Asymptotic Semicircular Laws Induced by p-Adic Number Fields ℚ p and C^*-Algebras over Primes p

Abstract

In this paper, we study asymptotic semicircular laws induced both by arbitrarily fixed $C^{*}$-probability spaces, and p-adic number fields ${\left\{{\mathbb{Q}}_p\right\}}_{p\in \mathcal{P}}$, as p→ ∞ in the set $\mathcal{P}$ of all primes.

1. Introduction

The main purposes of this paper are (i) to establish tensor product $C^{*}$ -probability spaces

$$(A{\otimes }_{\mathbb{C}}{\mathfrak{S}}_p,\psi \otimes {\phi }_j^p)$$

induced both by arbitrary unital $C^{*}$-probability spaces $(A,$ $\psi ),$ and by analytic structures $({\mathfrak{S}}_p,$ ${\phi }_j^p)$ acting on p-adic number fields ${\mathbb{Q}}_p$ for all primes p in the set $\mathcal{P}$ of all primes, where j ∈ $\mathbb{Z},$ (ii) to consider free-probabilistic structures of (i) affected both by the free probability on $(A,$ $\psi ),$ and by the number theory on ${\mathbb{Q}}_p$ for all p ∈ $\mathcal{P},$ (iii) to study asymptotic behaviors on the structures of (i) as p→ ∞ in $\mathcal{P},$ 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 [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22], respectively.

In [23], weighted-semicircular elements, and semicircular elements induced by p-adic number fields ${\mathbb{Q}}_p$ are considered by the author and Jorgensen, for each p ∈ $\mathcal{P}$, 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 ${\left\{{\mathbb{Q}}_p\right\}}_{p\in \mathcal{P}}$ 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 ${\left\{{\mathbb{Q}}_p\right\}}_{p\in \mathcal{P}},$ as $p\to \infty$. For example, they show that our analyses on ${\left\{{\mathbb{Q}}_p\right\}}_{p\in \mathcal{P}}$ 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 Section 3, Section 4, Section 5 and Section 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 ${\mathbb{Q}}_p,$ and free-probabilistic information of fixed unital $C^{*}$-probability spaces. Then, our free-probabilistic structure ${\mathfrak{LS}}_A$, a free product Banach ∗-probability space, is constructed, and the free probability on ${\mathfrak{LS}}_A$ is investigated in Section 8.

In Section 9, asymptotic behaviors on ${\mathfrak{LS}}_A$ are considered over $\mathcal{P},$ and they analyze the asymptotic semicircular laws on ${\mathfrak{LS}}_A$ over $\mathcal{P}$ 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., [5,12,13,14,26]), but also to related fields (e.g., [18,27,28,29,30,31]).

Here, combinatorial free probability is used (e.g., [15,16,17]). In the text, free moments, free cumulants, and the free product of ∗-probability spaces are considered without detailed introduction.

2.2. Analysis on ${\mathbb{Q}}_p$

For p-adic analysis and Adelic analysis, see [21,22]. We use definitions, concepts, and notations from there. Let p ∈ $\mathcal{P}$ be a prime, and let $\mathbb{Q}$ be the set of all rational numbers. Define a non-Archimedean norm ${\left|.\right|}_p,$ called the p-norm on $\mathbb{Q}$ by

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

for all x = $p^k\frac{a}{b}$ ∈ $\mathbb{Q},$ where $k,$ a ∈ $\mathbb{Z},$ and b ∈ $\mathbb{Z}$∖$\left\{0\right\}.$

The normed space ${\mathbb{Q}}_p$ is the maximal p-norm closures in $\mathbb{Q},$ i.e., the set ${\mathbb{Q}}_p$ forms a Banach space, for p ∈ $\mathcal{P}$ (e.g., [22]). Each element x of ${\mathbb{Q}}_p$ is uniquely expressed by

$$x={\sum }_{k=-N}^{\infty }{x}_k{p}^k,x_k\in \{0,1,...,p-1\},$$

for N ∈ $\mathbb{N}$, decomposed by

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

If x = ${\sum }_{k=0}^{\infty }{x}_k{p}^k$ in ${\mathbb{Q}}_p,$ then x is said to be a p-adic integer, and it satisfies ${\left|x\right|}_p$ ≤$1.$ Thus, one can define the unit disk ${\mathbb{Z}}_p$ of ${\mathbb{Q}}_p,$

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

For the p-adic addition and the p-adic multiplication in the sense of [22], the algebraic structure ${\mathbb{Q}}_p$ forms a field, and hence, ${\mathbb{Q}}_p$ is a Banach field.

Note that ${\mathbb{Q}}_p$ is also a measure space,

$${\mathbb{Q}}_p=\left({\mathbb{Q}}_p,\sigma \left({\mathbb{Q}}_p\right),{\mu }_p\right),$$

equipped with the $\sigma$-algebra $\sigma \left({\mathbb{Q}}_p\right)$ of ${\mathbb{Q}}_p$, and a left-and-right additive invariant Haar measure on ${\mu }_p,$ satisfying

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

If we take

$$U_k=p^k{\mathbb{Z}}_p=\{p^{k}x\in {\mathbb{Q}}_p:x\in {\mathbb{Z}}_p\},$$

(1)

in $\sigma \left({\mathbb{Q}}_p\right),$ for all k ∈ $\mathbb{Z},$ then these subsets $U_k$’s of (1) satisfy

$${\mathbb{Q}}_p={\displaystyle \underset{k\in \mathbb{Z}}{\cup }}{U}_k,$$

and

$${\mu }_p\left(U_k\right)=\frac{1}{p^k}={\mu }_p\left(x+U_k\right),$$

(2)

for all x ∈ ${\mathbb{Q}}_p,$ and

$$···\subset U_2\subset U_1\subset U_0={\mathbb{Z}}_p\subset U_{-1}\subset U_{-2}\subset ···,$$

i.e., the family ${\left\{U_k\right\}}_{k\in \mathbb{Z}}$ of (1) is a topological basis element of ${\mathbb{Q}}_p$ (e.g., [22]).

Define subsets ${\partial }_k$ ∈ $\sigma \left({\mathbb{Q}}_p\right)$ by

$${\partial }_k=U_k\setminus U_{k+1},$$

(3)

for all k ∈ $\mathbb{Z}.$

Such ${\mu }_p$-measurable subsets ${\partial }_k$ of (3) are called the k-th boundaries (of $U_k$) in ${\mathbb{Q}}_p,$ for all k ∈ $\mathbb{Z}.$ By (2) and (3),

$$\begin{gathered} {\mathbb{Q}}_p={\displaystyle \bigsqcup _{k\in \mathbb{Z}}}{\partial }_k, \\ {\mu }_p\left({\partial }_k\right)={\mu }_p\left(U_k\right)-{\mu }_p\left(U_{k+1}\right)=\frac{1}{p^k}-\frac{1}{p^{k+1}}, \end{gathered}$$

(4)

where ⊔ is the disjoint union, for all k ∈ $\mathbb{Z},$

Let ${\mathcal{M}}_p$ be an algebraic algebra,

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

(5a)

where ${\chi }_S$ are the usual characteristic functions of ${\mu }_p$-measurable subsets S of ${\mathbb{Q}}_p$. Thus, f ∈ ${\mathcal{M}}_p,$ if and only if

$$f={\displaystyle \sum _{S\in \sigma \left({\mathbb{Q}}_p\right)}}{t}_S{\chi }_S;t_S\in \mathbb{C},$$

(5b)

where ∑ is the finite sum. Note that the algebra ${\mathcal{M}}_p$ of (5a) is a ∗-algebra over $\mathbb{C},$ with its well-defined adjoint,

$${\left({\displaystyle \sum _{S\in \sigma \left(G_p\right)}}{t}_S{\chi }_S\right)}^{*}\stackrel{def}{=}{\displaystyle \sum _{S\in \sigma \left(G_p\right)}}\overline{t_S}{\chi }_S,$$

for $t_S$ ∈ $\mathbb{C}$ with their conjugates $\overline{t_S}$ in $\mathbb{C}.$

If f ∈ ${\mathcal{M}}_p$ is given as in (5b), then one defines the integral of f by

$${\int }_{{\mathbb{Q}}_p}fd{\mu }_p={\displaystyle \sum _{S\in \sigma \left({\mathbb{Q}}_p\right)}}{t}_S{\mu }_p\left(S\right).$$

(6a)

Remark that, by (5a), the integral (6a) is unbounded on ${\mathcal{M}}_p,$ i.e.,

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

(6b)

by (2).

Note that, by (4), for each S ∈ $\sigma \left({\mathbb{Q}}_p\right),$ there exists a corresponding subset ${\Lambda }_S$ of $\mathbb{Z},$

$${\Lambda }_S=\{j\in \mathbb{Z}:S\cap {\partial }_j\ne ⌀\},$$

(7)

satisfying

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

by (6a)

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

(8)

by (4), for the set ${\Lambda }_S$ of (7).

Remark again that the right-hand side of (8) can be ∞; for instance, ${\Lambda }_{{\mathbb{Q}}_p}$ = $\mathbb{Z},$ e.g., see (4), (6a) and (6b). By (8), one obtains the following proposition.

Proposition 1.

Let S ∈ $\sigma \left({\mathbb{Q}}_p\right),$ and let ${\chi }_S$ ∈ ${\mathcal{M}}_p.$ Then, there exists $r_j$ ∈ $\mathbb{R},$ such that

$$\begin{gathered} 0\le r_j=\frac{{\mu }_p(S\cap {\partial }_j)}{{\mu }_p\left({\partial }_j\right)}\le 1,\forall j\in {\Lambda }_S; \\ {\int }_{{\mathbb{Q}}_p}{\chi }_{S}d{\mu }_p={\displaystyle \sum _{j\in {\Lambda }_S}}{r}_j\left(\frac{1}{p^j}-\frac{1}{p^{j+1}}\right). \end{gathered}$$

(9)

3. Statistical Models on ${\mathcal{M}}_{\mathit{p}}$

In this section, fix p ∈ $\mathcal{P}$, and let ${\mathbb{Q}}_p$ be the p-adic number field, and let ${\mathcal{M}}_p$ be the ∗-algebra (5a). We here establish a suitable statistical model on ${\mathcal{M}}_p$ with free-probabilistic language.

Let $U_k$ be the basis elements (1), and ${\partial }_k,$ their boundaries (3) of ${\mathbb{Q}}_p,$ i.e.,

$$U_k=p^k{\mathbb{Z}}_p,$$

for all k ∈ $\mathbb{Z},$ and

$${\partial }_k=U_k\setminus U_{k+1};k\in \mathbb{Z}.$$

(10)

Define a linear functional ${\phi }_p$:${\mathcal{M}}_p$→$\mathbb{C}$ by the integration (6a), i.e.,

$${\phi }_p\left(f\right)={\int }_{{\mathbb{Q}}_p}fd{\mu }_p,$$

(11)

for all f ∈ ${\mathcal{M}}_p.$

Then, by (9), one obtains that ${\phi }_p\left({\chi }_{U_j}\right)$ = $\frac{1}{p^j},$ and ${\phi }_p\left({\chi }_{{\partial }_j}\right)$ = $\frac{1}{p^j}-\frac{1}{p^{j+1}},$ since ${\Lambda }_{U_j}$ = $\{k$ ∈ $\mathbb{Z}$: k ≥ $j\},$ and ${\Lambda }_{{\partial }_j}$ = $\left\{j\right\},$ for all j ∈ $\mathbb{Z},$ where ${\Lambda }_S$ are in the sense of (7) for all S ∈ $\sigma \left({\mathbb{Q}}_p\right).$

Definition 1.

The pair $\left({\mathcal{M}}_p,{\phi }_p\right)$ is called the p-adic (unbounded-)measure space for p ∈ $\mathcal{P},$ where ${\phi }_p$ is the linear functional (11) on ${\mathcal{M}}_p.$

Let ${\partial }_k$ be the k-th boundaries (10) of ${\mathbb{Q}}_p$, for all k ∈ $\mathbb{Z}.$ Then, for $k_1,$ $k_2$ ∈ $\mathbb{Z},$ one obtains that

$${\chi }_{{\partial }_{k_1}}{\chi }_{{\partial }_{k_2}}={\chi }_{{\partial }_{k_1}\cap {\partial }_{k_2}}={\delta }_{k_1,k_2}{\chi }_{{\partial }_{k_1}},$$

and hence,

$$\begin{array}{cc}{\phi }_p\left({\chi }_{{\partial }_{k_1}}{\chi }_{{\partial }_{k_2}}\right)\hfill & ={\delta }_{k_1,k_2}{\phi }_p\left({\chi }_{{\partial }_{k_1}}\right)\hfill \\ & ={\delta }_{k_1,k_2}\left(\frac{1}{p^{k_1}}-\frac{1}{p^{k_1+1}}\right).\hfill \end{array}$$

(12)

Proposition 2.

Let $(j_1,$ …, $j_N)$ ∈ ${\mathbb{Z}}^N,$ for N ∈ $\mathbb{N}.$ Then,

$$\stackrel{N}{{\displaystyle \underset{l=1}{\Pi }}}{\chi }_{{\partial }_{j_l}}={\delta }_{(j_1,...,j_N)}{\chi }_{{\partial }_{j_1}}in{\mathcal{M}}_p,$$

and hence,

$${\phi }_p\left(\stackrel{N}{{\displaystyle \underset{l=1}{\Pi }}}{\chi }_{{\partial }_{j_l}}\right)={\delta }_{(j_1,...,j_N)}\left(\frac{1}{p^{j_1}}-\frac{1}{p^{j_1+1}}\right),$$

(13)

where

$${\delta }_{(j_1,...,j_N)}=\left(\stackrel{N-1}{{\displaystyle \underset{l=1}{\Pi }}}{\delta }_{j_l,j_{l+1}}\right)\left({\delta }_{j_N,j_1}\right).$$

Proof. 

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

Recall that, for any S ∈ $\sigma \left({\mathbb{Q}}_p\right),$

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

(14)

for some $0\le r_j\le 1,$ for j ∈ ${\Lambda }_S,$ by (9). Thus, by (14), if $S_1,$ $S_2$ ∈ $\sigma \left({\mathbb{Q}}_p\right),$ then

$$\begin{array}{cc}{\chi }_{S_1}{\chi }_{S_2}& =\left({\displaystyle \sum _{k\in {\Lambda }_{S_1}}}{\chi }_{S_1\cap {\partial }_k}\right)\left({\displaystyle \sum _{j\in {\Lambda }_{S_2}}}{\chi }_{S_2\cap {\partial }_j}\right)\hfill \\ & ={\displaystyle \sum _{(k,j)\in {\Lambda }_{S_1}\times {\Lambda }_{S_2}}}\left({\chi }_{S_1\cap {\partial }_k}{\chi }_{S_2\cap {\partial }_j}\right)\hfill \\ & ={\displaystyle \sum _{(k,j)\in {\Lambda }_{S_1}\times {\Lambda }_{S_2}}}{\delta }_{k,j}{\chi }_{\left(S_1\cap S_2\right)\cap {\partial }_j}\hfill \\ & ={\displaystyle \sum _{j\in {\Lambda }_{S_1,S_2}}}{\chi }_{(S_1\cap S_2)\cap {\partial }_j},\hfill \end{array}$$

(15)

where

$${\Lambda }_{S_1,S_2}={\Lambda }_{S_1}\cap {\Lambda }_{S_2},$$

by (4).

Proposition 3.

Let $S_l$ ∈ $\sigma \left({\mathbb{Q}}_p\right),$ and let ${\chi }_{S_l}$ ∈ $\left({\mathcal{M}}_p,{\phi }_p\right),$ for l = $1,$ …, $N,$ for N ∈ $\mathbb{N}.$ Let

$${\Lambda }_{S_1,...,S_N}=\stackrel{N}{{\displaystyle \underset{l=1}{\cap }}}{\Lambda }_{S_l}in\mathbb{Z},$$

where ${\Lambda }_{S_l}$ are in the sense of (7), for l = $1,$ …, $N.$ Then, there exists $r_j$ ∈ $\mathbb{R},$ such that

$$0\le r_j\le 1in\mathbb{R},$$

for all j ∈ ${\Lambda }_{S_1,...,S_N},$ and

$${\phi }_p\left(\stackrel{N}{{\displaystyle \underset{l=1}{\Pi }}}{\chi }_{S_l}\right)={\displaystyle \sum _{j\in {\Lambda }_{S_1,\dots ,S_N}}}{r}_j\left(\frac{1}{p^j}-\frac{1}{p^{j+1}}\right).$$

(16)

Proof. 

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

4. Representation of $\left({\mathcal{M}}_{\mathit{p}},{\mathit{\phi }}_{\mathit{p}}\right)$

Fix a prime p ∈ $\mathcal{P}.$ Let $\left({\mathcal{M}}_p,{\phi }_p\right)$ be the p-adic measure space. By understanding ${\mathbb{Q}}_p$ as a measure space, construct the $L^2$-space,

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

(17)

over $\mathbb{C}.$ Then, this Hilbert space $H_p$ of (17) consists of all square-integrable elements of ${\mathcal{M}}_p,$ equipped with its inner product $<,{>}_2,$

$${〈f_1,f_2〉}_2\stackrel{def}{=}{\int }_{{\mathbb{Q}}_p}{f}_1{f}_2^{*}d{\mu }_p,$$

(18a)

for all $f_1,$ $f_2$ ∈ $H_p.$ Naturally, $H_p$ is has its $L^2$-norm ${∥.∥}_2$ on ${\mathcal{M}}_p,$

$${∥f∥}_2\stackrel{def}{=}\sqrt{{〈f,f〉}_2},$$

(18b)

for all f ∈ $H_p,$ where $<,{>}_2$ is the inner product (18a) on $H_p.$

Definition 2.

The Hilbert space $H_p$ of (17) is called the p-adic Hilbert space.

Our ∗-algebra ${\mathcal{M}}_p$ acts on the p-adic Hilbert space $H_p,$ via an action ${\alpha }^p,$

$${\alpha }^p\left(f\right)\left(h\right)=fh,\mathrm{for}\mathrm{all}h\in H_p,$$

(19a)

for all f ∈ ${\mathcal{M}}_p.$ i.e., the morphism ${\alpha }^p$ of (19a) is a ∗-homomorphism from ${\mathcal{M}}_p$ to the operator algebra $B\left(H_p\right),$ consisting of all Hilbert-space operators on $H_p.$ For instance,

$$\begin{array}{cc}{\alpha }^p\left({\chi }_{{\mathbb{Q}}_p}\right)\left({\displaystyle \sum _{S\in \sigma \left({\mathbb{Q}}_p\right)}}{t}_S{\chi }_S\right)\hfill & ={\displaystyle \sum _{S\in \sigma \left({\mathbb{Q}}_p\right)}}{t}_S{\chi }_{{\mathbb{Q}}_p\cap S}\hfill \\ & ={\displaystyle \sum _{S\in \sigma \left({\mathbb{Q}}_p\right)}}{t}_S{\chi }_S,\hfill \end{array}$$

(19b)

for all h = ${\displaystyle \sum _{S\in \sigma \left({\mathbb{Q}}_p\right)}}{t}_S{\chi }_S$ ∈ $H_p,$ with ${∥h∥}_2$<$\infty ,$ for ${\chi }_{{\mathbb{Q}}_p}$ ∈ ${\mathcal{M}}_p,$ even though ${\chi }_{{\mathbb{Q}}_p}$∉$H_p.$

Indeed, It is not difficult to check that

$$\begin{gathered} {\alpha }^p\left(f_1{f}_2\right)={\alpha }^p\left(f_1\right){\alpha }^p\left(f_2\right)on{H}_p,\forall f_1,f_2\in {\mathcal{M}}_p, \\ {\left({\alpha }^p\left(f\right)\right)}^{*}=\alpha \left(f^{*}\right)on{H}_p,\forall f\in {\mathcal{M}}_p. \end{gathered}$$

(20a)

Notation 1.

Denote ${\alpha }^p\left(f\right)$ by ${\alpha }_f^p,$ for all f ∈ ${\mathcal{M}}_p.$ In addition, for convenience, denote ${\alpha }_{{\chi }_S}^p$ simply by ${\alpha }_S^p,$ for all S ∈ $\sigma \left({\mathbb{Q}}_p\right).$

Note that, by (19b), one can have a well-defined operator ${\alpha }_{{\mathbb{Q}}_p}^p$ = ${\alpha }_{{\chi }_{{\mathbb{Q}}_p}}^p$ in $B\left(H_p\right)$, and it satisfies that

$${\alpha }_{{\mathbb{Q}}_p}^p\left(h\right)=h=1_{H_p}\left(h\right),\forall h\in H_p,$$

(20b)

where $1_{H_p}$ ∈ $B\left(H_p\right)$ is the identity operator on $H_p.$

Proposition 4.

The pair $(H_p,$ ${\alpha }^p)$ is a Hilbert-space representation of ${\mathcal{M}}_p.$

Proof. 

It suffices to show that ${\alpha }^p$ is an algebra-action of ${\mathcal{M}}_p$ on $H_p.$ However, this morphism ${\alpha }^p$ is a ∗-homomorphism from ${\mathcal{M}}_p$ into $B\left(H_p\right),$ by (20a). ☐

Definition 3.

The Hilbert-space representation $\left(H_p,{\alpha }^p\right)$ is called the p-adic representation of ${\mathcal{M}}_p.$

Depending on the p-adic representation $(H_p,$ ${\alpha }^p)$ of ${\mathcal{M}}_p,$ one can define the $C^{*}$-subalgebra $M_p$ of $B\left(H_p\right)$ as follows.

Definition 4.

Let $M_p$ be the operator-norm closure of ${\mathcal{M}}_p$,

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

(21)

in $B\left(H_p\right),$ where $\overline{X}$ are the operator-norm closures of subsets X of $B\left(H_p\right).$ This $C^{*}$-algebra $M_p$ is said to be the p-adic $C^{*}$-algebra of $\left({\mathcal{M}}_p,{\phi }_p\right).$

By (21), the p-adic $C^{*}$-algebra $M_p$ is a unital $C^{*}$-algebra contains its unity (or the unit, or the multiplication-identity) $1_{H_p}$ = ${\alpha }_{{\mathbb{Q}}_p}^p,$ by (20b).

5. Statistics on ${\mathit{M}}_{\mathit{p}}$

In this section, fix p ∈ $\mathcal{P},$ and let $M_p$ be the corresponding p-adic $C^{*}$-algebra of (21). Define a linear functional ${\phi }_j^p$:$M_p$→$\mathbb{C}$ by

$${\phi }_j^p\left(a\right)\stackrel{def}{=}{〈a\left({\chi }_{{\partial }_j}\right),{\chi }_{{\partial }_j}〉}_2,\forall a\in M_p,$$

(22a)

for ${\chi }_{{\partial }_j}$ ∈ $H_p,$ where $<,{>}_2$ is the inner product (4.2) on the p-adic Hilbert space $H_p$ of (4.1), and ${\partial }_j$ are the boundaries (3.1) of ${\mathbb{Q}}_p,$ for all j ∈ $\mathbb{Z}.$ It is not hard to check such a linear functional ${\phi }_j^p$ on $M_p$ is bounded, since

$$\begin{array}{cc}{\phi }_j^p\left({\alpha }_S^p\right)& ={〈{\alpha }_S^p\left({\chi }_{{\partial }_j}\right),{\chi }_{{\partial }_j}〉}_2={〈{\chi }_S{\chi }_{{\partial }_j},{\chi }_{{\partial }_j}〉}_2\hfill \\ & ={〈{\chi }_{S\cap {\partial }_j},{\chi }_{{\partial }_j}〉}_2={\int }_{{\mathbb{Q}}_p}{\chi }_{S\cap {\partial }_j}d{\mu }_p\hfill \\ & \le {\int }_{{\mathbb{Q}}_p}{\chi }_{{\partial }_j}d{\mu }_p={\mu }_p\left({\partial }_j\right)=\frac{1}{p^j}-\frac{1}{p^{j+1}},\hfill \end{array}$$

(22b)

for all S ∈ $\sigma \left({\mathbb{Q}}_p\right),$ for any fixed j ∈ $\mathbb{Z}.$

Definition 5.

Let ${\phi }_j^p$ be bounded linear functionals (22a) on the p-adic $C^{*}$-algebra $M_p,$ for all j ∈ $\mathbb{Z}.$ Then, the pairs $\left(M_p,{\phi }_j^p\right)$ are said to be the j-th p-adic $C^{*}$-measure spaces, for all j ∈ $\mathbb{Z}.$

Thus, one can get the system

$$\{(M_p,{\phi }_j^p):j\in \mathbb{Z}\}$$

of the j-th p-adic $C^{*}$-measure spaces $(M_p,$ ${\phi }_j^p)$’s.

Note that, for any fixed j ∈ $\mathbb{Z},$ and $(M_p,$ ${\phi }_j^p),$ the unity

$$1_{M_p}\stackrel{denote}{=}{1}_{H_p}={\alpha }_{{\mathbb{Q}}_p}^{p}of{M}_p$$

satisfies that

$$\begin{array}{cc}{\phi }_j^p\left(1_{M_p}\right)\hfill & ={〈{\chi }_{{\mathbb{Q}}_p\cap {\partial }_j},{\chi }_{{\partial }_j}〉}_2\hfill \\ & ={∥{\chi }_{{\partial }_j}∥}^2=\frac{1}{p^j}-\frac{1}{p^{j+1}}.\hfill \end{array}$$

(23)

Thus, the j-th p-adic $C^{*}$-measure space $(M_p,$ ${\phi }_j^p)$ is a bounded-measure space, but not a probability space, in general.

Proposition 5.

Let S ∈ $\sigma \left({\mathbb{Q}}_p\right),$ and ${\alpha }_S^p$ ∈ $\left(M_p,{\phi }_j^p\right),$ for a fixed j ∈ $\mathbb{Z}.$ Then, there exists $r_S$ ∈ $\mathbb{R},$ such that

$$0\le r_S\le 1in\mathbb{R},$$

and

$${\phi }_j^p\left({\left({\alpha }_S^p\right)}^n\right)=r_S\left(\frac{1}{p^j}-\frac{1}{p^{j+1}}\right);n\in \mathbb{N}.$$

(24)

Proof. 

Remark that the element ${\alpha }_S^p$ is a projection in $M_p,$ in the sense that:

$${\left({\alpha }_S^p\right)}^{*}={\alpha }_{\left({\chi }_S^{*}\right)}^p={\alpha }_S^p={\alpha }_{\left({\chi }_S\cap {\chi }_S\right)}^p={\left({\alpha }_S^p\right)}^2,\mathrm{in}{M}_p,$$

and hence,

$${\left({\alpha }_S^p\right)}^n={\alpha }_S^p,$$

for all n ∈ $\mathbb{N}.$ Thus, we obtain the formula (24) by (22b). ☐

As a corollary of (24), one obtains that, if ${\partial }_k$ is a k-th boundaries of ${\mathbb{Q}}_p,$ then

$${\phi }_j^p\left({\left({\alpha }_{{\partial }_k}^p\right)}^n\right)={\delta }_{j,k}\left(\frac{1}{p^j}-\frac{1}{p^{j+1}}\right),$$

(25)

for all n ∈ $\mathbb{N},$ for k ∈ $\mathbb{Z}.$

6. The ${\mathit{C}}^{*}$-Subalgebra ${\mathfrak{S}}_{\mathit{p}}$ of ${\mathit{M}}_{\mathit{p}}$

Let $M_p$ be the p-adic $C^{*}$-algebra for p ∈ $\mathcal{P}$. Let

$$P_{p,j}={\alpha }_{{\partial }_j}^p\in M_p,$$

(26)

for all j ∈ $\mathbb{Z}.$ By (24) and (25), these operators $P_{p,j}$ of (26) are projections on the p-adic Hilbert space $H_p$, in $M_p,$ for all p ∈ $\mathcal{P},$ j ∈ $\mathbb{Z}.$

Definition 6.

Let p ∈ $\mathcal{P},$ and let ${\mathfrak{S}}_p$ be the $C^{*}$-subalgebra

$${\mathfrak{S}}_p=C^{*}\left({\left\{P_{p,j}\right\}}_{j\in \mathbb{Z}}\right)=\overline{\mathbb{C}\left[{\left\{P_{p,j}\right\}}_{j\in \mathbb{Z}}\right]}of{M}_p,$$

(27)

where $P_{p,j}$ are in the sense of ((26)), for all j ∈ $\mathbb{Z}.$ We call ${\mathfrak{S}}_p,$ the p-adic boundary ($C^{*}$-)subalgebra of $M_p$.

Proposition 6.

If ${\mathfrak{S}}_p$ is the p-adic boundary subalgebra (27), then

$${\mathfrak{S}}_p\stackrel{*-\mathrm{iso}}{=}{\displaystyle \underset{j\in \mathbb{Z}}{\oplus }}\left(\mathbb{C}·P_{p,j}\right)\stackrel{*-\mathrm{iso}}{=}{\mathbb{C}}^{\oplus \left|\mathbb{Z}\right|},$$

(28)

in the p-adic $C^{*}$-algebra $M_p.$

Proof. 

It is enough to show that the generating operators ${\left\{P_{p,j}\right\}}_{j\in \mathbb{Z}}$ of ${\mathfrak{S}}_p$ are mutually orthogonal from each other. It is not hard to check that

$$P_{p,j_1}{P}_{p,j_2}={\alpha }^p\left({\chi }_{{\partial }_{j_1}^p\cap {\partial }_{j_2}^p}\right)={\delta }_{j_1,j_2}{\alpha }_{{\partial }_{j_1}^p}^p={\delta }_{j_1,j_2}{P}_{p,j_1},$$

in ${\mathfrak{S}}_p,$ for all $j_1,$ $j_2$ ∈ $\mathbb{Z}.$ Therefore, the structure theorem (28) is shown. ☐

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

$${\mathfrak{S}}_p\left(j\right)\stackrel{denote}{=}\left({\mathfrak{S}}_p,{\phi }_j^p\right),\forall j\in \mathbb{Z},$$

(29)

for p ∈ $\mathcal{P},$ where the linear functionals ${\phi }_j^p$ of (29) are the restrictions ${\phi }_j^p{\mid }_{{\mathfrak{S}}_p}$ of (22a), for all p ∈ $\mathcal{P},$ j ∈ $\mathbb{Z}.$

7. On the Tensor Product ${\mathit{C}}^{*}$-Probability Spaces $\left(\mathit{A}{\otimes }_{\mathbb{C}}{\mathfrak{S}}_{\mathit{p}},\mathit{\psi }\otimes {\mathit{\phi }}_{\mathit{j}}^{\mathit{p}}\right)$

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

$$\psi \left(1_A\right)=1,$$

where $1_A$ is the unity of a $C^{*}$-algebra $A.$ In addition, let

$${\mathfrak{S}}_p\left(j\right)=\left({\mathfrak{S}}_p,{\phi }_j^p\right)$$

(30)

be the p-adic $C^{*}$-measure spaces (29), for all p ∈ $\mathcal{P},$ j ∈ $\mathbb{Z}.$

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

$${\mathfrak{S}}_p^A\stackrel{def}{=}A{\otimes }_{\mathbb{C}}{\mathfrak{S}}_p,$$

(31)

and a linear functional ${\psi }_j^p$ on ${\mathfrak{S}}_p^A$ by a linear morphism satisfying

$${\psi }_j^p\left(a\otimes P_{p,k}\right)={\phi }_j^p\left(\psi \left(a\right)P_{p,k}\right),$$

(32)

for all a ∈ $(A,$ $\psi ),$ and k ∈ $\mathbb{Z}.$

Note that, by the structure theorem (28) of the p-adic boundary subalgebra ${\mathfrak{S}}_p,$

$${\mathfrak{S}}_p^A\stackrel{\ast -\mathrm{iso}}{=}A{\otimes }_{\mathbb{C}}\left({\mathbb{C}}^{\oplus \left|\mathbb{Z}\right|}\right)\stackrel{\ast -\mathrm{iso}}{=}{A}^{\oplus \left|\mathbb{Z}\right|},$$

(33)

by (31).

By (33), one can verify that a morphism ${\psi }_j^p$ of (32) is indeed a well-defined bounded linear functional on ${\mathfrak{S}}_p^A.$

Definition 7.

For any arbitrarily fixed p ∈ $\mathcal{P},$ j ∈ $\mathbb{Z},$ let ${\mathfrak{S}}_p^A$ be the tensor product $C^{*}$-algebra (31), and ${\psi }_j^p,$ the linear functional (32) on ${\mathfrak{S}}_p^A.$ Then, we call ${\mathfrak{S}}_p^A,$ the A-tensor p-adic boundary algebra. The corresponding structure,

$${\mathfrak{S}}_p^A\left(j\right)\stackrel{denote}{=}\left({\mathfrak{S}}_p^A,{\psi }_j^p\right)$$

(34)

is said to be the j-th p-adic A-(tensor $C^{*}$-probability-)space.

Note that, by (22a), (22b) and (32), the j-th p-adic A-space ${\mathfrak{S}}_p^A\left(j\right)$ of (34) is not a “unital” $C^{*}$-probability space, even though $(A,$ $\psi )$ is. Indeed, the $C^{*}$-algebra ${\mathfrak{S}}_p^A$ of (31) has its unity $1_A\otimes 1_{M_p},$ satisfying

$$\begin{array}{cc}{\psi }_j^p\left(1_A\otimes 1_{M_p}\right)\hfill & ={\phi }_j^p\left(\psi \left(1_A\right)1_{M_p}\right)\hfill \\ & =1·{\phi }_j^p\left(1_{M_p}\right)=\frac{1}{p^j}-\frac{1}{p^{j+1}},\hfill \end{array}$$

for j ∈ $\mathbb{Z}.$

Remark that, by (32),

$${\psi }_j^p\left(a\otimes P_{p,k}\right)=\psi \left(a\right){\phi }_j^p\left(P_{p,k}\right),$$

(35a)

for all a ∈ $(A,$ $\psi ),$ and k ∈ $\mathbb{Z}.$ Thus, by abusing notation, one may write the definition (32) by

$${\psi }_j^p=\psi \otimes {\phi }_j^p\mathrm{on}A{\otimes }_{\mathbb{C}}{\mathfrak{S}}_p={\mathfrak{S}}_p^A,$$

(35b)

in the sense of (35a), for all p ∈ $\mathcal{P},$ j ∈ $\mathbb{Z}.$

Proposition 7.

Let a ∈ $(A,$ $\psi ),$ and $P_{p,k}$, the k-th generating projection of ${\mathfrak{S}}_p,$ for all k ∈ $\mathbb{Z},$ and let $a\otimes P_{p,k}$ be the corresponding free random variable of the j-th p-adic A-space ${\mathfrak{S}}_p^A\left(j\right),$ for j ∈ $\mathbb{Z}$. Then,

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

(36)

for all n ∈ $\mathbb{N}$.

Proof. 

Let $T_{p,k}^a$ = $a\otimes P_{p,k}$ be a given free random variable of ${\mathfrak{S}}_p^A\left(j\right).$ Then,

$${\left(T_{p,k}^a\right)}^n={\left(a\otimes P_{p,k}\right)}^n=a^n\otimes P_{p,k}=T_{p,k}^{a^n},$$

and hence

$${\psi }_j^p\left({\left(T_{p,k}^a\right)}^n\right)={\psi }_j^p\left(T_{p,k}^{a^n}\right)$$

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

by (35a)

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

for all n ∈ $\mathbb{N}.$ Therefore, the free-distributional data (36) holds. ☐

Suppose a is a “self-adjoint” free random variable in $(A,$ $\psi )$ in the above proposition. Then, formula (36) completely characterizes the free distribution of $a\otimes P_{p,k}$ in the j-th p-adic A-space ${\mathfrak{S}}_p^A\left(j\right)$ of (34), i.e., the free distribution of $a\otimes P_{p,k}$ is characterized by the sequence,

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

for all p ∈ $\mathcal{P},$ and $j,$ k ∈ $\mathbb{Z}$ because $a\otimes P_{p,k}$ is self-adjoint in ${\mathfrak{S}}_p^A$ too.

It illustrates that the free probability on ${\mathfrak{S}}_p^A\left(j\right)$ is determined both by the free probability on $(A,$ $\psi ),$ and by the statistical data on ${\mathfrak{S}}_p\left(j\right)$ of (30) (implying p-adic analytic information), for p ∈ $\mathcal{P},$ j ∈ $\mathbb{Z}.$

Notation. 

From below, for convenience, let’s denote the free random variables $a\otimes P_{p,k}$ of ${\mathfrak{S}}_p^A\left(j\right),$ with a ∈ $(A,$ $\psi )$ and k ∈ $\mathbb{Z},$ by $T_{p,k}^a,$ i.e.,

$$T_{p,k}^a\stackrel{denote}{=}a\otimes P_{p,k},$$

for all p ∈ $\mathcal{P},$ j ∈ $\mathbb{Z}.$

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

$${\left(T_{p,k}^a\right)}^n=T_{p,k}^{a^n}\in {\mathfrak{S}}_p^A\left(j\right)$$

(37)

for all n ∈ $\mathbb{N}.$ More generally, the following free-distributional data is obtained.

Theorem 1.

Fix p ∈ $\mathcal{P},$ and j ∈ $\mathbb{Z},$ and let ${\mathfrak{S}}_p^A\left(j\right)$ be the j-th p-adic A-space (34). Let $T_{p,k_l}^{a_l}$ ∈ ${\mathfrak{S}}_p^A\left(j\right),$ for l = $1,$ …, $N,$ for N ∈ $\mathbb{N}.$ Then,

$${\psi }_j^p\left(\stackrel{N}{{\displaystyle \underset{l=1}{\Pi }}}{\left(T_{p,k_l}^{a_l}\right)}^{n_l}\right)=\left(\stackrel{N}{{\displaystyle \underset{l=1}{\Pi }}}{\delta }_{j,k_l}\right)\left(\frac{1}{p^j}-\frac{1}{p^{j+1}}\right)\psi \left(\stackrel{N}{{\displaystyle \underset{l=1}{\Pi }}}{a}_l^{n_l}\right),$$

(38)

for all $n_1,$ …, $n_N$ ∈ $\mathbb{N}.$

Proof. 

Let $T_{p,k_l}^{a_l}$ = $a_l\otimes P_{p,k_l}$ be free random variables of ${\mathfrak{S}}_p^A\left(j\right),$ for l = $1,$ …, $N.$ Then, by (37),

$${\left(T_{p,k_l}^{a_l}\right)}^{n_l}=T_{p,k_l}^{a_l^{n_l}}\in {\mathfrak{S}}_p^A\left(j\right),\mathrm{for}{n}_l\in \mathbb{N},$$

for all l = $1,$ …, $N.$ Thus,

$$T=\stackrel{N}{{\displaystyle \underset{l=1}{\Pi }}}{\left(T_{p,k_l}^{a_l}\right)}^{n_l}=\left(\stackrel{N}{{\displaystyle \underset{l=1}{\Pi }}}{a}_l^{n_l}\right)\otimes \left({\delta }_{j:k_1,...,k_N}{P}_{p,j}\right)$$

in ${\mathfrak{S}}_p^A\left(j\right),$ with

$${\delta }_{j:k_1,...,k_N}=\stackrel{N}{{\displaystyle \underset{l=1}{\Pi }}}{\delta }_{j,k_l}\in \{0,1\}.$$

Therefore,

$$\begin{array}{cc}{\psi }_j^p\left(T\right)\hfill & ={\delta }_{j:k_1,...,k_N}\psi \left(\stackrel{N}{{\displaystyle \underset{l=1}{\Pi }}}{a}_l^{n_l}\right){\phi }_j^p\left(P_{p,j}\right)\hfill \\ & ={\delta }_{j:k_1,...,k_N}\left(\frac{1}{p^j}-\frac{1}{p^{j+1}}\right)\psi \left(\stackrel{N}{{\displaystyle \underset{l=1}{\Pi }}}{a}_l^{n_l}\right),\hfill \end{array}$$

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 ∗-probability Spaces ${\mathfrak{LS}}_{\mathit{p},\mathit{j}}^{\mathit{A}}$

Let $(A,$ $\psi )$ be an arbitrarily fixed unital $C^{*}$-probability space, and let ${\mathfrak{S}}_p\left(j\right)$ be in the sense of (30), for all p ∈ $\mathcal{P},$ j ∈ $\mathbb{Z}.$ Then, one can construct the tensor product $C^{*}$-probability spaces, the j-th p-adic A-space,

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

of (34), for p ∈ $\mathcal{P},$ j ∈ $\mathbb{Z}.$

Throughout this section, we fix p ∈ $\mathcal{P},$ j ∈ $\mathbb{Z},$ and the corresponding j-th p-adic A-space ${\mathfrak{S}}_p^A\left(j\right).$ In addition, we keep using our notation $T_{p,k}^a$ for the free random variables $a\otimes P_{p,k}$ of ${\mathfrak{S}}^A\left(j\right),$ for all a ∈ $(A,$ $\psi )$ and k ∈ $\mathbb{Z},$ where $P_{p,k}$ are the generating projections (26) of the p-adic boundary subalgebra ${\mathfrak{S}}_p.$

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

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

(39)

Now, let $\varphi$ be the Euler totient function,

$$\varphi :\mathbb{N}\to \mathbb{C},$$

defined by

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

(40)

for all n ∈ $\mathbb{N},$ where $\left|X\right|$ are the cardinalities of sets $X,$ and gcd is the greatest common divisor.

By the definition (40),

$$\varphi \left(n\right)=n\left({\displaystyle \underset{q\in \mathcal{P},q\mid n}{\Pi }}\left(1-\frac{1}{q}\right)\right),$$

(41)

for all n ∈ $\mathbb{N},$ where “$q\mid n$” means “ q divides $n.$” Thus,

$$\varphi \left(q\right)=q-1=q\left(1-\frac{1}{q}\right),\forall q\in \mathcal{P},$$

(42)

by (40) and (41).

By (42), we have

$$\begin{array}{cc}{\phi }_j^p\left(P_{p,k}\right)\hfill & =\frac{{\delta }_{j,k}}{p^j}\left(1-\frac{1}{p}\right)\hfill \\ & =\frac{{\delta }_{j,k}\varphi \left(p\right)}{p^{j+1}},\hfill \end{array}$$

for $P_{p,k}$ ∈ ${\mathfrak{S}}_p,$ and hence,

$${\psi }_j^p\left(T_{p,k}^a\right)={\delta }_{j,k}\left(\frac{\varphi \left(p\right)}{p^{j+1}}\right)\psi \left(a\right),$$

(43)

for all $T_{p,k}^a$ ∈ ${\mathfrak{S}}_p^A\left(j\right),$ by (39).

Let’s consider the following estimates.

Lemma 1.

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

$${\displaystyle \underset{p\to \infty }{lim}}\frac{\varphi \left(p\right)}{p^{j+1}}=\left\{\begin{array}{cc}0,\hfill & ifj>0,\hfill \\ 1,\hfill & ifj=0,\hfill \\ \infty ,\mathrm{Undefined},\hfill & ifj<0,\hfill \end{array}\right.$$

(44)

for all j ∈ $\mathbb{Z}$, where “p→∞” means “p is getting bigger and bigger in $\mathcal{P}.$”

Proof. 

Observe that

$${\displaystyle \underset{p\to \infty }{lim}}\frac{\varphi \left(p\right)}{p}={\displaystyle \underset{p\to \infty }{lim}}\left(1-\frac{1}{p}\right)=1,$$

by (42). Thus, one can get that

$${\displaystyle \underset{p\to \infty }{lim}}\frac{\varphi \left(p\right)}{p^{j+1}}={\displaystyle \underset{p\to \infty }{lim}}\left(\frac{\varphi \left(p\right)}{p}\right)\left(\frac{1}{p^j}\right)={\displaystyle \underset{p\to \infty }{lim}}\frac{1}{p^j},$$

for j ∈ $\mathbb{Z}.$ Thus,

$${\displaystyle \underset{p\to \infty }{lim}}\frac{\varphi \left(p\right)}{p^{j+1}}={\displaystyle \underset{p\to \infty }{lim}}\frac{1}{p^j}=\left\{\begin{array}{cc}0,\hfill & \mathrm{if}j>0,\hfill \\ 1,\hfill & \mathrm{if}j=0,\hfill \\ {\displaystyle \underset{p\to \infty }{lim}}{p}^{\left|j\right|}=\infty ,\hfill & \mathrm{if}j<0,\hfill \end{array}\right.$$

where $\left|j\right|$ are the absolute values of j ∈ $\mathbb{Z}.$ Thus, the estimation (44) holds. ☐

8.1. Semicircular Elements

Let $(B,$ $\phi )$ 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 $\phi$ on $B.$

Definition 8.

A self-adjoint operator a ∈ B is said to be semicircular in $(B,$ $\phi ),$ if

$$\phi \left(a^n\right)={\omega }_n{c}_{\frac{n}{2}};n\in \mathbb{N},{\omega }_n=\left\{\begin{array}{cc}1,\hfill & \mathit{if}n\mathit{is}\mathit{even},\hfill \\ 0,\hfill & \mathit{if}n\mathit{is}\mathit{odd},\hfill \end{array}\right.$$

(45)

and $c_k$ are the k-th Catalan numbers,

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

for all k ∈ ${\mathbb{N}}_0$ = $\mathbb{N}$∪$\left\{0\right\}.$

By [15,16,17], if $k_n(...)$ is the free cumulant on B in terms of $\phi$, then a self-adjoint operator a is semicircular in $(B,$ $\phi ),$ if and only if

$$k_n\left(\underset{n-\mathrm{times}}{\underbrace{a,a,......,a}}\right)=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}n=2,\hfill \\ 0,\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

(46)

for all n ∈ $\mathbb{N}$. 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 $a_l$ are semicircular elements in topological ∗-probability spaces $(B_l,$ ${\phi }_l),$ for l = $1,$ $2,$ then the free distributions of $a_l$ are completely characterized by the free-moment sequences,

$${\left({\phi }_l\left(a_l^n\right)\right)}_{n=1}^{\infty },forl=1,2,$$

by the self-adjointness of $a_1$ and $a_2$; and by (45), one obtains that

$$\begin{array}{cc}{\left({\phi }_1\left(a_1^n\right)\right)}_{n=1}^{\infty }\hfill & ={\left({\omega }_n{c}_{\frac{n}{2}}\right)}_{n=1}^{\infty }\hfill \\ & =\left(0,c_1,0,c_2,0,c_3,...\right)\hfill \\ & ={\left({\phi }_2\left(a_2^n\right)\right)}_{n=1}^{\infty }.\hfill \end{array}$$

Equivalently, the free distributions of the semicircular elements $a_1$ and $a_2$ are characterized by the free-cumulant sequences,

$${\left(k_n^1(a_1,...,a_1)\right)}_{n=1}^{\infty }=\left(0,1,0,0,0,...\right)={\left(k_n^2(a_2,...,a_2)\right)}_{n=1}^{\infty },$$

by (46), where $k_n^l(...)$ are the free cumulants on $B_l$ in terms of ${\phi }_l$, 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

$${\left({\omega }_n{c}_{\frac{n}{2}}\right)}_{n=1}^{\infty },$$

(47)

or the free-cumulant sequence

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

Definition 9.

Let a be a semicircular element of a topological ∗-probability space $(B,$ $\phi ).$ The free distribution of a is called “the” semicircular law.

8.2. Tensor Product Banach ∗-algebra ${\mathfrak{LS}}_p^A$

Let ${\mathfrak{S}}_p^A\left(k\right)$ = $\left({\mathfrak{S}}_p^A,{\psi }_k^p\right)$ be the k-th p-adic A-space (34), for all p ∈ $\mathcal{P},$ k ∈ $\mathbb{Z}.$ Throughout this section, we fix p ∈ $\mathcal{P},$ k ∈ $\mathbb{Z}$, and ${\mathfrak{S}}_p^A\left(k\right).$ In addition, denote $a\otimes P_{p,j}$ by $T_{p,j}^a$ in ${\mathfrak{S}}_p^A\left(k\right),$ for all a ∈ $(A,$ $\psi )$ and j ∈ $\mathbb{Z}.$

Define now bounded linear transformations ${\mathbf{c}}_p^A$ and ${\mathbf{a}}_p^A$ “acting on the tensor product $C^{*}$-algebra ${\mathfrak{S}}_p^A,$” by linear morphisms satisfying,

$$\begin{gathered} {\mathbf{c}}_p^A\left(T_{p,j}^a\right)=T_{p,j+1}^a, \\ {\mathbf{a}}_p^A\left(T_{p,j}^a\right)=T_{p,j-1}^a, \end{gathered}$$

(48)

on ${\mathfrak{S}}_p,$ for all j ∈ $\mathbb{Z}.$

By the definitions (27) and (31), and by the structure theorem (33), the above linear morphisms ${\mathbf{c}}_p^A$ and ${\mathbf{a}}_p^A$ of (48) are well-defined on ${\mathfrak{S}}_p^A.$

By (48), one can understand ${\mathbf{c}}_p^A$ and ${\mathbf{a}}_p^A$ as bounded linear transformations contained in the operator space $B\left({\mathfrak{S}}_p^A\right)$ consisting of all bounded linear operators acting on ${\mathfrak{S}}_p^A,$ by regarding the $C^{*}$-algebra ${\mathfrak{S}}_p^A$ as a Banach space equipped with its $C^{*}$-norm (e.g., [32]). Under this sense, the operators ${\mathbf{c}}_p^A$ and ${\mathbf{a}}_p^A$ of (48) are well-defined Banach-space operators on ${\mathfrak{S}}_p^A.$

Definition 10.

The Banach-space operators ${\mathbf{c}}_p^A$ and ${\mathbf{a}}_p^A$ on ${\mathfrak{S}}_p^A,$ in the sense of (48), are called the A-tensor p-creation, respectively, the A-tensor p-annihilation on ${\mathfrak{S}}_p^A.$ Define a new Banach-space operator $l_p^A$ by

$${\mathbf{l}}_p^A={\mathbf{c}}_p^A+{\mathbf{a}}_p^{A}on{\mathfrak{S}}_p^A.$$

(49)

We call this operator ${\mathbf{l}}_p^A$, the A-tensor p-radial operator on ${\mathfrak{S}}_p^A.$

Let ${\mathbf{l}}_p^A$ be the A-tensor p-radial operator ${\mathbf{c}}_p^A+{\mathbf{a}}_p^A$ of (49) in $B\left({\mathfrak{S}}_p^A\right)$. Construct a closed subspace ${\mathfrak{L}}_p^A$ of $B\left({\mathfrak{S}}_p^A\right)$ by

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

(50)

equipped with the inherited operator-norm $∥.∥$ from the operator space $B\left({\mathfrak{S}}_p^A\right),$ defined by

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

where ${∥.∥}_{{\mathfrak{S}}_p^A}$ is the $C^{*}$-norm on the A-tensor p-adic algebra ${\mathfrak{S}}_p^A$ (e.g., [32]).

By the definition (50), the set ${\mathfrak{L}}_p^A$ is not only a closed subspace of $B\left({\mathfrak{S}}_p^A\right),$ but also an algebra over $\mathbb{C}.$ Thus, the subspace ${\mathfrak{L}}_p^A$ is a Banach algebra embedded in $B\left({\mathfrak{S}}_p^A\right).$

On the Banach algebra ${\mathfrak{L}}_p^A$ of (50), define a unary operation (∗) by

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

(51)

where $s_k$ ∈ $\mathbb{C},$ with their conjugates $\overline{s_k}$ ∈ $\mathbb{C}.$

Then, the operation (51) is a well-defined adjoint on ${\mathfrak{L}}_p^A$. Thus, equipped with the adjoint (51), this Banach algebra ${\mathfrak{L}}_p^A$ of (50) forms a Banach ∗-algebra in $B\left({\mathfrak{S}}_p^A\right).$ For example, all elements of ${\mathfrak{L}}_p^A$ are adjointable (in the sense of [32]) in $B\left({\mathfrak{S}}_p^A\right).$

Let ${\mathfrak{L}}_p^A$ be in the sense of (50). Construct now the tensor product Banach ∗-algebra ${\mathfrak{LS}}_p^A$ by

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

(52)

where ${\otimes }_{\mathbb{C}}$ is the tensor product of Banach ∗-algebras. Since ${\mathfrak{S}}_p^A$ is a $C^{*}$-algebra, it is a Banach ∗-algebra too.

Take now a generating element ${\left({\mathbf{l}}_p^A\right)}^n\otimes T_{p,j}^a,$ for some n ∈ ${\mathbb{N}}_0,$ and j ∈ $\mathbb{Z},$ where $T_{p,j}^a$ = $a\otimes P_{p,j}$ are in the sense of (37) in ${\mathfrak{S}}_p^A$, with axiomatization:

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

the identity operator on ${\mathfrak{S}}_p^A$ in $B\left({\mathfrak{S}}_p^A\right),$ satisfying

$$1_{{\mathfrak{S}}_p^A}\left(T\right)=T,$$

for all T ∈ ${\mathfrak{S}}_p^A.$ Define now a bounded linear morphism $E_p^A$:${\mathfrak{LS}}_p^A$→${\mathfrak{S}}_p^A$ by a linear transformation satisfying that:

$$E_p^A\left({\left({\mathbf{l}}_p^A\right)}^k\otimes T_{p,j}^a\right)=\frac{1}{\left[\frac{k}{2}\right]+1}{\left({\mathbf{l}}_p^A\right)}^k\left(T_{p,j}^a\right),$$

(53)

for all k ∈ ${\mathbb{N}}_0,$ j ∈ $\mathbb{Z},$ where $\left[\frac{k}{2}\right]$ is the minimal integer greater than or equal to $\frac{k}{2},$ for all k ∈ ${\mathbb{N}}_0$, for example,

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

By the cyclicity (50) of the tensor factor ${\mathfrak{L}}_p^A$ of ${\mathfrak{LS}}_p^A,$ and by the structure theorem (33) of the other tensor factor ${\mathfrak{S}}_p^A$ of ${\mathfrak{LS}}_p^A$, the above morphism $E_p^A$ of (53) is a well-defined bounded linear transformation from ${\mathfrak{LS}}_p^A$ onto ${\mathfrak{S}}_p^A$.

Now, consider how our A-tensor p-radial operator ${\mathbf{l}}_p^A={\mathbf{c}}_p^A+{\mathbf{a}}_p^A$ acts on ${\mathfrak{S}}_p^A.$ First, observe that: if ${\mathbf{c}}_p^A$ and ${\mathbf{a}}_p^A$ are the A-tensor p-creation, respectively, the A-tensor p-annihilation on ${\mathfrak{S}}_p^A,$ then

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

for all a ∈ $(A,$ $\psi ),$ and for all j ∈ $\mathbb{Z},$ p ∈ $\mathcal{P},$ and, hence,

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

(54)

Lemma 2.

Let ${\mathbf{c}}_p^A,$ ${\mathbf{a}}_p^A$ be the A-tensor p-creation, respectively, the A-tensor p-annihilation on ${\mathfrak{S}}_p^A.$ Then,

$$\begin{gathered} {\left({\mathbf{c}}_p^A\right)}^n{\left({\mathbf{a}}_p^A\right)}^n=1_{{\mathfrak{S}}_p^A}={\left({\mathbf{a}}_p^A\right)}^n{\left({\mathbf{c}}_p^A\right)}^n, \\ {\left({\mathbf{c}}_p^A\right)}^{n_1}{\left({\mathbf{a}}_p^A\right)}^{n_2}={\left({\mathbf{a}}_p^A\right)}^{n_2}{\left({\mathbf{c}}_p^A\right)}^{n_1}, \end{gathered}$$

(55)

on ${\mathfrak{S}}_p^A,$ for all $n,$ $n_1,$ $n_2$ ∈ $\mathbb{N}.$

Proof. 

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

By (55), one can get that

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

(56)

with identity:

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

for all n ∈ $\mathbb{N}$, where

$$\left(\begin{array}{c}n\\ k\end{array}\right)=\frac{n!}{k!(n-k)!},$$

for all $k\le n$ ∈ ${\mathbb{N}}_0.$ By (56), one obtains the following proposition.

Proposition 8.

Let ${\mathbf{l}}_p^A$ ∈ ${\mathfrak{L}}_p^A$ be the A-tensor p-radial operator on ${\mathfrak{S}}_p^A.$ Then,

$${\left({\mathbf{l}}_p^A\right)}^{2m-1}doesnotcontain{1}_{{\mathfrak{S}}_p^A}-term,and$$

(57)

$${\left({\mathbf{l}}_p^A\right)}^{2m}containsits{1}_{{\mathfrak{S}}_p^A}-term,\left(\begin{array}{c}2m\\ m\end{array}\right)·1_{{\mathfrak{S}}_p^A},$$

(58)

for all m ∈ $\mathbb{N}.$

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 $Q_{p,j}^a$ in ${\mathfrak{LS}}_p^A$

Fix p ∈ $\mathcal{P}$, and a unital $C^{*}$-probability space $(A,$ $\psi ),$ and let ${\mathfrak{LS}}_p^A$ be the Banach ∗-algebra (52). Let $E_p^A$:${\mathfrak{LS}}_p^A$→${\mathfrak{S}}_p^A$ be the linear transformation (53). Throughout this section, let

$$Q_{p,j}^a\stackrel{denote}{=}{\mathbf{l}}_p^A\otimes T_{p,j}^a\in {\mathfrak{LS}}_p^A,$$

(59)

for all j ∈ $\mathbb{Z},$ where $T_{p,j}^a$ = $a\otimes P_{p,j}$ ∈ ${\mathfrak{S}}_p^A$ are in the sense of (37) generating ${\mathfrak{S}}_p^A,$ for a ∈ $(A,$ $\psi ),$ and j ∈ $\mathbb{Z}.$ Observe that

$$\begin{array}{cc}{\left(Q_{p,j}^a\right)}^n\hfill & ={\left({\mathbf{l}}_p^A\otimes T_{p,j}^a\right)}^n\hfill \\ & ={\left({\mathbf{l}}_p^A\right)}^n\otimes {\left(T_{p,j}^a\right)}^n={\left({\mathbf{l}}_p^A\right)}^n\otimes T_{p,j}^{a^n},\hfill \end{array}$$

(60)

by (37), for all n ∈ $\mathbb{N},$ for all j ∈ $\mathbb{Z}$.

If $Q_{p,j}^a$ ∈ ${\mathfrak{LS}}_p^A$ is in the sense of (59) for j ∈ $\mathbb{Z},$ then

$$E_p^A\left({\left(Q_{p,j}^a\right)}^n\right)=\frac{1}{\left[\frac{n}{2}\right]+1}{\left({\mathbf{l}}_p^A\right)}^n\left(T_{p,j}^{a^n}\right),$$

(61)

by (53) and (60), for all n ∈ $\mathbb{N}.$

For any fixed j ∈ $\mathbb{Z},$ define a linear functional ${\tau }_j^p$ on ${\mathfrak{LS}}_p^A$ by

$${\tau }_j^p={\psi }_j^p\circ E_p^{A}on{\mathfrak{LS}}_p^A,$$

(62)

where ${\psi }_j^p$ = $\psi \otimes {\phi }_j^p$ is a linear functional (35a), or (35b) on ${\mathfrak{S}}_p^A$.

By the linearity of both ${\psi }_j^p$ and $E_p^A$, the morphism ${\tau }_j^p$ of (62) is a well-defined linear functional on ${\mathfrak{LS}}_p^A$ for j ∈ $\mathbb{Z}.$ Thus, the pair $\left({\mathfrak{LS}}_p^A,{\tau }_j^p\right)$ forms a Banach∗-probability space (e.g., [22]).

Definition 11.

The Banach ∗-probability spaces

$${\mathfrak{LS}}_{p,j}^A\stackrel{denote}{=}\left({\mathfrak{LS}}_p^A,{\tau }_j^p\right)$$

(63)

are called the A-tensor j-th p-adic (free-)filters, for all p ∈ $\mathcal{P},$ j ∈ $\mathbb{Z},$ where ${\tau }_j^p$ are in the sense of (62).

By (61) and (62), if $Q_{p,j}^a$ is in the sense of (59) in ${\mathfrak{LS}}_{p,j}^A$, then

$${\tau }_j^p\left({\left(Q_{p,j}^a\right)}^n\right)=\frac{1}{\left[\frac{n}{2}\right]+1}{\psi }_j^p\left({\left({\mathbf{l}}_p^A\right)}^n\left(T_{p,j}^{a^n}\right)\right),$$

(64)

for all n ∈ $\mathbb{N}$.

Theorem 2.

Let $Q_{p,k}^a$ = ${\mathbf{l}}_p^A$⊗$T_{p,k}^a$ = ${\mathbf{l}}_p^A\otimes \left(a\otimes P_{p,k}\right)$ be a free random variable (59) of the A-tensor j-th p-adic filter ${\mathfrak{LS}}_{p,j}^A$ of (63), for p ∈ $\mathcal{P},$ j ∈ $\mathbb{Z},$ for all k ∈ $\mathbb{Z}.$ Then,

$${\tau }_j^p\left({\left(Q_{p,k}^a\right)}^n\right)={\delta }_{j,k}{\omega }_n\psi \left(a^n\right)c_{\frac{n}{2}}\left(\frac{\varphi \left(p\right)}{p^{j+1}}\right),$$

(65)

where ${\omega }_n$ are in the sense of (45), for all n ∈ $\mathbb{N}.$

Proof. 

Let $Q_{p,j}^a$ be in the sense of (59) in ${\mathfrak{LS}}_{p,j}^A,$ for the fixed p ∈ $\mathcal{P}$ and j ∈ $\mathbb{Z}.$ Then,

$${\tau }_j^p\left({\left(Q_{p,j}^a\right)}^{2n-1}\right)={\psi }_j^p\left(E_p^A\left({\left(Q_{p,j}^a\right)}^{2n-1}\right)\right)$$

by (62)

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

by (64)

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

by (56)

$$=0,$$

by (57), for all n ∈ $\mathbb{N}.$

Observe now that, for any n ∈ $\mathbb{N},$

$${\tau }_j^p\left({\left(Q_{p,j}^a\right)}^{2n}\right)=\left(\frac{1}{\left[\frac{2n}{2}\right]+1}\right){\psi }_j^p\left({\left({\mathbf{l}}_p^A\right)}^{2n}\left(T_{p,j}^{a^{2n}}\right)\right)$$

by (64)

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

by (56)

$$=\left(\frac{1}{n+1}\right){\psi }_j^p\left(\left(\begin{array}{c}2n\\ n\end{array}\right)T_{p,j}^{a^{2n}}+\left[\mathrm{Rest}\mathrm{terms}\right]\right)$$

by (58)

$$=\frac{1}{n+1}\left(\begin{array}{c}2n\\ n\end{array}\right){\psi }_j^p\left(T_{p,j}^{a^{2n}}\right)=\frac{1}{n+1}\left(\begin{array}{c}2n\\ n\end{array}\right)\psi \left(a^{2n}\right)\left(\frac{\varphi \left(p\right)}{p^{j+1}}\right)$$

by (39) and (43)

$$=c_n\psi \left(a^{2n}\right)\left(\frac{\varphi \left(p\right)}{p^{j+1}}\right),$$

where $c_n$ are the n-th Catalan numbers.

If k≠ j in $\mathbb{Z},$ and if $Q_{p,k}^a$ are in the sense of (59) in ${\mathfrak{LS}}_{p,j}^A,$ then

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

for all n ∈ $\mathbb{N},$ by the definition (22a) of the linear functional ${\phi }_j^p$ on ${\mathfrak{S}}_p,$ inducing the linear functional ${\psi }_j^p$ = $\psi \otimes {\phi }_j^p$ on the tensor factor ${\mathfrak{S}}_p^A$ of ${\mathfrak{LS}}_{p,j}^A.$

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

Note that, if a is self-adjoint in $(A,$ $\psi ),$ then the generating operators $Q_{p,k}^a$ of the A-tensor j-th p-adic filter ${\mathfrak{LS}}_{p,j}^A$ are self-adjoint in ${\mathfrak{LS}}_p^A,$ since

$$\begin{array}{cc}{\left(Q_{p,k}^a\right)}^{*}\hfill & ={\left({\mathbf{l}}_p^A\otimes T_{p,k}^a\right)}^{*}={\left({\mathbf{l}}_p^A\right)}^{*}\otimes {\left(T_{p,k}^a\right)}^{*}\hfill \\ & ={\mathbf{l}}_p^A\otimes T_{p,k}^{a^{*}}=Q_{p,k}^a,\hfill \end{array}$$

for all k ∈ $\mathbb{Z},$ for p ∈ $\mathcal{P},$ j ∈ $\mathbb{Z},$ by (51).

Thus, if a is a self-adjoint free random variable of $(A,$ $\psi ),$ then the above formula (65) fully characterizes the free distributions (up to ${\tau }_j^p$) of the generating operators $Q_{p,k}^a$ of ${\mathfrak{LS}}_p^A,$ for all $k,$ j ∈ $\mathbb{Z},$ for p ∈ $\mathcal{P}.$

The free-distributional data (65) can be refined as follows: if p ∈ $\mathcal{P},$ j ∈ $\mathbb{Z}$, and if ${\mathfrak{LS}}_{p,j}^A$ is the corresponding A-tensor j-th p-adic filter (63), then

$${\tau }_j^p\left({\left(Q_{p,j}^a\right)}^n\right)={\omega }_n{c}_{\frac{n}{2}}\psi \left(a^n\right)\left(\frac{\varphi \left(p\right)}{p^{j+1}}\right),$$

(66)

for all n ∈ $\mathbb{N},$ and

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

(67)

for all n ∈ $\mathbb{N}$, whenever k≠ j in $\mathbb{Z},$ for all n ∈ $\mathbb{N}.$

Before we focus on non-zero free-distributional data (66) of $Q_{p,j}^a,$ let’s conclude the following result for ${\left\{Q_{p,k}^a\right\}}_{k\ne j\in \mathbb{Z}}$.

Corollary 1.

Let p ∈ $\mathcal{P},$ j ∈ $\mathbb{Z},$ and let ${\mathfrak{LS}}_{p,j}^A$ be the A-tensor j-th p-adic filter (63). Then, the generating operators

$$Q_{p,k}^a={\mathbf{l}}_p^A\otimes T_{p,j}^a={\mathbf{l}}_p^A\otimes \left(a\otimes P_{p,j}\right)\in {\mathfrak{LS}}_{p,j}^A$$

have the zero free distribution, whenever k ≠ j in $\mathbb{Z}.$

Proof. 

It is proven by (65) and (67). ☐

By the above corollary, we now restrict our interests to the “ j-th” generating operators $Q_{p,j}^a$ of (59) in the A-tensor “ j-th” p-adic filter ${\mathfrak{LS}}_{p,j}^A,$ for all p ∈ $\mathcal{P},$ j ∈ $\mathbb{Z},$ having non-zero free distributions determined by (66).

9. On the Free Product Banach ∗-probability Space ${\mathfrak{LS}}_{\mathit{A}}$

Throughout this section, let $(A,$ $\psi )$ be a fixed unital $C^{*}$-probability space, and let

$${\mathfrak{LS}}_{p,j}^A=\left({\mathfrak{LS}}_p^A,{\tau }_j^p\right)$$

(68)

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

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

are in the sense of (52), and ${\tau }_j^p$ are the linear functionals (62) on ${\mathfrak{LS}}_p^A,$ for all p ∈ $\mathcal{P},$ j ∈ $\mathbb{Z}$.

Let $Q_{p,k}^a$ = ${\mathbf{l}}_p^A\otimes T_{p,k}^a$ = ${\mathbf{l}}_p^A\otimes \left(a\otimes P_{p,k}\right)$ be the generating elements (59) of ${\mathfrak{LS}}_{p,j}^A$ of (68), for a ∈ $(A,$ $\psi ),$ p ∈ $\mathcal{P},$ and $k,$ j ∈ $\mathbb{Z}$. Then, these operators $Q_{p,k}^a$ of ${\mathfrak{LS}}_{p,j}^A$ have their free-distributional data,

$${\tau }_j^p\left({\left(Q_{p,k}^a\right)}^n\right)={\delta }_{j,k}{\omega }_n\psi \left(a^n\right)c_{\frac{n}{2}}\left(\frac{\varphi \left(p\right)}{p^{j+1}}\right),$$

(69)

for all n ∈ $\mathbb{N}$, by (65).

By (66) and (67), we here concentrate on the “ j-th” generating operators of ${\mathfrak{LS}}_{p,j}^A$ having non-zero free distributions (69) for all j ∈ $\mathbb{Z},$ for all p ∈ $\mathcal{P}.$

9.1. Free Product Banach ∗-probability Space $\left({\mathfrak{LS}}_A,\tau \right)$

By (68), we have the family

$$\left\{{\mathfrak{LS}}_{p,j}^A:p\in \mathcal{P},j\in \mathbb{Z}\right\}$$

of Banach ∗-probability spaces, consisting of the A-tensor j-th p-adic filters ${\mathfrak{LS}}_{p,j}^A.$

Define the free product Banach ∗-probability space,

$$\begin{array}{cc}\left({\mathfrak{LS}}_A,\tau \right)\hfill & \stackrel{def}{=}{\displaystyle \underset{p\in \mathcal{P},j\in \mathbb{Z}}{🟉}}{\mathfrak{LS}}_{p,j}^A,\hfill \\ & =\left({\displaystyle \underset{p\in \mathcal{P},j\in \mathbb{Z}}{🟉}}{\mathfrak{LS}}_p^A,{\displaystyle \underset{p\in \mathcal{P},j\in \mathbb{Z}}{🟉}}{\tau }_j^p\right)\hfill \end{array}$$

(70)

in the sense of [15,22].

By (70), the A-tensor j-th p-adic filters ${\mathfrak{LS}}_{p,j}$ of (68) are the free blocks of the Banach ∗-probability space $({\mathfrak{LS}}_A,$ $\tau )$ of (70).

All operators of the Banach ∗-algebra ${\mathfrak{LS}}_A$ in (70) are the Banach-topology limits of linear combinations of noncommutative free reduced words (under operator-multiplication) in

$${\displaystyle \bigsqcup _{p\in \mathcal{P},j\in \mathbb{Z}}}{\mathfrak{LS}}_{p,j}^A.$$

More precisely, since each free block ${\mathfrak{LS}}_{p,j}^A$ is generated by ${\left\{Q_{p,k}^a\right\}}_{a\in A,k\in \mathbb{Z}},$ for all p ∈ $\mathcal{P},$ j ∈ $\mathbb{Z},$ all elements of ${\mathfrak{LS}}_A$ are the Banach-topology limits of linear combinations of free words in

$${\displaystyle \bigsqcup _{p\in \mathcal{P},j\in \mathbb{Z}}}\{Q_{p,k}^a\in {\mathfrak{LS}}_{p,j}:a\in A,k\in \mathbb{Z}\}.$$

In particular, all noncommutative free words have their unique free “reduced” words (as operators of ${\mathfrak{LS}}_A$ under operator-multiplication) formed by

$$\stackrel{N}{\underset{l=1}{\Pi }}{\left(Q_{p_l,k_l}^{a_l}\right)}^{n_l},\mathrm{where}{Q}_{p_l,k_l}^{a_l}\in {\mathfrak{LS}}_{p_l,j_l}^A$$

in ${\mathfrak{LS}}_A,$ for all $a_1,$ …, $a_N$ ∈ $(A,$ $\psi ),$ and $n_1,$ …, $n_N$ ∈ $\mathbb{N},$ where either the N-tuple

$$\left(p_1,...,p_N\right),\mathrm{or}\left(j_1,...,j_N\right)$$

is alternating in $\mathcal{P},$ respectively, in $\mathbb{Z}$, in the sense that:

$$p_1\ne p_2,p_2\ne p_3,...,p_{N-1}\ne p_N\mathrm{in}\mathcal{P},$$

respectively,

$$j_1\ne j_2,j_2\ne j_3,...,j_{N-1}\ne j_N\mathrm{in}\mathbb{Z}$$

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

For example, a 5-tuple

$$\left(2,2,3,7,2\right)$$

is not alternating in $\mathcal{P},$ while a 5-tuple

$$\left(2,3,2,7,2\right)$$

is alternating in $\mathcal{P},$ etc.

By (70), if $Q_{p,j}^a$ are the j-th a-tensor generating operators of a free block ${\mathfrak{LS}}_{p,j}^A$ of the Banach ∗-probability space $({\mathfrak{LS}}_A,$ $\tau )$, for all j ∈ $\mathbb{Z},$ for p ∈ $\mathcal{P},$ j ∈ $\mathbb{Z},$ then ${\left(Q_{p,j}^a\right)}^n$ are contained in the same free block ${\mathfrak{LS}}_{p,j}^A$ of $\left({\mathfrak{LS}}_A,\tau \right),$ and, hence, they are free reduced words with their lengths-1, for all n ∈ $\mathbb{N}.$ Therefore, we have

$$\begin{array}{cc}\tau \left({\left(Q_{p,j}^a\right)}^n\right)\hfill & ={\tau }_j^p\left({\left(Q_{p,j}^a\right)}^n\right)\hfill \\ & ={\omega }_n{c}_{\frac{n}{2}}\psi \left(a^n\right)\left(\frac{\varphi \left(p\right)}{p^{j+1}}\right),\hfill \end{array}$$

(71)

for all n ∈ $\mathbb{N},$ by (69).

Definition 12.

The Banach ∗-probability space ${\mathfrak{LS}}_A$ $\stackrel{denote}{=}$ $\left({\mathfrak{LS}}_A,\tau \right)$ of (70) is called the A-tensor (free-)Adelic filterization of ${\left\{{\mathfrak{LS}}_{p,j}^A\right\}}_{p\in \mathcal{P},j\in \mathbb{Z}}$.

As we discussed at the beginning of Section 9, we now focus on studying free random variables of the A-tensor Adelic filterization ${\mathfrak{LS}}_A$ of (70) having “non-zero” free distributions.

Define a subset $\mathcal{U}$ of ${\mathfrak{LS}}_A$ by

$$\mathcal{U}=\left\{Q_{p,j}^{1_A}\in {\mathfrak{LS}}_{p,j}^A\left|\forall p\in \mathcal{P},j\in \mathbb{Z}\right.\right\}$$

(72)

in ${\mathfrak{LS}}_A,$ where $1_A$ is the unity of $A,$ and $Q_{p,j}^{1_A}$ are the “ j-th” $1_A$-tensor generating operators of ${\mathfrak{LS}}_A,$ in the free blocks ${\mathfrak{LS}}_{p,j}^A,$ for all p ∈ $\mathcal{P},$ j ∈ $\mathbb{Z}.$

Then, the elements $Q_{p,j}^{1_A}$ of $\mathcal{U}$ have their non-zero free distributions,

$${\left({\omega }_n{c}_{\frac{n}{2}}\psi \left(1_A^n\right)\left(\frac{\varphi \left(p\right)}{p^{j+1}}\right)\right)}_{n=1}^{\infty }={\left({\omega }_n{c}_{\frac{n}{2}}\left(\frac{\varphi \left(p\right)}{p^{j+1}}\right)\right)}_{n=1}^{\infty },$$

by (71), since

$$\psi \left(1_A^n\right)=\psi \left(1_A\right)=1,$$

for all n ∈ $\mathbb{N}.$ Now, define a Cartesian product set

$${\mathcal{U}}_A\stackrel{def}{=}A\times \mathcal{U},$$

(73a)

set-theoretically, where $\mathcal{U}$ is in the sense of (72).

Define a function $\mathsf{\Omega }$:${\mathcal{U}}_A$→${\mathfrak{LS}}_A$ by

$$\mathsf{\Omega }\left((a,Q_{p,j}^{1_A})\right)\stackrel{def}{=}{Q}_{p,j}^a\mathrm{in}{\mathfrak{LS}}_A,$$

(73b)

for all $(a,$ $Q_{p,j}^{1_A})$ ∈ ${\mathcal{U}}_A,$ where ${\mathcal{U}}_A$ is in the sense of (73a).

It is not difficult to check that this function $\mathsf{\Omega }$ of (73b) is a well-defined injective map. Moreover, it induces all j-th a-tensor generating elements $Q_{p,j}^a$ of ${\mathfrak{LS}}_{p,j}^a$ in ${\mathfrak{LS}}_A,$ for all p ∈ $\mathcal{P},$ and j ∈ $\mathbb{Z}.$

Define a Banach ∗-subalgebra ${\mathbb{LS}}_A$ of the A-tensor Adelic filterization ${\mathfrak{LS}}_A$ of (70) by

$${\mathbb{LS}}_A\stackrel{def}{=}\overline{\mathbb{C}\left[\mathsf{\Omega }\left({\mathcal{U}}_A\right)\right]}\mathrm{in}{\mathfrak{LS}}_A,$$

(74a)

where $\mathsf{\Omega }\left({\mathcal{U}}_A\right)$ is the subset of ${\mathfrak{LS}}_A,$ induced by (73a) and (73b), and $\overline{Y}$ mean the Banach-topology closures of subsets Y of ${\mathfrak{LS}}_A.$

Then, this Banach ∗-subalgebra ${\mathbb{LS}}_A$ of (74a) has a sub-structure,

$${\mathbb{LS}}_A\stackrel{denote}{=}\left({\mathbb{LS}}_A,\tau =\tau {\mid }_{{\mathbb{LS}}_A}\right)$$

(74b)

in the A-tensor Adelic filterization ${\mathfrak{LS}}_A.$

Theorem 3.

Let ${\mathbb{LS}}_A$ be the Banach ∗-algebra (74a) in the A-tensor Adelic filterization ${\mathfrak{LS}}_A.$ Then,

$$\begin{array}{cc}{\mathbb{LS}}_A\hfill & \stackrel{*-\mathrm{iso}}{=}\underset{p\in \mathcal{P},j\in \mathbb{Z}}{🟉}\overline{\mathbb{C}\left[\{Q_{p,j}^a:a\in (A,\psi \}\right]}\hfill \\ & \stackrel{*-\mathrm{iso}}{=}\overline{\mathbb{C}\left[{\displaystyle \underset{p\in \mathcal{P},j\in \mathbb{Z}}{🟉}}\{Q_{p,j}^a:a\in (A,\psi \}\right]},\hfill \end{array}$$

(75)

where $Q_{p,j}^a$ ∈ $\mathsf{\Omega }\left({\mathcal{U}}_A\right)$ 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 $\mathsf{\Omega }\left({\mathcal{U}}_A\right)$.

Proof. 

Let ${\mathbb{LS}}_A$ be the Banach ∗-subalgebra (74a) in ${\mathfrak{LS}}_A.$ Then,

$${\mathbb{LS}}_A=\overline{\mathbb{C}\left[{\{Q_{p,j}^a\in {\mathfrak{LS}}_{p,j}^A:a\in (A,\psi )\}}_{p\in \mathcal{P},j\in \mathbb{Z}}\right]}$$

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

$$\stackrel{*-\mathrm{iso}}{=}{\displaystyle \underset{p\in \mathcal{P},j\in \mathbb{Z}}{🟉}}\overline{\mathbb{C}\left[\{Q_{p,j}^a:a\in (A,\psi )\}\right]}$$

in ${\mathfrak{LS}}_A$, since all elements $Q_{p,j}^a$ ∈ $\mathsf{\Omega }\left({\mathcal{U}}_A\right)$ are chosen from mutually distinct free blocks ${\mathfrak{LS}}_{p,j}^A$ of the A-tensor Adelic filterization ${\mathfrak{LS}}_A$, and, hence, the operators $\{Q_{p,j}^a,$ $Q_{p,j}^{a^{*}}{\}}_{p\in \mathcal{P},j\in \mathbb{Z}}$ are free from each other in ${\mathfrak{LS}}_A,$ for any a ∈ $(A,$ $\psi ),$ for all p ∈ $\mathcal{P},$ j ∈ $\mathbb{Z},$ moreover,

$$\stackrel{*-\mathrm{iso}}{=}\overline{\mathbb{C}\left[{\displaystyle \underset{p\in \mathcal{P},j\in \mathbb{Z}}{🟉}}\{Q_{p,j}^a:a\in (A,\psi )\}\right]},$$

because all elements of ${\mathbb{LS}}_A$ are the (Banach-topology limits of) linear combinations of free words in $\mathsf{\Omega }\left({\mathcal{U}}_A\right),$ by the very above ∗-isomorphic relation. Indeed, for any noncommutative (pure-algebraic) free words in

$${\displaystyle \underset{p\in \mathcal{P},j\in \mathbb{Z}}{\cup }}\{Q_{p,j}^a:a\in (A,\psi )\}$$

have their unique free “reduced” words under operator-multiplication on ${\mathfrak{LS}}_A$, as operators of ${\mathbb{LS}}_A.$

Therefore, the structure theorem (75) holds. ☐

The above theorem characterizes the free-probabilistic structure of the Banach ∗-algebra ${\mathbb{LS}}_A$ of (74a) in the A-tensor Adelic filterization ${\mathfrak{LS}}_A.$ This structure theorem (75) demonstrates that the Banach ∗-probability space $({\mathbb{LS}}_A,$ $\tau )$ of (74b) is well-determined, having its natural inherited free probability from that on ${\mathfrak{LS}}_A$.

Definition 13.

Let $({\mathbb{LS}}_A,$ $\tau )$ be the Banach ∗-probability space (74b). Then, we call

$${\mathbb{LS}}_A\stackrel{denote}{=}({\mathbb{LS}}_A,\tau ),$$

the A-tensor (Adelic) sub-filterization of the A-tensor Adelic filterization ${\mathfrak{LS}}_A.$

By (69), (71), (72) and (75), one can verify that the free probability on the A-tensor sub-filterization ${\mathbb{LS}}_A$ provide “possible” non-zero free distributions on the A-tensor Adelic filterization ${\mathfrak{LS}}_A,$ up to free probability on $(A,$ $\psi ).$ i.e., if a ∈ $(A,$ $\psi )$ have their non-zero free distributions, then $Q_{p,j}^a$ ∈ ${\mathbb{LS}}_A$ have non-zero free distributions, and, hence, they have their non-zero free distributions on ${\mathfrak{LS}}_A.$

Theorem 4.

Let $Q_{p,j}^a$ ∈ $\mathsf{\Omega }\left({\mathcal{U}}_A\right)$ be free random variables of the A-tensor sub-filterization ${\mathbb{LS}}_A$, for a ∈ $(A,$ $\psi ),$ and p ∈ $\mathcal{P},$ and j ∈ $\mathbb{Z}.$ Then,

$$\begin{gathered} \tau \left({\left(Q_{p,j}^a\right)}^n\right)={\omega }_n{c}_{\frac{n}{2}}\psi \left(a^n\right)\left(\frac{\varphi \left(p\right)}{p^{j+1}}\right), \\ \tau \left({\left({\left(Q_{p,j}^a\right)}^{*}\right)}^n\right)={\omega }_n{c}_{\frac{n}{2}}\overline{\psi \left(a^n\right)}\left(\frac{\varphi \left(p\right)}{p^{j+1}}\right), \end{gathered}$$

(76)

for all n ∈ $\mathbb{N}.$

Proof. 

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

$$\begin{array}{cc}{\left(Q_{p,j}^a\right)}^{*}& ={\left({\mathbf{l}}_p^A\otimes T_{p,j}^a\right)}^{*}={\left({\mathbf{l}}_p^A\otimes \left(a\otimes P_{p,j}\right)\right)}^{*}\hfill \\ & ={\left({\mathbf{l}}_p^A\right)}^{*}\otimes {\left(a\otimes P_{p,j}\right)}^{*}={\mathbf{l}}_p^A\otimes \left(a^{*}\otimes P_{p,j}\right),\end{array}$$

and, hence,

$${\left(Q_{p,j}^a\right)}^{*}=Q_{p,j}^{a^{*}}in{\mathbb{LS}}_A,$$

(77)

for all $Q_{p,j}^a$ ∈ $\mathsf{\Omega }\left({\mathcal{U}}_A\right).$ Thus, one has

$${\left({\left(Q_{p,j}^a\right)}^{*}\right)}^n={\left(Q_{p,j}^{a^{*}}\right)}^n=Q_{p,j}^{{\left(a^{*}\right)}^n}=Q_{p,j}^{{\left(a^n\right)}^{*}}\mathrm{in}{\mathbb{LS}}_A,$$

by (77).

Thus, one has

$$\begin{array}{cc}\tau \left({\left({\left(Q_{p,j}^a\right)}^{*}\right)}^n\right)\hfill & ={\omega }_n{c}_{\frac{n}{2}}\psi \left({\left(a^n\right)}^{*}\right)\left(\frac{\varphi \left(p\right)}{p^{j+1}}\right)\hfill \\ & ={\omega }_n{c}_{\frac{n}{2}}\overline{\psi \left(a^n\right)}\left(\frac{\varphi \left(p\right)}{p^{j+1}}\right),\hfill \end{array}$$

by (71), for all n ∈ $\mathbb{N}.$ Therefore, the second formula of (76) holds too. ☐

9.2. Prime-Shifts on ${\mathbb{LS}}_A$

Let ${\mathbb{LS}}_A$ be the A-tensor sub-filterization (70) of the A-tensor Adelic filterization ${\mathfrak{LS}}_A.$ In this section, we define a certain ∗-homomorphism on ${\mathbb{LS}}_A$, and study asymptotic free-distributional data on ${\mathbb{LS}}_A$ (and hence those on ${\mathfrak{LS}}_A$) over primes.

Let $\mathcal{P}$ be the set of all primes in $\mathbb{N}$, regarded as a totally ordered set (in short, a TOset) for the usual ordering (≤), i.e.,

$$\mathcal{P}=\{q_1<q_2<q_3<q_4<···\},$$

(78)

with

$$q_1=2,q_2=3,q_3=5,q_4=7,q_5=11,...,\mathrm{etc}.$$

Define an injective function h:$\mathcal{P}$→$\mathcal{P}$ by

$$h\left(q_k\right)=q_{k+1};k\in \mathbb{N},$$

(79)

where $q_k$ are primes of (78), for all k ∈ $\mathbb{N}.$

Definition 14.

Let h be an injective function (79) on the TOset $\mathcal{P}$ of (78). We call h the shift on $\mathcal{P}.$

Let h be the shift (79) on the TOset $\mathcal{P},$ and let

$$h^{\left(n\right)}\stackrel{def}{=}\underset{n-\mathrm{times}}{\underbrace{h\circ h\circ h\circ ···\circ h}},on\mathcal{P},$$

(80)

for all n ∈ $\mathbb{N}$, where (∘) is the usual functional composition.

By the definitions (79) and (80),

$$h^{\left(n\right)}\left(q_k\right)=q_{k+n},$$

(81)

for all n ∈ $\mathbb{N},$ in $\mathcal{P}.$ For instance, $h^{\left(3\right)}\left(2\right)$ = $7,$ and $h^{\left(4\right)}\left(5\right)$ = $17,$ etc.

These injective functions $h^{\left(n\right)}$ of (80) are called the n-shifts on $\mathcal{P},$ for all n ∈ $\mathbb{N}.$

For the shift h on $\mathcal{P},$ one can define a ∗-homomorphism ${\pi }_h$ on the A-tensor sub-filterization ${\mathbb{LS}}_A$ by a bounded “multiplicative” linear transformation, satisfying that

$${\pi }_h\left(Q_{q_k,j}^a\right)=Q_{h\left(q_k\right),j}^a=Q_{q_{k+1},j}^a,$$

(82)

for all $Q_{q_k,j}$ ∈ $\mathsf{\Omega }\left({\mathcal{U}}_A\right),$ for all $q_k$ ∈ $\mathcal{P},$ for all j ∈ $\mathbb{Z},$ where h is the shift (79) on $\mathcal{P}.$

By (82), we have

$${\pi }_h\left(\stackrel{N}{{\displaystyle \underset{l=1}{\Pi }}}{\left(Q_{q_{k_l},j_l}^{a_l}\right)}^{n_l}\right)=\stackrel{N}{{\displaystyle \underset{l=1}{\Pi }}}{\left(Q_{h\left(q_{k_l}\right),j_l}^{a_l}\right)}^{n_l}=\stackrel{N}{{\displaystyle \underset{l=1}{\Pi }}}{\left(Q_{q_{k_l+1},j_l}^{a_l}\right)}^{n_l},$$

(83)

in ${\mathbb{LS}}_A,$ for all $Q_{q_{k_l},j_l}^a$ ∈ $\mathsf{\Omega }\left({\mathcal{U}}_A\right),$ for $q_{k_l}$ ∈ $\mathcal{P},$ $j_l$ ∈ $\mathbb{Z},$ for l = $1,$ …, $N,$ for N ∈ $\mathbb{N},$ where $n_1,$ …, $n_N$ ∈ $\mathbb{N}.$

Remark 1.

Note that the multiplicative linear transformation ${\pi }_h$ of (82) is indeed a ∗-homomorphism satisfying

$${\pi }_h\left(T^{*}\right)={\left({\pi }_h\left(T\right)\right)}^{*},$$

for all T ∈ ${\mathbb{LS}}_A$, because

$$\begin{array}{cc}{\pi }_h\left({\left(Q_{p,j}^a\right)}^{*}\right)\hfill & ={\pi }_h\left(Q_{p,j}^{a^{*}}\right)=Q_{h\left(p\right),j}^{a^{*}}\hfill \\ & ={\left(Q_{h\left(p\right),j}^a\right)}^{*}={\left({\pi }_h\left(Q_{p,j}^a\right)\right)}^{*},\hfill \end{array}$$

for all $Q_{p,j}^a$ ∈ $\mathsf{\Omega }\left({\mathcal{U}}_A\right).$

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

$${\pi }_h^n=\underset{n-\mathrm{times}}{\underbrace{{\pi }_h{\pi }_h{\pi }_h···{\pi }_h}},\mathrm{on}{\mathbb{LS}}_A,$$

(84)

the products (or compositions) of the n-copies of the ∗-homomorphism ${\pi }_h$ of (82), acting on ${\mathbb{LS}}_A.$ It is not difficult to check that

$$\begin{array}{cc}{\pi }_h^n\left(Q_{p,j}^a\right)\hfill & ={\pi }_h^{n-1}\left(Q_{h\left(p\right),j}^a\right)={\pi }_h^{n-2}\left(Q_{h^{\left(2\right)}\left(p\right),j}^a\right)\hfill \\ & =···={\pi }_h\left(Q_{h^{(n-1)}\left(p\right),j}^a\right)=Q_{h^{\left(n\right)}\left(p\right),j}^a,\hfill \end{array}$$

(85)

for all $Q_{p,j}^a$ ∈ $\mathsf{\Omega }\left({\mathcal{U}}_A\right)$ in ${\mathbb{LS}}_A,$ where $h^{\left(k\right)}$ are the k-shifts (80) on $\mathcal{P},$ for all k ∈ $\mathbb{N}.$

Definition 15.

Let ${\pi }_h$ be the ∗-homomorphism (82) on the A-tensor sub-filterization ${\mathbb{LS}}_A,$ and let ${\pi }_h^n$ be the products (84) acting on ${\mathbb{LS}}_A,$ for all n ∈ $\mathbb{N},$ with ${\pi }_h^1$ = ${\pi }_h.$ Then, we call ${\pi }_h^n,$ the n-prime-shift (∗-homomorphism) on ${\mathbb{LS}}_A,$ for all n ∈ $\mathbb{N}.$ In particular, the 1-prime-shift ${\pi }_h$ is simply said to be the prime-shift (∗-homomorphism) on ${\mathbb{LS}}_A.$

Thus, for any $Q_{q_k,j}^a$ ∈ $\mathsf{\Omega }\left({\mathcal{U}}_A\right)$ in ${\mathbb{LS}}_A,$ for $q_k$ ∈ $\mathcal{P}$ (in the sense of (78) with k ∈ $\mathbb{N}$), the n-prime-shift ${\pi }_h^n$ satisfies

$${\pi }_h^n\left(Q_{q_k,j}^a\right)=Q_{h^{\left(n\right)}\left(q_k\right),j}^a=Q_{q_{k+n},j}^a,$$

(86)

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

$${\pi }_h^n\left(\stackrel{N}{{\displaystyle \underset{l=1}{\Pi }}}{\left(Q_{q_{k_l},j_l}^{a_l}\right)}^{n_l}\right)=\stackrel{N}{{\displaystyle \underset{l=1}{\Pi }}}{\left(Q_{q_{k_l+n},j_l}^{a_l}\right)}^{n_l},$$

(87)

by (83) and (86), for all n ∈ $\mathbb{N}.$

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

$${\pi }_h^n={\pi }_{h^{\left(n\right)}}\mathrm{on}{\mathbb{LS}}_A,\mathrm{for}\mathrm{all}\in \mathbb{N},$$

where $h^{\left(n\right)}$ are the n-shifts (81) on the TOset $\mathcal{P}$.

Consider now the sequence

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

(88)

of the n-prime-shifts on ${\mathbb{LS}}_A.$

For any fixed T ∈ ${\mathbb{LS}}_A,$ the sequence $\Pi$ of (88) induces the sequence of operators,

$$\Pi \left(T\right)={\left({\pi }_h^n\left(T\right)\right)}_{n=1}^{\infty }=\left({\pi }_h\left(T\right),{\pi }_h^2\left(T\right),{\pi }_h^3\left(T\right),···\right)$$

in ${\mathbb{LS}}_A,$ and this sequence $\Pi \left(T\right)$ has its corresponding free-distributional data, represented by the following $\mathbb{C}$-sequence:

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

(89)

We are interested in the convergence of the $\mathbb{C}$-sequence $\tau (\Pi (T\left)\right)$ of (89), as n→$\infty .$

Either convergent or divergent, the $\mathbb{C}$-sequence $\tau (\Pi (T\left)\right)$ of (89), induced by any fixed operator T ∈ ${\mathbb{LS}}_A,$ shows the asymptotic free distributional data of the family ${\left\{{\pi }_h^n\left(T\right)\right\}}_{n=1}^{\infty }$⊂${\mathbb{LS}}_A,$ as n→ ∞ in $\mathbb{N},$ equivalently, as $q_n$→ ∞ in $\mathcal{P}.$

9.3. Asymptotic Behaviors in ${\mathbb{LS}}_A$ over $\mathcal{P}$

Recall that, by (44), we have

$${\displaystyle \underset{p\to \infty }{lim}}\frac{\varphi \left(p\right)}{p^{j+1}}=\left\{\begin{array}{cc}0,\hfill & \mathrm{if}j>0,\hfill \\ 1,\hfill & \mathrm{if}j=0,\hfill \\ \infty ,\mathrm{Undefined},\hfill & \mathrm{if}j<0,\hfill \end{array}\right.$$

(90)

for j ∈ $\mathbb{Z}.$

Recall also that there are bounded ∗-homomorphisms

$$\Pi ={\left({\pi }_h^n\right)}_{n=1}^{\infty },\mathrm{acting}\mathrm{on}{\mathbb{LS}}_A,$$

of (88), where ${\pi }_h^n$ are the n-prime shifts of (84), where h is the shift (79) on the TOset $\mathcal{P}$ of (78). Then, these ∗-homomorphisms of $\Pi$ satisfies

$${\displaystyle \underset{n\to \infty }{lim}}\left({\pi }_h^n\left(Q_{p,j}^a\right)\right)={\displaystyle \underset{n\to \infty }{lim}}\left(Q_{h^{\left(n\right)}\left(p\right),j}^a\right),$$

(91)

for all $Q_{p,j}^a$ ∈ $\mathsf{\Omega }\left({\mathcal{U}}_A\right)$ in ${\mathbb{LS}}_A,$ where $h^{\left(n\right)}$ are the n-shifts (80) on $\mathcal{P},$ for all n ∈ $\mathbb{N}.$

Thus, one can get that: if $\stackrel{N}{{\displaystyle \underset{l=1}{\Pi }}}{\left(Q_{p_l,j_l}^{a_l}\right)}^{n_l}$ is a free reduced words of ${\mathbb{LS}}_A$ in $\mathsf{\Omega }\left({\mathcal{U}}_A\right),$ then

$$\begin{array}{cc}{\displaystyle \underset{n\to \infty }{lim}}{\pi }_h^n\left(\stackrel{N}{{\displaystyle \underset{l=1}{\Pi }}}{\left(Q_{p_l,j_l}^{a_l}\right)}^{n_l}\right)& ={\displaystyle \underset{n\to \infty }{lim}}\left(\stackrel{N}{{\displaystyle \underset{l=1}{\Pi }}}{\pi }_h^n\left({\left(Q_{p_l,j_l}^{a_l}\right)}^{n_l}\right)\right)\hfill \\ & ={\displaystyle \underset{n\to \infty }{lim}}\left(\stackrel{N}{{\displaystyle \underset{l=1}{\Pi }}}{\left({\pi }_h^n\left(Q_{p_l,j_l}^{a_l}\right)\right)}^{n_l}\right)\hfill \end{array}$$

since ${\pi }_h^n$ are ∗-homomorphisms on ${\mathbb{LS}}_A$

$$={\displaystyle \underset{n\to \infty }{lim}}\left(\stackrel{N}{{\displaystyle \underset{l=1}{\Pi }}}{\left(Q_{h^{\left(n\right)}\left(p_l\right),j_l}^{a_l}\right)}^{n_l}\right)$$

by (91)

$$=\stackrel{N}{{\displaystyle \underset{l=1}{\Pi }}}\left({\displaystyle \underset{n\to \infty }{lim}}{\left(Q_{h^{\left(n\right)}\left(p_l\right),j_l}^{a_l}\right)}^{n_l}\right),$$

(92)

under the Banach-topology for ${\mathbb{LS}}_A,$ for all $Q_{p_l,j_l}^{a_l}$ ∈ $\mathsf{\Omega }\left({\mathcal{U}}_A\right)$, for $a_l$ ∈ $(A,$ $\psi ),$ $p_l$ ∈ $\mathcal{P},$ $j_l$ ∈ $\mathbb{Z},$ for l = $1,$ …, $N,$ for all N ∈ $\mathbb{N}.$

Notation 2.

(in short, N 2 from below) For convenience, we denote ${\displaystyle \underset{n\to \infty }{lim}}{\pi }_h^n$ symbolically by $\pi ,$ for the sequence $\Pi$ = ${\left({\pi }_h^n\right)}_{n=1}^{\infty }$ of (88).

Lemma 3.

Let $Q_{p_l,j_l}^{a_l}$ ∈ $\mathsf{\Omega }\left({\mathcal{U}}_A\right)$ be generators of the A-tensor sub-filterization ${\mathbb{LS}}_A,$ for l = $1,$ …, $N,$ for N ∈ $\mathbb{N}$. In addition, let Π be the sequence (88) acting on ${\mathbb{LS}}_A$. If π is in the sense of N 2, then

$$\begin{gathered} \pi \left(Q_{p_1,j_1}^{a_1}\right)={\displaystyle \underset{n\to \infty }{lim}}\left(Q_{\left(h^{\left(n\right)}\left(p_1\right)\right),j_1}^{a_1}\right), \\ \pi \left(\stackrel{N}{{\displaystyle \underset{l=1}{\Pi }}}{\left(Q_{p_l,j_l}^{a_l}\right)}^{n_l}\right)={\displaystyle \underset{n\to \infty }{lim}}\left(\stackrel{N}{{\displaystyle \underset{l=1}{\Pi }}}{\left(Q_{h^{\left(n\right)}\left(p_l\right),j_l}^{a_l}\right)}^{n_l}\right), \end{gathered}$$

(93)

for all $n_1,$ …, $n_N$ ∈ $\mathbb{N},$ where $h^{\left(n\right)}$ are the n-shifts (80) on $\mathcal{P}.$

Proof. 

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

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

$$\begin{gathered} \pi \left(Q_{p_1,j_1}^{a_1}\right)={\displaystyle \underset{p_1\to \infty }{lim}}{Q}_{p_1,j_1}^{a_1}, \\ \pi \left(\stackrel{N}{{\displaystyle \underset{l=1}{\Pi }}}{Q}_{p_l,j_l}^{n_l}\right)=\stackrel{N}{{\displaystyle \underset{l=1}{\Pi }}}\left({\displaystyle \underset{p_l\to \infty }{lim}}\left(Q_{p_l,j_l}^{n_l}\right)\right), \end{gathered}$$

(94a)

respectively, where “${\displaystyle \underset{q\to \infty }{lim}}$” for q ∈ $\mathcal{P}$ is in the sense of (44).

Such an understanding (94a) of the formula (93) is meaningful by the constructions (80) of n-shifts $h^{\left(n\right)}$ on $\mathcal{P}$. For example,

$${\displaystyle \underset{n\to \infty }{lim}}{h}^{\left(n\right)}\left(q\right)={\displaystyle \underset{p\to \infty }{lim}}p,forq\in \mathcal{P},$$

(94b)

where the right-hand side of (94b) means that: starting with $q,$ take bigger primes again and again in the TOset $\mathcal{P}$ 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 ${\tau }_0$ on ${\mathbb{LS}}_A$ with respect to the linear functional $\tau$ of (74a), by

$${\tau }_0\stackrel{def}{=}\tau \circ \pi on{\mathbb{LS}}_A,$$

(95)

where $\pi$ is in the sense of N 2.

Theorem 5.

Let ${\mathbb{LS}}_A$ = $({\mathbb{LS}}_A,$ $\tau )$ be the A-tensor sub-filterization (74b), and let ${\tau }_0$ = τ ∘ π be the new linear functional (95) on the Banach ∗-algebra ${\mathbb{LS}}_A$ of (74a). Then, for the generators

$${\left\{Q_{p,j}^a\right\}}_{p\in \mathcal{P}}\subset \mathsf{\Omega }\left({\mathcal{U}}_A\right)of{\mathbb{LS}}_A,$$

for an arbitrarily fixed a ∈ $(A,$ $\psi )$ and j ∈ $\mathbb{Z},$ we have that

$${\tau }_0\left({\left(Q_{p,j}^a\right)}^n\right)=\left\{\begin{array}{cc}0,\hfill & \mathrm{if}j>0,\hfill \\ {\omega }_n{c}_{\frac{n}{2}}\psi \left(a^n\right),\hfill & \mathrm{if}j=0,\hfill \\ \infty ,\mathrm{Undefined},\hfill & \mathrm{if}j<0,\hfill \end{array}\right.$$

(96)

for all n ∈ $\mathbb{N}.$

Proof. 

Let ${\left\{Q_{p,j}^a\right\}}_{p\in \mathcal{P}}$⊂$\mathsf{\Omega }\left({\mathcal{U}}_A\right)$ in ${\mathbb{LS}}_A,$ for fixed a ∈ $(A,$ $\psi )$ and j ∈ $\mathbb{Z}.$ Then,

$${\tau }_0\left({\left(Q_{p,j}^a\right)}^n\right)=\left(\tau \circ \pi \right)\left({\left(Q_{p,j}^a\right)}^n\right)=\tau \left({\displaystyle \underset{p\to \infty }{lim}}{\left(Q_{p,j}^a\right)}^n\right)$$

by (93) and (94a)

$$={\displaystyle \underset{p\to \infty }{lim}}\tau \left({\left(Q_{p,j}^a\right)}^n\right)$$

by the boundedness of $\tau$ for the (norm, or strong) topology for ${\mathbb{LS}}_A$

$$={\displaystyle \underset{p\to \infty }{lim}}{\tau }_j^p\left({\left(Q_{p,j}^a\right)}^n\right)={\displaystyle \underset{p\to \infty }{lim}}\left({\omega }_n{c}_{\frac{n}{2}}\psi \left(a^n\right)\left(\frac{\varphi \left(p\right)}{p^{j+1}}\right)\right)$$

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

$$=\left({\omega }_n{c}_{\frac{n}{2}}\psi \left(a^n\right)\right)\left({\displaystyle \underset{p\to \infty }{lim}}\frac{\varphi \left(p\right)}{p^{j+1}}\right)$$

$$=\left\{\begin{array}{cc}0,\hfill & \mathrm{if}j>0,\hfill \\ {\omega }_n{c}_{\frac{n}{2}}\psi \left(a^n\right),\hfill & \mathrm{if}j=0,\hfill \\ \infty ,\mathrm{Undefined},\hfill & \mathrm{if}j<0,\hfill \end{array}\right.$$

by (90), for each n ∈ $\mathbb{N}.$ Therefore, the free-distributional data (96) holds for ${\tau }_0$. ☐

By (96), we obtain the following corollary.

Corollary 2.

Let $Q_{p,0}^{1_A}$ ∈ $\mathsf{\Omega }\left({\mathcal{U}}_A\right)$ be free random variables of the A-tensor sub-filterization ${\mathbb{LS}}_A,$ for all p ∈ $\mathcal{P},$ where $1_A$ is the unity of $(A,$ $\psi ).$ Then, the asymptotic free distribution of the family

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

follows the semicircular law asymptotically as p→∞ in $\mathcal{P}.$

Proof. 

Let ${\mathcal{Q}}_0^{1_A}$ = ${\left\{Q_{p,0}^{1_A}\right\}}_{p\in \mathcal{P}}$⊂$\mathsf{\Omega }\left({\mathcal{U}}_A\right)$ in ${\mathbb{LS}}_A.$ Then, for the linear functional ${\tau }_0$ of (95) on ${\mathbb{LS}}_A,$

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

for all n ∈ $\mathbb{N}$, by (96), since

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

If p→ ∞ in $\mathcal{P},$ then the asymptotic free distribution of the family ${\mathcal{Q}}_0^{1_A}$ is the semicircular law by the self-adjointness of all $Q_{p,0}^{1_A}$’s, and by the semicircularity (45) and (47). ☐

Independent from (96), we obtain the following asymptotic free-distributional data on ${\mathbb{LS}}_A$.

Theorem 6.

Let $j_1,$ …, $j_N$ be “mutually distinct” in $\mathbb{Z},$ for N > 1 in $\mathbb{N},$ and hence the N-tuple

$$\left[j\right]=\left(j_1,...,j_N\right)\in {\mathbb{Z}}^N$$

is alternating in $\mathbb{Z}.$ In addition, let

$$\left[a\right]=(a_1,...,a_N)$$

be an arbitrarily fixed N-tuple of free random variables $a_1,$ …, $a_N$ of the unital $C^{*}$-probability space $(A,$ $\psi ),$ and let’s fix

$$\left[n\right]=(n_1,...,n_N)\in {\mathbb{N}}^N.$$

Now, define a family ${\mathcal{T}}_{\left[j\right]}^{\left[a\right],\left[n\right]}$ of free reduced words with their lengths-N,

$${\mathcal{T}}_{\left[j\right]}^{\left[a\right],\left[n\right]}=\left\{T=\stackrel{N}{{\displaystyle \underset{l=1}{\Pi }}}{\left(Q_{p_l,j_l}^{a_l}\right)}^{n_l}:p_1,...,p_N\in \mathcal{P}\right\},$$

(97)

in ${\mathbb{LS}}_A,$ for $Q_{p_l,j_l}^{a_l}$ ∈ $\mathsf{\Omega }\left({\mathcal{U}}_A\right),$ for all $p_l$ ∈ $\mathcal{P},$ where $a_l$ ∈ $\left[a\right],$ $j_l$ ∈ $\left[j\right],$ for l = $1,$ …, $N.$

For any free reduced words T ∈ ${\mathcal{T}}_{\left[j\right]}^{\left[a\right],\left[n\right]},$ if ${\tau }_0$ is the linear functional (95) on ${\mathbb{LS}}_A,$ then

$${\tau }_0\left(T\right)=\left\{\begin{array}{cc}0,\hfill & \mathrm{if}\sum _{l=1}^N{j}_l>1-N,\hfill \\ \stackrel{N}{{\displaystyle \underset{l=1}{\Pi }}}\left({\omega }_{n_l}{c}_{\frac{n_l}{2}}\psi \left(a^{n_l}\right)\right),\hfill & \mathrm{if}\sum _{l=1}^N{j}_l=1-N,\hfill \\ \infty ,\mathrm{Undefined},\hfill & \mathrm{if}\sum _{l=1}^N{j}_l<1-N,\hfill \end{array}\right.$$

(98)

for all n ∈ $\mathbb{N}.$

Proof. 

Let T ∈ ${\mathcal{T}}_{\left[j\right]}^{\left[a\right],\left[n\right]}$ be in the sense of (97) in the A-tensor sub-filterization ${\mathbb{LS}}_A.$ Then, these operators T form free reduced words with their lengths-N in ${\mathbb{LS}}_A,$ since $\left[j\right]$ is an alternating N-tuple of “mutually distinct” integers. Observe that

$${\tau }_0\left(T\right)=\tau \left(\pi \left(T\right)\right)=\tau \left(\stackrel{N}{{\displaystyle \underset{l=1}{\Pi }}}\left({\displaystyle \underset{p_l\to \infty }{lim}}{\left(Q_{p_l,j_l}^{a_l}\right)}^{n_l}\right)\right)$$

by (93) and (94a)

$$=\tau \left(\stackrel{N}{{\displaystyle \underset{l=1}{\Pi }}}\left({\displaystyle \underset{p\to \infty }{lim}}{\left(Q_{p,j_l}^{a_l}\right)}^{n_l}\right)\right)$$

because

$${\displaystyle \underset{p\to \infty }{lim}}p={\displaystyle \underset{n\to \infty }{lim}}{h}^{\left(n\right)}\left(p_l\right)={\displaystyle \underset{p_l\to \infty }{lim}}{p}_l,in\mathcal{P},$$

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

$$={\displaystyle \underset{p\to \infty }{lim}}\left(\tau \left({\left(\stackrel{N}{{\displaystyle \underset{l=1}{\Pi }}}{Q}_{p,j_l}^{a_l}\right)}^{n_l}\right)\right)$$

by the boundedness of $\tau$ for the (norm, or strong) topology for ${\mathbb{LS}}_A$

$$={\displaystyle \underset{p\to \infty }{lim}}\left(\stackrel{N}{{\displaystyle \underset{l=1}{\Pi }}}\left({\omega }_{n_l}{c}_{\frac{n_l}{2}}\psi \left(a_l^{n_l}\right)\left(\frac{\varphi \left(p\right)}{p^{j_l+1}}\right)\right)\right)$$

since $\left[j\right]$ consists of “mutually-distinct” integers, by the Möbius inversion

$$=\left(\stackrel{N}{{\displaystyle \underset{l=1}{\Pi }}}{\omega }_{n_l}{c}_{\frac{n_l}{2}}\psi \left(a_l^{n_l}\right)\right)\left({\displaystyle \underset{p\to \infty }{lim}}\left(\stackrel{N}{{\displaystyle \underset{l=1}{\Pi }}}\left(\frac{\varphi \left(p\right)}{p^{j_l+1}}\right)\right)\right)$$

$$=\left(\stackrel{N}{{\displaystyle \underset{l=1}{\Pi }}}{\omega }_{n_l}{c}_{\frac{n_l}{2}}\psi \left(a_l^{n_l}\right)\right)\left({\displaystyle \underset{p\to \infty }{lim}}\left(\frac{\varphi \left(p\right)}{p^{N+{\Sigma }_{l=1}^N{j}_l}}\right)\right)$$

$$=\left(\stackrel{N}{{\displaystyle \underset{l=1}{\Pi }}}{\omega }_{n_l}{c}_{\frac{n_l}{2}}\psi \left(a_l^{n_l}\right)\right)\left({\displaystyle \underset{p\to \infty }{lim}}\left(\frac{\varphi \left(p\right)}{p^{\left(N-1+{\Sigma }_{l=1}^N{j}_l\right)+1}}\right)\right)$$

$$=\left(\stackrel{N}{{\displaystyle \underset{l=1}{\Pi }}}{\omega }_{n_l}{c}_{\frac{n_l}{2}}\psi \left(a_l^{n_l}\right)\right)\left({\displaystyle \underset{p\to \infty }{lim}}\left(\frac{\varphi \left(p\right)}{p^{\left(N-1+{\Sigma }_{l=1}^N{j}_l\right)+1}}\right)\right)$$

$$=\left\{\begin{array}{cc}0\hfill & \mathrm{if}N-1+\sum _{l=1}^N{j}_l>0\hfill \\ \stackrel{N}{{\displaystyle \underset{l=1}{\Pi }}}\left({\omega }_{n_l}{c}_{\frac{n_l}{2}}\psi \left(a_l^{n_l}\right)\right)\hfill & \mathrm{if}N-1+\sum _{l=1}^N{j}_l=0\hfill \\ \infty \hfill & \mathrm{if}N-1+\sum _{l=1}^N{j}_l<0,\hfill \end{array}\right.$$

by (90), for all n ∈ $\mathbb{N}.$ Therefore, the family ${\mathcal{T}}_{\left[j\right]}^{\left[a\right],\left[n\right]}$ of (97) satisfies the asymptotic free-distributional data (98) in the A-tensor sub-filterization ${\mathbb{LS}}_A$ over $\mathcal{P}.$ ☐

The above two theorems illustrate the asymptotic free-probabilistic behaviors on the A-tensor sub-filterization ${\mathbb{LS}}_A$ over $\mathcal{P},$ by (96) and (98).

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

$${\mathcal{Q}}_0^{1_A}={\left\{Q_{p,0}^{1_A}\right\}}_{p\in \mathcal{P}}\subset {\mathbb{LS}}_A$$

has its asymptotic free distribution, the semicircular law in ${\mathbb{LS}}_A,$ 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 $(A,$ $\psi ).$ Assume that it satisfies

(i) 

$\psi \left(a\right)$ ∈ ${\mathbb{R}}^{\times }$ = $\mathbb{R}$∖$\left\{0\right\}$ in $\mathbb{C},$

(ii) 

$\psi \left(a^{2n}\right)$ = $\psi {\left(a\right)}^{2n},$ for all n ∈ $\mathbb{N}$.

Then, the family

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

(99)

follows the asymptotic semicircular law, in ${\mathbb{LS}}_A$ over $\mathcal{P}.$

Proof. 

Let a ∈ $(A,$ $\psi )$ be a self-adjoint free random variable satisfying two conditions (i) and (ii), and let ${\mathcal{X}}_0^a$ be the family (99) of the A-tensor sub-filterization ${\mathbb{LS}}_A.$ Then, all elements

$$X_{p,0}^a=\frac{1}{\psi \left(a\right)}{Q}_{p,0}^a={\mathbf{l}}_p^A\otimes \left(\left(\frac{1}{\psi \left(a\right)}a\right)\otimes P_{p,0}\right)of{\mathcal{X}}_0^a$$

are self-adjoint in ${\mathbb{LS}}_A,$ by the self-adjointness of $Q_{p,0}^a,$ and by the condition (i).

For any $X_{p,0}^a$ ∈ ${\mathcal{X}}_0^a,$ observe that

$$\begin{array}{cc}{\tau }_0\left({\left(X_{p,0}^a\right)}^n\right)& =\frac{1}{\psi {\left(a\right)}^n}{\tau }_0\left({\left(Q_{p,0}^a\right)}^n\right)\hfill \\ & =\frac{1}{\psi {\left(a\right)}^n}\left({\omega }_n{c}_{\frac{n}{2}}\psi \left(a^n\right)\right)\hfill \end{array}$$

by (96)

$$=\left({\omega }_n{c}_{\frac{n}{2}}\left(\frac{\psi \left(a^n\right)}{\psi \left(a^n\right)}\right)\right)$$

by the condition (ii)

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

for all n ∈ $\mathbb{N}.$ Therefore, the family ${\mathcal{X}}_0^a$ has its asymptotic semicircular law over $\mathcal{P},$ by (45). ☐

Similar to the construction of ${\mathcal{X}}_0^a$ of (99), if we construct the families ${\mathcal{X}}_j^a,$

$${\mathcal{X}}_j^a={\left\{\frac{1}{\psi \left(a\right)}{Q}_{p,j}^a:Q_{p,j}^a\in \mathsf{\Omega }\left({\mathcal{U}}_A\right)\right\}}_{p\in \mathcal{P}},$$

(100)

for a fixed a ∈ $(A,$ $\psi )$ satisfying the conditions (i) and (ii) of the above theorem, and, for a fixed j ∈ $\mathbb{Z},$ then one obtains the following corollary.

Corollary 3.

Fix a ∈ $(A,$ $\psi )$ satisfying the conditions (i) and (ii) of the above theorem. Let’s fix j ∈ $\mathbb{Z},$ and let ${\mathcal{X}}_j^a$ be the corresponding family (100) in the A-tensor sub-filterization ${\mathbb{LS}}_A$ = $\left({\mathbb{LS}}_A,\tau \right).$

$$Ifj=0,then{\mathcal{X}}_0^{a}hastheasymptoticsemicircularlawin{\mathbb{LS}}_A.$$

(101)

$$Ifj>0,then{\mathcal{X}}_j^{a}hasitsasymptoticfreedistribution,thezerofreedistribution,in{\mathbb{LS}}_A.$$

(102)

$$Ifj<0,thentheasymptoticfreedistributionof{\mathcal{X}}_j^{a}isundefinedin{\mathbb{LS}}_A.$$

(103)

Proof. 

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

By (96), if j > $0,$ then, for any T = $\frac{1}{\psi \left(a\right)}{Q}_{p,j}^a$ ∈ ${\mathcal{X}}_j^a,$ one has that

$${\tau }_0\left(T^n\right)=\frac{1}{\psi \left(a^n\right)}{\tau }_0\left({\left(Q_{p,j}^a\right)}^n\right)=0,$$

for all n ∈ $\mathbb{N}.$ Thus, the asymptotic free distribution of ${\mathcal{X}}_j^a$ is the zero free distribution in ${\mathbb{LS}}_A,$ as p → ∞ in $\mathcal{P}.$ Thus, the statement (102) holds.

Similarly, by (96), if j<$0,$ then the asymptotic free distribution ${\mathcal{X}}_j^a$ is undefined in ${\mathbb{LS}}_A$ over $\mathcal{P},$ equivalently, the statement (103) is shown. ☐

Motivated by (101), (102) and (103), we study the asymptotic semicircular law (over $\mathcal{P}$) on ${\mathbb{LS}}_A$ more in detail in Section 10 below.

10. Asymptotic Semicircular Laws on ${\mathbb{LS}}_{\mathit{A}}$ over $\mathcal{P}$

We here consider asymptotic semicircular laws on the A-tensor sub-filterization ${\mathbb{LS}}_A$ = $({\mathbb{LS}}_A,$ $\tau )$. In Section 9.3, we showed that the asymptotic free distribution of a family

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

(104)

is the semicircular law in ${\mathbb{LS}}_A$ as p→ ∞ in $\mathcal{P},$ for a fixed self-adjoint free random variable a ∈ $(A,$ $\psi )$ satisfying

(i)

$\psi \left(a\right)$ ∈ ${\mathbb{R}}^{\times },$ and

(ii)

$\psi \left(a^{2n}\right)$ = $\psi {\left(a\right)}^{2n},$ for all n ∈ $\mathbb{N}$.

As an example, the family

$${\mathcal{X}}_0^{1_A}=\{Q_{p,0}^{1_A}:p\in \mathcal{P}\}$$

(105)

follows the asymptotic semicircular law in ${\mathbb{LS}}_A$ over $\mathcal{P}.$

We now enlarge such asymptotic behaviors on ${\mathbb{LS}}_A$ up to certain ∗-isomorphisms.

Define bijective functions $g_{+}$ and $g_{-}$ on $\mathbb{Z}$ by

$$g_{+}\left(j\right)=j+1,and{g}_{-}\left(j\right)=j-1,$$

(106)

for all j ∈ $\mathbb{Z}.$

By (106), one can define bijective functions $g_{\pm }^{\left(n\right)}$ on $\mathbb{Z}$ by

$$g_{\pm }^{\left(n\right)}\stackrel{def}{=}\underset{n-\mathrm{times}}{\underbrace{g_{\pm }\circ g_{\pm }\circ g_{\pm }\circ ···\circ g_{\pm }}},$$

(107)

satisfying $g_{\pm }^{\left(1\right)}$ = $g_{\pm }$ on $\mathbb{Z}$, with axiomatization:

$$g_{\pm }^{\left(0\right)}=i{d}_{\mathbb{Z}},\mathrm{the}\mathrm{identity}\mathrm{function}\mathrm{on}\mathbb{Z},$$

for all n ∈ ${\mathbb{N}}_0$ = $\mathbb{N}$∪$\left\{0\right\}$. For example,

$$g_{\pm }^{\left(n\right)}\left(j\right)=j\pm n,$$

(108)

for all j ∈ $\mathbb{Z},$ for all n ∈ ${\mathbb{N}}_0.$

From the bijective functions $g_{\pm }^{\left(n\right)}$ of (107), define the bijective functions ${\left(g_{\pm }^o\right)}^{\left(n\right)}$ on the generator set $\mathsf{\Omega }\left({\mathcal{U}}_A\right)$ of (72) of the A-tensor sub-filterization ${\mathbb{LS}}_A$ by

$$\begin{array}{c}{\left(g_{+}^o\right)}^{\left(n\right)}\left(Q_{p,j}^a\right)=Q_{p,g_{+}^{\left(n\right)}\left(j\right)}^a=Q_{p,j+n}^a,\\ {\left(g_{-}^o\right)}^{\left(n\right)}\left(Q_{p,j}^a\right)=Q_{p,g_{-}^{\left(n\right)}\left(j\right)}^a=Q_{p,j-n}^a,\end{array}$$

(109)

with

$${\left(g_{\pm }^o\right)}^{\left(1\right)}=g_{\pm }^o,and{\left(g_{\pm }^o\right)}^{\left(0\right)}=id,$$

by (108), for all p ∈ $\mathcal{P}$ and j ∈ $\mathbb{Z},$ for all n ∈ ${\mathbb{N}}_0,$ where $id$ is the identity function on $\mathsf{\Omega }\left({\mathcal{U}}_A\right).$

By the construction (73a) of the generator set $\mathsf{\Omega }\left({\mathcal{U}}_A\right)$ of ${\mathbb{LS}}_A$ under (73b),

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

the functions ${\left(g_{\pm }^o\right)}^{\left(n\right)}$ of (109) are indeed well-defined bijections on $\mathsf{\Omega }\left({\mathcal{U}}_A\right),$ by the bijectivity of $g_{\pm }^{\left(n\right)}$ of (107).

Now, define bounded ∗-homomorphisms $G_{\pm }$ on ${\mathbb{LS}}_A$ by the bounded multiplicative linear transformations on ${\mathbb{LS}}_A$ satisfying that:

$$\begin{gathered} G_{+}\left(Q_{p,j}^a\right)=g_{+}^o\left(Q_{p,j}^a\right)=Q_{p,j+1}^a, \\ {G}_{-}\left(Q_{p,j}^a\right)=g_{-}^o\left(Q_{p,j}^a\right)=Q_{p,j-1}^a, \end{gathered}$$

(110)

in ${\mathbb{LS}}_A,$ by using the bijections $g_{\pm }^o$ of (109), for all $Q_{p,j}^a$ ∈ $\mathsf{\Omega }\left({\mathcal{U}}_A\right).$

More precisely, the morphisms $G_{\pm }$ of (110) satisfy that

$$\begin{array}{cc}{G}_{\pm }\left(\stackrel{N}{{\displaystyle \underset{l=1}{\Pi }}}{\left(Q_{p_l,j_l}^{a_l}\right)}^{n_l}\right)\hfill & =\stackrel{N}{{\displaystyle \underset{l=1}{\Pi }}}{g}_{\pm }^o\left({\left(Q_{p_l,j_l}^{a_l}\right)}^{n_l}\right)\hfill \\ & =\stackrel{N}{{\displaystyle \underset{l=1}{\Pi }}}{\left(Q_{p_l,j_l\pm 1}^{a_l}\right)}^{n_l}.\hfill \end{array}$$

(111a)

By (111a), one can get that

$$\begin{array}{cc}{G}_{\pm }\left({\left(\stackrel{N}{{\displaystyle \underset{l=1}{\Pi }}}{\left(Q_{p_l,j_l}^{a_l}\right)}^{n_l}\right)}^{*}\right)\hfill & =G_{\pm }\left(\stackrel{N}{{\displaystyle \underset{l=1}{\Pi }}}{\left(Q_{p_{N-l+1},j_{N-l+1}}^{a_{N-l+1}^{*}}\right)}^{n_{N-l+1}}\right)\hfill \\ & =\stackrel{N}{{\displaystyle \underset{l=1}{\Pi }}}{\left({\left(Q_{p_{N-l+1},\left(j_{N-l+1}\right)\pm 1}^{a_{N-l+1}}\right)}^{n_{N-l+1}}\right)}^{*}\hfill \\ & ={\left(\stackrel{N}{{\displaystyle \underset{l=1}{\Pi }}}{\left(Q_{p_l,j_l\pm 1}^{a_l}\right)}^{n_l}\right)}^{*}\hfill \\ & ={\left(G_{\pm }\left(\stackrel{N}{{\displaystyle \underset{l=1}{\Pi }}}{Q}_{p_l,j_l}^{n_l}\right)\right)}^{*}\hfill \end{array}$$

(111b)

for all $Q_{p_l,j_l}^{a_l}$ ∈ $\mathsf{\Omega }\left({\mathcal{U}}_A\right),$ for l = $1,$ …, $N,$ for N ∈ $\mathbb{N}.$

The formula (111a) are obtained by (110) and the multiplicativity of $G_{\pm }.$ The formulas in (111b), obtained from (111a), show that indeed $G_{\pm }$ are ∗-homomorphisms on ${\mathbb{LS}}_A,$ since

$$G_{\pm }\left(T^{*}\right)={\left(G_{\pm }\left(T\right)\right)}^{*},\forall T\in {\mathbb{LS}}_A.$$

By (110) and (111a),

$$\begin{array}{c}{G}_{\pm }^n\left(\stackrel{N}{{\displaystyle \underset{l=1}{\Pi }}}{\left(Q_{p_l,j_l}^{a_l}\right)}^{n_l}\right)=\stackrel{N}{{\displaystyle \underset{l=1}{\Pi }}}{\left(Q_{p_l,j_l\pm n}^{a_l}\right)}^{n_l},\\ G_{\pm }^n\left({\left(\stackrel{N}{{\displaystyle \underset{l=1}{\Pi }}}{\left(Q_{p_l,j_l}^{a_l}\right)}^{n_l}\right)}^{*}\right)={\left(G_{\pm }^n\left(\stackrel{N}{{\displaystyle \underset{l=1}{\Pi }}}{\left(Q_{p_l,j_l}^{a_l}\right)}^{n_l}\right)\right)}^{*},\end{array}$$

(112)

for all $Q_{p_l,j_l}^{a_l}$ ∈ $\mathsf{\Omega }\left({\mathcal{U}}_A\right),$ for l = $1,$ …, $N,$ for N ∈ $\mathbb{N},$ for all n ∈ ${\mathbb{N}}_0$.

Definition 16.

We call the bounded ∗-homomorphisms $G_{\pm }^n$ of (110), the n-(±)-integer-shifts on ${\mathbb{LS}}_A,$ for all n ∈ ${\mathbb{N}}_0.$

Based on the integer-shifting processes on ${\mathbb{LS}}_A,$ one can get the following asymptotic behavior on ${\mathbb{LS}}_A$ over $\mathcal{P}$.

Theorem 8.

Let ${\mathcal{X}}_j^a$ be a family (100) of the A-tensor sub-filterization ${\mathbb{LS}}_A,$ for any j ∈ $\mathbb{Z},$ where a is a fixed self-adjoint free random variable of $(A,$ $\psi )$ satisfying the additional conditions (i) and (ii) above. Then, there exists a $(-j)$-integer-shift $G_{-j}$ on ${\mathbb{LS}}_A,$ such that

$$G_{-j}=\left\{\begin{array}{cc}{G}_{-}^{\left|j\right|}=G_{-}^j\hfill & \mathrm{if}j\ge 0\mathrm{in}\mathbb{Z},\hfill \\ G_{+}^{\left|j\right|}=G_{+}^{-j}\hfill & \mathrm{if}j<0\mathrm{in}\mathbb{Z},\hfill \end{array}\right.$$

(113)

and

$${\tau }_0\left(G_j\left(T\right)\right)={\omega }_n{c}_{\frac{n}{2}},\forall n\in \mathbb{N},$$

(114)

for all T ∈ ${\mathcal{X}}_j^a,$ where $G_{\mp }^{\pm j}$ on the right-hand sides of (113) are the $\left|j\right|$-(∓)-integer shifts (110) on ${\mathbb{LS}}_A,$ and where ${\tau }_0$ = $\tau \circ \pi$ is the linear functional (95) on ${\mathbb{LS}}_A$.

Proof. 

Let ${\mathcal{X}}_j^a$ = $\left\{\frac{1}{\psi \left(a\right)}{Q}_{p,j}^a:p\in \mathcal{P}\right\}$ be a family (100) of ${\mathbb{LS}}_A,$ for a fixed j ∈ $\mathbb{Z},$ where a fixed self-adjoint free random variable a ∈ $(A,$ $\psi )$ satisfies the above additional conditions (i) and (ii).

Assume first that j ≥ 0 in $\mathbb{Z}.$ Then, one can take the $(-j)$-(−)-integer-shift $G_{-}^j$ of (110) on ${\mathbb{LS}}_A,$ satisfying

$$G_{-}^j\left(Q_{p,j}^a\right)=Q_{p,j-j}^a=Q_{p,0}^{a}in{\mathbb{LS}}_A,$$

for all $Q_{p,j}^a$ ∈ $\mathsf{\Omega }\left({\mathcal{U}}_A\right).$

Second, if j < 0 in $\mathbb{Z},$ then one can have the $\left|j\right|$-(+)-integer shift $G_{+}^{-j}$ of (110) on ${\mathbb{LS}}_A,$ satisfying that

$$G_{+}^{-j}\left(Q_{p,j}^a\right)=Q_{p,j+(-j)}^a=Q_{p,0}^{a}in{\mathbb{LS}}_A,$$

for all $Q_{p,j}^a$ ∈ $\mathsf{\Omega }\left({\mathcal{U}}_A\right).$

For example, for any $Q_{p,j}^a$ ∈ $\mathsf{\Omega }\left({\mathcal{U}}_A\right),$ we have the corresponding $(-j)$-integer-shift $G_{-j},$

$$G_{-j}=\left\{\begin{array}{cc}{G}_{-}^j\hfill & \mathrm{if}j\ge 0,\hfill \\ G_{+}^{-j}\hfill & \mathrm{if}j<0,\hfill \end{array}\right.$$

on ${\mathbb{LS}}_A$ in the sense of (113), such that

$$G_{-j}\left(Q_{p,j}^a\right)=Q_{p,0}^{a}in{\mathbb{LS}}_A,$$

for all p ∈ $\mathcal{P}.$

Then, for any $X_{p,j}^a$ = $\frac{1}{\psi \left(a\right)}{Q}_{p,j}^a$ ∈ ${\mathcal{X}}_j^a,$ we have that

$${\tau }_0\left(G_{-j}\left({\left(X_{p,j}^a\right)}^n\right)\right)={\tau }_0\left(\frac{1}{\psi {\left(a\right)}^n}{\left(G_{-j}\left(Q_{p,j}^a\right)\right)}^n\right),$$

since $G_{-j}$ is a ∗-homomorphism (113) on ${\mathbb{LS}}_A$

$$={\tau }_0\left(\frac{1}{\psi \left(a^n\right)}{\left(Q_{p,0}^a\right)}^n\right)={\omega }_n{c}_{\frac{n}{2}},$$

by (96) and (98), for all n ∈ $\mathbb{N}.$ Therefore, formula (114) holds true. ☐

By the above theorem, we obtain the following result.

Corollary 4.

Let ${\mathcal{X}}_j^a$ be a family (100) of the A-tensor sub-filterization ${\mathbb{LS}}_A,$ for j ∈ $\mathbb{Z},$ where a self-adjoint free random variable a ∈ $(A,$ $\psi )$ satisfies the conditions (i) and (ii). Then, the corresponding family

$${\mathcal{G}}_j^a=\left\{G_{-j}\left(X\right):X\in {\mathcal{X}}_j^a\right\}$$

(115)

has its asymptotic free distribution, the semicircular law, in ${\mathbb{LS}}_A$ over $\mathcal{P},$ where $G_{-j}$ is the ($-j$)-integer shift (113) on ${\mathbb{LS}}_A,$ for all j ∈ $\mathbb{Z}.$

Proof. 

The asymptotic semicircular law induced by the family ${\mathcal{G}}_j^a$ of (115) in ${\mathbb{LS}}_A$ is guaranteed by (114) and (45), for all j ∈ $\mathbb{Z}.$ ☐

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

Corollary 5.

Let ${\mathcal{X}}_j^{1_A}$ be in the sense of (100) in ${\mathbb{LS}}_A,$ where $1_A$ is the unity of $(A,$ $\psi ),$ and let

$${\mathcal{G}}_j^{1_A}=\left\{G_{-j}\left(X\right):X\in {\mathcal{X}}_j^{1_A}\right\}$$

be in the sense of (115), for all j ∈ $\mathbb{Z}.$ Then, the asymptotic free distributions of ${\mathcal{G}}_j^{1_A}$ are the semicircular law in ${\mathbb{LS}}_A$ over $\mathcal{P},$ for all j ∈ $\mathbb{Z}.$

Proof. 

The proof is done by Corollary 4. Indeed, the unity $1_A$ automatically satisfies the conditions (i) and (ii) in $(A,$ $\psi ).$ ☐

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

Theorem 9.

Let a ∈ $(A,$ $\psi )$ be a self-adjoint free random variable satisfying the conditions (i) and (ii), and let $p_0$ ∈ $\mathcal{P}$ be an arbitrarily fixed prime. Let

$${\mathcal{G}}_j^a[\ge p_0]\stackrel{def}{=}\left\{G_{-j}\left(X_{p,j}\right)\left|\begin{array}{c}{X}_{p,j}^a\in {\mathcal{X}}_j^a\mathrm{and}\\ p\ge p_0\mathrm{in}\mathcal{P}\end{array}\right.\right\},$$

where ${\mathcal{X}}_j^a$ is the family (100), and ${\mathcal{G}}_j^a$ is the family (115), for j ∈ $\mathbb{Z}.$ Then, the asymptotic free distribution of the family ${\mathcal{G}}_j^a[\ge p_0]$ is the semicircular law in ${\mathbb{LS}}_A.$

Proof. 

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

$$“p\to {\infty }^{”}\equiv “{\displaystyle \underset{n\to \infty }{lim}}{h}^n\left(2\right);2\in \mathcal{P}{,}^{”}$$

in the proof of Theorem 8 to

$$“p\to {\infty }^{”}\equiv “{\displaystyle \underset{n\to \infty }{lim}}{h}^n\left(p_0\right);p_0\in \mathcal{P}{,}^{”}$$

where (≡) means “being symbolically same”. ☐