*Article* **Asymptotic Semicircular Laws Induced by** *p***-Adic Number Fields** Q*<sup>p</sup>* **and** *C∗***-Algebras over Primes** *p*

## **Ilwoo Cho**

Department of Mathematics & Statistics, Saint Ambrose University, 421 Ambrose Hall, 518 W. Locust St., Davenport, IA 52803, USA; choilwoo@sau.edu

Received: 11 April 2019; Accepted: 4 June 2019; Published: 20 June 2019

**Abstract:** In this paper, we study asymptotic semicircular laws induced both by arbitrarily fixed *C*∗-probability spaces, and *p*-adic number fields {Q*p*}*p*∈P, as *p* → ∞ in the set P of all primes.

**Keywords:** free probability; *p*-adic number fields Q*p*; Banach ∗-probability spaces; *C*∗-algebras; semicircular elements; the semicircular law; asymptotic semicircular laws

## **1. Introduction**

The main purposes of this paper are (i) to establish *tensor product C*∗*-probability spaces*

(*<sup>A</sup>* <sup>⊗</sup><sup>C</sup> <sup>S</sup>*p*, *<sup>ψ</sup>* <sup>⊗</sup> *<sup>ϕ</sup><sup>p</sup> j* )

induced both by arbitrary unital *<sup>C</sup>*∗-probability spaces (*A*, *<sup>ψ</sup>*), and by analytic structures (S*p*, *<sup>ϕ</sup><sup>p</sup> j* ) acting on *p*-*adic number fields* Q*<sup>p</sup>* for all primes *p* in the set P of all *primes*, where *j* ∈ Z, (ii) to consider free-probabilistic structures of (i) affected both by the free probability on (*A*, *ψ*), and by the number theory on Q*<sup>p</sup>* for all *p* ∈ P, (iii) to study *asymptotic behaviors* on the structures of (i) as *p* → ∞ in P, based on the results of (ii), and (iv), and then investigate *asymptotic semicircular laws* from the free-distributional data of (iii).

Our main results illustrate cross-connections among *number theory*, *representation theory*, *operator theory*, *operator algebra theory*, and *stochastic analysis*, via *free probability theory*.

## *1.1. Preview and Motivation*

Relations between primes and *operators* have been studied in various different approaches. In [1], we studied how primes act on *operator algebras* induced by *dynamical systems* on *p*-*adic*, and *Adelic* objects. Meanwhile, in [2], primes are acting as *linear functionals* on *arithmetic functions*, characterized by *Krein-space operators*.

For number theory and free probability theory, see [3–22], respectively.

In [23], *weighted-semicircular elements*, and *semicircular elements* induced by *p*-*adic number fields* Q*<sup>p</sup>* are considered by the author and Jorgensen, for each *p* ∈ P, statistically. In [24], the author extended the constructions of *weighted-semicircular elements* of [23] under *free product* of [15,22]. The main results of [24] demonstrate that the (weighted-)semicircular law(s) of [23] is (are) well-determined free-probability-theoretically. As an application, the *free stochastic calculus* was considered in [6].

Independent from the above series of works, we considered *asymptotic semicircular laws* induced by {Q*p*}*p*∈P in [1]. The constructions of [1] are highly motivated by those of [6,23,24], but they are totally different not only conceptually, but also theoretically. Thus, even though the main results of [1] seem similar to those of [6,24], they indicate-and-emphasize "asymptotic" semicircularity induced by {Q*p*}*p*∈P, as *p* → ∞. For example, they show that our analyses on {Q*p*}*p*∈P not only provide natural semicircularity but also asymptotic semicircularity under free probability theory.

In this paper, we study *asymptotic-semicircular laws* over "both" primes and *unital C*∗*-probability spaces*. Since we generalize the asymptotic semicircularity of [25] up to *C*∗-algebra-tensor, the patterns and results of this paper would be similar to those of [25], but generalize-or-universalize them.

## *1.2. Overview*

In Section 2, fundamental concepts and backgrounds are introduced. In Sections 3–6, suitable free-probabilistic models are considered, where they contain *p*-adic number-theoretic information, for our purposes.

In Section 7, we establish-and-study *C*∗-probability spaces containing both analytic data from Q*p*, and free-probabilistic information of fixed unital *C*∗-probability spaces. Then, our free-probabilistic structure LS*A*, a free product Banach <sup>∗</sup>-probability space, is constructed, and the free probability on LS*<sup>A</sup>* is investigated in Section 8.

In Section 9, asymptotic behaviors on LS*<sup>A</sup>* are considered over <sup>P</sup>, and they analyze the asymptotic semicircular laws on LS*<sup>A</sup>* over <sup>P</sup> in Section 10.

## **2. Preliminaries**

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

#### *2.1. Free Probability*

See [15,22] (and the cited papers therein) for basic free probability theory. Roughly speaking, *free probability* is the noncommutative operator-algebraic extension of measure theory (containing probability theory) and statistical analysis. As an independent branch of operator algebra theory, it is applied not only to mathematical analysis (e.g., [5,12–14,26]), but also to related fields (e.g., [18,27–31]).

Here, combinatorial free probability is used (e.g., [15–17]). In the text, *free moments*, *free cumulants*, and the *free product of* ∗*-probability spaces* are considered without detailed introduction.

## *2.2. Analysis on* Q*<sup>p</sup>*

For *p*-*adic analysis* and *Adelic analysis*, see [21,22]. We use definitions, concepts, and notations from there. Let *p* ∈ P be a prime, and let Q be the set of all *rational numbers*. Define a *non-Archimedean norm* |.|*<sup>p</sup>* , called the *p-norm on* Q by

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

for all *x* = *p<sup>k</sup> <sup>a</sup> <sup>b</sup>* ∈ Q, where *k*, *a* ∈ Z, and *b* ∈ Z \ {0}.

The normed space Q*<sup>p</sup>* is the maximal *p*-norm closures in Q, i.e., the set Q*<sup>p</sup>* forms a *Banach space*, for *p* ∈ P (e.g., [22]). Each element *x* of Q*<sup>p</sup>* is uniquely expressed by

$$\mathbf{x} = \sum\_{k=-N}^{\infty} \mathbf{x}\_k p^k \text{ } \mathbf{x}\_k \in \{0, 1, \dots, p-1\}\text{ }$$

for *N* ∈ N, decomposed by

$$\mathbf{x} = \sum\_{l=-N}^{-1} \mathbf{x}\_l p^l + \sum\_{k=0}^{\infty} \mathbf{x}\_k p^k.$$

If *x* = ∑<sup>∞</sup> *<sup>k</sup>*=<sup>0</sup> *xk <sup>p</sup><sup>k</sup>* in <sup>Q</sup>*p*, then *<sup>x</sup>* is said to be a *<sup>p</sup>*-*adic integer*, and it satisfies <sup>|</sup>*x*|*<sup>p</sup>* <sup>≤</sup> 1. Thus, one can define the *unit disk* Z*<sup>p</sup> of* Q*p*,

$$\mathbb{Z}\_p = \{ \mathbf{x} \in \mathbb{Q}\_p : |\mathbf{x}|\_p \le 1 \}.$$

For the *p*-*adic addition* and the *p*-*adic multiplication* in the sense of [22], the algebraic structure Q*<sup>p</sup>* forms a *field*, and hence, Q*<sup>p</sup>* is a *Banach field*.

Note that Q*<sup>p</sup>* is also a *measure space*,

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

equipped with the *σ*-*algebra σ*(Q*p*) *of* Q*p*, and a left-and-right additive invariant *Haar measure* on *μp*, satisfying

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

If we take

$$\mathcal{U}\_k = p^k \mathbb{Z}\_p = \{ p^k \mathbf{x} \in \mathbb{Q}\_p : \mathbf{x} \in \mathbb{Z}\_p \}, \tag{1}$$
  $\text{in } \sigma\left(\mathbb{Q}\_p\right)$ , for all  $k \in \mathbb{Z}$ , then these subsets  $\mathcal{U}\_k$ 's of (1) satisfy}

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

and

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

for all *x* ∈ Q*p*, and

$$\cdots \cdot \subset \mathcal{U}\_2 \subset \mathcal{U}\_1 \subset \mathcal{U}\_0 = \mathbb{Z}\_p \subset \mathcal{U}\_{-1} \subset \mathcal{U}\_{-2} \subset \cdots \dots$$

i.e., the family {*Uk*}*k*∈<sup>Z</sup> of (1) is a *topological basis element of* Q*<sup>p</sup>* (e.g., [22]).

Define subsets *∂<sup>k</sup>* ∈ *σ*(Q*p*) by

$$\partial\_k = \mathcal{U}\_k \; \backslash \mathcal{U}\_{k+1} \; \vert \tag{3}$$

for all *k* ∈ Z.

Such *μp*-measurable subsets *∂<sup>k</sup>* of (3) are called the *k-th boundaries* (*of Uk*) in Q*p*, for all *k* ∈ Z. By (2) and (3),

$$\begin{aligned} \mathbb{Q}\_{\mathcal{P}} &= \underset{k \in \mathbb{Z}}{\text{ $\mathcal{Q}\_{k'}$ }} \partial\_{k'} \\\\ \mu\_{\mathcal{P}} \left( \partial\_{k} \right) &= \mu\_{\mathcal{P}} \left( \mathcal{U}\_{k} \right) - \mu\_{\mathcal{P}} \left( \mathcal{U}\_{k+1} \right) = \frac{1}{p^{k}} - \frac{1}{p^{k+1}} \end{aligned} \tag{4}$$

where is the *disjoint union*, for all *k* ∈ Z,

Let M*<sup>p</sup>* be an algebraic *algebra*,

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

where *χ<sup>S</sup>* are the usual *characteristic functions* of *μp*-measurable subsets *S* of Q*p*. Thus, *f* ∈ M*p*, if and only if

$$f = \sum\_{\mathbf{S} \in \sigma \left( \mathbb{Q}\_p \right)} t\_{\mathbf{S}} \chi\_{\mathbf{S}}; t\_{\mathbf{S}} \in \mathbb{C}, \tag{5b}$$

where ∑ is the *finite sum*. Note that the algebra M*<sup>p</sup>* of (5a) is a ∗-*algebra over* C, with its well-defined *adjoint*,

$$\left(\sum\_{S \in \sigma(G\_p)} t\_S \chi\_S\right)^\* \stackrel{def}{=} \sum\_{S \in \sigma(G\_p)} \overline{t\_S} \,\chi\_{S \wedge \sigma}$$

for *tS* ∈ C with their *conjugates tS* in C.

If *f* ∈ M*<sup>p</sup>* is given as in (5b), then one defines the *integral of f* by

$$\int\_{\mathbb{Q}\_p} f \, d\mu\_p = \sum\_{S \in \sigma(\mathbb{Q}\_p)} t\_S \, \mu\_p(S). \tag{6a}$$

Remark that, by (5a), the integral (6a) is unbounded on M*p*, i.e.,

$$\int\_{\mathbb{Q}\_p} \chi\_{\mathbb{Q}\_p} d\mu\_p = \mu\_p \left( \mathbb{Q}\_p \right) = \infty,\tag{6b}$$

*Symmetry* **2019**, *11*, 819

by (2).

Note that, by (4), for each *S* ∈ *σ*(Q*p*), there exists a corresponding subset Λ*<sup>S</sup>* of Z,

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

satisfying

$$\begin{aligned} \int\_{\mathbb{Q}\_p} \chi\_S \, d\mu\_p &= \int\_{\mathbb{Q}\_p} \sum\_{j \in \Lambda\_S} \chi\_{S \cap \partial\_j} \, d\mu\_p, \\ &= \sum\_{j \in \Lambda\_S} \mu\_p \left( S \cap \partial\_j \right) \end{aligned}$$

by (6a)

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

by (4), for the set Λ*<sup>S</sup>* of (7).

Remark again that the right-hand side of (8) can be ∞; for instance, ΛQ*<sup>p</sup>* = Z, e.g., see (4), (6a) and (6b). By (8), one obtains the following proposition.

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

$$\begin{aligned} 0 \le r\_{\dot{\jmath}} &= \frac{\mu\_p(\mathbb{S} \cap \partial\_{\dot{\jmath}})}{\mu\_p(\overline{\partial\_{\dot{\jmath}}})} \le 1, \forall \dot{\jmath} \in \Lambda\_S; \\\\ \int\_{\mathbb{Q}\_p} \chi\_S \, d\mu\_p &= \sum\_{\dot{j} \in \Lambda\_S} r\_{\dot{\jmath}} \left( \frac{1}{p^{\dot{\jmath}}} - \frac{1}{p^{\dot{\jmath} + 1}} \right). \end{aligned} \tag{9}$$

## **3. Statistical Models on** *M<sup>p</sup>*

In this section, fix *p* ∈ P, and let Q*<sup>p</sup>* be the *p*-adic number field, and let M*<sup>p</sup>* be the ∗-algebra (5a). We here establish a suitable statistical model on M*<sup>p</sup>* with free-probabilistic language.

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

$$\mathcal{U}\_k = p^k \mathbb{Z}\_{p^\star}$$

for all *k* ∈ Z, and

$$
\partial\_k = \mathcal{U}\_k \nmid \mathcal{U}\_{k+1}; k \in \mathbb{Z}. \tag{10}
$$

Define a linear functional *ϕ<sup>p</sup>* : M*<sup>p</sup>* → C by the *integration* (6a), i.e.,

$$\mathcal{p}\_p\left(f\right) = \int\_{\mathbb{Q}\_p} f \, d\mu\_{p\prime} \tag{11}$$

for all *f* ∈ M*p*.

Then, by (9), one obtains that *ϕ<sup>p</sup> χUj* = <sup>1</sup> *<sup>p</sup><sup>j</sup>* , and *<sup>ϕ</sup><sup>p</sup> χ∂j* = <sup>1</sup> *<sup>p</sup><sup>j</sup>* <sup>−</sup> <sup>1</sup> *<sup>p</sup>j*+<sup>1</sup> , since <sup>Λ</sup>*Uj* = {*<sup>k</sup>* ∈ Z : *k* ≥ *j*}, and Λ*∂<sup>j</sup>* = {*j*}, for all *j* ∈ Z, where Λ*<sup>S</sup>* are in the sense of (7) for all *S* ∈ *σ*(Q*p*).

**Definition 1.** *The pair* M*p*, *ϕ<sup>p</sup> is called the p-adic (unbounded-)measure space for p* ∈ P, *where ϕ<sup>p</sup> is the linear functional (11) on* M*p*.

Let *∂<sup>k</sup>* be the *k*-th boundaries (10) of Q*p*, for all *k* ∈ Z. Then, for *k*1, *k*<sup>2</sup> ∈ Z, one obtains that

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

and hence,

*ϕp χ∂<sup>k</sup>*<sup>1</sup> *χ∂<sup>k</sup>*<sup>2</sup> = *δk*1,*k*<sup>2</sup> *ϕ<sup>p</sup> χ∂<sup>k</sup>*<sup>1</sup> = *δk*1,*k*<sup>2</sup> - 1 *<sup>p</sup>k*<sup>1</sup> <sup>−</sup> <sup>1</sup> *pk*1+<sup>1</sup> . (12)

,

*Symmetry* **2019**, *11*, 819

**Proposition 2.** *Let* (*j*1, *..., jN*) ∈ Z*N*, *for N* ∈ N. *Then,*

$$\prod\_{l=1}^{N} \chi\_{\mathfrak{d}\_{\hat{j}\_l}} = \delta\_{(\hat{j}\_1, \dots, \hat{j}\_N)} \chi\_{\mathfrak{d}\_{\hat{j}\_1}} \text{ in } \mathcal{M}\_{p\times p}$$

*and hence,*

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

*where*

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

**Proof.** The computation (13) is shown by the induction on (12).

Recall that, for any *S* ∈ *σ* Q*<sup>p</sup>* ,

$$\varphi\_p\left(\chi\_S\right) = \sum\_{j \in \Lambda\_S} r\_j \left(\frac{1}{p^j} - \frac{1}{p^{j+1}}\right) \, \, \, \, \tag{14}$$

for some 0 ≤ *rj* ≤ 1, for *j* ∈ Λ*S*, by (9). Thus, by (14), if *S*1, *S*<sup>2</sup> ∈ *σ* Q*<sup>p</sup>* , then

$$\begin{split} \chi\_{S\_1 \wr S\_2} &= \left( \sum\_{k \in \Lambda\_{S\_1}} \chi\_{S\_1 \cap \mathfrak{d}\_k} \right) \left( \sum\_{j \in \Lambda\_{S\_2}} \chi\_{S\_2 \cap \mathfrak{d}\_j} \right) \\ &= \sum\_{(k,j) \in \Lambda\_{S\_1} \times \Lambda\_{S\_2}} \left( \chi\_{S\_1 \cap \mathfrak{d}\_k} \chi\_{S\_2 \cap \mathfrak{d}\_j} \right) \\ &= \sum\_{(k,j) \in \Lambda\_{S\_1} \times \Lambda\_{S\_2}} \delta\_{k,j} \, \mathcal{X}\_{(S\_1 \cap S\_2) \cap \mathfrak{d}\_j} \\ &= \sum\_{j \in \Lambda\_{S\_1}, j\_2} \mathcal{X}\_{(S\_1 \cap S\_2) \cap \mathfrak{d}\_{j'}} \tag{15} \end{split} \tag{15}$$

where

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

by (4).

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

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

*where* Λ*Sl are in the sense of (7), for l* = 1, *..., N*. *Then, there exists rj* ∈ R, *such that*

$$0 \le r\_{\hat{\jmath}} \le 1 \text{ іn } \mathbb{R},$$

*for all j* ∈ Λ*S*1,...,*SN* , *and*

$$\mathfrak{sp}\_p\left(\prod\_{l=1}^N \chi\_{S\_l}\right) = \sum\_{j \in \Lambda\_{S\_1,\dots,S\_N}} r\_j \left(\frac{1}{p^j} - \frac{1}{p^{j+1}}\right) \,. \tag{16}$$

**Proof.** The proof of (16) is done by the induction on (15), and by (13).

#### **4. Representation of** *Mp***,** *ϕ<sup>p</sup>*

Fix a prime *p* ∈ P. Let M*p*, *ϕ<sup>p</sup>* be the *p*-adic measure space. By understanding Q*<sup>p</sup>* as a measure space, construct the *L*2-*space*,

$$H\_p \stackrel{def}{=} L^2\left(\mathbb{Q}\_p, \sigma(\mathbb{Q}\_p), \,\mu\_p\right) = L^2\left(\mathbb{Q}\_p\right),\tag{17}$$

over C. Then, this *Hilbert space Hp* of (17) consists of all square-integrable elements of M*p*, equipped with its *inner product* <, >2,

$$\langle f\_1, f\_2 \rangle\_2 \stackrel{def}{=} \int\_{\mathbb{Q}\_p} f\_1 \, f\_2^\* \, d\mu\_{p^\*} \tag{18a}$$

for all *<sup>f</sup>*1, *<sup>f</sup>*<sup>2</sup> <sup>∈</sup> *Hp*. Naturally, *Hp* is has its *<sup>L</sup>*2-*norm* .<sup>2</sup> on <sup>M</sup>*p*,

$$\|f\|\_{2} \stackrel{def}{=} \sqrt{\langle f, f \rangle\_{2'}} \tag{18b}$$

for all *f* ∈ *Hp*, where <, ><sup>2</sup> is the inner product (18a) on *Hp*.

**Definition 2.** *The Hilbert space Hp of (17) is called the p-adic Hilbert space.*

Our ∗-algebra M*<sup>p</sup>* acts on the *<sup>p</sup>*-adic Hilbert space *Hp*, via an action *<sup>α</sup>p*,

$$a^p(f)\left(h\right) = fh,\text{ for all }h \in H\_{p^\*} \tag{19a}$$

for all *<sup>f</sup>* ∈ M*p*. i.e., the morphism *<sup>α</sup><sup>p</sup>* of (19a) is a ∗-homomorphism from M*<sup>p</sup>* to the *operator algebra B*(*Hp*), consisting of all Hilbert-space operators on *Hp*. For instance,

$$\begin{aligned} \pi^{\mathbb{P}}\left(\chi\_{\mathbb{Q}\_{\mathbb{P}}}\right)\left(\sum\_{\mathcal{S}\in\sigma\left(\mathbb{Q}\_{\mathbb{P}}\right)}t\_{\mathcal{S}}\chi\_{\mathcal{S}}\right) &= \sum\_{\mathcal{S}\in\sigma\left(\mathbb{Q}\_{\mathbb{P}}\right)}t\_{\mathcal{S}}\chi\_{\mathbb{Q}\_{\mathbb{P}}\cap\mathcal{S}} \\\\ &= \sum\_{\mathcal{S}\in\sigma\left(\mathbb{Q}\_{\mathbb{P}}\right)}t\_{\mathcal{S}}\chi\_{\mathcal{S}}\end{aligned} \tag{19b}$$

for all *h* = ∑ *S*∈*σ*(Q*p*) *tSχ<sup>S</sup>* ∈ *Hp*, with *h*<sup>2</sup> < ∞, for *χ*Q*<sup>p</sup>* ∈ M*p*, even though *χ*Q*<sup>p</sup>* ∈/ *Hp*.

Indeed, It is not difficult to check that

*<sup>α</sup>p*(*f*<sup>1</sup> *<sup>f</sup>*2) = *<sup>α</sup>p*(*f*1)*αp*(*f*2) on *Hp*, ∀ *<sup>f</sup>*1, *<sup>f</sup>*<sup>2</sup> ∈ M*p*, (20a)

$$\left(\mathfrak{a}^p(f)\right)^\* = \mathfrak{a}(f^\*) \text{ on } H\_{p\prime} \,\forall f \in \mathcal{M}\_p.$$

**Notation 1.** Denote *αp*(*f*) by *α<sup>p</sup> <sup>f</sup>* , for all *<sup>f</sup>* ∈ M*p*. In addition, for convenience, denote *<sup>α</sup><sup>p</sup> <sup>χ</sup><sup>S</sup>* simply by *<sup>α</sup><sup>p</sup> S*, for all *S* ∈ *σ* Q*<sup>p</sup>* .

Note that, by (19b), one can have a well-defined operator *α<sup>p</sup>* <sup>Q</sup>*<sup>p</sup>* <sup>=</sup> *<sup>α</sup><sup>p</sup> <sup>χ</sup>*Q*<sup>p</sup>* in *B*(*Hp*), and it satisfies that

$$a\_{\mathbb{Q}\_p}^p(h) = h = \mathbf{1}\_{H\_p}(h) \; , \forall h \in H\_{p\prime} \tag{20b}$$

where 1*Hp* ∈ *B*(*Hp*) is the identity operator on *Hp*.

**Proposition 4.** *The pair* (*Hp*, *<sup>α</sup>p*) *is a Hilbert-space representation of* M*p*.

**Proof.** It suffices to show that *<sup>α</sup><sup>p</sup>* is an algebra-action of M*<sup>p</sup>* on *Hp*. However, this morphism *<sup>α</sup><sup>p</sup>* is a ∗-homomorphism from M*<sup>p</sup>* into *B*(*Hp*), by (20a).

**Definition 3.** *The Hilbert-space representation Hp*, *αp is called the p-adic representation of* M*p*.

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

**Definition 4.** *Let Mp be the operator-norm closure of* M*p,*

$$\mathcal{M}\_p \stackrel{def}{=} \overline{a^p \left(\mathcal{M}\_p\right)} = \overline{\mathbb{C}\left[a\_f^p : f \in \mathcal{M}\_p\right]} \tag{21}$$

*in B*(*Hp*), *where X are the operator-norm closures of subsets X of B*(*Hp*). *This C*∗*-algebra Mp is said to be the p-adic C*∗*-algebra of* M*p*, *ϕ<sup>p</sup>* .

By (21), the *p*-adic *C*∗-algebra *Mp* is a unital *C*∗-algebra contains its *unity* (or the unit, or the multiplication-identity) 1*Hp* <sup>=</sup> *<sup>α</sup><sup>p</sup>* Q*<sup>p</sup>* , by (20b).

## **5. Statistics on** *Mp*

In this section, fix *p* ∈ P, and let *Mp* be the corresponding *p*-adic *C*∗-algebra of (21). Define a linear functional *ϕ<sup>p</sup> <sup>j</sup>* : *Mp* → C by

$$\left|\!\!\!\!p\_{j}^{p}\left(a\right)\stackrel{def}{=}\left\langle a\left(\chi\_{\partial\_{j}}\right),\ \chi\_{\partial\_{j}}\right\rangle\_{2},\ \forall a\in M\_{p\prime} \tag{22a}$$

for *χ∂<sup>j</sup>* ∈ *Hp*, where <, ><sup>2</sup> is the inner product (4.2) on the *p*-adic Hilbert space *Hp* of (4.1), and *∂<sup>j</sup>* are the boundaries (3.1) of <sup>Q</sup>*p*, for all *<sup>j</sup>* <sup>∈</sup> <sup>Z</sup>. It is not hard to check such a linear functional *<sup>ϕ</sup><sup>p</sup> <sup>j</sup>* on *Mp* is bounded, since

$$\begin{split} \left< \boldsymbol{a}\_{\boldsymbol{\upbeta}}^{p} \left( \boldsymbol{a}\_{\boldsymbol{\upbeta}}^{p} \right) = \left< \boldsymbol{a}\_{\boldsymbol{\upbeta}}^{p} \left( \boldsymbol{\upchi}\_{\boldsymbol{\upbeta}} \right), \, \boldsymbol{\upchi}\_{\boldsymbol{\upbeta}\_{\boldsymbol{\upbeta}}} \right>\_{2} = \left< \boldsymbol{\upchi}\_{S \cap \boldsymbol{\upbeta}\_{\boldsymbol{\upbeta}}} \, \boldsymbol{\upchi}\_{\boldsymbol{\upbeta}\_{\boldsymbol{\upbeta}}} \right>\_{2} \\ = & \left< \boldsymbol{\upchi}\_{S \cap \boldsymbol{\upbeta}\_{\boldsymbol{\upbeta}}} \, \boldsymbol{\upchi}\_{\boldsymbol{\upbeta}\_{\boldsymbol{\upbeta}}} \right>\_{2} = \int\_{\mathbb{Q}\_{p}} \boldsymbol{\upchi}\_{S \cap \boldsymbol{\upbeta}\_{\boldsymbol{\upbeta}}} d\boldsymbol{\upmu}\_{p} \\ \leq & \int\_{\mathbb{Q}\_{p}} \boldsymbol{\upchi}\_{\boldsymbol{\upbeta}\_{\boldsymbol{\upbeta}}} d\boldsymbol{\upmu}\_{p} = \boldsymbol{\upmu}\_{p} \left( \boldsymbol{\uptheta}\_{\boldsymbol{\upbeta}} \right) = \frac{1}{p^{j}} - \frac{1}{p^{j+1}}. \end{split} \tag{22b}$$

for all *S* ∈ *σ*(Q*p*), for any fixed *j* ∈ Z.

**Definition 5.** *Let ϕ<sup>p</sup> <sup>j</sup> be bounded linear functionals (22a) on the p-adic C*∗*-algebra Mp*, *for all j* ∈ Z. *Then, the pairs Mp*, *<sup>ϕ</sup><sup>p</sup> j are said to be the j-th p-adic C*∗*-measure spaces, for all j* ∈ Z.

Thus, one can get the system

{(*Mp*, *<sup>ϕ</sup><sup>p</sup> <sup>j</sup>* ) : *j* ∈ Z}

of the *<sup>j</sup>*-th *<sup>p</sup>*-adic *<sup>C</sup>*∗-measure spaces (*Mp*, *<sup>ϕ</sup><sup>p</sup> <sup>j</sup>* )'s.

Note that, for any fixed *<sup>j</sup>* <sup>∈</sup> <sup>Z</sup>, and (*Mp*, *<sup>ϕ</sup><sup>p</sup> <sup>j</sup>* ), the unity

$$1\_{M\_p} \stackrel{donde}{=} 1\_{H\_p} = \mathfrak{a}\_{\mathbb{Q}\_p}^p \text{ of } M\_p!$$

satisfies that

$$\begin{aligned} \left. \sigma\_{\mathcal{I}}^{p} \left( \mathbf{1}\_{M\_{\mathcal{P}}} \right) \right| &= \left\langle \chi\_{\mathbb{Q}\_{\mathcal{P}} \cap \mathfrak{d}\_{\mathcal{I}} \prime} \chi\_{\mathfrak{d}\_{\mathcal{I}}} \right\rangle\_{2} \\ &= \left\| \chi\_{\mathfrak{d}\_{\mathcal{I}}} \right\|^{2} = \frac{1}{p^{j}} - \frac{1}{p^{i+1}}. \end{aligned} \tag{23}$$

Thus, the *<sup>j</sup>*-th *<sup>p</sup>*-adic *<sup>C</sup>*∗-measure space (*Mp*, *<sup>ϕ</sup><sup>p</sup> <sup>j</sup>* ) is a bounded-measure space, but not a probability space, in general.

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

$$0 \le r\_S \le 1 \text{ in } \mathbb{R},$$

*and*

$$\rho\_j^p\left(\left(a\_S^p\right)^n\right) = r\_S\left(\frac{1}{p^l} - \frac{1}{p^{l+1}}\right); n \in \mathbb{N}.\tag{24}$$

**Proof.** Remark that the element *α<sup>p</sup> <sup>S</sup>* is a projection in *Mp*, in the sense that:

$$\left(\mathfrak{a}\_{\mathcal{S}}^{p}\right)^{\*} = \mathfrak{a}\_{\left(\chi\_{\mathcal{S}}^{\*}\right)}^{p} = \mathfrak{a}\_{\mathcal{S}}^{p} = \mathfrak{a}\_{\left(\chi\_{\mathcal{S}} \cap \chi\_{\mathcal{S}}\right)}^{p} = \left(\mathfrak{a}\_{\mathcal{S}}^{p}\right)^{2}{} \text{ in } M\_{p\mathcal{S}}$$

and hence,

$$\left(\mathfrak{a}\_{\mathbf{S}}^{p}\right)^{n} = \mathfrak{a}\_{\mathbf{S'}}^{p}$$

for all *n* ∈ N. Thus, we obtain the formula (24) by (22b).

As a corollary of (24), one obtains that, if *∂<sup>k</sup>* is a *k*-th boundaries of Q*p*, then

$$\boldsymbol{\rho}\_{\rangle}^{p}\left(\left(\boldsymbol{a}\_{\partial\_{k}}^{p}\right)^{n}\right) = \delta\_{\boldsymbol{\beta},k}\left(\frac{1}{p^{\dagger}} - \frac{1}{p^{\dagger+1}}\right),\tag{25}$$

for all *n* ∈ N, for *k* ∈ Z.

## **6. The** *C∗***-Subalgebra** S*<sup>p</sup>* **of** *Mp*

Let *Mp* be the *p*-adic *C*∗-algebra for *p* ∈ P. Let

$$P\_{p,j} = a\_{\partial\_j}^p \in M\_{p, \prime} \tag{26}$$

for all *j* ∈ Z. By (24) and (25), these operators *Pp*,*<sup>j</sup>* of (26) are *projections* on the *p*-adic Hilbert space *Hp*, in *Mp*, for all *p* ∈ P, *j* ∈ Z.

**Definition 6.** *Let p* ∈ P, *and let* <sup>S</sup>*<sup>p</sup> be the C*∗*-subalgebra*

$$\mathfrak{S}\_{\mathcal{P}} = \mathbb{C}^\* \left( \{ P\_{p, \mathbf{j}} \}\_{\mathbf{j} \in \mathbb{Z}} \right) = \overline{\mathbb{C} \left[ \{ P\_{p, \mathbf{j}} \}\_{\mathbf{j} \in \mathbb{Z}} \right]} \text{ of } \mathcal{M}\_{\mathcal{P}} \tag{27}$$

*where Pp*,*<sup>j</sup> are in the sense of ((26)), for all j* <sup>∈</sup> <sup>Z</sup>. *We call* <sup>S</sup>*p*, *the p-adic boundary (C*∗*-)subalgebra of Mp.*

**Proposition 6.** *If* <sup>S</sup>*<sup>p</sup> is the p-adic boundary subalgebra (27), then*

$$\mathfrak{S}\_p \stackrel{\ast \text{ iso}}{=} \underset{j \in \mathbb{Z}}{\ominus} \left(\mathbb{C} \cdot P\_{p,j}\right) \stackrel{\ast \text{ iso}}{=} \mathbb{C}^{\ominus \left|\mathbb{Z}\right|},\tag{28}$$

*in the p-adic C*∗*-algebra Mp*.

**Proof.** It is enough to show that the generating operators {*Pp*,*j*}*j*∈<sup>Z</sup> of <sup>S</sup>*<sup>p</sup>* are mutually orthogonal from each other. It is not hard to check that

$$P\_{p,j\_1}P\_{p,j\_2} = \mathfrak{a}^p \left(\chi\_{\mathfrak{d}^p\_{j\_1}\cap\mathfrak{d}^p\_{j\_2}}\right) = \delta\_{j\_1,j\_2}\mathfrak{a}^p\_{\mathfrak{d}^p\_{j\_1}} = \delta\_{j\_1,j\_2}P\_{p,j\_1,j\_2}$$

in <sup>S</sup>*p*, for all *<sup>j</sup>*1, *<sup>j</sup>*<sup>2</sup> <sup>∈</sup> <sup>Z</sup>. Therefore, the structure theorem (28) is shown.

By (27), one can define the measure spaces,

$$\mathfrak{S}\_p(j) \stackrel{donate}{=} \left( \mathfrak{S}\_{p'} \begin{array}{c} \mathfrak{q}\_j^p \end{array} \right), \forall j \in \mathbb{Z}, \tag{29}$$

for *<sup>p</sup>* ∈ P, where the linear functionals *<sup>ϕ</sup><sup>p</sup> <sup>j</sup>* of (29) are the restrictions *<sup>ϕ</sup><sup>p</sup> <sup>j</sup>* |S*<sup>p</sup>* of (22a), for all *p* ∈ P, *j* ∈ Z.

#### **7. On the Tensor Product** *C∗***-Probability Spaces** *<sup>A</sup> <sup>⊗</sup>*<sup>C</sup> <sup>S</sup>*p***,** *<sup>ψ</sup> <sup>⊗</sup>ϕ<sup>p</sup> j*

In this section, we define and study our main objects of this paper. Let (*A*, *ψ*) be an arbitrary unital *C*∗-probability space (e.g., [22]), satisfying

$$
\psi(1\_A) = 1\_{\prime}
$$

where 1*<sup>A</sup>* is the unity of a *C*∗-algebra *A*. In addition, let

$$\mathfrak{S}\_p(j) = \left(\mathfrak{S}\_{p\prime} \begin{array}{c} \mathfrak{q}^p\_j \end{array}\right) \tag{30}$$

be the *p*-adic *C*∗-measure spaces (29), for all *p* ∈ P, *j* ∈ Z.

$$\text{Fix now a unital } \mathbb{C}^\*\text{-probability space } (A, \psi)\text{, and } p \in \mathcal{P}, j \in \mathbb{Z}. \text{ Define a tensor product } \mathbb{C}^\*\text{-algebra}$$

$$\mathfrak{S}\_p^A \stackrel{def}{=} A \otimes\_{\mathbb{C}} \mathfrak{S}\_{p'} \tag{31}$$

and a linear functional *ψ<sup>p</sup> <sup>j</sup>* on <sup>S</sup>*<sup>A</sup> <sup>p</sup>* by a linear morphism satisfying

$$
\psi\_j^p \left( a \otimes P\_{p,k} \right) = \varphi\_j^p \left( \psi(a) P\_{p,k} \right),
\tag{32}
$$

for all *a* ∈ (*A*, *ψ*), and *k* ∈ Z.

Note that, by the structure theorem (28) of the *<sup>p</sup>*-adic boundary subalgebra <sup>S</sup>*p*,

$$\mathfrak{S}\_p^A \stackrel{\*\text{-iṣo}}{=} A \otimes\_{\mathbb{C}} \left( \mathbb{C}^{\oplus \lfloor \mathbb{Z} \rfloor} \right) \stackrel{\*\text{-iṣo}}{=} A^{\oplus \lfloor \mathbb{Z} \rfloor},\tag{33}$$

by (31).

By (33), one can verify that a morphism *ψ<sup>p</sup> <sup>j</sup>* of (32) is indeed a well-defined bounded linear functional on S*<sup>A</sup> p* .

**Definition 7.** *For any arbitrarily fixed <sup>p</sup>* ∈ P, *<sup>j</sup>* <sup>∈</sup> <sup>Z</sup>, *let* <sup>S</sup>*<sup>A</sup> <sup>p</sup> be the tensor product <sup>C</sup>*∗*-algebra (31), and <sup>ψ</sup><sup>p</sup> j* , *the linear functional (32) on* S*<sup>A</sup> <sup>p</sup>* . *Then, we call* <sup>S</sup>*<sup>A</sup> <sup>p</sup>* , *the A-tensor p-adic boundary algebra. The corresponding structure,*

$$\left(\mathfrak{S}\_p^A(j)\right) \stackrel{dcnot}{=} \left(\mathfrak{S}\_p^A, \,\,\Psi\_j^p\right) \tag{34}$$

*is said to be the j-th p-adic A-(tensor C*∗*-probability-)space.*

Note that, by (22a), (22b) and (32), the *<sup>j</sup>*-th *<sup>p</sup>*-adic *<sup>A</sup>*-space S*<sup>A</sup> <sup>p</sup>* (*j*) of (34) is not a "unital" *<sup>C</sup>*∗-probability space, even though (*A*, *<sup>ψ</sup>*) is. Indeed, the *<sup>C</sup>*∗-algebra S*<sup>A</sup> <sup>p</sup>* of (31) has its unity 1*<sup>A</sup>* ⊗ 1*Mp* , satisfying

$$\begin{aligned} \psi\_j^p \left( \mathbf{1}\_A \otimes \mathbf{1}\_{M\_p} \right) &= q\_j^p \left( \psi(\mathbf{1}\_A) \mathbf{1}\_{M\_p} \right) \\ &= \mathbf{1} \cdot q\_j^p (\mathbf{1}\_{M\_p}) = \frac{1}{p^\circ} - \frac{1}{p^{j+1}}. \end{aligned}$$

for *j* ∈ Z.

Remark that, by (32),

$$\left(\Psi\_{\boldsymbol{j}}^{p}\left(\boldsymbol{a}\otimes\boldsymbol{P}\_{p,\boldsymbol{k}}\right)=\psi(\boldsymbol{a})\ \boldsymbol{\varrho}\_{\boldsymbol{j}}^{p}\left(\boldsymbol{P}\_{p,\boldsymbol{k}}\right)\;\tag{35a}$$

for all *a* ∈ (*A*, *ψ*), and *k* ∈ Z. Thus, by abusing notation, one may write the definition (32) by

$$
\psi\_j^p = \psi \otimes \sigma\_j^p \text{ on } A \otimes\_\mathbb{C} \mathfrak{S}\_p = \mathfrak{S}\_p^A,\tag{35b}
$$

in the sense of (35a), for all *p* ∈ P, *j* ∈ Z.

**Proposition 7.** *Let <sup>a</sup>* <sup>∈</sup> (*A*, *<sup>ψ</sup>*), *and Pp*,*k, the k-th generating projection of* <sup>S</sup>*p*, *for all <sup>k</sup>* <sup>∈</sup> <sup>Z</sup>, *and let <sup>a</sup>* <sup>⊗</sup> *Pp*,*<sup>k</sup> be the corresponding free random variable of the j-th p-adic A-space* S*<sup>A</sup> <sup>p</sup>* (*j*), *for j* ∈ Z*. Then,*

$$\psi\_j^p \left( \left( a \otimes P\_{p,k} \right)^n \right) = \delta\_{j,k} \,\psi(a^n) \left( \frac{1}{p^j} - \frac{1}{p^{j+1}} \right) \,\,\, \tag{36}$$

*for all n* ∈ N*.*

**Proof.** Let *T<sup>a</sup> <sup>p</sup>*,*<sup>k</sup>* <sup>=</sup> *<sup>a</sup>* <sup>⊗</sup> *Pp*,*<sup>k</sup>* be a given free random variable of <sup>S</sup>*<sup>A</sup> <sup>p</sup>* (*j*). Then,

$$\left(T^{a}\_{p,k}\right)^{n} = \left(a \otimes P\_{p,k}\right)^{n} = a^{n} \otimes P\_{p,k} = T^{q^{n}}\_{p,k^{\*}}$$

and hence

$$\begin{aligned} \psi\_j^p \left( \left( T\_{p,k}^a \right)^n \right) &= \psi\_j^p \left( T\_{p,k}^{a^n} \right) \\ = \psi(a^n) \, \vert \, \psi\_j^p \left( P\_{p,k} \right) &= \psi(a^n) \left( \delta\_{j,k} \left( \frac{1}{p^j} - \frac{1}{p^{j+1}} \right) \right) \end{aligned}$$

by (35a)

$$\delta = \delta\_{j,k} \psi(a^n) \left( \frac{1}{p^\dagger} - \frac{1}{p^{j+1}} \right) \dots$$

for all *n* ∈ N. Therefore, the free-distributional data (36) holds.

Suppose *a* is a "self-adjoint" free random variable in (*A*, *ψ*) in the above proposition. Then, formula (36) completely characterizes the free distribution of *<sup>a</sup>* <sup>⊗</sup> *Pp*,*<sup>k</sup>* in the *<sup>j</sup>*-th *<sup>p</sup>*-adic *<sup>A</sup>*-space <sup>S</sup>*<sup>A</sup> <sup>p</sup>* (*j*) of (34), i.e., the free distribution of *a* ⊗ *Pp*,*<sup>k</sup>* is characterized by the sequence,

$$\left(\delta\_{j,k}\psi(a^n)\left(\frac{1}{p^\circ}-\frac{1}{p^{\circ\frac{\cdot}{1}}}\right)\right)\_{n=1}^{\infty}$$

for all *<sup>p</sup>* ∈ P, and *<sup>j</sup>*, *<sup>k</sup>* <sup>∈</sup> <sup>Z</sup> because *<sup>a</sup>* <sup>⊗</sup> *Pp*,*<sup>k</sup>* is self-adjoint in <sup>S</sup>*<sup>A</sup> <sup>p</sup>* too.

It illustrates that the free probability on S*<sup>A</sup> <sup>p</sup>* (*j*) is determined both by the free probability on (*A*, *<sup>ψ</sup>*), and by the statistical data on S*p*(*j*) of (30) (implying *<sup>p</sup>*-adic analytic information), for *<sup>p</sup>* ∈ P, *<sup>j</sup>* <sup>∈</sup> <sup>Z</sup>. **Notation.** From below, for convenience, let's denote the free random variables *<sup>a</sup>* <sup>⊗</sup> *Pp*,*<sup>k</sup>* of <sup>S</sup>*<sup>A</sup> <sup>p</sup>* (*j*), with *<sup>a</sup>* ∈ (*A*, *<sup>ψ</sup>*) and *<sup>k</sup>* ∈ Z, by *<sup>T</sup><sup>a</sup> <sup>p</sup>*,*k*, i.e.,

$$T\_{p,k}^{a} \stackrel{demote}{=} a \otimes P\_{p,k'}$$

for all *p* ∈ P, *j* ∈ Z.

In the proof of (36), it is observed that

$$\left(T\_{p,k}^{a}\right)^{n} = T\_{p,k}^{a^{n}} \in \mathfrak{S}\_{p}^{A}(j) \tag{37}$$

for all *n* ∈ N. More generally, the following free-distributional data is obtained.

**Theorem 1.** *Fix <sup>p</sup>* ∈ P, *and <sup>j</sup>* <sup>∈</sup> <sup>Z</sup>, *and let* <sup>S</sup>*<sup>A</sup> <sup>p</sup>* (*j*) *be the j-th p-adic A-space (34). Let <sup>T</sup>al <sup>p</sup>*,*kl* <sup>∈</sup> <sup>S</sup>*<sup>A</sup> <sup>p</sup>* (*j*), *for l* = 1, *..., N*, *for N* ∈ N. *Then,*

$$
\psi\_j^p \left( \prod\_{l=1}^N \left( T\_{p,k\_l}^{a\_l} \right)^{n\_l} \right) = \left( \prod\_{l=1}^N \delta\_{j,k\_l} \right) \left( \frac{1}{p^j} - \frac{1}{p^{j+1}} \right) \psi \left( \prod\_{l=1}^N a\_l^{n\_l} \right), \tag{38}
$$

*for all n*1, *..., nN* ∈ N.

**Proof.** Let *Tal <sup>p</sup>*,*kl* <sup>=</sup> *al* <sup>⊗</sup> *Pp*,*kl* be free random variables of <sup>S</sup>*<sup>A</sup> <sup>p</sup>* (*j*), for *l* = 1, ..., *N*. Then, by (37),

$$\left(T\_{p,k\_l}^{a\_l}\right)^{n\_l} = T\_{p,k\_l}^{a\_l^{n\_l}} \in \mathfrak{S}\_p^A(j), \text{ for } n\_l \in \mathbb{N}\_\star$$

for all *l* = 1, ..., *N*. Thus,

*Symmetry* **2019**, *11*, 819

$$T = \prod\_{l=1}^{N} \left( T\_{p,k\_l}^{a\_l} \right)^{n\_l} = \left( \prod\_{l=1}^{N} a\_l^{n\_l} \right) \otimes \left( \delta\_{j;k\_1,\dots,k\_N} P\_{p,j} \right)^{n\_l}$$

in S*<sup>A</sup> <sup>p</sup>* (*j*), with

$$
\delta\_{j:k\_1,\dots,k\_N} = \prod\_{l=1}^N \delta\_{j,k\_l} \in \{0,1\}.
$$

Therefore,

$$\begin{aligned} \boldsymbol{\psi}\_{\boldsymbol{j}}^{p}(\boldsymbol{T}) &= \delta\_{\boldsymbol{j}:k\_{1},\ldots,k\_{N}} \boldsymbol{\psi} \left( \prod\_{l=1}^{N} a\_{l}^{n\_{l}} \right) \boldsymbol{\upvarphi}\_{\boldsymbol{j}}^{p} \left( \boldsymbol{P}\_{p,\boldsymbol{j}} \right) \\\\ &= \delta\_{\boldsymbol{j}:k\_{1},\ldots,k\_{N}} \left( \frac{1}{p^{\boldsymbol{j}}} - \frac{1}{p^{\boldsymbol{j}+1}} \right) \boldsymbol{\upvarphi} \left( \prod\_{l=1}^{N} a\_{l}^{n\_{l}} \right) , \end{aligned}$$

by (35a). Thus, the joint free-distributional data (38) holds.

Definitely, if *N* = 1 in (38), one obtains the formula (36).

#### **8. On the Banach** *<sup>∗</sup>***-Probability Spaces** LS*<sup>A</sup> p***,***j*

Let (*A*, *<sup>ψ</sup>*) be an arbitrarily fixed unital *<sup>C</sup>*∗-probability space, and let S*p*(*j*) be in the sense of (30), for all *p* ∈ P, *j* ∈ Z. Then, one can construct the tensor product *C*∗-probability spaces, the *j*-th *p*-adic *A*-space,

$$\mathfrak{S}\_p^A(j) = \left(\mathfrak{S}\_p^A, \,\,\psi\_j^p\right) = \left(A \otimes\_{\mathbb{C}} \mathfrak{S}\_{p'}, \,\,\psi \otimes \,\,\phi\_j^p\right).$$

of (34), for *p* ∈ P, *j* ∈ Z.

Throughout this section, we fix *<sup>p</sup>* ∈ P, *<sup>j</sup>* <sup>∈</sup> <sup>Z</sup>, and the corresponding *<sup>j</sup>*-th *<sup>p</sup>*-adic *<sup>A</sup>*-space S*<sup>A</sup> <sup>p</sup>* (*j*). In addition, we keep using our notation *T<sup>a</sup> <sup>p</sup>*,*<sup>k</sup>* for the free random variables *<sup>a</sup>* <sup>⊗</sup> *Pp*,*<sup>k</sup>* of <sup>S</sup>*A*(*j*), for all *<sup>a</sup>* <sup>∈</sup> (*A*, *<sup>ψ</sup>*) and *<sup>k</sup>* <sup>∈</sup> <sup>Z</sup>, where *Pp*,*<sup>k</sup>* are the generating projections (26) of the *<sup>p</sup>*-adic boundary subalgebra <sup>S</sup>*p*.

Recall that, by (36) and (38),

$$
\psi\_j^p \left( T\_{p,k}^a \right) = \delta\_{j,k} \psi(a) \left( \frac{1}{p^\dagger} - \frac{1}{p^{j+1}} \right), \forall k \in \mathbb{Z}. \tag{39}
$$

Now, let *φ* be the *Euler totient function*,

$$
\Phi: \mathbb{R} \to \mathbb{C}\_{\prime}
$$

defined by

$$\phi(n) = \left| \{ k \in \mathbb{N} : k \le n, \text{ } \gcd(n, k) = 1 \} \right|, \tag{40}$$

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

By the definition (40),

$$\phi(n) = n \left( \underset{q \in \mathcal{P}, \, q \vert n}{\Pi} \left( 1 - \frac{1}{q} \right) \right),\tag{41}$$

for all *n* ∈ N, where "*q* | *n*" means "*q divides n*." Thus,

$$
\phi(q) = q - 1 = q \left( 1 - \frac{1}{q} \right), \forall q \in \mathcal{P}\_{\prime} \tag{42}
$$

by (40) and (41).

By (42), we have

$$\begin{aligned} q\_j^p \left( P\_{p,k} \right) &= \frac{\delta\_{j,k}}{p!} \left( 1 - \frac{1}{p} \right), \\ &= \frac{\delta\_{j,k} \Phi(p)}{p^{j+1}}, \end{aligned}$$

for *Pp*,*<sup>k</sup>* <sup>∈</sup> <sup>S</sup>*p*, and hence,

$$
\psi\_{\rangle}^{p}\left(T\_{p,k}^{a}\right) = \delta\_{j,k}\left(\frac{\phi(p)}{p^{\circ+1}}\right)\psi(a),\tag{43}
$$

for all *T<sup>a</sup> <sup>p</sup>*,*<sup>k</sup>* <sup>∈</sup> <sup>S</sup>*<sup>A</sup> <sup>p</sup>* (*j*), by (39).

Let's consider the following estimates.

**Lemma 1.** *Let φ be the Euler totient function (40). Then,*

$$\lim\_{p \to \infty} \frac{\phi(p)}{p^{j+1}} = \begin{cases} 0, & \text{if } j > 0, \\ 1, & \text{if } j = 0, \\ \infty, \text{ lIndefined}, & \text{if } j < 0, \end{cases} \tag{44}$$

*for all j* ∈ Z*, where "p* → ∞*" means "p is getting bigger and bigger in* P.*"*

**Proof.** Observe that

$$\lim\_{p \to \infty} \frac{\phi(p)}{p} = \lim\_{p \to \infty} \left( 1 - \frac{1}{p} \right) = 1,$$

by (42). Thus, one can get that

$$\lim\_{p \to \infty} \frac{\phi(p)}{p^{\circ+1}} = \lim\_{p \to \infty} \left(\frac{\phi(p)}{p}\right)\left(\frac{1}{p^{\circ}}\right) = \lim\_{p \to \infty} \frac{1}{p^{\circ}}\lambda$$

for *j* ∈ Z. Thus,

$$\lim\_{p \to \infty} \frac{\phi(p)}{p^{j+1}} = \lim\_{p \to \infty} \frac{1}{p^j} = \begin{cases} 0, & \text{if } j > 0, \\ 1, & \text{if } j = 0, \\ \lim\_{p \to \infty} p^{|j|} = \infty, & \text{if } j < 0, \end{cases}$$

where |*j*| are the absolute values of *j* ∈ Z. Thus, the estimation (44) holds.

#### *8.1. Semicircular Elements*

Let (*B*, *ϕ*) be an arbitrary *topological* ∗-*probability space* (*C*∗-probability space, or *W*∗-probability space, or Banach ∗-probability space, etc.) equipped with a topological ∗-algebra *B* (*C*∗-algebra, resp., *W*∗-algebra, resp., Banach ∗-algebra), and a linear functional *ϕ* on *B*.

**Definition 8.** *A self-adjoint operator a* ∈ *B is said to be semicircular in* (*B*, *ϕ*), *if*

$$\varphi\left(a^{\mathfrak{n}}\right) = \omega\_{\mathfrak{n}}c\_{\mathfrak{k}}; n \in \mathbb{N}, \omega\_{\mathfrak{n}} = \begin{cases} 1, & \text{if } n \text{ is even,} \\\ 0, & \text{if } n \text{ is odd,} \end{cases} \tag{45}$$

*and ck are the k-th Catalan numbers,*

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

*for all k* ∈ N<sup>0</sup> = N ∪ {0}.

By [15–17], if *kn*(...) is the *free cumulant on B in terms of ϕ*, then a self-adjoint operator *a* is *semicircular* in (*B*, *ϕ*), if and only if

$$k\_n \left( \underbrace{a\_\prime \, a\_\prime \dots \,, a}\_{n \text{-times}} \right) = \begin{cases} 1, & \text{if } n = 2, \\ 0, & \text{otherwise}, \end{cases} \tag{46}$$

for all *n* ∈ N. The above characterization (46) of the semicircularity (45) holds by the *Möbius inversion of* [15]. For example, definition (45) and the characterization (46) give equivalent free distributions, *the semicircular law*.

If *al* are semicircular elements in topological ∗-probability spaces (*Bl*, *ϕl*), for *l* = 1, 2, then the free distributions of *al* are completely characterized by the free-moment sequences,

$$\left(\wp\_l(a\_l^n)\right)\_{n=1}^\infty, \text{ for } l=1,2,\dots$$

by the self-adjointness of *a*<sup>1</sup> and *a*2; and by (45), one obtains that

$$\begin{array}{rcl} \left(\boldsymbol{\varrho}\_{1}(\boldsymbol{a}\_{1}^{n})\right)\_{n=1}^{\infty} &= \left(\omega\_{\boldsymbol{n}}\boldsymbol{c}\_{\frac{n}{2}}\right)\_{n=1}^{\infty} \\ &= \left(\boldsymbol{0}, \ c\_{1}, \ \boldsymbol{0}, \ c\_{2}, \ \boldsymbol{0}, \ c\_{3}, \ \ldots\right) \\ &= \left(\boldsymbol{\varrho}\_{2}(\boldsymbol{a}\_{2}^{n})\right)\_{n=1}^{\infty}. \end{array}$$

Equivalently, the free distributions of the semicircular elements *a*<sup>1</sup> and *a*<sup>2</sup> are characterized by the free-cumulant sequences,

$$\left(k\_n^1(a\_1,\ldots,a\_1)\right)\_{n=1}^\infty = \left(0,\ 1,\ 0,\ 0,\ 0,\ \ldots\right) = \left(k\_n^2(a\_2,\ldots,a\_2)\right)\_{n=1}^\infty.$$

by (46), where *k<sup>l</sup> <sup>n</sup>*(...) are the free cumulants on *Bl* in terms of *ϕl*, for all *l* = 1, 2.

It shows the universality of free distributions of semicircular elements. For example, the free distributions of any semicircular elements are universally characterized by either the free-moment sequence

$$\left(\omega\_n c\_{\frac{\Psi}{\Psi}}\right)\_{n=1}^{\infty} \tag{47}$$

or the free-cumulant sequence

(0, 1, 0, 0, ...).

**Definition 9.** *Let a be a semicircular element of a topological* ∗*-probability space* (*B*, *ϕ*). *The free distribution of a is called "the" semicircular law.*

#### *8.2. Tensor Product Banach* <sup>∗</sup>*-Algebra* LS*<sup>A</sup> p*

Let S*<sup>A</sup> <sup>p</sup>* (*k*) = S*A <sup>p</sup>* , *<sup>ψ</sup><sup>p</sup> k* be the *k*-th *p*-adic *A*-space (34), for all *p* ∈ P, *k* ∈ Z. Throughout this section, we fix *<sup>p</sup>* ∈ P, *<sup>k</sup>* <sup>∈</sup> <sup>Z</sup>, and S*<sup>A</sup> <sup>p</sup>* (*k*). In addition, denote *<sup>a</sup>* ⊗ *Pp*,*<sup>j</sup>* by *<sup>T</sup><sup>a</sup> <sup>p</sup>*,*<sup>j</sup>* in <sup>S</sup>*<sup>A</sup> <sup>p</sup>* (*k*), for all *a* ∈ (*A*, *ψ*) and *j* ∈ Z.

Define now bounded linear transformations **c***<sup>A</sup> <sup>p</sup>* and **a***<sup>A</sup> <sup>p</sup>* "acting on the tensor product *C*∗-algebra S*A <sup>p</sup>* ," by linear morphisms satisfying,

$$\begin{aligned} \mathbf{c}\_p^A \left( T\_{p,j}^a \right) &= T\_{p,j+1'}^a \\\\ \mathbf{a}\_p^A \left( T\_{p,j}^a \right) &= T\_{p,j-1'}^a \end{aligned} \tag{48}$$

on S*p*, for all *<sup>j</sup>* <sup>∈</sup> <sup>Z</sup>.

By the definitions (27) and (31), and by the structure theorem (33), the above linear morphisms **c***<sup>A</sup> p* and **a***<sup>A</sup> <sup>p</sup>* of (48) are well-defined on <sup>S</sup>*<sup>A</sup> p* .

*p*,*j*

*p*

By (48), one can understand **c***<sup>A</sup> <sup>p</sup>* and **a***<sup>A</sup> <sup>p</sup>* as bounded linear transformations contained in the *operator space <sup>B</sup>*(S*<sup>A</sup> <sup>p</sup>* ) consisting of all bounded linear operators acting on <sup>S</sup>*<sup>A</sup> <sup>p</sup>* , by regarding the *<sup>C</sup>*∗-algebra <sup>S</sup>*<sup>A</sup> p* as a *Banach space* equipped with its *C*∗-norm (e.g., [32]). Under this sense, the operators **c***<sup>A</sup> <sup>p</sup>* and **a***<sup>A</sup> <sup>p</sup>* of (48) are well-defined *Banach-space operators on* S*<sup>A</sup> p* .

**Definition 10.** *The Banach-space operators* **c***<sup>A</sup> <sup>p</sup> and* **a***<sup>A</sup> <sup>p</sup> on* <sup>S</sup>*<sup>A</sup> <sup>p</sup>* , *in the sense of (48), are called the A-tensor p-creation, respectively, the A-tensor p-annihilation on* S*<sup>A</sup> <sup>p</sup>* . *Define a new Banach-space operator l<sup>A</sup> <sup>p</sup> by*

$$\mathbf{d}\_p^A = \mathbf{c}\_p^A + \mathbf{a}\_p^A \text{ on } \mathfrak{S}\_p^A. \tag{49}$$

*We call this operator* **l***<sup>A</sup> <sup>p</sup> , the A-tensor p-radial operator on* <sup>S</sup>*<sup>A</sup> p* .

Let **l***<sup>A</sup> <sup>p</sup>* be the *A*-tensor *p*-radial operator **c***<sup>A</sup> <sup>p</sup>* + **a***<sup>A</sup> <sup>p</sup>* of (49) in *<sup>B</sup>*(S*<sup>A</sup> <sup>p</sup>* ). Construct a *closed subspace* <sup>L</sup>*<sup>A</sup> p* of *<sup>B</sup>*(S*<sup>A</sup> <sup>p</sup>* ) by

$$\mathfrak{L}\_p^A = \overline{\mathbb{C}[\{\mathbf{l}\_p^A\}]} \subset B(\mathfrak{S}\_p^A), \tag{50}$$

equipped with the inherited *operator-norm* . from the operator space *<sup>B</sup>*(S*<sup>A</sup> <sup>p</sup>* ), defined by

$$||T|| = \sup\{ ||Tx||\_{\mathfrak{S}\_p^A} : \mathfrak{x} \in \mathfrak{S}\_p^A \text{ s.t. } ||x||\_{\mathfrak{S}\_p^A} = 1 \},$$

where .S*<sup>A</sup> <sup>p</sup>* is the *<sup>C</sup>*∗-norm on the *<sup>A</sup>*-tensor *<sup>p</sup>*-adic algebra <sup>S</sup>*<sup>A</sup> <sup>p</sup>* (e.g., [32]).

By the definition (50), the set L*<sup>A</sup> <sup>p</sup>* is not only a closed subspace of *<sup>B</sup>*(S*<sup>A</sup> <sup>p</sup>* ), but also an algebra over <sup>C</sup>. Thus, the subspace L*<sup>A</sup> <sup>p</sup>* is a Banach algebra embedded in *<sup>B</sup>*(S*<sup>A</sup> <sup>p</sup>* ).

On the Banach algebra L*<sup>A</sup> <sup>p</sup>* of (50), define a unary operation (∗) by

$$\left(\sum\_{k=0}^{\infty} \mathbf{s}\_k \left(\mathbf{l}\_p^A\right)^k\right)^\* = \sum\_{k=0}^{\infty} \overline{\mathbf{s}\_k} \left(\mathbf{l}\_p^A\right)^k \text{ in } \mathfrak{L}\_p^A\tag{51}$$

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

Then, the operation (51) is a well-defined *adjoint on* L*<sup>A</sup> <sup>p</sup>* . Thus, equipped with the adjoint (51), this Banach algebra L*<sup>A</sup> <sup>p</sup>* of (50) forms a *Banach* <sup>∗</sup>*-algebra* in *<sup>B</sup>*(S*<sup>A</sup> <sup>p</sup>* ). For example, all elements of <sup>L</sup>*<sup>A</sup> <sup>p</sup>* are adjointable (in the sense of [32]) in *<sup>B</sup>*(S*<sup>A</sup> <sup>p</sup>* ).

Let L*<sup>A</sup> <sup>p</sup>* be in the sense of (50). Construct now the tensor product Banach <sup>∗</sup>-algebra LS*<sup>A</sup> <sup>p</sup>* by

$$\mathfrak{L}\mathfrak{S}\_p^A \stackrel{def}{=} \mathfrak{L}\_p^A \otimes\_{\mathbb{C}} \mathfrak{S}\_p^A = \mathfrak{L}\_p^A \otimes\_{\mathbb{C}} \left(A \otimes\_{\mathbb{C}} \mathfrak{S}\_p\right),\tag{52}$$

where <sup>⊗</sup><sup>C</sup> is the *tensor product* of Banach <sup>∗</sup>-algebras. Since <sup>S</sup>*<sup>A</sup> <sup>p</sup>* is a *C*∗-algebra, it is a Banach ∗-algebra too.

Take now a generating element **l***A p n* ⊗ *<sup>T</sup><sup>a</sup> p*,*j* , for some *<sup>n</sup>* ∈ N0, and *<sup>j</sup>* ∈ Z, where *<sup>T</sup><sup>a</sup> <sup>p</sup>*,*<sup>j</sup>* = *a* ⊗ *Pp*,*<sup>j</sup>* are in the sense of (37) in S*<sup>A</sup> <sup>p</sup>* , with axiomatization:

$$\left(\mathbf{1}\_p^A\right)^0 = \mathbf{1}\_{\circledast\_p^{A,\prime}}$$

the *identity operator on* S*<sup>A</sup> <sup>p</sup> in B* S*A p* , satisfying

$$1\_{\otimes\_p^A}(T) = T\_{\prime}$$

for all *<sup>T</sup>* <sup>∈</sup> S*<sup>A</sup> <sup>p</sup>* . Define now a bounded linear morphism *E<sup>A</sup> <sup>p</sup>* : LS*<sup>A</sup> <sup>p</sup>* <sup>→</sup> <sup>S</sup>*<sup>A</sup> <sup>p</sup>* by a linear transformation satisfying that:

$$E\_p^A\left(\left(\mathbf{I}\_p^A\right)^k \otimes T\_{p,j}^a\right) = \frac{1}{\lfloor\frac{k}{2}\rfloor + 1} \left(\mathbf{I}\_p^A\right)^k (T\_{p,j}^a)\_\prime \tag{53}$$

for all *<sup>k</sup>* <sup>∈</sup> <sup>N</sup>0, *<sup>j</sup>* <sup>∈</sup> <sup>Z</sup>, where *k* 2 is the *minimal integer greater than or equal to <sup>k</sup>* <sup>2</sup> , for all *k* ∈ N0, for example,

$$
\begin{bmatrix} \frac{3}{2} \end{bmatrix} = 2 = \begin{bmatrix} \frac{4}{2} \end{bmatrix} \dots
$$

*Symmetry* **2019**, *11*, 819

By the cyclicity (50) of the tensor factor L*<sup>A</sup> <sup>p</sup>* of LS*<sup>A</sup> <sup>p</sup>* , and by the structure theorem (33) of the other tensor factor <sup>S</sup>*<sup>A</sup> <sup>p</sup>* of LS*<sup>A</sup> <sup>p</sup>* , the above morphism *E<sup>A</sup> <sup>p</sup>* of (53) is a well-defined bounded linear transformation from LS*<sup>A</sup> <sup>p</sup>* onto <sup>S</sup>*<sup>A</sup> p* .

Now, consider how our *A*-tensor *p*-radial operator **l***<sup>A</sup> <sup>p</sup>* = **c***<sup>A</sup> <sup>p</sup>* + **a***<sup>A</sup> <sup>p</sup>* acts on <sup>S</sup>*<sup>A</sup> <sup>p</sup>* . First, observe that: if **c***<sup>A</sup> <sup>p</sup>* and **a***<sup>A</sup> <sup>p</sup>* are the *<sup>A</sup>*-tensor *<sup>p</sup>*-creation, respectively, the *<sup>A</sup>*-tensor *<sup>p</sup>*-annihilation on <sup>S</sup>*<sup>A</sup> <sup>p</sup>* , then

$$\mathbf{c}\_p^A \mathbf{a}\_p^A \left( T^a\_{p,j} \right) = T^a\_{p,j} = \mathbf{a}\_p^A \mathbf{c}\_p^A \left( T^a\_{p,j} \right).$$

for all *a* ∈ (*A*, *ψ*), and for all *j* ∈ Z, *p* ∈ P, and, hence,

$$\mathbf{c}\_p^A \mathbf{a}\_p^A = \mathbf{1}\_{\mathfrak{S}\_p^A} = \mathbf{a}\_p^A \mathbf{c}\_p^A \text{ on } \mathfrak{S}\_p^A. \tag{54}$$

**Lemma 2.** *Let* **c***<sup>A</sup> <sup>p</sup>* , **a***<sup>A</sup> <sup>p</sup> be the A-tensor p-creation, respectively, the A-tensor p-annihilation on* <sup>S</sup>*<sup>A</sup> <sup>p</sup>* . *Then,*

$$\begin{aligned} \left(\mathbf{c}\_p^A\right)^n \left(\mathbf{a}\_p^A\right)^n &= \mathbf{1}\_{\mathbb{S}\_p^A} = \left(\mathbf{a}\_p^A\right)^n \left(\mathbf{c}\_p^A\right)^n, \\\\ \left(\mathbf{c}\_p^A\right)^{n\_1} \left(\mathbf{a}\_p^A\right)^{n\_2} &= \left(\mathbf{a}\_p^A\right)^{n\_2} \left(\mathbf{c}\_p^A\right)^{n\_1}, \end{aligned} \tag{55}$$

*on* S*<sup>A</sup> <sup>p</sup>* , *for all n*, *n*1, *n*<sup>2</sup> ∈ N.

**Proof.** The formulas in (55) hold by induction on (54).

By (55), one can get that

$$\left(\mathbf{l}\_p^A\right)^n = \left(\mathbf{c}\_p^A + \mathbf{a}\_p^A\right)^n = \sum\_{k=0}^n \binom{n}{k} \left(\mathbf{c}\_p^A\right)^k \left(\mathbf{a}\_p^A\right)^{n-k},\tag{56}$$

with identity:

$$\left(\mathbf{c}\_p^A\right)^0 = \mathbf{1}\_{\mathfrak{S}\_p^A} = \left(\mathbf{a}\_p^A\right)^0,$$

for all *n* ∈ N, where

$$
\binom{n}{k} = \frac{n!}{k!(n-k)!},
$$

for all *k* ≤ *n* ∈ N0. By (56), one obtains the following proposition.

**Proposition 8.** *Let* **l***<sup>A</sup> <sup>p</sup>* <sup>∈</sup> <sup>L</sup>*<sup>A</sup> <sup>p</sup> be the A-tensor p-radial operator on* <sup>S</sup>*<sup>A</sup> <sup>p</sup>* . *Then,*

$$\left(\mathbf{1}\_p^A\right)^{2m-1} \text{ does not contain } \mathbf{1}\_{\mathbb{Q}\_p^A}\text{-term, and }\tag{57}$$

$$\left(\mathbf{1}\_p^A\right)^{2m} \text{ contains its } \mathbf{1}\_{\mathfrak{S}\_p^A}\text{-term}\_{\prime}\left(\begin{array}{c} 2m\\m\end{array}\right) \cdot \mathbf{1}\_{\mathfrak{S}\_p^A}\tag{58}$$

*for all m* ∈ N.

**Proof.** The proofs of (57) and (58) are done by straightforward computations of (56) with the help of (55).

#### *8.3. Free-Probabilistic Information of Q<sup>a</sup> <sup>p</sup>*,*<sup>j</sup> in* LS*<sup>A</sup> p*

Fix *<sup>p</sup>* ∈ P, and a unital *<sup>C</sup>*∗-probability space (*A*, *<sup>ψ</sup>*), and let LS*<sup>A</sup> <sup>p</sup>* be the Banach ∗-algebra (52). Let *E<sup>A</sup> <sup>p</sup>* : LS*<sup>A</sup> <sup>p</sup>* <sup>→</sup> <sup>S</sup>*<sup>A</sup> <sup>p</sup>* be the linear transformation (53). Throughout this section, let

$$\mathcal{Q}\_{p,j}^{a} \stackrel{demote}{=} \mathbf{l}\_p^A \odot T\_{p,j}^a \in \mathfrak{QC}\_{p,}^A \tag{59}$$

for all *<sup>j</sup>* ∈ Z, where *<sup>T</sup><sup>a</sup> <sup>p</sup>*,*<sup>j</sup>* <sup>=</sup> *<sup>a</sup>* <sup>⊗</sup> *Pp*,*<sup>j</sup>* <sup>∈</sup> <sup>S</sup>*<sup>A</sup> <sup>p</sup>* are in the sense of (37) generating <sup>S</sup>*<sup>A</sup> <sup>p</sup>* , for *a* ∈ (*A*, *ψ*), and *j* ∈ Z. Observe that

$$\begin{aligned} \left(\mathbb{Q}\_{p,j}^{a}\right)^{n} &= \left(\mathbb{I}\_{p}^{A}\otimes T\_{p,j}^{a}\right)^{n} \\ &= \left(\mathbb{I}\_{p}^{A}\right)^{n}\otimes \left(T\_{p,j}^{a}\right)^{n} = \left(\mathbb{I}\_{p}^{A}\right)^{n}\otimes T\_{p,j'}^{a^{n}} \end{aligned} \tag{60}$$

by (37), for all *n* ∈ N, for all *j* ∈ Z.

If *Q<sup>a</sup> <sup>p</sup>*,*<sup>j</sup>* <sup>∈</sup> LS*<sup>A</sup> <sup>p</sup>* is in the sense of (59) for *j* ∈ Z, then

$$E\_p^A\left(\left(Q\_{p,j}^a\right)^n\right) = \frac{1}{\left[\frac{n}{2}\right] + 1} \left(\mathbf{I}\_p^A\right)^n \left(T\_{p,j}^{a^n}\right),\tag{61}$$

by (53) and (60), for all *n* ∈ N.

For any fixed *<sup>j</sup>* <sup>∈</sup> <sup>Z</sup>, define a linear functional *<sup>τ</sup><sup>p</sup> <sup>j</sup>* on LS*<sup>A</sup> <sup>p</sup>* by

$$\mathfrak{tr}\_{\mathfrak{p}}^{p} = \mathfrak{p}\_{\mathfrak{p}}^{p} \circ E\_{\mathfrak{p}}^{A} \text{ on } \mathfrak{QC}\_{\mathfrak{p}}^{A} \text{.}\tag{62}$$

where *ψ<sup>p</sup> <sup>j</sup>* <sup>=</sup> *<sup>ψ</sup>* <sup>⊗</sup> *<sup>ϕ</sup><sup>p</sup> <sup>j</sup>* is a linear functional (35a), or (35b) on <sup>S</sup>*<sup>A</sup> p* .

By the linearity of both *ψ<sup>p</sup> <sup>j</sup>* and *<sup>E</sup><sup>A</sup> <sup>p</sup>* , the morphism *<sup>τ</sup><sup>p</sup> <sup>j</sup>* of (62) is a well-defined linear functional on LS*<sup>A</sup> <sup>p</sup>* for *<sup>j</sup>* <sup>∈</sup> <sup>Z</sup>. Thus, the pair LS*<sup>A</sup> <sup>p</sup>* , *<sup>τ</sup><sup>p</sup> j* forms a *Banach* ∗-*probability space* (e.g., [22]).

**Definition 11.** *The Banach* ∗*-probability spaces*

$$\mathfrak{L}\mathfrak{S}\mathfrak{F}\_{p,j}^{A} \stackrel{donde}{=} \left(\mathfrak{L}\mathfrak{S}\mathfrak{F}\_{p}^{A}, \,\,\tau\_{j}^{p}\right) \tag{63}$$

*are called the A-tensor j-th p-adic (free-)filters, for all p* ∈ P, *<sup>j</sup>* <sup>∈</sup> <sup>Z</sup>, *where <sup>τ</sup><sup>p</sup> <sup>j</sup> are in the sense of (62).*

By (61) and (62), if *Q<sup>a</sup> <sup>p</sup>*,*<sup>j</sup>* is in the sense of (59) in LS*<sup>A</sup> p*,*j* , then

$$\pi\_{\!}^{p}\left(\left(Q\_{p,\!\!\!/ }^{a}\right)^{n}\right) = \frac{1}{\left[\frac{n}{2}\right] + 1} \,\,\psi\_{\!\!\!/ }^{p}\left((\mathbf{I}\_{p}^{A})^{n}\left(T\_{p,\!\!\!/ }^{a}\right)\right),\tag{64}$$

for all *n* ∈ N.

**Theorem 2.** *Let Q<sup>a</sup> <sup>p</sup>*,*<sup>k</sup>* = **<sup>l</sup>***<sup>A</sup> <sup>p</sup>* ⊗ *<sup>T</sup><sup>a</sup> <sup>p</sup>*,*<sup>k</sup>* = **<sup>l</sup>***<sup>A</sup> p* ⊗ *a* ⊗ *Pp*,*<sup>k</sup> be a free random variable (59) of the A-tensor j-th p-adic filter* LS*<sup>A</sup> <sup>p</sup>*,*<sup>j</sup> of (63), for p* ∈ P, *j* ∈ Z, *for all k* ∈ Z. *Then,*

$$\pi\_j^p\left(\left(Q\_{p,k}^a\right)^n\right) = \delta\_{j,k}\omega\_n\psi(a^n)\varepsilon\_{\frac{n}{2}}\left(\frac{\phi(p)}{p^{j+1}}\right),\tag{65}$$

*where ω<sup>n</sup> are in the sense of (45), for all n* ∈ N.

**Proof.** Let *Q<sup>a</sup> <sup>p</sup>*,*<sup>j</sup>* be in the sense of (59) in LS*<sup>A</sup> p*,*j* , for the fixed *p* ∈ P and *j* ∈ Z. Then,

$$\pi\_j^p\left(\left(Q^a\_{p,j}\right)^{2n-1}\right) = \psi\_j^p\left(E\_p^A\left(\left(Q^a\_{p,j}\right)^{2n-1}\right)\right).$$

by (62)

$$\left(\frac{1}{\left[\frac{2n-1}{2}\right]+1}\right)\psi\_j^p\left((\mathbf{1}\_p^A)^{2n-1}\left(T\_{p,j}^{a^{2n-1}}\right)\right)$$

by (64)

by (56)

$$= \left(\frac{1}{\left[\frac{2n-1}{2}\right]+1}\right)\psi\_j^p \left(\begin{pmatrix} 2n-1\\k\end{pmatrix} (\mathbf{c}\_p^A)^k (\mathbf{a}\_p^A)^{2n-1-k}\right) \left(T\_{p,j}^{a^{2n-1}}\right)\right)^{\frac{1}{2}}$$

= 0,

by (57), for all *n* ∈ N.

Observe now that, for any *n* ∈ N,

$$\pi\_j^p\left(\left(Q\_{p,j}^a\right)^{2u}\right) = \left(\frac{1}{\left[\frac{2u}{2}\right]+1}\right)\psi\_j^p\left((\mathbf{1}\_p^A)^{2u}\left(T\_{p,j}^{a^{2u}}\right)\right)$$

by (64)

$$= \left(\frac{1}{n+1}\right) \psi\_j^p \left( \begin{pmatrix} \sum\_{k=0}^{2n} \\ k \end{pmatrix} (\mathbf{c}\_p^A)^k (\mathbf{a}\_p^A)^{2n-k} \right) \left( T\_{p,j}^{a^{2n}} \right))$$

by (56)

$$\theta = \left(\frac{1}{n+1}\right)\psi\_j^p \left(\left(\begin{array}{c} 2n \\ n \end{array}\right)T\_{p,j}^{\text{ar}} + \left[\text{Rest terms}\right]\right)$$

by (58)

$$\psi = \frac{1}{n+1} \left( \begin{array}{c} 2n \\ n \end{array} \right) \psi\_j^p \left( \begin{array}{c} T\_{p,j}^{\text{a}^{2n}} \end{array} \right) = \frac{1}{n+1} \left( \begin{array}{c} 2n \\ n \end{array} \right) \psi(a^{2n}) \left( \frac{\phi(p)}{p^{i+1}} \right)^{\frac{1}{i}}$$

by (39) and (43)

$$\;=c\_n\psi(a^{2n})\left(\frac{\phi(p)}{p^{\prime+1}}\right).$$

where *cn* are the *n*-th Catalan numbers.

If *<sup>k</sup>* = *<sup>j</sup>* in Z, and if *<sup>Q</sup><sup>a</sup> <sup>p</sup>*,*<sup>k</sup>* are in the sense of (59) in LS*<sup>A</sup> p*,*j* , then

$$
\tau\_j^p \left( \left( Q\_{p,k}^a \right)^n \right) = 0,
$$

for all *<sup>n</sup>* <sup>∈</sup> <sup>N</sup>, by the definition (22a) of the linear functional *<sup>ϕ</sup><sup>p</sup> <sup>j</sup>* on <sup>S</sup>*p*, inducing the linear functional *ψp <sup>j</sup>* <sup>=</sup> *<sup>ψ</sup>* <sup>⊗</sup> *<sup>ϕ</sup><sup>p</sup> <sup>j</sup>* on the tensor factor <sup>S</sup>*<sup>A</sup> <sup>p</sup>* of LS*<sup>A</sup> p*,*j* .

Therefore, the free-distributional data (65) holds true.

Note that, if *a* is self-adjoint in (*A*, *ψ*), then the generating operators *Q<sup>a</sup> <sup>p</sup>*,*<sup>k</sup>* of the *A*-tensor *j*-th *<sup>p</sup>*-adic filter LS*<sup>A</sup> <sup>p</sup>*,*<sup>j</sup>* are self-adjoint in LS*<sup>A</sup> <sup>p</sup>* , since

$$\begin{aligned} \left(\mathbf{Q}\_{p,k}^{a}\right)^{\*} &= \left(\mathbf{l}\_{p}^{A} \otimes T\_{p,k}^{a}\right)^{\*} = (\mathbf{l}\_{p}^{A})^{\*} \otimes \left(T\_{p,k}^{a}\right)^{\*}\\ &= \mathbf{l}\_{p}^{A} \otimes T\_{p,k}^{a} = \mathbf{Q}\_{p,k'}^{a} \end{aligned}$$

for all *k* ∈ Z, for *p* ∈ P, *j* ∈ Z, by (51).

Thus, if *a* is a self-adjoint free random variable of (*A*, *ψ*), then the above formula (65) fully characterizes the free distributions (up to *τ<sup>p</sup> <sup>j</sup>* ) of the generating operators *<sup>Q</sup><sup>a</sup> <sup>p</sup>*,*<sup>k</sup>* of LS*<sup>A</sup> <sup>p</sup>* , for all *k*, *j* ∈ Z, for *p* ∈ P.

The free-distributional data (65) can be refined as follows: if *<sup>p</sup>* ∈ P, *<sup>j</sup>* <sup>∈</sup> <sup>Z</sup> , and if LS*<sup>A</sup> <sup>p</sup>*,*<sup>j</sup>* is the corresponding *A*-tensor *j*-th *p*-adic filter (63), then

$$\pi\_j^p\left(\left(Q\_{p,j}^a\right)^n\right) = \omega\_n c\_{\frac{n}{2}} \psi(a^n) \left(\frac{\phi(p)}{p^{j+1}}\right),\tag{66}$$

for all *n* ∈ N, and

$$\tau^p\_j \left( \left( \mathbb{Q}^a\_{p,k} \right)^n \right) = 0,\tag{67}$$

for all *n* ∈ N, whenever *k* = *j* in Z, for all *n* ∈ N.

Before we focus on non-zero free-distributional data (66) of *Q<sup>a</sup> p*,*j* , let's conclude the following result for {*Q<sup>a</sup> <sup>p</sup>*,*k*}*k*=*j*∈Z.

**Corollary 1.** *Let <sup>p</sup>* ∈ P, *<sup>j</sup>* <sup>∈</sup> <sup>Z</sup>, *and let* LS*<sup>A</sup> <sup>p</sup>*,*<sup>j</sup> be the A-tensor j-th p-adic filter (63). Then, the generating operators*

$$Q\_{p,k}^{a} = \mathbf{l}\_p^A \otimes T\_{p,j}^a = \mathbf{l}\_p^A \otimes (a \otimes P\_{p,j}) \in \mathfrak{LSet}\_{p,j}^A$$

*have the zero free distribution, whenever k* = *j in* Z.

## **Proof.** It is proven by (65) and (67).

By the above corollary, we now restrict our interests to the "*j*-th" generating operators *Q<sup>a</sup> <sup>p</sup>*,*<sup>j</sup>* of (59) in the *<sup>A</sup>*-tensor "*j*-th" *<sup>p</sup>*-adic filter LS*<sup>A</sup> p*,*j* , for all *p* ∈ P, *j* ∈ Z, having non-zero free distributions determined by (66).

## **9. On the Free Product Banach** *∗***-Probability Space** LS*<sup>A</sup>*

Throughout this section, let (*A*, *ψ*) be a fixed unital *C*∗-probability space, and let

$$\mathfrak{LC}\_{p,j}^{A} = \left(\mathfrak{LC}\_{p}^{A}, \,\, \tau\_{j}^{p}\right) \tag{68}$$

be *A*-tensor *j*-th *p*-adic filters, where

$$
\mathfrak{L}\mathfrak{S}\_p^A = \mathfrak{L}\_p^A \otimes\_{\mathbb{C}} \mathfrak{S}\_p^A = \mathfrak{L}\_p^A \otimes\_{\mathbb{C}} \left(\mathcal{A} \otimes\_{\mathbb{C}} \mathfrak{S}\_p\right),
$$

are in the sense of (52), and *τ<sup>p</sup> <sup>j</sup>* are the linear functionals (62) on LS*<sup>A</sup> <sup>p</sup>* , for all *p* ∈ P, *j* ∈ Z.

Let *Q<sup>a</sup> <sup>p</sup>*,*<sup>k</sup>* = **<sup>l</sup>***<sup>A</sup> <sup>p</sup>* ⊗ *<sup>T</sup><sup>a</sup> <sup>p</sup>*,*<sup>k</sup>* = **<sup>l</sup>***<sup>A</sup> p* ⊗ *a* ⊗ *Pp*,*<sup>k</sup>* be the generating elements (59) of LS*<sup>A</sup> <sup>p</sup>*,*<sup>j</sup>* of (68), for *a* ∈ (*A*, *<sup>ψ</sup>*), *<sup>p</sup>* ∈ P, and *<sup>k</sup>*, *<sup>j</sup>* ∈ Z. Then, these operators *<sup>Q</sup><sup>a</sup> <sup>p</sup>*,*<sup>k</sup>* of LS*<sup>A</sup> <sup>p</sup>*,*<sup>j</sup>* have their free-distributional data,

$$\pi\_j^p\left(\left(Q\_{p,k}^a\right)^n\right) = \delta\_{j,k}\omega\_n\psi(a^n)c\_{\frac{n}{2}}\left(\frac{\phi(p)}{p^{l+1}}\right),\tag{69}$$

for all *n* ∈ N, by (65).

By (66) and (67), we here concentrate on the "*j*-th" generating operators of LS*<sup>A</sup> <sup>p</sup>*,*<sup>j</sup>* having non-zero free distributions (69) for all *j* ∈ Z, for all *p* ∈ P.

*9.1. Free Product Banach* <sup>∗</sup>*-Probability Space* (LS*A*, *<sup>τ</sup>*)

By (68), we have the family

$$\left\{ \mathfrak{L}\mathfrak{S}\_{p\_{\vec{J}}}^{A} : p \in \mathcal{P}, \ j \in \mathbb{Z} \right\}$$

of Banach <sup>∗</sup>-probability spaces, consisting of the *<sup>A</sup>*-tensor *<sup>j</sup>*-th *<sup>p</sup>*-adic filters LS*<sup>A</sup> p*,*j* .

Define the *free product Banach* ∗*-probability space*,

$$\begin{aligned} \left(\mathfrak{L}\mathfrak{S}\_{A\prime}\,\,\,\tau\right) & \stackrel{def}{=} \limits\_{p\in\mathcal{P}\_{\prime}} \star\_{j\in\mathbb{Z}} \mathfrak{L}\mathfrak{S}\_{p,j\prime}^{A} \\\\ &= \left(\limits\_{p\in\mathcal{P}\_{\prime}, j\in\mathbb{Z}} \mathfrak{L}\mathfrak{S}\_{p\prime}^{A} \,\,\_{p\in\mathcal{P}\_{\prime}, j\in\mathbb{Z}} \mathfrak{T}\_{j}^{p}\right) \end{aligned} \tag{70}$$

in the sense of [15,22].

By (70), the *<sup>A</sup>*-tensor *<sup>j</sup>*-th *<sup>p</sup>*-adic filters LS*p*,*<sup>j</sup>* of (68) are the *free blocks* of the Banach <sup>∗</sup>-probability space (LS*A*, *<sup>τ</sup>*) of (70).

All operators of the Banach <sup>∗</sup>-algebra LS*<sup>A</sup>* in (70) are the Banach-topology limits of linear combinations of noncommutative free reduced words (under operator-multiplication) in

$$\mathop{\sqcup}\_{p \in \mathcal{P}\_{\prime}, j \in \mathbb{Z}} \mathfrak{W}\_{p,j}^{A}.$$

More precisely, since each free block LS*<sup>A</sup> <sup>p</sup>*,*<sup>j</sup>* is generated by {*Q<sup>a</sup> <sup>p</sup>*,*k*}*a*∈*A*,*k*∈Z, for all *<sup>p</sup>* ∈ P, *<sup>j</sup>* ∈ Z, all elements of LS*<sup>A</sup>* are the Banach-topology limits of linear combinations of free words in

$$\mathop{\sqcup}\_{p \in \mathcal{P}, j \in \mathbb{Z}} \{ Q^{a}\_{p,k} \in \mathfrak{L} \mathfrak{S}\_{p,j} : a \in A, k \in \mathbb{Z} \}.$$

In particular, all noncommutative free words have their unique free "reduced" words (as operators of LS*<sup>A</sup>* under operator-multiplication) formed by

$$\prod\_{l=1}^N \left( \mathbb{Q}\_{p\_l, k\_l}^{a\_l} \right)^{n\_l} \text{, where } \mathbb{Q}\_{p\_l, k\_l}^{a\_l} \in \mathfrak{L} \mathfrak{S}\_{p\_l, \hat{j}\_l}^A$$

in LS*A*, for all *<sup>a</sup>*1, ..., *aN* <sup>∈</sup> (*A*, *<sup>ψ</sup>*), and *<sup>n</sup>*1, ..., *nN* <sup>∈</sup> <sup>N</sup>, where either the *<sup>N</sup>*-tuple

$$(p\_1, \ldots, p\_N) \text{ or } (j\_1, \ldots, j\_N)$$

is alternating in P, respectively, in Z, in the sense that:

$$p\_1 \neq p\_2 \ p\_2 \neq p\_3 \dots \ p\_{N-1} \neq p\_N \text{ in } \mathcal{P}\_{\prime 1}$$

respectively,

$$j\_1 \neq j\_2, j\_2 \neq j\_3, \dots, j\_{N-1} \neq j\_N \text{ in } \mathbb{Z}\_2$$

(e.g., see [22]).

For example, a 5-tuple

(2, 2, 3, 7, 2)

is not alternating in P, while a 5-tuple

(2, 3, 2, 7, 2)

is alternating in P, etc.

By (70), if *Q<sup>a</sup> <sup>p</sup>*,*<sup>j</sup>* are the *<sup>j</sup>*-th *<sup>a</sup>*-tensor generating operators of a free block LS*<sup>A</sup> <sup>p</sup>*,*<sup>j</sup>* of the Banach <sup>∗</sup>-probability space (LS*A*, *<sup>τ</sup>*), for all *<sup>j</sup>* <sup>∈</sup> <sup>Z</sup>, for *<sup>p</sup>* ∈ P, *<sup>j</sup>* <sup>∈</sup> <sup>Z</sup>, then *Qa p*,*j n* are contained in the same free block LS*<sup>A</sup> <sup>p</sup>*,*<sup>j</sup>* of (LS*A*, *<sup>τ</sup>*), and, hence, they are free reduced words with their lengths-1, for all *<sup>n</sup>* <sup>∈</sup> N. Therefore, we have

$$\begin{aligned} \left(\pi\left(\left(Q^a\_{p,j}\right)^n\right)\_- = \pi\_j^p \left(\left(Q^a\_{p,j}\right)^n\right)\right) \\ &= \omega\_n c\_{\frac{n}{2}} \psi(a^n) \left(\frac{\phi(p)}{p^{j+1}}\right), \end{aligned} \tag{71}$$

for all *n* ∈ N, by (69).

**Definition 12.** *The Banach* <sup>∗</sup>*-probability space* LS*<sup>A</sup> denote* <sup>=</sup> (LS*A*, *<sup>τ</sup>*) *of (70) is called the A-tensor (free-)Adelic filterization of* {LS*<sup>A</sup> p*,*j* }*p*∈P,*j*∈Z*.*

As we discussed at the beginning of Section 9, we now focus on studying free random variables of the *<sup>A</sup>*-tensor Adelic filterization LS*<sup>A</sup>* of (70) having "non-zero" free distributions.

Define a subset <sup>U</sup> of LS*<sup>A</sup>* by

$$\mathcal{U} = \left\{ \mathbb{Q}\_{p,j}^{1\_A} \in \mathfrak{L} \mathfrak{S}\_{p,j}^A \, | \, \forall p \in \mathcal{P}, \, j \in \mathbb{Z} \right\} \tag{72}$$

in LS*A*, where 1*<sup>A</sup>* is the unity of *<sup>A</sup>*, and *<sup>Q</sup>*1*<sup>A</sup> <sup>p</sup>*,*<sup>j</sup>* are the "*j*-th" 1*A*-tensor generating operators of LS*A*, in the free blocks LS*<sup>A</sup> p*,*j* , for all *p* ∈ P, *j* ∈ Z.

Then, the elements *Q*1*<sup>A</sup> <sup>p</sup>*,*<sup>j</sup>* of U have their non-zero free distributions,

$$\left(\omega\_n c\_{\frac{n}{2}} \psi(1\_A^n) \left(\frac{\Phi(p)}{p^{\ell+1}}\right)\right)\_{n=1}^{\infty} = \left(\omega\_n c\_{\frac{n}{2}} \left(\frac{\Phi(p)}{p^{\ell+1}}\right)\right)\_{n=1}^{\infty}.$$

by (71), since

$$
\psi(1\_A^n) = \psi(1\_A) = 1\_r
$$

for all *n* ∈ N. Now, define a Cartesian product set

$$\mathcal{U}\_A \stackrel{def}{=} A \times \mathcal{U}\_A \tag{73a}$$

set-theoretically, where U is in the sense of (72).

Define a function <sup>Ω</sup> : <sup>U</sup>*<sup>A</sup>* <sup>→</sup> LS*<sup>A</sup>* by

$$
\Omega\left(\left(a,\,Q\_{p,j}^{1}\right)\right) \stackrel{def}{=} Q\_{p,j}^{a} \text{ in } \mathfrak{LSet}\_{A\prime} \tag{73b}
$$

for all (*a*, *Q*1*<sup>A</sup> p*,*j* ) ∈ U*A*, where U*<sup>A</sup>* is in the sense of (73a).

It is not difficult to check that this function Ω of (73b) is a well-defined injective map. Moreover, it induces all *j*-th *a*-tensor generating elements *Q<sup>a</sup> <sup>p</sup>*,*<sup>j</sup>* of LS*<sup>a</sup> <sup>p</sup>*,*<sup>j</sup>* in LS*A*, for all *<sup>p</sup>* ∈ P, and *<sup>j</sup>* <sup>∈</sup> <sup>Z</sup>.

Define a Banach <sup>∗</sup>-subalgebra LS*<sup>A</sup>* of the *<sup>A</sup>*-tensor Adelic filterization LS*<sup>A</sup>* of (70) by

$$\mathbb{TS}\_A \stackrel{def}{=} \overline{\mathbb{C}\left[\Omega\left(\mathcal{U}\_A\right)\right]} \text{ in } \mathfrak{HS}\_{A\_{\prime}}\tag{74a}$$

where <sup>Ω</sup>(U*A*) is the subset of LS*A*, induced by (73a) and (73b), and *<sup>Y</sup>* mean the Banach-topology closures of subsets *<sup>Y</sup>* of LS*A*.

Then, this Banach ∗-subalgebra LS*<sup>A</sup>* of (74a) has a sub-structure,

$$\mathbb{L}\mathbb{S}\_A \stackrel{\text{defute}}{=} \left(\mathbb{L}\mathbb{S}\_{A\prime} \,\,\,\tau = \tau \mid\_{\mathbb{L}\mathbb{S}\_A}\right) \tag{74b}$$

in the *<sup>A</sup>*-tensor Adelic filterization LS*A*.

**Theorem 3.** *Let* LS*<sup>A</sup> be the Banach* <sup>∗</sup>*-algebra (74a) in the A-tensor Adelic filterization* LS*A*. *Then,*

$$\begin{array}{l} \mathbb{L}\mathbb{S}\_{A} \stackrel{\ast \text{-iso}}{=} \begin{array}{l} \star \stackrel{\ast \text{-iso}}{} \\ p \in \mathcal{P}, \, j \in \mathbb{Z} \end{array} \overline{\mathbb{C}\left[ \{ \mathcal{Q}\_{p,j}^{a} : a \in (A, \,\,\,\Psi) \} \right]} \\\\ \stackrel{\ast \text{-iso}}{} \overline{\mathbb{C}\left[ \begin{array}{l} \star \\ p \in \mathcal{P}, \, j \in \mathbb{Z} \end{array} \{ \mathcal{Q}\_{p,j}^{a} : a \in (A, \,\,\,\Psi) \} \right]} \end{array} \tag{75}$$

*where Q<sup>a</sup> <sup>p</sup>*,*<sup>j</sup>* ∈ Ω(U*A*) *of (73b). Here, () in the first* ∗*-isomorphic relation in (75) is the free-probability-theoretic free product determined by the linear functional τ of (70), or of (74b) (e.g., [15,22]), and () in the second* ∗*-isomorphic relation in (75) is the pure-algebraic free product generating noncommutative free words in* Ω(U*A*)*.*

**Proof.** Let LS*<sup>A</sup>* be the Banach <sup>∗</sup>-subalgebra (74a) in LS*A*. Then,

$$\mathbb{LB}\_A = \mathbb{C}\left[\{Q^a\_{p,j} \in \mathfrak{QC}^A\_{p,j} : a \in (A, \,\,\Psi)\}\_{p \in \mathcal{P}, \, j \in \mathbb{Z}}\right],$$

by (73a), (73b) and (74a)

$$\overset{\ast\text{:}\mathbf{i}\mathbf{s}}{\overset{\ast\text{:}\mathbf{i}\mathbf{s}}{\overset{\ast}{}}}\_{p\in\mathcal{P},\,\mathbf{j}\in\mathbb{Z}}\overline{\mathbb{C}\left[\left\{\mathbf{Q}^{a}\_{p\_{\mathbf{j}}}:a\in(A\_{\prime},\Psi)\right\}\right]}$$

in LS*A*, since all elements *<sup>Q</sup><sup>a</sup> <sup>p</sup>*,*<sup>j</sup>* <sup>∈</sup> <sup>Ω</sup> (U*A*) are chosen from mutually distinct free blocks LS*<sup>A</sup> <sup>p</sup>*,*<sup>j</sup>* of the *<sup>A</sup>*-tensor Adelic filterization LS*A*, and, hence, the operators {*Q<sup>a</sup> p*,*j* , *Qa*<sup>∗</sup> *p*,*j* }*p*∈P, *<sup>j</sup>*∈<sup>Z</sup> are free from each other in LS*A*, for any *<sup>a</sup>* <sup>∈</sup> (*A*, *<sup>ψ</sup>*), for all *<sup>p</sup>* ∈ P, *<sup>j</sup>* <sup>∈</sup> <sup>Z</sup>, moreover,

$$\stackrel{\ast\text{-iso}}{=} \mathbb{C}\left[\_{p \in \mathcal{P}\_{\prime}} \star\_{\text{j} \in \mathbb{Z}} \{ \mathbf{Q}^{a}\_{p, \dot{\mathbf{j}}} : a \in (A\_{\prime}, \Psi) \} \right] \prime$$

because all elements of LS*<sup>A</sup>* are the (Banach-topology limits of) linear combinations of free words in Ω(U*A*), by the very above ∗-isomorphic relation. Indeed, for any noncommutative (pure-algebraic) free words in

$$\bigcup\_{p \in \mathcal{P}, j \in \mathbb{Z}} \{ \mathcal{Q}\_{p,j}^a : a \in (\mathcal{A}, \psi) \},$$

have their unique free "reduced" words under operator-multiplication on LS*A*, as operators of LS*A*.

Therefore, the structure theorem (75) holds.

The above theorem characterizes the free-probabilistic structure of the Banach ∗-algebra LS*<sup>A</sup>* of (74a) in the *<sup>A</sup>*-tensor Adelic filterization LS*A*. This structure theorem (75) demonstrates that the Banach ∗-probability space (LS*A*, *τ*) of (74b) is well-determined, having its natural inherited free probability from that on LS*A*.

**Definition 13.** *Let* (LS*A*, *τ*) *be the Banach* ∗*-probability space (74b). Then, we call*

$$\mathsf{L}\mathfrak{B}\_A \stackrel{donde}{=} (\mathsf{L}\mathfrak{B}\_{A\prime}\mathfrak{r})\_{\prime\prime}$$

*the A-tensor (Adelic) sub-filterization of the A-tensor Adelic filterization* LS*A*.

By (69), (71), (72) and (75), one can verify that the free probability on the *A*-tensor sub-filterization LS*<sup>A</sup>* provide "possible" non-zero free distributions on the *<sup>A</sup>*-tensor Adelic filterization LS*A*, up to free probability on (*A*, *<sup>ψ</sup>*). i.e., if *<sup>a</sup>* ∈ (*A*, *<sup>ψ</sup>*) have their non-zero free distributions, then *<sup>Q</sup><sup>a</sup> <sup>p</sup>*,*<sup>j</sup>* ∈ LS*<sup>A</sup>* have non-zero free distributions, and, hence, they have their non-zero free distributions on LS*A*.

**Theorem 4.** *Let Q<sup>a</sup> <sup>p</sup>*,*<sup>j</sup>* ∈ Ω(U*A*) *be free random variables of the A-tensor sub-filterization* LS*A, for a* ∈ (*A*, *ψ*), *and p* ∈ P, *and j* ∈ Z. *Then,*

$$\begin{aligned} \tau\left(\left(Q^{a}\_{p,j}\right)^{n}\right) &= \omega\_{n}c\_{\frac{n}{2}}\psi(a^{n})\left(\frac{\phi(p)}{p^{j+1}}\right), \\\\ \tau\left(\left(\left(Q^{a}\_{p,j}\right)^{\*}\right)^{n}\right) &= \omega\_{n}c\_{\frac{n}{2}}\overline{\psi(a^{n})}\left(\frac{\phi(p)}{p^{j+1}}\right), \end{aligned} \tag{76}$$

*for all n* ∈ N.

**Proof.** The first formula of (76) is shown by (71). Thus, it suffices to prove the second formula of (76) holds. Note that

$$\begin{aligned} \left(\mathsf{Q}\_{p,j}^{a}\right)^{\*} &= \left(\mathsf{I}\_{p}^{A} \otimes \mathsf{I}\_{p,j}^{a}\right)^{\*} = \left(\mathsf{I}\_{p}^{A} \otimes \left(a \otimes P\_{p,j}\right)\right)^{\*}\\ &= \left(\mathsf{I}\_{p}^{A}\right)^{\*} \otimes \left(a \otimes P\_{p,j}\right)^{\*} = \mathsf{I}\_{p}^{A} \otimes \left(a^{\*} \otimes P\_{p,j}\right), \end{aligned}$$

and, hence,

$$\left(\mathbb{Q}\_{p,j}^{a}\right)^{\*} = \mathbb{Q}\_{p,j}^{a^\*} \text{ in } \mathbb{L}\mathbb{S}\_{A\prime} \tag{77}$$

for all *Q<sup>a</sup> <sup>p</sup>*,*<sup>j</sup>* ∈ Ω (U*A*). Thus, one has

$$\left(\left(\mathbb{Q}^{a}\_{p,j}\right)^{\*}\right)^{n} = \left(\mathbb{Q}^{a^{\*}}\_{p,j}\right)^{n} = \mathbb{Q}^{(a^{\*})^{n}}\_{p,j} = \mathbb{Q}^{(a^{n})^{\*}}\_{p,j} \text{ in } \mathbb{L}\mathbb{S}\_{A,n}$$

by (77).

Thus, one has

$$\begin{aligned} \pi \left( \left( \left( Q^{a}\_{p,j} \right)^{\*} \right)^{n} \right)^{n} &= \omega\_{n} c\_{\frac{n}{2}} \psi \left( (a^{n})^{\*} \right) \left( \frac{\phi(p)}{p^{\prime + 1}} \right), \\\\ &= \omega\_{n} c\_{\frac{n}{2}} \overline{\psi(a^{n})} \left( \frac{\phi(p)}{p^{\prime + 1}} \right), \end{aligned}$$

by (71), for all *n* ∈ N. Therefore, the second formula of (76) holds too.

## *9.2. Prime-Shifts on* LS*<sup>A</sup>*

Let LS*<sup>A</sup>* be the *<sup>A</sup>*-tensor sub-filterization (70) of the *<sup>A</sup>*-tensor Adelic filterization LS*A*. In this section, we define a certain ∗-homomorphism on LS*A*, and study asymptotic free-distributional data on LS*<sup>A</sup>* (and hence those on LS*A*) over primes.

Let P be the set of all primes in N, regarded as a *totally ordered set* (in short, a TOset) for the usual ordering (≤), i.e.,

$$\mathcal{P} = \{q\_1 < q\_2 < q\_3 < q\_4 < \cdots \},\tag{78}$$

with

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

Define an injective function *h* : P→P by

$$h\left(q\_k\right) = q\_{k+1}; k \in \mathbb{N},\tag{79}$$

where *qk* are primes of (78), for all *k* ∈ N.

**Definition 14.** *Let h be an injective function (79) on the TOset* P *of (78). We call h the shift on* P.

Let *h* be the shift (79) on the TOset P, and let

$$h^{(n)} \stackrel{def}{=} \underbrace{\mathfrak{h} \circ h \circ h \circ \cdots \circ h}\_{n \text{-times}} \text{ on } \mathcal{P},\tag{80}$$

for all *n* ∈ N, where (◦) is the usual functional *composition*.

By the definitions (79) and (80),

$$h^{(n)}\left(q\_k\right) = q\_{k+n'} \tag{81}$$

for all *<sup>n</sup>* <sup>∈</sup> <sup>N</sup>, in <sup>P</sup>. For instance, *<sup>h</sup>*(3)(2) = 7, and *<sup>h</sup>*(4)(5) = 17, etc.

These injective functions *<sup>h</sup>*(*n*) of (80) are called the *<sup>n</sup>*-*shifts* on <sup>P</sup>, for all *<sup>n</sup>* <sup>∈</sup> <sup>N</sup>.

For the shift *h* on P, one can define a ∗-*homomorphism π<sup>h</sup>* on the *A*-tensor sub-filterization LS*<sup>A</sup>* by a bounded "multiplicative" linear transformation, satisfying that

$$
\pi\_h \left( Q^{a}\_{q\_k, j} \right) = Q^{a}\_{h(q\_k), j} = Q^{a}\_{q\_{k+1}, j'} \tag{82}
$$

for all *Qqk*,*<sup>j</sup>* ∈ Ω(U*A*), for all *qk* ∈ P, for all *j* ∈ Z, where *h* is the shift (79) on P.

By (82), we have

$$\pi\_{\boldsymbol{h}}\left(\prod\_{l=1}^{N}\left(\boldsymbol{Q}\_{q\_{l\_{l}},j\_{l}}^{\boldsymbol{a}\boldsymbol{\eta}}\right)^{n\_{l}}\right) = \prod\_{l=1}^{N}\left(\boldsymbol{Q}\_{h(q\_{l\_{l}}),j\_{l}}^{\boldsymbol{a}\boldsymbol{\eta}}\right)^{n\_{l}} = \prod\_{l=1}^{N}\left(\boldsymbol{Q}\_{q\_{l\_{l}+1},j\_{l}}^{\boldsymbol{a}\boldsymbol{\eta}}\right)^{n\_{l}}\tag{83}$$

in LS*A*, for all *Q<sup>a</sup> qkl* ,*jl* ∈ <sup>Ω</sup>(U*A*), for *qkl* ∈ P, *jl* ∈ Z, for *<sup>l</sup>* = 1, ..., *<sup>N</sup>*, for *<sup>N</sup>* ∈ N, where *<sup>n</sup>*1, ..., *nN* ∈ N.

**Remark 1.** *Note that the multiplicative linear transformation π<sup>h</sup> of (82) is indeed a* ∗*-homomorphism satisfying*

$$
\pi\_{\mathrm{h}}\left(T^\*\right) = \left(\pi\_{\mathrm{h}}(T)\right)^\*,
$$

*for all T* ∈ LS*A, because*

$$\begin{aligned} \pi\_h\left(\left(Q^a\_{p,j}\right)^\*\right)^\* &= \pi\_h\left(Q^{a^\*}\_{p,j}\right) = Q^{a^\*}\_{h(p),j} \\ &= \left(Q^a\_{h(p),j}\right)^\* = \left(\pi\_h\left(Q^a\_{p,j}\right)\right)^\*. \end{aligned}$$

*for all Q<sup>a</sup> <sup>p</sup>*,*<sup>j</sup>* ∈ Ω (U*A*).

In addition, by (82), we obtain the ∗-homomorphisms,

$$
\pi\_h^n = \underbrace{\pi\_h \pi\_h \pi\_h \cdot \cdots \pi\_h}\_{n \text{-times}} \text{ on } \mathbb{L}\mathbb{S}\_{A\prime} \tag{84}
$$

the products (or compositions) of the *n*-copies of the ∗-homomorphism *π<sup>h</sup>* of (82), acting on LS*A*. It is not difficult to check that

$$\begin{aligned} \pi\_h^n \left( Q\_{p,j}^a \right) &= \pi\_h^{n-1} \left( Q\_{h(p),j}^a \right) = \pi\_h^{n-2} \left( Q\_{h^{(2)}(p),j}^a \right) \\ &= \cdot \cdot \cdot = \pi\_h \left( Q\_{h^{(n-1)}(p),j}^a \right) = Q\_{h^{(n)}(p),j'}^a \end{aligned} \tag{85}$$

for all *Q<sup>a</sup> <sup>p</sup>*,*<sup>j</sup>* <sup>∈</sup> <sup>Ω</sup>(U*A*) in LS*A*, where *<sup>h</sup>*(*k*) are the *<sup>k</sup>*-shifts (80) on <sup>P</sup>, for all *<sup>k</sup>* <sup>∈</sup> <sup>N</sup>.

**Definition 15.** *Let <sup>π</sup><sup>h</sup> be the* ∗*-homomorphism (82) on the A-tensor sub-filterization* LS*A*, *and let <sup>π</sup><sup>n</sup> <sup>h</sup> be the products (84) acting on* LS*A*, *for all <sup>n</sup>* ∈ N, *with <sup>π</sup>*<sup>1</sup> *<sup>h</sup>* = *<sup>π</sup>h*. *Then, we call <sup>π</sup><sup>n</sup> <sup>h</sup>* , *the n-prime-shift (*∗*-homomorphism) on* LS*A*, *for all n* ∈ N. *In particular, the* 1*-prime-shift π<sup>h</sup> is simply said to be the prime-shift (*∗*-homomorphism) on* LS*A*.

Thus, for any *Q<sup>a</sup> qk*,*<sup>j</sup>* ∈ <sup>Ω</sup>(U*A*) in LS*A*, for *qk* ∈ P (in the sense of (78) with *<sup>k</sup>* ∈ N), the *<sup>n</sup>*-prime-shift *πn <sup>h</sup>* satisfies

$$
\pi^{n}\_{h}\left(\mathcal{Q}^{a}\_{q\_{k},j}\right) = \mathcal{Q}^{a}\_{h^{(n)}(q\_{k}),j} = \mathcal{Q}^{a}\_{q\_{k+n,j'}} \tag{86}
$$

by (81) and (85), and, hence,

$$
\pi\_h^n \left( \prod\_{l=1}^N \left( Q\_{q\_{l\_l, j\_l}}^{q\_l} \right)^{n\_l} \right) = \prod\_{l=1}^N \left( Q\_{q\_{l\_l + n, j\_l}}^{q\_l} \right)^{n\_l}, \tag{87}
$$

by (83) and (86), for all *n* ∈ N.

By (86) and (87), one may write as follows;

*πn <sup>h</sup>* = *πh*(*n*) on LS*A*, for all *n* ∈ N,

where *<sup>h</sup>*(*n*) are the *<sup>n</sup>*-shifts (81) on the TOset <sup>P</sup>.

Consider now the sequence

$$
\Pi = \left(\pi\_h^n\right)\_{n=1}^\infty \tag{88}
$$

of the *n*-prime-shifts on LS*A*.

For any fixed *T* ∈ LS*A*, the sequence Π of (88) induces the sequence of operators,

Π(*T*) = *πn <sup>h</sup>* (*T*) ∞ *<sup>n</sup>*=<sup>1</sup> = *πh*(*T*), *π*<sup>2</sup> *<sup>h</sup>*(*T*), *<sup>π</sup>*<sup>3</sup> *<sup>h</sup>*(*T*), · ·· in LS*A*, and this sequence Π(*T*) has its corresponding free-distributional data, represented by the following C-sequence:

$$\pi(\Pi(T)) = \left(\pi\left(\pi\_h^n(T)\right)\right)\_{n=1}^\infty. \tag{89}$$

We are interested in the convergence of the C-sequence *τ*(Π(*T*)) of (89), as *n* → ∞.

Either convergent or divergent, the C-sequence *τ*(Π(*T*)) of (89), induced by any fixed operator *T* ∈ LS*A*, shows the asymptotic free distributional data of the family {*π<sup>n</sup> <sup>h</sup>* (*T*)}<sup>∞</sup> *<sup>n</sup>*=<sup>1</sup> ⊂ LS*A*, as *n* → ∞ in N, equivalently, as *qn* → ∞ in P.

## *9.3. Asymptotic Behaviors in* LS*<sup>A</sup> over* P

Recall that, by (44), we have

$$\lim\_{p \to \infty} \frac{\phi(p)}{p^{j+1}} = \begin{cases} 0, & \text{if } j > 0, \\ 1, & \text{if } j = 0, \\ \infty, \text{ Undefined,} & \text{if } j < 0, \end{cases} \tag{90}$$

for *j* ∈ Z.

Recall also that there are bounded ∗-homomorphisms

Π = *πn h* ∞ *<sup>n</sup>*=<sup>1</sup> , acting on LS*A*,

of (88), where *π<sup>n</sup> <sup>h</sup>* are the *n*-*prime shifts* of (84), where *h* is the shift (79) on the TOset P of (78). Then, these ∗-homomorphisms of Π satisfies

$$\lim\_{n \to \infty} \left( \pi\_h^n \left( Q\_{p,j}^a \right) \right) = \lim\_{n \to \infty} \left( Q\_{h^{(n)}(p),j}^a \right), \tag{91}$$

for all *Q<sup>a</sup> <sup>p</sup>*,*<sup>j</sup>* <sup>∈</sup> <sup>Ω</sup>(U*A*) in LS*A*, where *<sup>h</sup>*(*n*) are the *<sup>n</sup>*-shifts (80) on <sup>P</sup>, for all *<sup>n</sup>* <sup>∈</sup> <sup>N</sup>.

Thus, one can get that: if *<sup>N</sup>* Π *l*=1 *Qal pl*,*jl nl* is a free reduced words of LS*<sup>A</sup>* in Ω (U*A*), then

$$\begin{aligned} \lim\_{n \to \infty} \pi\_h^n \left( \prod\_{l=1}^N \left( Q\_{p\_l, j\_l}^{a\_l} \right)^{n\_l} \right) &= \lim\_{n \to \infty} \left( \prod\_{l=1}^N \pi\_h^n \left( \left( Q\_{p\_l, j\_l}^{a\_l} \right)^{n\_l} \right) \right) \\ &= \lim\_{n \to \infty} \left( \prod\_{l=1}^N \left( \pi\_h^n \left( Q\_{p\_l, j\_l}^{a\_l} \right) \right)^{n\_l} \right) \end{aligned}$$

since *π<sup>n</sup> <sup>h</sup>* are ∗-homomorphisms on LS*<sup>A</sup>*

$$=\lim\_{n\to\infty} \left(\prod\_{l=1}^{N} \left(Q^{a\_l}\_{\boldsymbol{h}^{(n)}(p\_l), j\_l}\right)^{n\_l}\right)^{\varepsilon}$$

by (91)

$$=\prod\_{l=1}^{N} \left( \lim\_{n \to \infty} \left( \mathbb{Q}\_{h^{(n)}(p\_l), j\_l}^{a\_l} \right)^{n\_l} \right),\tag{92}$$

under the Banach-topology for LS*A*, for all *<sup>Q</sup>al pl*,*jl* ∈ <sup>Ω</sup>(U*A*), for *al* ∈ (*A*, *<sup>ψ</sup>*), *pl* ∈ P, *jl* ∈ Z, for *<sup>l</sup>* = 1, ..., *N*, for all *N* ∈ N.

**Notation 2.** (in short, **N 2** from below) For convenience, we denote lim*n*→∞*π<sup>n</sup> <sup>h</sup>* symbolically by *π*, for the sequence Π = *πn h* ∞ *<sup>n</sup>*=<sup>1</sup> of (88).

**Lemma 3.** *Let Qal pl*,*jl* ∈ <sup>Ω</sup>(U*A*) *be generators of the A-tensor sub-filterization* LS*A*, *for <sup>l</sup>* = 1, *..., <sup>N</sup>*, *for <sup>N</sup>* ∈ N*. In addition, let* Π *be the sequence (88) acting on* LS*A. If π is in the sense of N 2, then*

$$\pi\left(Q^{a\_1}\_{p\_1,j\_1}\right) = \lim\_{n \to \infty} \left(Q^{a\_1}\_{\left(h^{(n)}(p\_1)\right),j\_1}\right),$$

*(93)*

$$\pi\left(\prod\_{l=1}^N \left(\mathbb{Q}\_{p\_l,\bar{j}\_l}^{a\_l}\right)^{n\_l}\right) = \lim\_{n \to \infty} \left(\prod\_{l=1}^N \left(\mathbb{Q}\_{\mathbf{h}^{(n)}(p\_l),\bar{j}\_l}^{a\_l}\right)^{n\_l}\right).$$

*for all n*1, *..., nN* <sup>∈</sup> <sup>N</sup>, *where h*(*n*) *are the n-shifts (80) on* <sup>P</sup>.

**Proof.** The proof of (93) is done by (91) and (92).

By abusing notation, one may/can understand the above formula (93) as follows

$$\begin{aligned} \pi \left( Q\_{p\_1, j\_1}^{a\_1} \right) &= \lim\_{p\_1 \to \infty} Q\_{p\_1, j\_1}^{a\_1} \\\\ \left( \prod\_{l=1}^N Q\_{p\_l, j\_l}^{n\_l} \right) &= \prod\_{l=1}^N \left( \lim\_{p\_l \to \infty} \left( Q\_{p\_l, j\_l}^{n\_l} \right) \right), \end{aligned} \tag{94a}$$

respectively, where " lim*q*→∞" for *<sup>q</sup>* ∈ P is in the sense of (44).

*π*

Such an understanding (94a) of the formula (93) is meaningful by the constructions (80) of *n*-shifts *<sup>h</sup>*(*n*) on <sup>P</sup>. For example,

$$\lim\_{n \to \infty} h^{(n)}(q) = \lim\_{p \to \infty} p\_\prime \text{ for } q \in \mathcal{P}, \tag{94b}$$

where the right-hand side of (94b) means that: starting with *q*, take bigger primes again and again in the TOset P of (78).

**Assumption and Notation:** From below, for convenience, the notations in (94a) are used for (93), if there is no confusion.

We now define a new (unbounded) linear functional *τ*<sup>0</sup> on LS*<sup>A</sup>* with respect to the linear functional *τ* of (74a), by

$$
\pi \mathfrak{o} \stackrel{def}{=} \mathfrak{x} \circ \mathfrak{x} \text{ on } \mathbb{L}\mathbb{S}\_{A\prime} \tag{95}
$$

where *π* is in the sense of **N 2**.

**Theorem 5.** *Let* LS*<sup>A</sup>* = (LS*A*, *τ*) *be the A-tensor sub-filterization (74b), and let τ*<sup>0</sup> = *τ* ◦ *π be the new linear functional (95) on the Banach* ∗*-algebra* LS*<sup>A</sup> of (74a). Then, for the generators*

$$\{Q^{a}\_{p\_{j}j}\}\_{p \in \mathcal{P}} \subset \Omega(\mathcal{U}\_{A}) \text{ of } \mathbb{L}\mathbb{S}\_{A\mathcal{A}}$$

*for an arbitrarily fixed a* ∈ (*A*, *ψ*) *and j* ∈ Z, *we have that*

$$\pi\_0\left(\left(Q^{\underline{a}}\_{p,j}\right)^n\right) = \begin{cases} 0, & \text{if } j > 0, \\ \omega\_{\mathbb{M}} c\_{\frac{\pi}{2}} \psi(a^n), & \text{if } j = 0, \\ \infty, & \text{Lndefined}, \qquad \text{if } j < 0, \end{cases} \tag{96}$$

*for all n* ∈ N.

**Proof.** Let {*Q<sup>a</sup> p*,*j* }*p*∈P ⊂ Ω(U*A*) in LS*A*, for fixed *a* ∈ (*A*, *ψ*) and *j* ∈ Z. Then,

$$\pi\_0\left(\left(Q^{\mathfrak{a}}\_{p,j}\right)^n\right) = \left(\pi \circ \pi\right)\left(\left(Q^{\mathfrak{a}}\_{p,j}\right)^n\right) = \pi\left(\lim\_{p \to \infty} \left(Q^{\mathfrak{a}}\_{p,j}\right)^n\right).$$

by (93) and (94a)

$$= \lim\_{p \to \infty} \tau \left( \left( Q^{a}\_{p,j} \right)^{n} \right)^{p}$$

by the boundedness of *τ* for the (norm, or strong) topology for LS*<sup>A</sup>*

$$1 = \lim\_{p \to \infty} \pi\_{\not\!\!/ }^p \left( \left( Q\_{p, \not\!\!/ }^a \right)^n \right) = \lim\_{p \to \infty} \left( \omega\_n c\_{\not\!\!/ } \psi(a^n) \left( \frac{\phi(p)}{p^{\not\!\!/ } ^\pm} \right) \right)$$

by (70), (75) and (77)

$$\begin{aligned} &= \left(\omega\_{\mathfrak{m}} c\_{\frac{\mathfrak{n}}{2}} \psi(a^{\mathfrak{n}})\right) \left(\lim\_{p \to \infty} \frac{\phi(p)}{p^{\mathfrak{l}+1}}\right) \\ &= \begin{cases} 0, & \text{if } j > 0, \\ \omega\_{\mathfrak{m}} c\_{\frac{\mathfrak{n}}{2}} \psi(a^{\mathfrak{n}}), & \text{if } j = 0, \\ \infty, & \text{undefined}, \end{cases} \end{aligned}$$

by (90), for each *n* ∈ N. Therefore, the free-distributional data (96) holds for *τ*0.

By (96), we obtain the following corollary.

**Corollary 2.** *Let Q*1*<sup>A</sup> <sup>p</sup>*,0 ∈ Ω(U*A*) *be free random variables of the A-tensor sub-filterization* LS*A*, *for all p* ∈ P, *where* 1*<sup>A</sup> is the unity of* (*A*, *ψ*). *Then, the asymptotic free distribution of the family*

$$\mathcal{Q}\_0^{1\_A} = \{ \mathcal{Q}\_{p,0}^{1\_A} \in \Omega(\mathcal{U}\_A) \}\_{p \in \mathcal{P}^\*}$$

*follows the semicircular law asymptotically as p* → ∞ *in* P.

**Proof.** Let <sup>Q</sup>1*<sup>A</sup>* <sup>0</sup> <sup>=</sup> {*Q*1*<sup>A</sup> <sup>p</sup>*,0}*p*∈P ⊂ Ω(U*A*) in LS*A*. Then, for the linear functional *τ*<sup>0</sup> of (95) on LS*A*,

$$
\pi\_0 \left( \left( Q\_{p,0}^{1\_A} \right)^n \right) = \omega\_n c\_{\frac{n}{2},n}
$$

for all *n* ∈ N, by (96), since

$$
\psi(1\_A^n) = \psi(1\_A) = 1; n \in \mathbb{N}.
$$

If *<sup>p</sup>* <sup>→</sup> <sup>∞</sup> in <sup>P</sup>, then the asymptotic free distribution of the family <sup>Q</sup>1*<sup>A</sup>* <sup>0</sup> is the semicircular law by the self-adjointness of all *Q*1*<sup>A</sup> <sup>p</sup>*,0's, and by the semicircularity (45) and (47).

Independent from (96), we obtain the following asymptotic free-distributional data on LS*A*.

**Theorem 6.** *Let j*1, *..., jN be "mutually distinct" in* Z, *for N* > 1 *in* N, *and hence the N-tuple*

$$[j] = (j\_{1'} \dots \not{j\_N}) \in \mathbb{Z}^N$$

*is alternating in* Z. *In addition, let*

$$[a] = (a\_{1\prime} \dots \, a\_N)$$

*be an arbitrarily fixed N-tuple of free random variables a*1, *..., aN of the unital C*∗*-probability space* (*A*, *ψ*), *and let's fix*

$$[n] = (n\_1, \ldots, n\_N) \in \mathbb{N}^N.$$

*Now, define a family* <sup>T</sup> [*a*],[*n*] [*j*] *of free reduced words with their lengths-N,*

$$\mathcal{T}\_{[j]}^{[a],[n]} = \left\{ T = \bigcap\_{l=1}^{N} \left( Q\_{p\_l, j\_l}^{a\_l} \right)^{n\_l} : p\_1, \dots, p\_N \in \mathcal{P} \right\}, \tag{97}$$

*in* LS*A*, *for Qal pl*,*jl* ∈ <sup>Ω</sup> (U*A*), *for all pl* ∈ P, *where al* ∈ [*a*], *jl* ∈ [*j*], *for l* = 1, *..., N*.

*For any free reduced words T* ∈ T [*a*],[*n*] [*j*] , *if τ*<sup>0</sup> *is the linear functional (95) on* LS*A*, *then*

$$\pi\_0\left(T\right) = \begin{cases} 0, & \text{if } \sum\_{l=1}^N j\_l > 1 - N, \\ \prod\_{l=1}^N \left(\omega\_{\text{lv}} c\_{\frac{\pi\_l}{2}} \Psi(a^{\text{lv}})\right), & \text{if } \sum\_{l=1}^N j\_l = 1 - N, \\ \infty, & \text{if } \sum\_{l=1}^N j\_l < 1 - N, \end{cases} \tag{98}$$

*for all n* ∈ N.

**Proof.** Let *<sup>T</sup>* ∈ T [*a*],[*n*] [*j*] be in the sense of (97) in the *A*-tensor sub-filterization LS*A*. Then, these operators *T* form free reduced words with their lengths-*N* in LS*A*, since [*j*] is an alternating *N*-tuple of "mutually distinct" integers. Observe that

$$\pi\_0\left(T\right) = \pi\left(\pi(T)\right) = \pi\left(\prod\_{l=1}^N \left(\lim\_{p\_l \to \infty} \left(\mathbb{Q}\_{p\_l, j\_l}^{a\_l}\right)^{m\_l}\right)\right)^{\frac{1}{m\_l}}$$

by (93) and (94a)

$$=\tau\left(\prod\_{l=1}^N \left(\lim\_{p\to\infty} \left(Q^{a\_l}\_{p,\dot{h}}\right)^{n\_l}\right)\right)^2$$

because

$$\lim\_{p \to \infty} p = \lim\_{n \to \infty} h^{(n)}(p\_l) = \lim\_{p\_l \to \infty} p\_{l'} \text{ in } \mathcal{P}\_{\prime l}$$

in the sense of (44), for all *l* = 1, ..., *N*, and, hence, it goes to

$$=\lim\_{p\to\infty} \left( \pi \left( \left( \prod\_{l=1}^N Q\_{p,j\_l}^{a\_l} \right)^{n\_l} \right) \right)$$

by the boundedness of *τ* for the (norm, or strong) topology for LS*<sup>A</sup>*

$$=\lim\_{p\to\infty} \left( \prod\_{l=1}^{N} \left( \omega\_{n\_l} c\_{\frac{n\_l}{2}} \psi(a\_l^{n\_l}) \left( \frac{\Phi(p)}{p^{n\_l+1}} \right) \right) \right)$$

since [*j*] consists of "mutually-distinct" integers, by the Möbius inversion

= - *N* Π *l*=1 *ωnl c nl* 2 *ψ*(*a nl l* ) - lim *p*→∞ - *N* Π *l*=1 - *φ*(*p*) *pj <sup>l</sup>*+<sup>1</sup> = - *N* Π *l*=1 *ωnl c nl* 2 *ψ*(*a nl l* ) - lim *p*→∞ - *φ*(*p*) *pN*+Σ*<sup>N</sup> <sup>l</sup>*=<sup>1</sup> *<sup>j</sup> l* = - *N* Π *l*=1 *ωnl c nl* 2 *ψ*(*a nl l* ) lim *p*→∞ *φ*(*p*) *p*(*<sup>N</sup>*−1+Σ*<sup>N</sup> <sup>l</sup>*=<sup>1</sup> *<sup>j</sup> <sup>l</sup>*)+<sup>1</sup> = - *N* Π *l*=1 *ωnl c nl* 2 *ψ*(*a nl l* ) lim *p*→∞ *φ*(*p*) *p*(*<sup>N</sup>*−1+Σ*<sup>N</sup> <sup>l</sup>*=<sup>1</sup> *<sup>j</sup> <sup>l</sup>*)+<sup>1</sup> = ⎧ ⎪⎪⎨ ⎪⎪⎩ 0 if *<sup>N</sup>* <sup>−</sup> <sup>1</sup> <sup>+</sup> <sup>∑</sup>*<sup>N</sup> <sup>l</sup>*=<sup>1</sup> *jl* > 0 *N* Π *l*=1 *ωnl c nl* 2 *ψ*(*a nl l* ) if *<sup>N</sup>* <sup>−</sup> <sup>1</sup> <sup>+</sup> <sup>∑</sup>*<sup>N</sup> <sup>l</sup>*=<sup>1</sup> *jl* = 0 <sup>∞</sup> if *<sup>N</sup>* <sup>−</sup> <sup>1</sup> <sup>+</sup> <sup>∑</sup>*<sup>N</sup> <sup>l</sup>*=<sup>1</sup> *jl* < 0,

by (90), for all *<sup>n</sup>* <sup>∈</sup> <sup>N</sup>. Therefore, the family <sup>T</sup> [*a*],[*n*] [*j*] of (97) satisfies the asymptotic free-distributional data (98) in the *A*-tensor sub-filterization LS*<sup>A</sup>* over P.

The above two theorems illustrate the asymptotic free-probabilistic behaviors on the *A*-tensor sub-filterization LS*<sup>A</sup>* over P, by (96) and (98).

As a corollary of (96), we showed that the family

$$\mathcal{Q}\_0^{1\_A} = \{ \mathcal{Q}\_{p,0}^{1\_A} \}\_{p \in \mathcal{P}} \subset \mathbb{L} \mathbb{S}\_A$$

has its asymptotic free distribution, the semicircular law in LS*A*, as *p* → ∞. More generally, the following theorem is obtained.

**Theorem 7.** *Let a be a self-adjoint free random variable of our unital C*∗*-probability space* (*A*, *ψ*). *Assume that it satisfies*

*Symmetry* **2019**, *11*, 819

$$\begin{array}{ll}(i) & \psi(a) \in \mathbb{R}^{\times} = \mathbb{R} \\ (ii) & \psi(a^{2n}) = \psi(a)^{2n}, for all \ n \in \mathbb{N}.\end{array}$$

*Then, the family*

$$\mathcal{X}\_0^a = \left\{ X\_{p,0}^a = \frac{1}{\Psi(a)} Q\_{p,0}^a : p \in \mathcal{P} \right\} \tag{99}$$

*follows the asymptotic semicircular law, in* LS*<sup>A</sup> over* P.

**Proof.** Let *a* ∈ (*A*, *ψ*) be a self-adjoint free random variable satisfying two conditions (i) and (ii), and let X *<sup>a</sup>* <sup>0</sup> be the family (99) of the *A*-tensor sub-filterization LS*A*. Then, all elements

$$X\_{p,0}^{a} = \frac{1}{\Psi(a)} Q\_{p,0}^{a} = \mathbf{l}\_p^A \odot \left( \left( \frac{1}{\Psi(a)} a \right) \odot P\_{p,0} \right) \text{ of } \mathcal{X}\_0^{a}$$

are self-adjoint in LS*A*, by the self-adjointness of *Q<sup>a</sup> <sup>p</sup>*,0, and by the condition (i).

For any *X<sup>a</sup> <sup>p</sup>*,0 ∈ X *<sup>a</sup>* <sup>0</sup> , observe that

$$\begin{aligned} \pi\_0\left(\left(X\_{p,0}^a\right)^n\right) &= \frac{1}{\psi(a)^n} \,\, \_0\left(\left(Q\_{p,0}^a\right)^n\right) \\ &= \frac{1}{\psi(a)^n} \left(\omega\_W c\_{\frac{n}{2}} \psi(a^n)\right) \end{aligned}$$

by (96)

$$= \left(\omega\_{\mathfrak{n}}\mathcal{L}\_{\frac{\mathfrak{n}}{2}}\left(\frac{\psi(\mathfrak{a}^{\mathfrak{n}})}{\psi(\mathfrak{a}^{\mathfrak{n}})}\right)\right)$$

by the condition (ii)

$$=\omega\_n c\_{\frac{n}{2}'} $$

for all *<sup>n</sup>* ∈ N. Therefore, the family X *<sup>a</sup>* <sup>0</sup> has its asymptotic semicircular law over P, by (45).

Similar to the construction of X *<sup>a</sup>* <sup>0</sup> of (99), if we construct the families X *<sup>a</sup> j* ,

$$\mathcal{X}\_{j}^{a} = \left\{ \frac{1}{\psi(a)} \mathcal{Q}\_{p,j}^{a} : \mathcal{Q}\_{p,j}^{a} \in \Omega \left(\mathcal{U}\_{A}\right) \right\}\_{p \in \mathcal{P}} \tag{100}$$

for a fixed *a* ∈ (*A*, *ψ*) satisfying the conditions (i) and (ii) of the above theorem, and, for a fixed *j* ∈ Z, then one obtains the following corollary.

**Corollary 3.** *Fix a* ∈ (*A*, *ψ*) *satisfying the conditions (i) and (ii) of the above theorem. Let's fix j* ∈ Z, *and let* X *a <sup>j</sup> be the corresponding family (100) in the A-tensor sub-filterization* LS*<sup>A</sup>* = (LS*A*, *τ*).

$$\text{If } fj = 0 \text{, then } \mathcal{X}\_0^a \text{ has the asymptotic semicircular law in } \mathbb{E}\mathbb{S}\_A. \tag{101}$$

*Ifj* > 0, *then* X *<sup>a</sup> <sup>j</sup> has its asymptotic free distribution, the zero free distribution, in* LS*A*. *(102) Ifj* < 0, *then the asymptotic free distribution of* X *<sup>a</sup> <sup>j</sup> is undefined in* LS*A*. *(103)*

**Proof.** The proof of (101) is done by (99).

By (96), if *j* > 0, then, for any *T* = <sup>1</sup> *ψ*(*a*)*Q<sup>a</sup> <sup>p</sup>*,*<sup>j</sup>* ∈ X *<sup>a</sup> <sup>j</sup>* , one has that

$$\pi\_0\left(T^n\right) = \frac{1}{\Psi(a^n)}\pi\_0\left(\left(Q^a\_{p,j}\right)^n\right) = 0.7$$

for all *<sup>n</sup>* ∈ N. Thus, the asymptotic free distribution of X *<sup>a</sup> <sup>j</sup>* is the zero free distribution in LS*A*, as *p* → ∞ in P. Thus, the statement (102) holds.

Similarly, by (96), if *<sup>j</sup>* < 0, then the asymptotic free distribution X *<sup>a</sup> <sup>j</sup>* is undefined in LS*<sup>A</sup>* over P, equivalently, the statement (103) is shown.

Motivated by (101), (102) and (103), we study the asymptotic semicircular law (over P) on LS*<sup>A</sup>* more in detail in Section 10 below.

## **10. Asymptotic Semicircular Laws on** LS*<sup>A</sup>* **over** *P*

We here consider asymptotic semicircular laws on the *A*-tensor sub-filterization LS*<sup>A</sup>* = (LS*A*, *τ*). In Section 9.3, we showed that the asymptotic free distribution of a family

$$\mathcal{X}\_0^a = \{ \frac{1}{\psi(a)} Q\_{p,0}^a : p \in \mathcal{P} \} \tag{104}$$

is the semicircular law in LS*<sup>A</sup>* as *p* → ∞ in P, for a fixed self-adjoint free random variable *a* ∈ (*A*, *ψ*) satisfying


As an example, the family

$$\mathcal{X}\_0^{1\_A} = \{ \mathcal{Q}\_{p,0}^{1\_A} : p \in \mathcal{P} \} \tag{105}$$

follows the asymptotic semicircular law in LS*<sup>A</sup>* over P.

We now enlarge such asymptotic behaviors on LS*<sup>A</sup>* up to certain ∗-isomorphisms.

Define bijective functions *g*<sup>+</sup> and *g*<sup>−</sup> on Z by

$$\text{g}\_{+}(j) = j + 1, \text{ and } \text{g}\_{-}(j) = j - 1,\tag{106}$$

for all *j* ∈ Z.

By (106), one can define bijective functions *g* (*n*) <sup>±</sup> on <sup>Z</sup> by

$$\mathcal{S}^{(n)}\_{\pm} \stackrel{def}{=} \underbrace{\mathcal{S}^{\pm} \circ \mathcal{S}^{\pm} \circ \mathcal{S}^{\pm} \circ \cdots \circ \mathcal{S}^{\pm}}\_{n \text{-times}} \tag{107}$$

satisfying *g* (1) <sup>±</sup> <sup>=</sup> *<sup>g</sup>*<sup>±</sup> on <sup>Z</sup>, with axiomatization:

> *g* (0) <sup>±</sup> <sup>=</sup> *id*Z, the identity function on <sup>Z</sup>,

for all *n* ∈ N<sup>0</sup> = N ∪ {0}. For example,

$$\mathcal{g}^{(n)}\_{\pm}(j) = j \pm n,\tag{108}$$

for all *j* ∈ Z, for all *n* ∈ N0.

From the bijective functions *g* (*n*) <sup>±</sup> of (107), define the bijective functions *go* ± (*n*) on the generator set Ω(U*A*) of (72) of the *A*-tensor sub-filterization LS*<sup>A</sup>* by

$$\begin{aligned} \left(\mathcal{S}\_{+}^{o}\right)^{(n)} \left(\mathcal{Q}\_{p,j}^{a}\right) &= \mathcal{Q}\_{p,\mathcal{S}\_{+}^{(n)}(j)}^{a} = \mathcal{Q}\_{p,j+n'}^{a} \\\\ \left(\mathcal{S}\_{-}^{o}\right)^{(n)} \left(\mathcal{Q}\_{p,j}^{a}\right) &= \mathcal{Q}\_{p,\mathcal{S}\_{-}^{(n)}(j)}^{a} = \mathcal{Q}\_{p,j-n'}^{a} \end{aligned} \tag{109}$$

with

$$\left(\mathcal{g}^{\boldsymbol{\sigma}}\_{\pm}\right)^{(1)} = \mathcal{g}^{\boldsymbol{\sigma}}\_{\pm \prime} \text{ and } \left(\mathcal{g}^{\boldsymbol{\sigma}}\_{\pm}\right)^{(0)} = \mathrm{id}\_{\prime}$$

*p*,*g*

<sup>−</sup> (*j*)

by (108), for all *p* ∈ P and *j* ∈ Z, for all *n* ∈ N0, where *id* is the identity function on Ω(U*A*).

By the construction (73a) of the generator set Ω(U*A*) of LS*<sup>A</sup>* under (73b),

$$
\Omega(\mathcal{U}\_A) = \bigsqcup\_{p \in \mathcal{P}} \{ \mathcal{Q}\_{p,j}^a : a \in A, j \in \mathbb{Z} \},
$$

the functions *go* ± (*n*) of (109) are indeed well-defined bijections on <sup>Ω</sup>(U*A*), by the bijectivity of *<sup>g</sup>* (*n*) ± of (107).

Now, define bounded ∗-homomorphisms *G*<sup>±</sup> on LS*<sup>A</sup>* by the bounded multiplicative linear transformations on LS*<sup>A</sup>* satisfying that:

$$\begin{aligned} \mathcal{G}\_{+}\left(\mathcal{Q}\_{p,j}^{a}\right) &= \mathcal{g}\_{+}^{a}\left(\mathcal{Q}\_{p,j}^{a}\right) = \mathcal{Q}\_{p,j+1'}^{a} \\\\ \mathcal{G}\_{-}\left(\mathcal{Q}\_{p,j}^{a}\right) &= \mathcal{g}\_{-}^{a}\left(\mathcal{Q}\_{p,j}^{a}\right) = \mathcal{Q}\_{p,j-1'}^{a} \end{aligned} \tag{110}$$

in LS*A*, by using the bijections *g<sup>o</sup>* <sup>±</sup> of (109), for all *<sup>Q</sup><sup>a</sup> <sup>p</sup>*,*<sup>j</sup>* ∈ Ω(U*A*). More precisely, the morphisms *G*<sup>±</sup> of (110) satisfy that

$$\begin{aligned} \mathbf{G}\_{\pm} \left( \prod\_{l=1}^{N} \left( \mathbf{Q}\_{p\_l, j\_l}^{a\_l} \right)^{n\_l} \right) &= \prod\_{l=1}^{N} \mathbf{g}\_{\pm}^{a} \left( \left( \mathbf{Q}\_{p\_l, j\_l}^{a\_l} \right)^{n\_l} \right) \\ &= \prod\_{l=1}^{N} \left( \mathbf{Q}\_{p\_l, j\_l \pm 1}^{a\_l} \right)^{n\_l} . \end{aligned} \tag{111a}$$

By (111a), one can get that

$$\begin{split} \mathbf{G}\_{\pm} \left( \left( \prod\_{l=1}^{N} \left( \mathbf{Q}\_{p\_l, j\_l}^{\mathbf{a}\_l} \right)^{n\_l} \right)^{\*} \right)^{\*} &= \mathbf{G}\_{\pm} \left( \prod\_{l=1}^{N} \left( \mathbf{Q}\_{p\_{N-l+1}, j\_{N-l+1}}^{\mathbf{a}\_{N-l+1}} \right)^{n\_{N-l+1}} \right) \\ &= \prod\_{l=1}^{N} \left( \left( \mathbf{Q}\_{p\_{N-l+1}, \left( j\_{N-l+1} \right) \pm 1}^{\mathbf{a}\_{N-l+1}} \right)^{n\_{N-l+1}} \right)^{\*} \\ &= \left( \prod\_{l=1}^{N} \left( \mathbf{Q}\_{p\_l, j\_l \pm 1}^{\mathbf{a}\_l} \right)^{n\_l} \right)^{\*} \\ &= \left( \mathbf{G}\_{\pm} \left( \prod\_{l=1}^{N} \mathbf{Q}\_{p\_l, j\_l}^{n\_l} \right) \right)^{\*} \end{split} \tag{111b}$$

for all *Qal pl*,*jl* ∈ <sup>Ω</sup>(U*A*), for *<sup>l</sup>* = 1, ..., *<sup>N</sup>*, for *<sup>N</sup>* ∈ N.

The formula (111a) are obtained by (110) and the multiplicativity of *G*±. The formulas in (111b), obtained from (111a), show that indeed *G*<sup>±</sup> are ∗-homomorphisms on LS*A*, since

$$\mathcal{G}\_{\pm} \left( T^\* \right) = \left( \mathcal{G}\_{\pm} \left( T \right) \right)^\*, \forall T \in \mathbb{L} \mathbb{S}\_A.$$

By (110) and (111a),

$$\begin{aligned} G\_{\pm}^{\boldsymbol{n}} \left( \prod\_{l=1}^{N} \left( \mathbb{Q}\_{p\_l, j\_l}^{a\_l} \right)^{n\_l} \right) &= \prod\_{l=1}^{N} \left( \mathbb{Q}\_{p\_l, j\_l \pm n}^{a\_l} \right)^{n\_l}, \\\\ \left( \left( \prod\_{l=1}^{N} \left( \mathbb{Q}\_{p\_l, j\_l}^{a\_l} \right)^{n\_l} \right)^{\*} \right) &= \left( G\_{\pm}^{\boldsymbol{n}} \left( \prod\_{l=1}^{N} \left( \mathbb{Q}\_{p\_l, j\_l}^{a\_l} \right)^{n\_l} \right) \right)^{\*}, \end{aligned} \tag{112}$$

for all *Qal pl*,*jl* ∈ <sup>Ω</sup>(U*A*), for *<sup>l</sup>* = 1, ..., *<sup>N</sup>*, for *<sup>N</sup>* ∈ N, for all *<sup>n</sup>* ∈ N0.

*Gn* ±

**Definition 16.** *We call the bounded* ∗*-homomorphisms <sup>G</sup><sup>n</sup>* <sup>±</sup> *of (110), the n-(*±*)-integer-shifts on* LS*A*, *for all n* ∈ N0.

Based on the integer-shifting processes on LS*A*, one can get the following asymptotic behavior on LS*<sup>A</sup>* over P.

**Theorem 8.** *Let* X *<sup>a</sup> <sup>j</sup> be a family (100) of the A-tensor sub-filterization* LS*A*, *for any j* ∈ Z, *where a is a fixed self-adjoint free random variable of* (*A*, *ψ*) *satisfying the additional conditions (i) and (ii) above. Then, there exists a* (−*j*)*-integer-shift G*−*<sup>j</sup> on* LS*A*, *such that*

$$\mathcal{G}\_{-j} = \begin{cases} \mathcal{G}\_{-}^{|j|} = \mathcal{G}\_{-}^{j} & \text{if } j \ge 0 \text{ in } \mathbb{Z}, \\\mathcal{G}\_{+}^{|j|} = \mathcal{G}\_{+}^{-j} & \text{if } j < 0 \text{ in } \mathbb{Z}, \end{cases} \tag{113}$$

*and*

$$
\pi\_0\left(G\_{\vec{\lambda}}(T)\right) = \omega\_n c\_{\frac{n}{2}'} \,\forall n \in \mathbb{N},\tag{114}
$$

*for all <sup>T</sup>* ∈ X *<sup>a</sup> <sup>j</sup>* , *where <sup>G</sup>*±*<sup>j</sup>* <sup>∓</sup> *on the right-hand sides of (113) are the* <sup>|</sup>*j*|*-(*∓*)-integer shifts (110) on* LS*A*, *and where τ*<sup>0</sup> = *τ* ◦ *π is the linear functional (95) on* LS*A.*

**Proof.** Let X *<sup>a</sup> <sup>j</sup>* = 1 *ψ*(*a*)*Q<sup>a</sup> <sup>p</sup>*,*<sup>j</sup>* : *<sup>p</sup>* ∈ P be a family (100) of LS*A*, for a fixed *j* ∈ Z, where a fixed self-adjoint free random variable *a* ∈ (*A*, *ψ*) satisfies the above additional conditions (i) and (ii).

Assume first that *<sup>j</sup>* <sup>≥</sup> 0 in <sup>Z</sup>. Then, one can take the (−*j*)-(−)-integer-shift *<sup>G</sup><sup>j</sup>* <sup>−</sup> of (110) on LS*A*, satisfying

$$\mathcal{G}^{j}\_{-}\left(\mathcal{Q}^{a}\_{p,j}\right) = \mathcal{Q}^{a}\_{p,j-j} = \mathcal{Q}^{a}\_{p,0} \text{ in } \mathbb{L}\mathbb{S}\_{A\prime}$$

for all *Q<sup>a</sup> <sup>p</sup>*,*<sup>j</sup>* ∈ Ω (U*A*).

> Second, if *<sup>j</sup>* <sup>&</sup>lt; 0 in <sup>Z</sup>, then one can have the <sup>|</sup>*j*|-(+)-integer shift *<sup>G</sup>*−*<sup>j</sup>* <sup>+</sup> of (110) on LS*A*, satisfying that

$$\mathcal{G}\_+^{-j}\left(\mathcal{Q}\_{p,j}^a\right) = \mathcal{Q}\_{p,j+(-j)}^a = \mathcal{Q}\_{p,0}^a \text{ in } \mathbb{E}\mathbb{S}\_{A\prime}$$

for all *Q<sup>a</sup> <sup>p</sup>*,*<sup>j</sup>* ∈ Ω (U*A*).

For example, for any *Q<sup>a</sup> <sup>p</sup>*,*<sup>j</sup>* ∈ Ω(U*A*), we have the corresponding (−*j*)-integer-shift *G*−*j*,

$$G\_{-j} = \begin{cases} \ G\_-^j & \text{if } j \ge 0, \\\ G\_+^{-j} & \text{if } j < 0, \end{cases}$$

on LS*<sup>A</sup>* in the sense of (113), such that

$$G\_{-j} \left( Q\_{p,j}^a \right) = Q\_{p,0}^a \text{ in } \mathbb{LB}\_{A'} $$

for all *p* ∈ P.

Then, for any *X<sup>a</sup> <sup>p</sup>*,*<sup>j</sup>* <sup>=</sup> <sup>1</sup> *ψ*(*a*)*Q<sup>a</sup> <sup>p</sup>*,*<sup>j</sup>* ∈ X *<sup>a</sup> <sup>j</sup>* , we have that

$$\begin{aligned} \pi\_0\left(G\_{-j}\left(\left(X\_{p,j}^a\right)^n\right)\right) &= \pi\_0\left(\frac{1}{\psi(a)^n}\left(G\_{-j}(Q\_{p,j}^a)\right)^n\right), \\ \vdots &= \pi\_{(1,1,2)}\dots\pi\_{,n} \end{aligned}$$

since *G*−*<sup>j</sup>* is a ∗-homomorphism (113) on LS*<sup>A</sup>*

$$\omega\_0 \left( \frac{1}{\Psi(a^n)} \left( \mathbb{Q}\_{p,0}^a \right)^n \right) = \omega\_n c\_{\frac{n}{2}} \omega$$

by (96) and (98), for all *n* ∈ N. Therefore, formula (114) holds true.

By the above theorem, we obtain the following result.

**Corollary 4.** *Let* X *<sup>a</sup> <sup>j</sup> be a family (100) of the A-tensor sub-filterization* LS*A*, *for j* ∈ Z, *where a self-adjoint free random variable a* ∈ (*A*, *ψ*) *satisfies the conditions (i) and (ii). Then, the corresponding family*

$$\mathcal{G}\_{\dot{\jmath}}^{a} = \left\{ G\_{-\dot{\jmath}} \left( X \right) : X \in \mathcal{A}\_{\dot{\jmath}}^{a} \right\} \tag{115}$$

*has its asymptotic free distribution, the semicircular law, in* LS*<sup>A</sup> over* P, *where G*−*<sup>j</sup> is the (*−*j)-integer shift (113) on* LS*A*, *for all j* ∈ Z.

**Proof.** The asymptotic semicircular law induced by the family G*<sup>a</sup> <sup>j</sup>* of (115) in LS*<sup>A</sup>* is guaranteed by (114) and (45), for all *j* ∈ Z.

By the above corollary, the following result is immediately obtained.

**Corollary 5.** *Let* <sup>X</sup> <sup>1</sup>*<sup>A</sup> <sup>j</sup> be in the sense of (100) in* LS*A*, *where* 1*<sup>A</sup> is the unity of* (*A*, *ψ*), *and let*

$$\mathcal{G}\_{\boldsymbol{j}}^{1\_{\mathcal{A}}} = \left\{ \mathcal{G}\_{-\boldsymbol{j}}(\boldsymbol{X}) : \boldsymbol{X} \in \mathcal{X}\_{\boldsymbol{j}}^{1\_{\mathcal{A}}} \right\}.$$

*be in the sense of (115), for all j* <sup>∈</sup> <sup>Z</sup>. *Then, the asymptotic free distributions of* <sup>G</sup>1*<sup>A</sup> <sup>j</sup> are the semicircular law in* LS*<sup>A</sup> over* P, *for all j* ∈ Z.

**Proof.** The proof is done by Corollary 4. Indeed, the unity 1*<sup>A</sup>* automatically satisfies the conditions (i) and (ii) in (*A*, *ψ*).

More general to Theorem 8, we obtain the following result too.

**Theorem 9.** *Let a* ∈ (*A*, *ψ*) *be a self-adjoint free random variable satisfying the conditions (i) and (ii), and let p*<sup>0</sup> ∈ P *be an arbitrarily fixed prime. Let*

$$\mathcal{G}\_j^a \left[ \geq p\_0 \right] \stackrel{def}{=} \left\{ G\_{-j} \left( X\_{p,j} \right) \; \middle| \; \begin{array}{l} X\_{p,j}^a \in \mathcal{X}\_j^a \; and \\ p \geq p\_0 \; in \; \mathcal{P} \end{array} \right\} \; \prime$$

*where* X *<sup>a</sup> <sup>j</sup> is the family (100), and* G*<sup>a</sup> <sup>j</sup> is the family (115), for j* ∈ Z. *Then, the asymptotic free distribution of the family* G*<sup>a</sup> <sup>j</sup>* [≥ *p*0] *is the semicircular law in* LS*A*.

**Proof.** The proof of this theorem is similar to that of Theorem 8. One can simply replace

$$\mu^{\prime\prime}p \to \infty^{\prime} \equiv \mathop{\rm ren}\limits\_{n \to \infty} h^n(2); 2 \in \mathcal{P}\_{\prime}^{\prime\prime}$$

in the proof of Theorem 8 to

$$\text{``}\, ^\nu p \to \infty \prime \equiv \text{``}\lim\_{n \to \infty} h^n(p\_0) ; \, p\_0 \in \mathcal{P} \,, \, ^\nu \theta$$

where (≡) means "being symbolically same".

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

**Conflicts of Interest:** The author declares no conflict of interest.

#### **References**


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