \, j \in \mathbb{Z}}$$

of LS, consisting of "*j*th" generating elements *Qp*,*<sup>j</sup>* of the "*j*th" *p*-adic filters LS*p*(*j*), which are the free blocks of LS, for all *j* ∈ Z, for all *p* ∈ P.

**Lemma 2.** *Let* Q *be the above family in the free Adelic filtration* LS. *Then, all elements Qp*,*<sup>j</sup> of* Q *are p*<sup>2</sup>(*j*+<sup>1</sup>)*-semicircular in the free Adelic filtration* LS.

**Proof.** Since all self-adjoint elements *Qp*,*<sup>j</sup>* of the family Q are chosen from mutually-distinct *free blocks* LS*p*(*j*) of LS, they are *p*<sup>2</sup>(*j*+<sup>1</sup>)-semicircular in LS*p*(*j*). Indeed, since every element *Qp*,*<sup>j</sup>* ∈ Q is from a free block LS*p*(*j*), the powers *Qnp*,*<sup>j</sup>* are free reduced words with their lengths-*N* in LS*p*(*j*) in LS. Therefore, each element *Qp*,*<sup>j</sup>* ∈ Q satisfies that:

$$\pi^0\left(\mathbb{Q}^n\_{p,j}\right) = \pi^0\_{p,j}\left(\mathbb{Q}^n\_{p,j}\right) = \omega\_n p^{n(j+1)}\mathbb{c}\_{\frac{n}{2}\prime}$$

equivalently,

$$\begin{aligned} k\_n^0 \left( \mathcal{Q}\_{p,j} \; \cdots \; \mathcal{Q}\_{p,j} \right) &= k\_n^{0,p,j} \left( \mathcal{Q}\_{p,j} \; \cdots \; \mathcal{Q}\_{p,j} \right) \\ &= \begin{cases} p^{2(j+1)} & \text{if } n = 2 \\ 0 & \text{otherwise} \end{cases} \end{aligned}$$

for all *n* ∈ N, by (62) and (63), where *<sup>k</sup>*0*n*(...) is the free cumulant on LS in terms of *τ*0. Therefore, all elements *Qp*,*<sup>j</sup>* ∈ Q are *p*<sup>2</sup>(*j*+<sup>1</sup>)-semicircular in LS, for all *p* ∈ P, *j* ∈ Z.

Furthermore, since all *p*<sup>2</sup>(*j*+<sup>1</sup>)-semicircular elements *Qp*,*<sup>j</sup>* ∈ Q are taken from the mutually-distinct free blocks LS*p*(*j*) of LS, they are mutually free from each other in the free Adelic filtration LS of (66), for all *p* ∈ P, *j* ∈ Z.

Recall that a subset *S* = {*at*}*<sup>t</sup>*∈<sup>Δ</sup> of an arbitrary (topological or pure-algebraic) ∗-probability space (*<sup>A</sup>*, *ϕ*) is said to be a *free family*, if, for any pair (*<sup>t</sup>*1, *<sup>t</sup>*2) ∈ Δ<sup>2</sup> of *t*1 = *t*2 in a countable (finite or infinite) index set Δ, the corresponding free random variables *at*1 and *at*2 are free in (*<sup>A</sup>*, *ϕ*) (e.g., [7,14]).

**Definition 13.** *Let S* = {*at*}*<sup>t</sup>*∈<sup>Δ</sup> *be a free family in an arbitrary topological* ∗*-probability space* (*<sup>A</sup>*, *ϕ*). *This family S is said to be a free (weighted-)semicircular family, if it is not only a free family, but also a set consisting of all (weighted-)semicircular elements at in* (*<sup>A</sup>*, *ϕ*), *for all t* ∈ Δ.

Therefore, by the construction (66) of the free Adelic filtration LS, we obtain the following result.

**Theorem 2.** *Let* LS *be the free Adelic filtration* (66)*, and let:*

$$\mathcal{Q} = \{Q\_{p,j} \in \mathfrak{LSet}\_p(j)\}\_{p \in \mathcal{P}, \, j \in \mathbb{Z}} \subset \mathfrak{LSet},\tag{67}$$

*where* LS*p*(*j*) *are the jth p-adic filters, the free blocks of* LS. *Then, this family* Q *of* (67) *is a free weighted-semicircular family in* LS.

**Proof.** Let Q be a subset (67) in LS. Then, all elements *Qp*,*<sup>j</sup>* of Q are *p*<sup>2</sup>(*j*+<sup>1</sup>)-semicircular in LS by the above lemma, for all *p* ∈ P, *j* ∈ Z. Furthermore, all elements *Qp*,*<sup>j</sup>* of Q are mutually free from each other in LS, because they are contained in the mutually-distinct free blocks LS*p*(*j*) of LS, for all *p* ∈ P, *j* ∈ Z. Therefore, the family Q of (67) is a free weighted-semicircular family in LS.

Now, take elements:

$$\Theta\_{p,j} \stackrel{def}{=} \frac{1}{p^{j+1}} \mathcal{Q}\_{p,j}, \forall p \in \mathcal{P}, j \in \mathbb{Z},\tag{68}$$

in LS, where *Qp*,*<sup>j</sup>* ∈ Q, where Q is the free weighted-semicircular family (67) in the free Adelic filtration LS.

Then, by the self-adjointness of *Qp*,*j*, these operators <sup>Θ</sup>*p*,*<sup>j</sup>* of (68) are self-adjoint in LS, as well, because:

$$p^{j+1} \in \mathbb{Q} \subset \mathbb{R} \text{ in } \mathbb{C}\_{\prime}$$

satisfying *pj*+<sup>1</sup> = *pj*+1, for all *p* ∈ P, *j* ∈ Z.

Furthermore, one obtains the following free-cumulant computation; if *<sup>k</sup>*0*n*(...) is the free cumulant on LS in terms of *τ*0, then:

$$\begin{array}{rcl}k\_n^0 \left(\Theta\_{p,j}, \ldots, \Theta\_{p,j}\right) &=& k\_n^{0, p, j} \left(\frac{1}{p^{j+1}} Q\_{p,j}, \ldots, \frac{1}{p^{j+1}} Q\_{p,j}\right) \\ &=& \left(\frac{1}{p^{j+1}}\right)^n k\_n^{0, p, j} \left(Q\_{p,j}, \ldots, Q\_{p,j}\right), \end{array} \tag{69}$$

by the *bimodule-map property* of the free cumulant (e.g., see [14]), for all *n* ∈ N, where *<sup>k</sup>*0,*p*,*<sup>j</sup> n* (...) are the free cumulants (63) on the free blocks LS*p*(*j*) in terms of the linear functionals *τ*0*p*,*j* of (60) on LS*p*, for all *p* ∈ P, *j* ∈ Z.

**Theorem 3.** *Let* <sup>Θ</sup>*p*,*<sup>j</sup>* = 1 *pj*+<sup>1</sup> *Qp*,*<sup>j</sup> be free random variables* (68) *of the free Adelic filtration* LS, *for Qp*,*<sup>j</sup>* ∈ Q. *Then, the family:*

$$\Theta = \left\{ \Theta\_{p,j} \in \mathfrak{L}\mathfrak{S}\_p(j) : p \in \mathcal{P}, \ j \in \mathbb{Z} \right\} \tag{70}$$

*forms a free semicircular family in* LS.

**Proof.** Consider that:

$$\begin{aligned} &k\_{\mathfrak{n}}^{0}\left(\Theta\_{p,j\_{\varproj}}\ldots\Theta\_{p,j}\right) = \left(\frac{1}{p^{j+1}}\right)^{\mathfrak{n}} k\_{\mathfrak{n}}^{0,p,j}\left(Q\_{p,j\_{\varproj}}\ldots\bigtriangleup\_{p,j}\right) \text{ by (69)}\\ &= \begin{cases} \left(\frac{1}{p^{j+1}}\right)^{2} k\_{2}^{0,p,j}\left(Q\_{p,j\_{\varproj}}\ldots Q\_{p,j}\right) & \text{if } n=2\\\\ \left(\frac{1}{p^{j+1}}\right)^{\mathfrak{n}} k\_{\mathfrak{n}}^{0,p,j}\left(Q\_{p,j\_{\varproj}}\ldots Q\_{p,j}\right) = 0 & \text{otherwise.} \end{cases} \end{aligned}$$

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

$$\mathbf{x} = \begin{cases} \left(\frac{1}{p^{j+1}}\right)^2 \left(p^{j+1}\right)^2 = 1 & \text{if } n = 2\\ 0 & \text{otherwise,} \end{cases} \tag{71}$$

for all *n* ∈ N.

By the free-cumulant computation (71), these self-adjoint free random variables <sup>Θ</sup>*p*,*<sup>j</sup>* ∈ LS*p*(*j*) are semicircular in LS by (43), for all *p* ∈ P, *j* ∈ Z.

Furthermore, the family Θ of (70) forms a free family in LS, because all elements <sup>Θ</sup>*p*,*<sup>j</sup>* are the scalar-multiples of *Qp*,*<sup>j</sup>* ∈ Q, contained in mutually-distinct free blocks LS*p*(*j*) of LS, for all *j* ∈ Z, *p* ∈ P.

Therefore, this family Θ of (70) is a free semicircular family in LS. Now, define a Banach ∗-subalgebra LS of LS by:

$$\mathbb{L}\mathfrak{S} \stackrel{def}{=} \overline{\mathbb{C}[\mathfrak{Q}]} in \mathfrak{S}\mathfrak{S},\tag{72}$$

where Q is the free weighted-semicircular family (67) and *Y* means the Banach-topology closures of subsets *Y* of LS.

Then, one can obtain the following structure theorem for the Banach ∗-algebra LS of (72) in LS.

**Theorem 4.** *Let* LS *be the Banach* ∗*-subalgebra* (72) *of the free Adelic filtration* LS *generated by the free weighted-semicircular family* Q *of* (67)*. Then:*

$$\mathbb{L}\mathbb{S} = \overline{\mathbb{C}\left[\Theta\right]} in \mathfrak{V} \mathfrak{S},\tag{73}$$

*where* Θ *is the free semicircular family* (70) *and where "*=*" means "being identically same as sets." Moreover,*

$$\mathbb{L}\mathbb{S} \stackrel{\ast \text{ iso}}{=} \underset{p \in \mathcal{P}, j \in \mathbb{Z}}{\text{\textbullet}} \overline{\mathbb{C}\left[\{Q\_{p,j}\}\right]} \stackrel{\ast \text{ iso}}{=} \mathbb{C}\left[\underset{p \in \mathcal{P}, j \in \mathbb{Z}}{\text{\textbullet}} \{Q\_{p,j}\}\right],\tag{74}$$

*in* LS, *where "*<sup>∗</sup>*-iso* = *" means "being Banach-*∗*-isomorphic," and:*

C{*Qp*,*j*} *are Banach* ∗*-subalgebras of* LS*p*(*j*),

*for all p* ∈ P, *j* ∈ Z, *in* LS.

*Here, (*-*) in the first* ∗*-isomorphic relation of* (74) *is the (free-probability-theoretic) free product of [14,15], and (*-*) in the second* ∗*-isomorphic relation of* (74) *is the (pure-algebraic) free product (generating noncommutative algebraic free words in the family* Q*).*

**Proof.** Let LS be the Banach ∗-subalgebra (72) of LS. Then, all generating operators *Qp*,*<sup>j</sup>* ∈ Q of LS are contained in mutually-distinct free blocks LS*p*(*j*) of LS, and hence, the Banach ∗-subalgebras C {*Qp*,*j*} of LS are contained in the free blocks LS*p*(*j*), for all *p* ∈ P, *j* ∈ Z. Therefore, as embedded sub-structures of LS, they are free from each other. Equivalently,

$$\mathsf{LE3} \stackrel{\ast \text{-iso}}{=} \underset{p \in \mathcal{P}, j \in \mathbb{Z}}{\mathsf{C}} \; \overrightarrow{\{\{Q\_{p,j}\}\}} in \mathfrak{S} \mathfrak{S}'} \tag{75}$$

by (66).

Since every free block C {*Qp*,*j*} of the Banach ∗-algebra LS of (75) is generated by a single self-adjoint (weighted-semicircular) element, every operator *T* of LS is a limit of linear combinations of free words in the free family Q of (67), which form noncommutative free "reduced" words (in the sense of [14,15]), as operators in LS of (75). Note that every (pure-algebraic) free word in Q has a unique free reduced word in LS, under operator-multiplication on LS (and hence, on LS). Therefore, the ∗-isomorphic relation (75) guarantees that:

$$\mathsf{LZ} \stackrel{\*\text{-iso}}{=} \overline{\mathbb{C}\left[\underset{p \in \mathcal{P}\_{\prime}}{\star}\_{j \in \mathbb{Z}} \{Q\_{p,j}\}\right]} \, ^{\prime} \tag{76}$$

where the free product (-) in (76) is pure-algebraic.

Remark that, indeed, the relation (76) holds well, because all weighted-semicircular elements of Q are self-adjoint; if:

$$T = \prod\_{l=1}^{N} \mathbb{Q}\_{p\_l, j\_l}^{n\_l} \in \mathbb{L} \mathbb{S}$$

is a free (reduced) word (as an operator), then:

$$T^\* = \prod\_{l=1}^N \mathbb{Q}\_{p\_{N-l+1}j\_{N-l+1}}^{n\_{N-l+1}} \in \mathbb{L}\mathbb{B}$$

is a free word of LS in Q, as well. Therefore, by (75) and (76), the structure theorem (74) holds true. Note now that *Qp*,*<sup>j</sup>*∈ Q satisfy:

$$Q\_{p,j} = p^{j+1} \Theta\_{p,j'} \text{ for all } p \in \mathcal{P}, j \in \mathbb{Z}\_{\prime}$$

where <sup>Θ</sup>*p*,*<sup>j</sup>* are semicircular elements in the family Θ of (70). Therefore, the free blocks of (75) satisfy that:

$$\overline{\mathbb{C}\left[\{Q\_{p,j}\}\right]} = \overline{\mathbb{C}\left[\{p^{j+1}\Theta\_{p,j}\}\right]} = \overline{\mathbb{C}\left[\{\Theta\_{p,j}\}\right]},\tag{77}$$

for all *p* ∈ P, *j* ∈ Z.

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

$$\mathbb{LE} \triangleq \mathop{\ast}\_{p \in \mathcal{P}, \, j \in \mathbb{Z}} \star \overbrace{\mathbb{C} \left[ \{ \Theta\_{p,j} \} \right]} \tag{78}$$

by (75) and (77).

> With similar arguments of (75), we have:

$$\mathbb{LN} = \overline{\mathbb{C}[\Theta]}, \text{set}-theoretically,\tag{79}$$

by (78).

> Therefore, the identity (73) holds true by (79).

As a sub-structure, one can restrict the linear functional *τ*0 of (66) on LS to that on LS, i.e., one can obtain the Banach ∗-probability space,

$$\left(\mathbb{L}\mathbb{S}, \; \tau^0 \stackrel{donde}{=} \; \tau^0 \mid\_{\mathbb{L}\mathbb{C}}\right). \tag{80}$$

**Definition 14.** *Let* (LS, *τ*<sup>0</sup>) *be the Banach* ∗*-probability space* (80)*. Then, we call* (LS, *τ*<sup>0</sup>) *the semicircular (free Adelic sub-)filtration of* LS.

Note that, by (66), all elements of the semicircular filtration (LS, *τ*<sup>0</sup>) provide "possible" non-vanishing free distributions in the free Adelic filtration LS. Especially, all free reduced words of LS in the generator set {*Qp*,*j*}*p*∈*P*,*j*∈<sup>Z</sup> have non-zero free distributions only if they are contained in (LS, *<sup>τ</sup>*<sup>0</sup>). Therefore, studying free-distributional data on (LS, *τ*<sup>0</sup>) is to study possible non-zero free-distributional data on LS.

### **9. Truncated Linear Functionals on** LS

In *number theory*, one of the most interesting, but difficult topics is to find a number of primes or a density of primes contained in closed intervals [*<sup>t</sup>*1, *t*2] of the real numbers R (e.g., [3,6,21,22]). Since the theory is deep, we will not discuss more about it here. Hhowever, motivated by the theory, we consider certain "suitable" *truncated linear functionals* on our semicircular filtration (LS, *τ*<sup>0</sup>) of (80) in the free Adelic filtration LS of (66).

**Notation:** From below, we will use the following notations to distinguish their structural differences;

> LS *denote* = the Banach ∗-subalgebra (72) of LS,

LS0 *denote* = the semicircular filtration -LS, *τ*<sup>0</sup> of (80).


### *9.1. Linear Functionals* {*<sup>τ</sup>*(*t*)}*<sup>t</sup>*∈<sup>R</sup> *on* LS

Let LS0 be the semicircular filtration (LS, *τ*<sup>0</sup>) of the free Adelic filtration LS. Furthermore, let Q and Θ be the free weighted-semicircular family (67), respectively, the free semicircular family (70) of LS,generating LS by(73)and(74).Weheretruncate *τ*0 on LS forafixedrealnumber *t* ∈ R.

First, recall and remark that:

$$\tau^0 = \underset{p \in \mathcal{P}\_{\text{-}}, j \in \mathbb{Z}}{\star} \tau^0\_{p,j} \text{ on } \mathbb{L}\mathbb{S}',$$

by (66) and (80). Therefore, one can sectionize *τ*0 over P, as follows;

$$
\tau^0 = \underset{p \in \mathcal{P}}{\star} \tau^0\_p \text{ on } \mathbb{L}\mathbb{S},
$$

with:

$$\tau\_p^0 = \underset{j \in \mathbb{Z}}{\star} \tau\_{p,j}^0 \\ on \mathbb{L} \mathbb{S}\_{p \prime} \\ for all \, p \in \mathcal{P}, \tag{81}$$

where:

$$\mathbb{L}\mathbb{S}\_p \stackrel{def}{=} \underset{j \in \mathbb{Z}}{\star} \overline{\mathbb{C}[\{\Theta\_{p,j}\}]} \subset \mathbb{L}\mathbb{S} \subset \mathfrak{L}\mathfrak{S},\tag{82}$$

for each *p* ∈ P, under (74).

> From below, we understand the Banach ∗-subalgebras LS*p* of LS as free-probabilistic sub-structures,

$$\mathbb{LS}\_{(p)} \stackrel{dueute}{=} \left( \mathbb{LS}\_{p\prime}, \tau\_p^0 \right), for all \, p \in \mathcal{P}. \tag{83}$$

**Lemma 3.** *Let* LS*pl be in the sense of* (82) *in the semicircular filtration* LS0, *for l* = 1, 2. *Then,* LS*p*1 *and* LS*p*2 *are free in* LS0, *if and only if p*1 = *p*2 *in* P.

**Proof.** The proof is directly done by (81) and (82). Indeed,

$$\begin{aligned} \mathbb{L}\mathbb{S}^{\mathbb{L}} &= \mathop{\star}\_{p \in \mathcal{P}, j \in \mathbb{Z}} \overline{\mathbb{C}\left[\{\Theta\_{p,j}\}\right]} \\ &= \mathop{\star}\_{p \in \mathcal{P}} \left( \mathop{\star}\_{j \in \mathbb{Z}} \overline{\mathbb{C}\left[\{\Theta\_{p,j}\}\right]} \right) = \mathop{\star}\_{p \in \mathcal{P}} \mathbb{L}\mathbb{S}\_{p\prime} \end{aligned}$$

by (80) and (82).

Therefore, LS*p*1and LS*p*2are free in LS0, if and only if *p*1 = *p*2 in P.

Fix now *t* ∈ R, and define a new linear functional *<sup>τ</sup>*(*t*) on LS by:

$$\pi\_{(t)} \stackrel{def}{=} \begin{cases} \begin{array}{l} \star \tau\_p^0 \\ p \le t \end{array} & \text{on } \begin{array}{l} \star \text{L\ $}\_p \subset \text{L\$ } \\\\ O & \text{on } \text{L\ $} \end{array} \Big| \begin{array}{l} \text{L\$ }\_p \subset \text{L\ $} \\\\ \begin{array}{l} \star \text{L\$ }\_p \end{array} \Big| \\\\ \end{array} \tag{84}$$

where *τ*0*p* are the linear functionals (81) on the Banach ∗-subalgebras LS*p* of (82) in LS0, for all *p* ∈ P, and *O* means the *zero linear functional* on LS, satisfying that:

$$O(T) = 0, \text{ for all } T \in \mathbb{L}\mathbb{S}.$$

For convenience, if there is no confusion, we simply write the definition (84) as:

$$
\pi\_{(t)} \stackrel{donde}{=} \underset{p \le t}{\star} \pi\_p^0. \tag{85}
$$

By the definition (84) (with a simpler expression (85)), one can easily verify that, if *t* < 2 in R, then the corresponding linear functional *<sup>τ</sup>*(*t*) is identical to the zero linear functional *O* on LS. To avoid such triviality, one may refine *<sup>τ</sup>*(*t*) of (84) by:

$$\pi\_{(t)} \stackrel{def}{=} \begin{cases} \quad \pi\_{(t)} \text{ of (84)} & \text{if } t \ge 2 \\ \text{O} & \text{if } t < 2, \end{cases} \tag{86}$$

for all *t* ∈ R.

In the following text, *<sup>τ</sup>*(*t*) mean the linear functionals in (86), satisfying (84) whenever *t* ≥ 2, for all *t* ∈ R. In fact, we are not interested in the cases where *t* < 2.

For example,

$$
\pi\_{(\frac{\omega\_5}{2})} = O, \tau\_{(2.1003)} = \tau\_{2'}^0 \, \_\prime \text{and} \\
\tau\_{(6)} = \tau\_2^0 \star \tau\_3^0 \star \tau\_5^0 \, \_\prime \text{and}
$$

on LS, under (85), etc.

**Theorem 5.** *Let Qp*,*<sup>j</sup>* ∈ Q *and* <sup>Θ</sup>*p*,*<sup>j</sup>* ∈ Θ *in the semicircular filtration* LS0*, for p* ∈ P, *j* ∈ Z, *and let t* ∈ R *and <sup>τ</sup>*(*t*), *the corresponding linear functional* (86) *on* LS*. Then:*

$$\pi\_{(t)}\left(Q\_{p,j}^{n}\right) = \begin{cases} \,\,\omega\_{n}p^{2(j+1)}\,^{c}\mathbb{1} & \text{if } t \ge p\\ 0 & \text{if } t < p \end{cases}$$

*and:*

$$\pi\_{(t)}\left(\Theta\_{p,j}^{\mathbb{N}}\right) = \begin{cases} \ \omega\_n \varepsilon\_{\frac{\mathbb{N}}{2}} & \text{if } t \ge p \\ 0 & \text{if } t < p\_r \end{cases} \tag{87}$$

*for all n* ∈ N.

**Proof.** By the *p*<sup>2</sup>(*j*+<sup>1</sup>)-semicircularity of *Qp*,*<sup>j</sup>* ∈ Q, the semicircularity of <sup>Θ</sup>*p*,*<sup>j</sup>* ∈ Θ in the semicircular filtration LS0, and by the definition (86), if *t* ≥ *p* in R, then:

$$\begin{aligned} \pi\_{(t)}\left(\mathcal{Q}\_{p,j}^{\mathfrak{n}}\right) &= \pi\_p^0 \left(\mathcal{Q}\_{p,j}^{\mathfrak{n}}\right) = \pi\_{p,j}^0 \left(\mathcal{Q}\_{p,j}^{\mathfrak{n}}\right) \\ &= \omega\_n p^{2\binom{j+1}{j+1}} \mathcal{C}\_{\frac{\mathfrak{n}}{2}'} \end{aligned}$$

and:

$$\begin{aligned} \pi\_{(t)}\left(\Theta^{\mathfrak{n}}\_{p,j}\right) &= \pi\_p^0 \left(\Theta^{\mathfrak{n}}\_{p,j}\right) = \pi\_{p,j}^0 \left(\Theta^{\mathfrak{n}}\_{p,j}\right) \\ &= \omega\_{\mathfrak{n}} \mathfrak{c}\_{\frac{\mathfrak{n}}{2}} \end{aligned}$$

by (62), (71), and (81), for all *n* ∈ N. 

If *t* < *p*, then:

$$\tau\_{(t)} = \underset{2 \le q < t < p \text{ in } \mathcal{P}}{\star} \tau\_q^0 \text{ or } O\_\prime \text{ on } \mathbb{L}\mathbb{S}.$$

Therefore, in such cases,

$$\pi\_{(t)}\left(\mathbb{Q}^{\mathfrak{n}}\_{p,j}\right) = \pi\_{(t)}\left(\mathbb{G}^{\mathfrak{n}}\_{p,j}\right) = 0, \text{ for all } n \in \mathbb{N}\_{\omega}$$

by (84), (85), and (86).

> Therefore, the free-distributional data (87) for the linear functional *<sup>τ</sup>*(*t*) hold on LS.

The above theorem shows how the original free-probabilistic information on the semicircular filtration LS0 is affected by the new free-probabilistic models on LS, under "truncated" linear functionals *<sup>τ</sup>*(*t*) of *τ*0 on LS, for *t* ∈ R.

**Definition 15.** *Let <sup>τ</sup>*(*t*) *be the linear functionals* (86) *on* LS, *for t* ∈ R. *Then, the corresponding new Banach* ∗*-probability spaces,*

$$\mathbb{L}\mathbb{S}\_{(t)} \stackrel{dconte}{=} \left(\mathbb{L}\mathbb{S}, \,\,\tau\_{(t)}\right),\tag{88}$$

*are called the semicircular t-(truncated-)filtrations of* LS *(or, of* LS0*).*

Note that if *t* is "suitable" in the sense that "*<sup>τ</sup>*(*t*) = *O* on LS," then the free-probabilistic structure LS(*t*)of (88) is meaningful.

**Notation and Assumption 9.1** (**NA 9.1**, from below): In the following, we will say "*t* ∈ R is suitable," if the semicircular *t*-filtration "LS(*t*) of (88) is meaningful," in the sense that: *<sup>τ</sup>*(*t*) = *O* fully on LS. -

Now, let us consider the following concepts.

**Definition 16.** *Let* (*Ak*, *ϕk*) *be Banach* ∗*-probability spaces (or C*<sup>∗</sup>*-probability spaces, or W*<sup>∗</sup>*-probability spaces, etc.), for k* = 1, 2. *A Banach* ∗*-probability space* (*<sup>A</sup>*1, *ϕ*1) *is said to be free-homomorphic to a Banach* ∗*-probability space* (*<sup>A</sup>*2, *ϕ*2), *if there exists a bounded* ∗*-homomorphism:*

$$
\Phi: A\_1 \to A\_{2'}
$$

*such that:*

$$
\varphi\_2\left(\Phi(a)\right) = \varphi\_1\left(a\right),
$$

*for all a* ∈ *A*1. *Such a* ∗*-homomorphism* Φ *is called a free-homomorphism.*

*If* Φ *is both a* ∗*-isomorphism and a free-homomorphism, then* Φ *is said to be a free-isomorphism, and we say that* (*<sup>A</sup>*1, *ϕ*1) *and* (*<sup>A</sup>*2, *ϕ*2) *are free-isomorphic. Such a free-isomorphic relation is nothing but the equivalence in the sense of Voiculescu (e.g., [15]).*

By (87), we obtain the following free-probabilistic-structural theorem.

**Theorem 6.** *Let* LS*q* = *j*∈Z C {*Qq*,*j*} *be Banach* ∗*-subalgebras* (82) *of* LS, *for all q* ∈ P. *Let t* ∈ R *be suitable in the sense of NA 9.1 and* LS(*t*) *be the corresponding semicircular t-filtration* (88)*. Construct a Banach* ∗*-probability space* LS*<sup>t</sup> by a Banach* ∗*-probabilistic sub-structure of the semicircular filtration* LS0*,*

$$\mathbb{L}\mathbb{S}^{t} \stackrel{def}{=} \mathop{\star}\_{p \le t} \left( \mathbb{L}\mathbb{S}\_{p} \, \middle| \, \pi^{0}\_{p} \right) = \left( \mathop{\star}\_{p \le t} \mathbb{L}\mathbb{S}\_{p} \, \middle| \, \mathop{\star}\_{p \le t} \pi^{0}\_{p} \right),\tag{89}$$

*where τ*0*p* = *<sup>j</sup>*∈Z*τ*0*p*,*<sup>j</sup> are in the sense of* (81)*. Then:*

$$\mathbb{L}\mathbb{S}^t is free-homomorphism \mathbb{L}\mathbb{S}\_{(t)}.\tag{90}$$

**Proof.** Let LS(*t*) be the semicircular *t*-filtration (88) of LS, and let LS*<sup>t</sup>* be a Banach ∗-probability space (89), for a suitably fixed *t* ∈ R.

Define a bounded linear morphism:

$$
\Phi\_t: \mathbb{L}\mathbb{S}^t \to \mathbb{L}\mathbb{S}\_{(t)}.
$$

by the natural embedding map,

$$\Phi\_l\left(T\right) = \operatorname{Tin\mathbb{Z}}\_{(t)}\!\_{\prime} for\;\operatorname{all}\!T\in\mathbb{L}\mathbb{S}^{\dagger}.\tag{91}$$

Then, this morphism Φ*t* is an injective bounded ∗-homomorphism from LS*<sup>t</sup>* into LS(*t*), by (72), (75), (82), (89), and (91).

Therefore, one obtains that:

$$\pi\_{(t)}\left(\Phi(T)\right) = \pi\_{(t)}(T) = \left(\_{p \le t \text{ in } \mathcal{P}} \pi\_p^0\right)(T) = \pi^t(T),$$

for all *T* ∈ LS*<sup>t</sup>*, by (87).

It shows that the Banach ∗-probability space LS*<sup>t</sup>* of (89) is free-homomorphic to the semicircular *t*-filtration LS(*t*) of (88). Therefore, the statement (90) holds under the free-homomorphism Φ*t* of (91).

The above theorem shows that the Banach ∗-probability spaces LS*<sup>t</sup>* of (89) are free-homomorphic to the semicircular *t*-filtrations LS(*t*) of (88), for all *t* ∈ R. Note that it "seems" they are not free-isomorphic, because:

$$\left(\mathop{\star}\_{q \le t \text{ in } \mathcal{P}} \mathbb{E} \mathbb{S}\_q\right) \overset{\subset}{\asymp} \left(\mathop{\star}\_{p \in \mathcal{P}} \mathbb{E} \mathbb{S}\_p\right) = \mathbb{E} \mathbb{S}'$$

set-theoretically, for *t* ∈ R. However, we are not sure at this moment that they are free-isomorphic or not, because we have the similar difficulties discussed in [19].

**Remark 3.** *The famous main result of [19] says that: if L*(*Fn*) *are the free group factors (group von Neumann algebras) of the free groups Fn with n-generators, for all:*

$$n \in \mathbb{N}\_{>1}^{\infty} = (\mathbb{N} \backslash \{1\}) \cup \{\infty\}^{\prime}$$

*then either (I) or (II) holds true, where:*

> *(I) L*(*Fn*) ∗*-iso* = *<sup>L</sup>*(*<sup>F</sup>*∞), *for all n* ∈ <sup>N</sup><sup>∞</sup>>1,

*(II) <sup>L</sup>*(*Fn*1 ) ∗*-iso* = *<sup>L</sup>*(*Fn*2 ), *if and only if n*1 = *n*2 ∈ <sup>N</sup><sup>∞</sup>>1,

*where "*<sup>∗</sup>*-iso* = *" means "being W*∗*-isomorphic." Depending on the author's knowledge, he does not know which one is true at this moment.*

*We here have similar troubles. Under the similar difficulties, we are not sure at this moment that* LS*<sup>t</sup> and* LS(*t*) *(or* LS*<sup>t</sup> and* LS*) are* ∗*-isomorphic or not (and hence, free-isomorphic or not).*

*However, definitely,* LS*<sup>t</sup> is free-homomorphic "into"* LS(*t*) *in the semicircular filtration* LS0, *by the above theorem.*

The above free-homomorphic relation (90) lets us understand all

"non-zero" free distributions of free reduced words of LS(*t*) as those of LS*<sup>t</sup>*, for all *t* ∈ R, by the injectivity of a free-homomorphism Φ*t* of (91).

**Corollary 2.** *All free reduced words T of the semicircular t-filtration* LS(*t*) *in* Q ∪ Θ, *having non-zero free distributions, are contained in the Banach* ∗*-probability space* LS*<sup>t</sup> of* (89)*, whenever t is suitable. The converse holds true, as well.*

**Proof.** The proof of this characterization is done by (87), (89), and (90). In particular, the injectivity of the free-homomorphism Φ*t* of (91) guarantees that this characterization holds.

Therefore, whenever we consider a non-zero free-distribution having free reduced words *T* of semicircular *t*-filtrations LS(*t*), they are regarded as free random variables of the Banach ∗-probability spaces LS*<sup>t</sup>* of (89), for all suitable *t* ∈ R.

*9.2. Truncated Linear Functionals <sup>τ</sup>t*1<*t*2 *on* LS

In this section, we generalize the semicircular *t*-filtrations LS(*t*) by defining so-called *truncated linear functionals* on the Banach ∗-algebra LS.

Throughout this section, let [*<sup>t</sup>*1, *t*2] be a *closed interval* in R, satisfying:

$$|t\_1 - t\_2| \neq 0, \text{ for } t\_1 < t\_2 \in \mathbb{R}.$$

For such a fixed closed interval [*<sup>t</sup>*1, *<sup>t</sup>*2], define the corresponding linear functional *<sup>τ</sup>t*1<*t*2 on the semicircular filtration LS by:

$$\tau\_{t\_1$$

where *τ*0*p* are the linear functionals (81) on the Banach ∗-subalgebras LS*p* of (82) in LS, for *p* ∈ P. Similar to Section 9.1, if there is no confusion, then we simply write the definition (92) as:

$$
\pi\_{t\_1$$

To make the linear functionals *<sup>τ</sup>t*1<*t*2 of (92) be non-zero-linear functionals on LS, the interval [*<sup>t</sup>*1, *t*2] must be taken "suitably." For example,

$$\tau\_{t\_1$$

and:

$$\pi\_{8 \le 10} = O\_\prime \text{ } \pi\_{14 \le 16} = O\_\prime \text{ and } \pi\_{\frac{3}{2} \le \frac{3}{2}} = O\_\prime \text{ etc.}$$

but:

$$
\tau\_{\frac{3}{2} < 8} = \tau\_{(8)} = \tau\_2^0 \star \tau\_3^0 \star \tau\_5^0 \star \tau\_7^0
$$

and:

$$
\tau\_{7<14} = \tau\_{7}^{0} \star \tau\_{11}^{0} \star \tau\_{13'}^{0},
$$

under (93) on LS.

It is not difficult to check that the definition (92) of truncated linear functionals *<sup>τ</sup>t*1<*t*2 covers the definition of linear functionals *<sup>τ</sup>*(*t*) of (86). In particular, *<sup>τ</sup>*(*t*) is "suitable" in the sense of **NA 9.1**, then:

$$
\tau\_{(t)} = \tau\_{2 \le t} = \tau\_{s \le t\prime} \text{ for all } 2 \ge s \in \mathbb{R}.
$$

For our purposes, we will axiomatize:

$$
\tau\_{p$$

notationally, where *τ*0*p* are the linear functionals (81), for all *p* ∈ P, under (93). Remark that the very above axiomatized notations *<sup>τ</sup>p*<*p* will be used only when *p* are primes.

**Definition 17.** *Let* [*<sup>t</sup>*1, *t*2] *be a given interval in* R *and <sup>τ</sup>t*1<*t*2 *, the corresponding linear functional* (92) *on* LS. *Then, we call it the* [*<sup>t</sup>*1, *<sup>t</sup>*2]*(-truncated)-linear functional on* LS. *The corresponding Banach* ∗*-probability space:*

$$\mathbb{L}\mathbb{Z}\_{t\_1 < t\_2} = (\mathbb{L}\mathbb{Z}, \mathbb{1}\_{t\_1 < t\_2})\tag{94}$$

*is said to be the semicircular a* [*<sup>t</sup>*1, *<sup>t</sup>*2]*(-truncated)-filtration.*

As we discussed in the above paragraphs, the semicircular [*<sup>t</sup>*1, *<sup>t</sup>*2]-filtration LS*<sup>t</sup>*1<*t*2 of (94) will be "meaningful," if *t*1 < *t*2 are suitable in R, as in **NA 9.1**.

**Notation and Assumption 9.2** (**NA 9.2**, from below): In the rest of this paper, if we write "*t*1 < *t*2 are suitable," then this means "LS*<sup>t</sup>*1<*t*2 is meaningful," in the sense that *<sup>τ</sup>t*1<*t*2 = *O* fully on LS, with additional axiomatization:

$$\mathfrak{r}\_{p$$

in the sense of (93). -

**Theorem 7.** *Let t*1 ≤ 2 *and t*2 *be suitable in* R *in the sense of NA 9.1.*

$$\text{The semicircular}[t\_1, t\_2] \text{-fibration} \mathbb{L} \mathbb{S}\_{t\_1 < t\_2} \text{ is not only suitable in the sense of } \mathsf{NA}. \mathsf{9.2}, \text{ but also, it is free-isomorphic to the semicircular} \text{-fibration } \mathbb{L} \mathbb{S}\_{(t\_2)} \text{ of (88).}$$

$$The\ Banach\ \ast-probability\ space\ \mathbb{L}\mathbb{S}^{t\_2}\ of\ (89)\ is\ free-homomorphism\ to\ \mathbb{L}\mathbb{S}\_{t\_1$$

**Proof.** Suppose *t*1 ≤ 2, and *t*2 are suitable in R in the sense of **NA 9.1**. Then, *t*1 < *t*2 are suitable in R in the sense of **NA 9.2**. Therefore, both the semicircular *t*2-filtration LS(*<sup>t</sup>*2) and the semicircular [*<sup>t</sup>*1, *<sup>t</sup>*2]-filtration LS*<sup>t</sup>*1<*t*2are meaningful.

Since *t*1 is assumed to be less than or equal to two, the linear functional *<sup>τ</sup>t*1<*t*2 = *<sup>τ</sup>*(*<sup>t</sup>*2), by (86) and (92), including the case where *τ*2<2 = *τ*02, in the sense of (93). Therefore,

$$\mathbb{L}\mathbb{S}\_{t\_1 \le t\_2} = (\mathbb{L}\mathbb{S}, \ \pi\_{t\_1 \le t\_2}) = \left(\mathbb{L}\mathbb{S}, \ \pi\_{(t\_2)}\right) = \mathbb{L}\mathbb{S}^{(t\_2)}.$$

Therefore, LS*<sup>t</sup>*1<*t*2 and LS(*<sup>t</sup>*2) are free-isomorphic under the identity map on LS, acting as a free-isomorphism. Therefore, the statement (95) holds.

By (90), the Banach ∗-probability space LS*<sup>t</sup>*2 of (89) is free-homomorphic to LS(*<sup>t</sup>*2). Therefore, under the hypothesis, LS*<sup>t</sup>*2 is free-homomorphic to LS*<sup>t</sup>*1<*t*2 by (95). Equivalently, the statement (96) holds.

The above theorem characterizes the free-probabilistic structures for semicircular [*<sup>t</sup>*1, *<sup>t</sup>*2]-filtrations LS*<sup>t</sup>*1<*t*2 , whenever *t*1 ≤ 2, and *t*2 are suitable, by (95) and (96). Therefore, we now restrict our interests to the cases where:

> *t*1 ≥ 2 in R.

Therefore, we focus on the semicircular [*<sup>t</sup>*1, *<sup>t</sup>*2]-filtration LS*<sup>t</sup>*1<*t*2 , where:

$$2 \le t\_1 < t\_2 \text{ are suitable in } \mathbb{R}\_+$$

in the sense of **NA 9.2**.

**Theorem 8.** *Let* 2 ≤ *t*1 < *t*2 *be suitable in* R, *and let* LS*<sup>t</sup>*1<*t*2 *be the semicircular* [*<sup>t</sup>*1, *<sup>t</sup>*2]*-filtration* (94)*. Then, the Banach* ∗*-probability space:*

$$\mathbb{L}\mathbb{S}^{t\_1$$

*equipped with its linear functional τt*1<*t*<sup>2</sup> = - *<sup>t</sup>*1≤*p*≤*t*2*τ*0*<sup>p</sup>* , *is free-homomorphic to* LS*<sup>t</sup>*1<*t*2 *in* LS, *i.e., if* 2 ≤ *t*1 < *t*2 *are suitable in* R,

$$\mathbb{L}\mathbb{S}^{t\_1 < t\_2} \text{ is free-homomorphism to } \mathbb{L}\mathbb{S}\_{t\_1 < t\_2} \text{ in } \mathbb{L}\mathbb{S}\_0. \tag{98}$$

**Proof.** Let LS*<sup>t</sup>*1<*t*2 be in the sense of (97) in the semicircular filtration LS0, i.e., 

$$\mathbb{L}\mathbb{S}^{t\_1$$

as a free-probabilistic sub-structure of the semicircular filtration LS0.

By (94), one can define the embedding map Φ from LS*<sup>t</sup>*1<*t*2 into LS, satisfying:

$$
\Phi(T) = T,\text{ for all } T \in \mathbb{L} \mathbb{S}^{t\_1 \times t\_2}.
$$

Then, for any *T* ∈ LS*<sup>t</sup>*1<*t*2 , one can ge<sup>t</sup> that:

$$
\pi^{t\_1 \prec t\_2}(T) = \mathfrak{n}\_{1 \prec t\_2}(T) = \pi^0(T).
$$

Therefore, the Banach ∗-probability space LS*<sup>t</sup>*1<*t*2 is free-homomorphic to LS*<sup>t</sup>*1<*t*2 in LS. Therefore, the relation (98) holds.

Remark again that we are not sure if LS*<sup>t</sup>*1<*t*2 and LS*<sup>t</sup>*1<*t*2 are free-isomorphic, or not, at this moment (see Remark 9.1 above). However, similar to (90), one can verify that all free reduced words *T* of LS*<sup>t</sup>*1<*t*2 have non-zero free distributions embedded in LS*<sup>t</sup>*1<*t*2 , and conversely, all free reduced words of LS*<sup>t</sup>*1<*t*2 having non-zero free distributions are contained in LS*<sup>t</sup>*1<*t*2 .

**Corollary 3.** *Let T be a free reduced word of the semicircular* [*<sup>t</sup>*1, *<sup>t</sup>*2]*-filtration* LS*<sup>t</sup>*1<*t*2 *in* Q ∪ Θ, *and assume that the free distribution of T is non-zero for <sup>τ</sup>t*1<*t*2 *. Then, T is an element of the Banach* ∗*-probability space* LS*<sup>t</sup>*1<*t*2 *of* (97)*. The converse holds true.* -

*9.3. More about Free-Probabilistic Information on* LS*<sup>t</sup>*1<*t*2

In this section, we discuss more about free-probabilistic information in semicircular [*<sup>t</sup>*1, *<sup>t</sup>*2]-filtrations LS*<sup>t</sup>*1<*t*2 , for *t*1 < *t*2 ∈ R (which are not necessarily suitable in the sense of **NA 9.2**). First,letusmentionaboutthefollowingtrivialcases.

**Proposition 9.** *Let* LS*<sup>t</sup>*1<*t*2 *be the semicircular* [*<sup>t</sup>*1, *<sup>t</sup>*2]*-filtration for t*1 < *t*2 *in* R.

*If t*2 < 2 *in* R*, then all elements of* LS*<sup>t</sup>*1<*t*2 *have the zero free distribution.* (99)

*Let t*1, *t*2 ≥ 2 *in* R*. If the closed interval* [*<sup>t</sup>*1, *t*2] *does not contain a prime in* R*, then all elements of* LS*<sup>t</sup>*1<*t*2 *have the zero free distribution*. (100)

**Proof.** The proofs of the statements (99) and (100) are done immediately by (90), (95), (96), and (98).

Even though the above results (99) and (100), themselves, are trivial, they illustrate how our original (non-zero) free-distributional data on the semicircular filtration LS0 are distorted under our "unsuitable" truncations.

Now, suppose *t*1 < *t*2 are suitable in R, and:

$$t\_1 \to \infty \text{ in } \mathbb{R}\_\prime$$

in the sense that: *t*1 is big "enough" in R. The existence of such suitable intervals [*<sup>t</sup>*1, *t*2] in R is guaranteed by the *prime number theorem* (e.g., [5,6]).

More precisely, let us collect all suitable pairs (*<sup>t</sup>*1, *<sup>t</sup>*2) in R2, i.e.,

$$\{(t\_1, t\_2) \in \mathbb{R}^2 : t\_1 < t\_2 \text{ are suitable in } \mathbb{R}\}\_\nu$$

and consider its boundary.

First, consider that if *p* → ∞ in P (under the usual total ordering on P, inherited by that on R), then: 

$$\lim\_{p \to \infty \text{ in } \mathcal{P}} p^{2(j+1)} = \begin{cases} 0 & \text{if } j < -1 \\ 1 & \text{if } j = -1 \\ \infty \text{, undefined} & \text{if } j > -1, \end{cases} \tag{101}$$

for an arbitrarily-fixed *j* ∈ Z.

**Theorem 9.** *Let* (*tn*) ∞ *<sup>n</sup>*=1*and* (*sn*)<sup>∞</sup> *<sup>n</sup>*=1*be monotonically "strictly"-increasing* R*-sequences, satisfying:*

> *tn* < *sn are suitable in* R*,*

*for all n* ∈ N. *By the suitability, there exists at least one prime pn* ∈ P, *such that:*

$$t\_n \le p\_n \le s\_{n\prime} for all n \in \mathbb{N},\tag{102}$$

*where the corresponding* R*-sequence* (*pn*)<sup>∞</sup>*n*=<sup>1</sup> *is monotonically increasing.*

*Let Qpn*,*<sup>j</sup> be the corresponding p*<sup>2</sup>(*j*+<sup>1</sup>) *n -semicircular element in the free weighted-semicircular family* Q*, as a free random variable of the semicircular* [*tn*, *sn*]*-filtration* LS*tn*<*sn* , *where pn are the primes of* (102)*, for all n* ∈ N, *for any j* ∈ Z. *Then:*

$$\lim\_{n \to \infty} \left( \pi\_{t\_n < s\_n} \left( Q\_{p\_n, j}^k \right) \right) = \begin{cases} 0 & \text{if } j < -1 \\ \omega\_k c\_{\frac{k}{2}} & \text{if } j = -1 \\ \infty & \text{if } j > -1, \end{cases} \tag{103}$$

*for all k* ∈ N.

**Proof.** Suppose *pn* are the primes satisfying (102) for given suitable:

$$t\_n < s\_n \text{ in } \mathbb{R},$$

in the sense of **NA 9.2**, for all *n* ∈ N. Then, for the *p*<sup>2</sup>(*j*+<sup>1</sup>) *n* -semicircular elements *Qpn*,*<sup>j</sup>* ∈ Q (in LS0), one has that:

$$\tau\_{t\_n < s\_n} \left( Q\_{p\_{n,j}}^k \right) = \begin{pmatrix} \star & \star\_q \tau\_q^0 \end{pmatrix} \left( Q\_{p\_{n,j}}^k \right) = \tau\_{p\_n}^0 \left( Q\_{p\_{n,j}}^k \right)$$

$$= \tau\_{p\_{n,j}}^0 \left( Q\_{p\_{n,j}}^k \right) = \omega\_k p\_n^{2(j+1)} c\_{\frac{k}{2}} \tag{104}$$

forall *k* ∈ N.

by (102)

> Thus, we have that:

$$\lim\_{n \to \infty} \left( \pi\_{t\_n < s\_n} \left( \mathbb{Q}\_{p\_n, j}^k \right) \right) = \lim\_{n \to \infty} \left( \omega\_k p\_n^{2(j+1)} c\_{\frac{k}{2}} \right),$$

by (104);

$$=\lim\_{p\to\infty} \left(\omega\_k p^{2(j+1)} \mathfrak{c}\_{\frac{k}{2}}\right) = \left(\omega\_k \mathfrak{c}\_{\frac{k}{2}}\right) \left(\lim\_{p\to\infty} p^{2(j+1)}\right),$$

$$=\begin{cases} 0 & \text{if } j < -1\\ \omega\_k \mathfrak{c}\_{\frac{k}{2}} & \text{if } j = -1\\ \infty & \text{if } j > -1, \end{cases}$$

by (101), for all *k* ∈ N. Therefore, the estimation (103) holds.

The above estimation (103) illustrates the asymptotic free-distributional data of our *p*<sup>2</sup>(*j*+<sup>1</sup>)-semicircular elements {*Qp*,*<sup>j</sup>* ∈ Q}*p*∈P (for a fixed *j* ∈ Z), under our suitable truncations, as *p* → ∞ in P.

**Corollary 4.** *Let t*1 < *t*2 *be suitable in* R *under NA 9.2, t*1 *be suitably big (i.e., t*1 → ∞*) in* R, *and j* ≤ −1 *be arbitrarily fixed in* Z. *Then, there exists t*0 ∈ R, *such that:*
