Banach-Space Operators Acting on Semicircular Elements Induced by p -Adic Number Fields over Primes p

: In this paper, we study certain Banach-space operators acting on the Banach *-probability space ( LS , τ 0 ) generated by semicircular elements Θ p , j induced by p -adic number ﬁelds Q p over the set P of all primes p . Our main results characterize the operator-theoretic properties of such operators, and then study how ( LS , τ 0 ).


Introduction
The main purposes of this paper are to study certain Banach-space operators acting on the Banach * -algebra LS generated by mutually free, infinitely-many semicircular elements induced by measurable functions on the p-adic number fields Q p , for primes p in the set P of all primes. By regarding the Banach * -algebra LS as a Banach space, we construct-and-consider certain Banach-space operators acting on LS.
In particular, we are interested in the case where these operators are generated by certain * -homomorphisms in the homomorphism semigroup Hom(LS), induced by shifting processes on the Cartesian product set P = P × Z, where Z is the set of all integers. Note that our shifting processes here are well-defined by understanding the sets P and Z as totally ordered sets (in short, TOsets) under the usual inequality (≤). Our main results categorize such Banach-space operators as in the usual Hilbert-space operator (spectral) theory. Artificially, but naturally, we study self-adjointness, projection-property, normality, isometry-property, and unitarity of these operators acting on the semicircular law (see Section 13 below). In addition, they show that some of such Banach-space operators preserves the free probability on LS, and hence the semicircular law (which is the free distributions of the free generators of LS) is preserved by the action of the operators; meanwhile, some of such operators distort the semicircular law, whose distortions are characterizable (see Section 14 below).

Preview and Motivation
Connections between primes and operators have been considered in different approaches (e.g., [1][2][3][4][5][6][7]). For instance, we consider relations between analysis on Q p , and (weighted-) semicircular elements, in [8][9][10]. In addition, the main results of them are globalized in [3], i.e., connections between LS, as an algebraic sub-structure of the homomorphism semigroup Hom(LS) of LS (see Sections 11 and 12). Finally, we study Banach-space operators generated by σ(LS), contained in the operator space B(LS) (in the sense of [2]), and investigate operator-theoretic properties of them (see Section 13), and then consider how such operators deform the original free-distributional data on LS (see Section 14).

Preliminaries
In this section, we briefly mention about backgrounds of our proceeding works.
In particular, we will use combinatorial free probability of Speicher (e.g., [11][12][13]). In text, without introducing detailed definitions and combinatorial backgrounds, free moments and free cumulants of operators will be computed. In addition, we use free product of * -probability spaces, without precise introduction.

Analysis on Q p
For more about p-adic number fields, and corresponding analyses, see [24]. We will use the same definitions, terminology, and notations of [24]. Let Q p be the p-adic number fields for p ∈ P. Recall that Q p are the maximal p-norm-topology closures in the normed space (Q, |.| p ) of all rational numbers, where |.| p are the non-Archimedean norms, called p-norms on Q, for all p ∈ P.
For any fixed p ∈ P, the Banach space Q p forms a field algebraically under the p-adic addition and the p-adic multiplication of [24], i.e., Q p is a Banach field.
In addition, such a Banach field Q p is understood as a measure space equipped with the left-and-right additive-invariant Haar measure µ p on the σ-algebra σ Q p , satisfying that where Z p is the unit disk of Q p , consisting of all p-adic integers of Q p , for all p ∈ P (e.g., [24,25]). As a topological space, the p-adic number field Q p contains its basis elements satisfying µ p (U k ) = 1 p k , for all k ∈ Z. (e.g., [24]). By regarding Q p as a measure space, one can establish a * -algebra M p over C as a * -algebra, where the sum ∑ is the finite sum, and χ S are the usual characteristic functions of S. On M p , one can naturally define a linear functional ϕ p by the integral, i.e., Remark 1. By (2), this linear functional ϕ p is unbounded on M p . Indeed, the algebra M p contains its unity (or the multiplication-identity, or the unit) χ Q p , satisfying that Define now subsets ∂ k of Q p by where U k are the basis elements (1) of Q p . We call these µ p -measurable subsets ∂ k of (3), the k-th boundaries (of U k ), for all k ∈ Z. By the basis property of the subsets {U k } k∈Z , one obtains that where means the disjoint union. In addition, one has for all k ∈ Z. Note that, by (4), if S ∈ σ(Q p ), then there exists a subset Λ S of Z, such that Λ S = {j ∈ Z : S ∩ ∂ j = ∅}. (6) Proposition 1. Let S ∈ σ(Q p ), and let χ S ∈ M p . Then, there exists r j ∈ R, such that where Λ S is in the sense of (6).

Free-Probabilistic Models on M p
Throughout this section, fix a prime p ∈ P, and the corresponding p-adic number field Q p , and let M p be the * -algebra induced by Q p . In this section, let us establish a suitable free-probabilistic model on M p , implying the number-theoretic data.
Let U k = p k Z p be the basis elements (1), and ∂ k , their boundaries (3) of Q p , i.e., Recall the linear functional ϕ p of (2) on M p , Then, by (7) and (9), one obtains that ϕ p χ U j = 1 p j , and ϕ p χ ∂ j = 1 since Λ U j = {k ∈ Z : k ≥ j}, and Λ ∂ j = {j}, for all j ∈ Z. Let ∂ k be the k-th boundaries (8) of Q p , for all k ∈ Z. Then, for k 1 , k 2 ∈ Z, one obtains that where δ is the Kronecker delta, and, hence, by Labels (10) and (11). If S 1 , S 2 ∈ σ Q p , then it is not difficult to have where Proposition 2. Let S l ∈ σ(Q p ), and let χ S l ∈ M p , ϕ p , for l = 1, . . . , N, for N ∈ N. Let where Λ S l are in the sense of Label (7), for l = 1, . . . , N. Then, there exists r j ∈ R, such that Proof. The Formula (14) is proven by (7), (12) and (13).

Representations of M p , ϕ p
Fix a prime p ∈ P. Construct the L 2 -Hilbert space, over C, equipped with its inner product <, > 2 , for all f 1 , f 2 ∈ H p , inducing the L 2 -norm, where <, > 2 is the inner product (16) on H p . Remark that the unity χ Q p ∈ M p is not contained in H p , since Definition 1. We call the Hilbert space H p of (15), the p-adic Hilbert space.
By Definition 1 of the p-adic Hilbert space H p , our * -algebra M p acts on H p , via an algebra-action α p , for all f ∈ M p . For example, the morphism α p of (17) is a * -homomorphism from M p to the operator algebra B(H p ) of all operators on H p . Indeed, for any f ∈ M p , the image α p ( f ) is a well-defined bounded multiplication operator on H p with its symbol f , satisfying , for all f ∈ M p . In addition, for convenience, denote α p χ S simply by α p S , for all S ∈ σ Q p .
Note that the unity χ Q p of M p act on H p (under the action α p of (17)) as the identity operator 1 H p of the operator algebra B(H p ), in the sense that: where X . mean the operator-norm closures of subsets X of B(H p ). This C * -algebra M p of (18) is called the p-adic C * -algebra of M p , ϕ p .

Free-Probabilistic Models on M p
Throughout this section, let's fix a prime p ∈ P. Let M p be the p-adic C * -algebra of (18). Define a linear functional ϕ p j : M p → C by a linear morphism, for all j ∈ Z, where <, > 2 is the inner product (16) on the p-adic Hilbert space H p of (15). Note that these linear functionals ϕ p j are bounded on M p because for any arbitrarily fixed j ∈ Z.
Definition 3. Let j ∈ Z, and let ϕ p j be the linear functional (19) on the p-adic C * -algebra M p . Then, the pair M p , ϕ p j is said to be the j-th p-measure space. Now, fix j ∈ Z, and take the j-th p-measure space M p , ϕ p j . For S ∈ σ Q p , and an element α p S ∈ M p , one has that for some 0 ≤ r S ≤ 1 in R.

Proposition 4.
Let ∂ k be the k-th boundaries (8) of Q p , for all k ∈ Z. Then, for all n ∈ N, for k ∈ Z.
Proof. Note that α p ∂ k are projections in M p in the sense that: Thus, the Formula (21) holds by (20), for all n ∈ N, for k ∈ Z.
6. Semigroup C * -Subalgebra S p of M p Let M p be the p-adic C * -algebra for p ∈ P. Take projections induced by boundaries ∂ j of Q p , for all j ∈ Z.
Definition 4. Fix p ∈ P. Let S p be the C * -subalgebra where P p,j are projections (22), for all j ∈ Z. We call this C * -subalgebra S p , the p-adic boundary (C * -)subalgebra of M p .
Every p-adic boundary subalgebra S p satisfies the following structure theorem.
Proposition 5. Let S p be the p-adic boundary subalgebra (23) of M p . Then, in M p .
Proof. It suffices to show that the generating projections {P p,j } j∈Z of S p are mutually orthogonal from each other. For any j 1 , j 2 ∈ Z, in S p . Therefore, the structure theorem (24) holds.

Statistical Data Determined by {P p,j } p∈P, j∈Z
Let M p , ϕ p j be the j-th p-measure space for j ∈ Z, and let S p be the boundary subalgebra (23) of M p , satisfying the structure theorem (24). Throughout this section, fix a prime p. On the pair by (21), where P p,k are the projections (22), generating S p . Now, let φ be the Euler totient function, which is an arithmetic function for all n ∈ N, where |X| mean the cardinalities of sets X, and gcd is the greatest common divisor. By (26), one has Thus, one can get that by (25) and (27), for j ∈ Z. Motivated by (28), define new linear functionals τ p j : S p → C, by linear morphisms, inducing new measure-theoretic structure, where τ p j are in the sense of (29). (30), and let P p,k be the generating projections (22) of S p , for all k ∈ Z. Then, Proof. The Formula (31) is proven by (28) and (29).

Weighted-Semicircular Elements
Let (A, ϕ) be an arbitrary topological * -probability space (C * -probability space, or W * -probability space, or Banach * -probability space, etc.), consisting of a topological * -algebra A (C * -algebra, resp., W * -algebra, resp., Banach * -algebra), and a linear functional ϕ on A. As usual in operator theory, an operator a ∈ (A, ϕ) is said to be self-adjoint, if a = a * in A, where a * is the adjoint of a.

Definition 5.
A self-adjoint operator a ∈ (A, ϕ) is said to be semicircular in (A, ϕ), if ϕ(a n ) = ω n c n 2 , f or all n ∈ N, with ω n = 1, if n is even, 0, if n is odd, for all n ∈ N, where c k are the k-th Catalan numbers, It is well-known that, if k n (. . .) is the free cumulant on A in terms of ϕ (in the sense of [11][12][13]), then a self-adjoint operator a is semicircular in (A, ϕ), if and only if k n   a, a, ......, a n-times for all n ∈ N (e.g., [11,13]). The above characterization (33) of the semicircularity (32) is obtained by the Möbius inversion of [12]. Thus, we use the semicircularity (32) and its characterization (33) alternatively. Motivated by (33), we define the following generalized concept of the semicircularity (32).

Definition 6.
Let a ∈ (A, ϕ) be a self-adjoint operator. It is said to be weighted-semicircular in (A, ϕ) with its weight t 0 (in short, t 0 -semicircular), if there exists t 0 ∈ C × = C \ {0}, such that k n   a, a, ...., a n-times for all n ∈ N, where k n (. . .) is the free cumulant on A in terms of ϕ.
By definition (34), and by the Möbius inversion of [12], one obtains the following free-moment characterization of (34): A self-adjoint operator a is t 0 -semicircular in (A, ϕ), if and only if there exists t 0 ∈ C × , such that ϕ(a n ) = ω n t where ω n and c n 2 are in the sense of (32), for all n ∈ N. (see [8] for details).

Tensor Product Banach * -Algebra LS p
Let S p (k) = S p , τ p k be a pair (30) for p ∈ P, k ∈ Z. Define now bounded linear transformations c p and a p "acting on the C * -algebra S p ," by linear morphisms satisfying, c p P p,j = P p,j+1 , and a p P p,j = P p,j−1 , on S p , for all j ∈ Z. They are well-defined on S p by (24).
By (36), one can understand c p and a p as Banach-space operators contained in the operator space B(S p ), consisting of all bounded linear operators acting on S p , by regarding S p as a Banach space equipped with its C * -norm (e.g., [2]).

Definition 7.
The Banach-space operators c p and a p of (36) are called the p-creation, respectively, the p-annihilation on S p . Define a new Banach-space operator l p by We call this operator l p of (37), the p-radial operator on S p .
Let l p be the p-radial operator (37) in B(S p ). Construct a closed subspace L p of B(S p ) by where Y are the operator-norm closures of subsets Y of the operator space B(S p ). By definition (38), L p is not only a subspace of B(S p ), but also an algebra. In addition, hence, it is a well-defined Banach algebra. On this Banach algebra L p , define a unary operation ( * ) by with axiomatization: where s k ∈ C, with their conjugates s k ∈ C. Then, the operation (39) is a well-defined adjoint on L p (e.g., [2,8,10]). Thus, equipped with the adjoint (39), this Banach algebra L p of (38) forms a Banach * -algebra embedded in B(S p ).
Let L p be the above Banach * -algebra acting on S p . Construct now the tensor product Banach * -algebra LS p by where ⊗ C is the tensor product of Banach * -algebras. Note that operators l n p ⊗ P p,j generate LS p , for all n ∈ N 0 , and j ∈ Z, where P p,j are the projections (22) of S p , by the definition (38) of the tensor factor L p of LS p , and the structure theorem (24) of the other tensor factor S p of LS p .
Define a linear morphism E p : LS p → S p by a linear transformation satisfying that: for all k ∈ N 0 , j ∈ Z, where k 2 is the minimal integer greater than or equal to k 2 , for all k ∈ N 0 . By (24), (38) and (40), this morphism E p of (41) is indeed a well-defined linear transformation. Now, consider how our p-radial operator l p = c p + a p acts on S p . First, observe that c p a p P p,j = P p,j = a p c p P p,j , for all j ∈ Z, p ∈ P. Thus, one has c n 1 p a n 2 p = a n 2 p c by (42), for all n, n 1 , n 2 ∈ N. Thus, we have l n p = c p + a p n = ∑ n k=0 n k c k p a n−k p on S p , with identity: Proposition 7. Let l p ∈ L p be the p-radial operator on S p . Then, (i) l 2m−1 p does not contain 1 S p -term, and (ii) l 2m p contains its 1 S p -term, Proof. The proofs of (i) and (ii) in Proposition 7 are done by straightforward computations of (44) under (43). See [8] for details.

Generating Operators {Q p,j } j∈Z of LS p
Fix p ∈ P, and let LS p be the tensor product Banach * -algebra (40), and let E p : LS p → S p be the linear transformation (41). Throughout this section, let for j ∈ Z, where P p,j are projections (22) generating S p . Observe that Q n p,j = l p ⊗ P p,j n = l n p ⊗ P n p,j = l n p ⊗ P p,j , for all n ∈ N, for all j ∈ Z. Thus, these operators {Q p,j } j∈Z of (45) generate the Banach * -algebra LS p , by (40). Consider now that, if Q p,j ∈ LS p is in the sense of (45) for j ∈ Z, then by (41) and (46), for all n ∈ N. For any fixed j ∈ Z, define a linear functional τ 0 p,j on LS p by where τ p j is a linear functional (29) on S p . The pair LS p , τ 0 p,j forms a new measure-theoretic structure. By (47) and (48), for all n ∈ N. Lemma 1. Fix j ∈ Z, and the pair LS p , τ 0 p,j . Let Q p,k = l p ⊗ P p,k ∈ LS p , τ 0 p,j , for all k ∈ Z. Then, τ 0 p,j Q n p,k = δ j,k ω n p 2(j+1) for all n ∈ N, where ω n are in the sense of (32).

Weighted-Semicircularity on LS
Let LS p and τ 0 p,j be in the sense of (40), respectively, (48). Then, one has the corresponding measure-theoretic pairs, for all p ∈ P, j ∈ Z. Let Q p,k = l p ⊗ P p,k be the generating elements (45) of LS p (j), for p ∈ P, k ∈ Z. Then, by (50), one has for all p ∈ P, j ∈ Z, for all n ∈ N, where ω n are in the sense of (32).
Let LS be the free Adelic filterization (53). Take a subset Q, of LS.
Theorem 1. The operators Q p,j of the family Q of (54) are p 2(j+1) -semicircular in the free Adelic filterization LS, for all p ∈ P, j ∈ Z. More precisely, Proof. Observe first that the operators Q p,j ∈ Q are self-adjoint in LS, since By (54), every operator Q p,j ∈ Q is taken from a free block LS p (j), and, hence, Q n p,j are contained in the same block LS p (j), as free reduced words of LS with their length-1, for all n ∈ N, for all p ∈ P, j ∈ Z. Therefore, by (53), τ 0 Q n p,j = τ 0 p,j Q n p,j = ω n p n(j+1) c n 2 , for all n ∈ N, by (52). In addition, by the Möbius inversion of [12], one has is the free cumulant on LS p (j) in terms of τ 0 p,j , for all p ∈ P, j ∈ Z. Therefore, by (34) and (35), these self-adjoint operators Q p,j ∈ Q are p 2(j+1) -semicircular in LS.
By the above weighted-semicircularity on LS, one obtains the following semicircularity on LS.
Theorem 2. Let Q p,j ∈ Q in the free Adelic filterization LS of (53), where Q is the family (54), for p ∈ P, j ∈ Z. Then, the operators are semicircular in LS, satisfying for all n ∈ N.
Proof. Let Θ p,j = 1 p j+1 Q p,j be in the sense of (55), where Q p,j ∈ Q are the p 2(j+1) -semicircular elements of LS, for p ∈ P, j ∈ Z. By the self-adjointness of Q p,j , the operator Θ p,j is self-adjoint in LS, too, because 1 p j+1 ∈ R × in C, for all P, j ∈ Z.
Since such a self-adjoint operator Θ p,j ∈ X is contained in the free block LS p (j) of LS, the operators Θ n p,j are contained in the same free block LS p (j) in LS, for all n ∈ N, as free reduced words with their lengths-1. Thus, one has that for all n ∈ N. Therefore, by (32) and (33), the operator Θ p,j is semicircular in LS.
be a subset of LS consisting of the operators Θ p,j of (55) induced by the family Q of (54).
Recall that a subset S of an arbitrary * -probability space (A, ϕ) is said to be a free family, if all elements of S are mutually free from each other in (A, ϕ) (e.g., [11,15]).

Definition 9.
A free family S is said to be a free (weighted-)semicircular family, if every element of S is (weighted)semicircular in a topological * -probability space (A, ϕ).

Free-Semicircular Adelic Filterization LS
Let LS be the free Adelic filterization (53), and let Q be the free weighted-semicircular family (59), and X , the free semicircular family (60) in LS. We now focus on the Banach * -subalgebra LS of LS generated by the free family X , where X are the Banach-topology closures of subsets X of LS. By (61), we obtain the corresponding Banach * -probability space, as a free-probabilistic sub-structure of LS.
Let LS be the semicircular Adelic filterization (62). Then, it satisfies the following structure theorem in LS.

Theorem 3.
Let LS be the semicircular Adelic filterization (62) of the free Adelic filterization LS. Then, the Banach * -algebra LS satisfies that in LS, where the free product ( ) in the first isomorphic relation of (63) means the free-probability-theoretic free product of [11,15] (with respect to the linear functional τ 0 of (62)), and the free product ( ) in the second isomorphic relation of (63) is the pure-algebraic free product inducing noncommutative free words in X .
In addition, one obtains the following set-identity.

Shifts on P Acting on LS
In this section, we study certain * -homomorphisms acting on LS induced by shift processes on P.

Shifts on P
Let P be the set of all primes in N. Since the set N of all natural numbers is a totally ordered set (or, TOset) under the usual inequality (≤), the subset P is a TOset under (≤) too. Without loss of generality, one can index P orderly by with Define now a function g : P → P by For the function g of (68), we define g n : P → P by , with axiomatization: g 0 = id P , the identity map on P, for all n ∈ N 0 , where (•) is the usual functional composition. By (69), one can have g n (q k ) = q k+n in P, for all k ∈ N, for all n ∈ N 0 .
Definition 11. Let g n be in the sense of (69) for all n ∈ N 0 . Then, these functions g n on P are said to be n-shifts on P, for all n ∈ N 0 . The 1-shift g = g 1 of (68) is simply called the shift on P.

Prime-Shift * -Homomorphisms on LS
Let LS be the semicircular Adelic filterization (63) generated by the free semicircular family X of (60), and let g n be the n-shifts (69) on the TOset P of (67), for n ∈ N 0 . Define a bounded "multiplicative" linear transformation satisfying for all p ∈ P, j ∈ Z, where g = g 1 is the shift on P. This multiplicative linear transformation G of (70) is well-defined by (61) and (63), since all generators Θ p,j ∈ X of LS are self-adjoint. Thus, if S = N Π l=1 Θ n l p l ,j l , and X = N Π l=1 Q n l p l ,j l are operator products of LS in X , respectively, in Q, for n 1 , . . . , n N ∈ N, then and, hence, in LS, by (71a). It is not difficult to check that for all t ∈ C, and Θ p,j ∈ X . It implies that G (T * ) = (G(T)) * , for all T ∈ LS, by (63), (71a) and (71b). Thus, this bounded multiplicative linear transformation G is adjoint-preserving, and, hence, it is a well-defined * -homomorphism on LS.
For the * -homomorphism G of (70), one can have the iterated products (or compositions) G n of (n-copies of) G, as new * -homomorphisms on LS, with G 1 = G, for all n ∈ N 0 , with identity More precisely, the morphisms G n satisfy for all Θ p,j ∈ X , in LS, where g n are the n-shifts (69) on P.
Definition 12. The * -homomorphism G of (70) on the semicircular Adelic filterization LS is called the prime-shift ( * -homomorphism) on LS. In addition, the n-th powers G n of (72) are called the n-prime-shift( * -homomorphism)s on LS, for all n ∈ N 0 .
By (72), we obtain the following result.
Theorem 4. Let S = Θ m p,j , and X = Q m p,j in LS, for Θ p,j ∈ X , and Q p,j ∈ Q, for m ∈ N. Then, for all k ∈ N, where G n are the n-shift on P, for all n ∈ N 0 .
Proof. Let S and X be given as above for a fixed m ∈ N. Then, they are free reduced words with their lengths-1 in LS. Moreover, by (71a), (71b) and (72), for all k ∈ N, where q = g n (p) ∈ P, for n ∈ N 0 . Therefore, one obtains the first free-distributional data in (73) by the semicircularity of Θ q,j ∈ X . In addition, the second free-distributional data in (73) is obtained because by (50) and (71b) for all k ∈ N, for all n ∈ N 0 . Therefore, the formula (73) holds.
The above free-distributional data (73) illustrate that the n-prime shifts G n of (72) preserve the free distributions of free reduced words of LS in Q ∪ X because the free distributions of free generators are preserved by G n , for all n ∈ N 0 .

Free-Homomorphisms on LS
In this section, motivated by (73), we consider free-homomorphic relations on LS under n-prime shifts G n , for n ∈ N 0 . Definition 13. Let (A 1 , ϕ 1 ), and (A 2 , ϕ 2 ) be topological * -probability spaces. Suppose there exists a bounded * -homomorphism Φ : A 1 → A 2 , and assume that By the free-homomorphic relation (74), one can get the following result.
Theorem 5. The n-prime shifts G n of (72) are free-homomorphisms on LS, for all n ∈ N 0 .
Proof. For any arbitrarily fixed n ∈ N 0 , take the n-prime shift G n on LS. Then, for any free reduced words S of LS in the free semicircular family X , one can get that for all k ∈ N, for all n ∈ N 0 , by (73) and the Möbius inversion. Indeed, the free distributions of free generators Θ p,j ∈ X of LS are preserved by acting G n , by (73).
Remark that, if S = N Π l=1 Θ n l p l ,j l is a free reduced word with its length-N, where either (p 1 , . . . , p N ) is alternating in P, or (j 1 , . . . , j N ) is alternating in Z, for n 1 , . . . , n N ∈ N, then the adjoint S * of S is again a free reduced word with the same length-N, by the self-adjointness of Θ p l ,j l , for all l = 1, ..., N. Therefore, the formula (75) holds by (73) under the Möbius inversion of [12], for all n ∈ N 0 . By (61) and (63), all operators T of LS are the limits of linear combinations of free reduced words in X . Thus, the free distributions of all summands of the operators T of LS are preserved by the n-prime shifts G n by (75), for all n ∈ N 0 . Therefore, the free distributions of T are identical to the free distributions of G n (T), for all n ∈ N 0 , for all T ∈ LS. For example, the condition (74) is satisfied under the action of G n on LS. Equivalently, the n-prime shifts G n are free-homomorphisms on LS, for all n ∈ N 0 .
The above theorem says that our n-prime shifts G n are not only * -homomorphisms, but also free-homomorphisms on LS, for all n ∈ N 0 .

Shifts on Z Acting on LS
In this section, we consider certain shifting processes h ± on Z, and the corresponding * -homomorphisms β ± on the semicircular Adelic filterization LS. 10.1. Shifts h ± on Z Let Z be the set of all integers as usual. Define functions h + and h − on Z by the bijections on Z, for all n ∈ Z. By the definition (76), one can have where id Z is the identity map on Z, i.e.,
Let h ± be the (±)-shifts (76) on Z. Define the functions h n ± on Z by for all n ∈ N 0 , with axiomatization: By the bijectivity (77) of h ± , these functions h n ± of (78) are bijective on Z, too, for all n ∈ N 0 .

Definition 15.
Let h n ± be the bijections (78), for all n ∈ N 0 , where h ± are the (±)-shifts (77) on Z. We call h n ± , the n-(±)-shifts on Z, for all n ∈ N 0 .
The n-(±)-shifts h n ± of (78) are directly understood as bijections, for all n ∈ N 0 .
For example, the morphisms β n ± are * -homomorphisms on LS, for all n ∈ N 0 .
where Q p,j ∈ Q, Θ p,j ∈ X , and m ∈ N. Then, for all k ∈ N, for all (e, n) ∈ N ± 0 .
Proof. Let T = Q m q,k and S = Θ m q,k be the free reduced words with their lengths-1 in LS, for all q ∈ P, k ∈ Z, for m ∈ N. Then, by the q 2(k+1) -semicircularity of Q q,k ∈ Q, and by the semicircularity of Θ q,k ∈ X , the operators Q m q,k and Θ m q,k are self-adjoint in LS, for all q ∈ P, k ∈ Z, and m ∈ N. Thus, if T = Q m p,j and S = Θ m p,j are in the sense of (81), then Thus, τ 0 (β n e (X)) k = τ 0 (β n e (X)) * k , and τ 0 (β n e (S)) k = τ 0 (β n e (S)) * k , for all k ∈ N. Observe now that τ 0 β n e (X) k = p km(j+1) τ 0 p,jen Θ km p,jen and τ 0 β n e (S) k = τ 0 p,jen Θ km p,jen = ω km c km by the semicircularity of Θ p,jen ∈ X in LS, for all k ∈ N, and (e, n) ∈ N ± 0 . Therefore, the formulas in (82) hold by (83) and (84).
The free-distributional data (82) demonstrate that the integer-shifts β n e preserve the free distributions on LS because they preserve the free distributions of generators of LS in X (∪Q), for all (e, n) ∈ N ± 0 .
Theorem 6. Let β n e be an integer-shift on LS, for (e, n) ∈ N ± 0 . Then, they are free-isomorphisms on the semicircular Adelic filterization LS.
Proof. Let (e, n) ∈ N ± 0 , and let β n e be an integer-shift on LS. Then, it is a well-defined * -isomorphism on LS because it is a generator-preserving, bijective * -homomorphism. Indeed, the restriction β n e | X is a bijection, as a function on the generator set X of LS. The bijectivity of β n e | X is guaranteed by the bijectivity of the n-(±)-shifts h n ± on Z. Moreover, by (82), the * -isomorphism β n e preserves the free distributions of free generators of LS. Thus, the free distributions of all free reduced words of LS in X are preserved by β n e , by (82) and the Möbius inversion of [12]. Therefore, it preserves the free probability on LS to that on β n e (LS) = LS. Equivalently, the morphisms β n e are free-isomorphisms on LS, by (74), for all (e, n) ∈ N ± 0 .

Shifts on P = P × Z
Now, define the Cartesian product set P, P de f = P × Z.
(87) Let g n be the n-shifts on P, and let h n e be the n-(e)-shifts on Z, for n ∈ N 0 , and (e, n) ∈ N ± 0 , with axiomatization: Define now shifts on the set P of (87) by s n 1 (e,n 2 ) for all n 1 , n 2 ∈ N 0 , and e ∈ {±}. For example, for any (p, j) ∈ P, s n 1 (e,n 2 ) (p, j) = g n 1 (p), h n 2 e (j) = (g n 1 (p), jen 2 ) in P. It is not difficult to check that such functions s n 1 (e,n 2 ) are injections on P, since g n 1 are injections on P, and h n 2 e are bijections on Z.
Definition 17. Let s n 1 (e,n 2 ) be injections (88) on the set P of (87), for n 1 ∈ N 0 , and (e, n 2 ) ∈ N ± 0 , with identity, where id P means the identity map on P. Then, these injections s n 1 (e,n 2 ) are called the shift(-function)s on P.

Prime-Integer Shifts on LS
Let P be the Cartesian product set (87), and let s n 1 (e,n 2 ) be shifts (88) on P. Then, for such shifts s n 1 (e,n 2 ) , one can construct the corresponding * -homomorphisms σ n 1 (e,n 2 ) on the semicircular Adelic filterization LS, defined by σ n 1 (e,n 2 ) = G n 1 β n 2 e on LS, for all n 1 ∈ N 0 , and (e, n 2 ) ∈ N ± 0 , where G n 1 are the n 1 -prime shifts, and β n 2 e are n 2 -(e)-integer shifts on LS. Since G n 1 are free-homomorphisms, and β n 2 e are free-isomorphisms on LS, the morphism σ n 1 (e,n 2 ) of (89) are indeed well-defined * -homomorphisms on LS.
The formula (92) shows that the product (or composition), inherited from that on Hom(LS), is closed on the set σ(LS). Thus, one can consider σ(LS) as an algebraic sub-structure (σ(LS), ·) in Hom(LS).
The above structure theorem characterizes the algebraic structure of σ(LS) as a commutative monoid embedded in Hom(LS).

Free-Distributional Data on LS Affected by σ(LS)
Recall that the prime-shifts G n are injective free-homomorphisms on LS by (75), for all n ∈ N 0 ; and the integer-shifts β n e are free-isomorphisms on LS by (82), for all (e, n) ∈ N ± 0 . Thus, it is not difficult to verify that the pi-shifts σ n (e,k) ∈ σ(LS) preserves the free probability on LS, for all n ∈ N 0 and (e, k) ∈ N ± 0 .
Theorem 8. Let Q p,j ∈ Q, and Θ p,j ∈ X in LS, for p ∈ P, j ∈ Z, and let σ denote = σ n (e,k) be a pi-shift in the pi-shift monoid σ(LS), for n ∈ N 0 , (e, k) ∈ N ± 0 . Then, σ(Q p,j ) is p 2(j+1) -semicircular in LS, and σ(Θ p,j ) is semicircular in LS. For example, All pi-shi f ts in σ(LS) are injective f ree-homomorphism on LS. (99) Proof. Let σ ∈ σ(LS) be a pi-shift given as above. Then, for all s ∈ N. Therefore, σ Q p,j (resp., σ(Θ p,j )) is p 2(j+1) -semicircular (resp., semicircular) in LS, too. By (100) and the Möbius inversion of [12], the free distributions of all free reduced words of LS in the free semicircular family X are preserved by pi-shifts of σ(LS). It shows that all pi-shifts preserves the free probability on LS. Therefore, the statement (99) holds.

Prime-Integer-Shift Operators on LS
Let LS be the semicircular Adelic filterization, and σ(LS), the pi-shift monoid (91) consisting of all pi-shifts σ n (e,k) , which are injective free-homomorphisms on LS by (99), for all n ∈ N 0 , (e, k) ∈ N ± 0 . In this section, we understand the Banach * -algebra LS as a Banach space, and construct the operator space B(LS) consisting of all bounded linear transformations on this Banach space LS (e.g., [2]).
Since all monoidal elements σ n (e,n) ∈ σ(LS) are injective * -homomorphisms, they are bounded linear transformations on LS, i.e., they can be regarded as Banach-space operators in B(LS). For example, Now, let's consider the following bounded "multiplicative" linear transformation G * on LS, defined by for all semicircular elements Θ p,j ∈ X . For instance, etc., for all j ∈ Z. Observe that, for any t ∈ C, for all Θ p,j ∈ X , implying that i.e., this multiplicative linear transformation G * of (102) is a * -homomorphism on LS, i.e., G * ∈ Hom(LS).
Thus, one can define the n-iterated products (G * ) n of G * in Hom(LS), for all n ∈ N. For example, the * -homomorphism (G * ) 3 in the sense of (103) satisfies that for all p ∈ P. Thus, it is non-zero if and only if p > 5. Here, the quantity 5 is the "3-rd" prime in the TOset P of (67).

Proposition 9.
Let (G * ) n be a * -homomorphism (103), and let G n be the n-prime shift (72) in Hom(LS), for n ∈ N 0 . Then, (G * ) n G n = 1 LS , on LS, and for all semicircular elements Θ p,j ∈ X , for all n ∈ N.

Now, let
with axiomatization: where (G * ) n are in the sense of (103), for all n ∈ N. Now, define the subset G of Hom(LS), consisting of all our n-prime shifts G n of (72), for all n ∈ N 0 , i.e., G = {G n : n ∈ N 0 }.
Note that both the sets G * 0 of (105), and G of (106) form semigroups in Hom(LS). Indeed, (103) and (104), and (G n 1 G n 2 ) G n 3 = G n 1 +n 2 +n 3 = G n 1 (G n 2 G n 3 ) , by (106). Thus, one can define the direct product semigroup G 0 of G * 0 and G, under the inherited operation (·) on Hom(LS). For example, this algebraic sub-structure G 0 of (107) satisfies as a sub-semigroup of Hom(LS). Indeed, the operation (·) on G 0 satisfies that if n 1 < n 2 , and if n 1 = n 2 , and if ∃q ∈ P, such that g n 2 (q) = p in P G n 1 −n 2 Θ p,j if n 1 > n 2 , and if ∃q ∈ P, such that g n 2 (q) = p in P G n 1 Θ q,j if n 1 < n 2 , and if ∃q ∈ P, such that by (103) and (104). Since this sub-semigroup G 0 of (107) contains 0 LS , having no (·)-inverse in G 0 , it is a sub-semigroup of Hom(LS), which is not a group. Moreover, this semigroup G 0 is "not" commutative by (108).

Define now the subset ∑(LS) of Hom(LS) by
where β n e are the n-(e)-integer shifts on LS, for all (e, n) ∈ N ± 0 . Note that our pi-shift monoid σ(LS) is contained in the set ∑(LS) of (109). Note also that the elements are free-homomorphisms, but the elements cannot be free-homomorphisms by (102), (106) and (108), in general. Especially, if n = 0 in N 0 , then they are * -homomorphisms, which are not free-homomorphisms. Observe now that if G * ∈ G * 0 ⊂ G 0 , and, hence, if G * β n e , β n e G * ∈ Hom(LS), for all Θ p,j ∈ X , implying that by (63). Therefore, for all n 1 , n 2 ∈ N, e ∈ {±}, by the induction on (110).
Note again that the pi-shift-operator semigroup ∑(LS) is contained in Hom(LS), and, hence, it is contained in the operator space B(LS) by (101), as operators acting on the Banach space LS. Therefore, one can construct the subspace S (∑(LS)) of the operator space B(LS) spanned by ∑(LS), By the above theorem, such an algebra A a of (113) is well-defined in B(LS). Thus, one can define the topological closure A (∑(LS)) of A a by where Y are the topological closures of subsets Y of B (LS) , where A a is an algebra (113). Then, this sub-structure A of (114) is not only a closed subspace of B(LS), but also a well-defined Banach algebra embedded in the operator space B(LS). By the definition (112), every pi-shift operators T ∈ S ⊂ A have its expression,

Definition 22. Let
with coefficients, where ∑ is a finite sum (under topology). By (113) and (114), all pi-shift operators of A are the limits of linear combinations of "products" of such operators T in the sense of (115). For example, the operators formed by T of (115) generate the pi-shift operator algebra A.

Proposition 10.
Let A be the pi-shift-operator algebra (114) generated by the pi-shift-operator semigroup ∑(LS) in the operator space B(LS), and let ( * ) be the unary operation (117) on A. Then, the operation ( * ) is a well-defined adjoint on A. Equivalently, every pi-shift operator T ∈ A is adjointable (in the sense of [2]) in B(LS).
By the above proposition, one can understand the Banach algebra A is a well-determined Banach * -algebra embedded in B(LS).
in A.
Note in (124) and (125) that on A, by (117). Thus, if a pi-shift operator T is decomposed to be (123), then the adjoint T * of T is decomposed to be Proof. Let T = T * + t 0 1 LS + T 1 ∈ S ⊂ A. Then, T * = T * * + t 0 1 LS + T * 1 by (107) = T * 1 + t 0 1 LS + T * * , in the sense of (123) by (126) , in S ⊂ A. Thus, one has if and only if T * 1 = T * in A, and t 0 ∈ R.
Let T ∈ S ⊂ A be a pi-shift operator (122). Then, one may write T as follows: with  Example 1. Let T 1 , T 2 , T 3 ∈ S ⊂ A be pi-shift operators as follows: Then, they are self-adjoint in A, by (127) and (130).
Let's consider general cases.
Theorem 11. Let U ∈ A be an arbitrary pi-shift operator. Then, the operators for t 1 , t 2 , t 3 ∈ R, are self-adjoint in A. Moreover, if U 1 and U 2 are self-adjoint pi-shift operators of A, then s 1 U 1 + s 2 U 2 is self-adjoint in A, too, for all s 1 , s 2 ∈ R in C.
Proof. Let U ∈ A be an arbitrary pi-shift operator (including the case where U ∈ A \ S), and let t 1 , t 2 , t 3 ∈ R. Then, the pi-shift operators T 1 , T 2 and T 3 satisfy that in A. Therefore, they are self-adjoint in A. Now, assume that U 1 and U 2 are self-adjoint in A, and let s 1 , s 2 ∈ R. If in A, and hence, it is self-adjoint in A.
Now, let's focus on pi-shift operators of A, contained in the subspace S of (112) in B(LS). Let T ∈ S ⊂ A be a pi-shift operator (128) with its decomposition (123). Such an operator T is self-adjoint in A, if and only if by (127). Observe first that In the above computation (131), one can realize immediately that, if T 1 = σ n (e,k) with n = 0, then (132) This observation (132) motivates us to verify the following result. Lemma 4. Let T ∈ S ⊂ A be a non-zero, self-adjoint, finite-sum pi-shift operator, T = ∑ N s=1 t s σ * n s (−e s ,k s ) + t 0 1 LS + ∑ N s=1 t s σ 1n s (e s ,k s ) , for N ∈ N. If t N = 0 in C, then T is not a projection in A. The converse also holds true too.
Proof. Suppose T is given as above. Then, by (127) and (130), this operator T is self-adjoint in A. Thus, to check the projection-property for such T, it suffices to check T 2 = T in A.
However, if t N = 0 in C, for N ∈ N, then the pi-shift operator T 2 contains non-zero summands t N 2 σ * (2n N ) (−e N ,2k N ) and t 2 N σ 2n N (e N ,2k N ) in A, by (131), i.e., T 2 = T in A. Thus, if t N = 0 in C, for N ∈ N, then T cannot be a projection in A.
Conversely, suppose T is a projection in A. Then, T 2 = T in A. If t N = 0 in C, for N ∈ N, then T 2 = T in A, by (131). It contradicts our assumption that T is a projection in A.
The above lemma shows that if a non-zero, self-adjoint, finite-sum pi-shift operator is self-adjoint, then T cannot be a projection in A in the sense of (ii) in Definition 23.
Proof. By the above lemma, a given self-adjoint finite-sum pi-shift operator T is a projection, if and only if Then, such a pi-shift operator T is a projection in A, if and only if if and only if if and only if T = 0 · 1 LS = 0 LS , or T = 1 LS , in A. Therefore, the projection-property (133) holds for T ∈ S in A.
The above theorem illustrates that the only "finite-sum" projections of S are either 0 LS or 1 LS in A. Meanwhile, one can obtain the following projection-property on A \ S too.
Proof. Let P 1 and P 2 be given as above. Note that P 1 = σ * σ = 1 LS in A. Indeed, Thus, it satisfies the projection-property, by (133). Observe now that P * 2 = (σσ * ) * = σ * * σ * = σσ * = P 2 , in A, and, hence, it is self-adjoint in A. Moreover, Therefore, the self-adjoint pi-shift operator P 2 is a projection in A in the sense of (ii) in Definition 23.
The above two theorems show that: (i) the finite-sum pi-shift operator T ∈ S is a projection in A, if and only if either T = 0 LS , or T = 1 LS ; and the pi-shift operators formed by σ * σ and σσ * are projections in A, for all generating operators σ ∈ ∑(LS) of the pi-shift-operator algebra A. Now, let σ = σ n (e,k) ∈ σ(LS) ⊂ ∑(LS) in the pi-shift-operator algebra A. Assume that n = 0 in N 0 . Then, it is not hard to show that σ * σ = σσ * on LS. Indeed, if p is an n p -th prime in the TOset P of (67), for some n p ∈ N, and if n ≥ n p in N, then for the corresponding semicircular element Θ p,j ∈ X .
The above relation (134) directly demonstrates that our pi-shift operators are not normal in A in the sense of (iii) in Definition 23, in general. Remark that, if n = 0 in N 0 , then Thus, in such a case, this pi-shift operator σ can be normal in A, because for all (e, k) ∈ N ± 0 .
(i) σ n is normal in A in the sense of (iii) in Definition 23, if and only if n = 0 in N 0 .
(ii) σ 0 is a unitary in A in the sense of (v) in Definition 23.
(iii) σ n is a unitary in A in the sense of (v) in Definition 23, if and only if n = 0 in N 0 .
(iv) σ 0 , and σ 0 * are isometries in A in the sense of (iv) in Definition 23.
Proof. Let σ n ∈ σ(LS) ⊂ A be given as above. By (135), if n = 0 in N 0 , then σ n is normal in A. Suppose now σ n is normal, where n = 0 in N 0 . Since n = 0, one has (σ n ) * σ n = σ n (σ n ) * on LS, by (134). Equivalently, σ n is not normal in A. It contradicts our assumption that it is normal. Therefore, the statement (i) in Theorem 14 holds.
The relation (135) shows not only that σ 0 is normal, but also it is unitary in A. Thus, the statement (ii) in Theorem 14 is shown.
By using the similar arguments of the proof of (i) in Theorem 14, the statement (iii) in Theorem 14 holds by (ii) in Theorem 14.
The statement (iv) in Theorem 14 is proven by (ii) and (iii) in Theorem 14 and the isometry-property (iv) in Definition 23 in A. Now, let's focus on the normality (iii) in Definition 23 for pi-shift operators T ∈ S in A. Let T ∈ S be a finite-sum pi-shift operator, decomposed to be T = T * + t 0 1 LS + T 1 , as in (123).
By (i) in Theorem 14, one can easily verify that T is not normal in A, in general.

Lemma 5.
Let T ∈ S be a pi-shift operator, Then, it is normal in A in the sense of (iii) in Definition 23.
Observe now that in A. Therefore, this pi-shift operator T is normal in A, by (138).
The above lemma shows that pi-shift operators T ∈ S formed by (136) are normal in A, in the sense of (iii) in Definition 23. How about the converse of this lemma for the elements of S in A? One can verify that the converse does not hold in general. Indeed, from the self-adjoint characterization (127), there are normal pi-shift operators which are not of the forms (136) because the self-adjointness (i) in Definition 23 implies the normality (iii) in Definition 23. Generally, one can get the following result.

Lemma 6.
There are normal pi-shift operators in the sense of (iii) in Definition 23 in S ⊂ A, which are not formed by (136). For example, the converse of the above lemma does not hold true.
Proof. First, note that, if a pi-shift operator T is self-adjoint in A in the sense of (i) in Definition 23, then it is normal in A, by definition. Thus, it is proven by construction. By (127) and (130), if Then, it is self-adjoint, and, hence, normal in A. Clearly, T is not of the form (136) in S ⊂ A.
Then, how about the following restricted case? We will restrict our interests to the cases where a "normal" pi-shift operator T ∈ S is "not self-adjoint" in A.

Lemma 7.
Let T ∈ S ⊂ A be a "non-zero" pi-shift operator (122), and suppose T is "not" self-adjoint in A. If T is normal in A, then T is formed by (136).
Proof. Suppose either T = σ n (e,k) , or σ * n (e,k) in A, for some n ∈ N 0 , (e, k) ∈ N ± 0 . Then, such a finite-sum pi-shift operator T is not normal in general, by (i) in Theorem 14.
Assume that T contains at least one non-zero summand s 0 σ n 0 (e 0 ,k 0 ) , for some n 0 ∈ N, (e 0 , k 0 ) ∈ N ± 0 , with s 0 = 0 in C, and r ∈ {1, * }. In addition, assume that T does not contain a summand s 0 σ * n 0 (−e,k 0 ) in (122). One can take such a summand of T, since T = 0 LS is assumed not to be self-adjoint, by (127). For example, T * 1 = T * in A, in the sense of (123).
Then, by (134) and (i) in Theorem 14, the operator T cannot be normal in A, i.e., if T ∈ S is a non-zero, non-self-adjoint pi-shift operator satisfying the above condition, then T * T = TT * on LS.
Conversely, let T ∈ S ⊂ A be a non-zero, non-self-adjoint operator, and assume that T is not formed by (136). Then, it is not normal in A. Equivalently, if a non-zero, non-self-adjoint pi-shift operator T ∈ S is normal in A, then it is formed by (136).
By the above three lemmas, we obtain the following normality condition in the pi-shift-operator algebra A.
Proof. Note that T ∈ S is assumed to be non-zero, and non-self-adjoint in A.
(⇐) If T is in the sense of (139), then it is normal in A by (138). (⇒) If T is not formed by (139), then T is not normal under conditions, by the very above lemma.
The above condition (139) characterizes the normality (iii) in Definition 23 of non-zero, non-self-adjoint, finite-sum pi-shift operators of S in A.
By the unitarity (v) in Definition 23 in the pi-shift-operator algebra A, if a pi-shift operator T is unitary, then it must be normal in A in the sense of (iii) in Definition 23. Thus, we restrict our interests to the unitarity (v) in Definition 23 for normal pi-shift operators. Furthermore, let's assume that T is normal, but "not" self-adjoint.
For example, to consider the pure-unitarity for (v) in Definition 23, we restrict our interests to non-self-adjoint, normal pi-shift operators in A. By (139) Therefore, T is unitary in A in the sense of (v) in Definition 23.
(⇒) Suppose that T is unitary in A, and assume that the condition (140) does not hold for T. Then, by (138), It contradicts our assumption that T is unitary.

Distorted Free Probability on LS by A (∑(LS))
Let LS be the semicircular Adelic filterization, and let A denote = A (∑(LS)) be the pi-shift-operator algebra in the operator space B(LS). In Section 13, we considered operator-theoretic properties on A. Here, we study how such operators distort the original free-distributional data on LS.
Recall first that the subset σ(LS) (which is the pi-shift monoid) of the pi-shift-operator semigroup ∑(LS) (generating A) are consisting of injective free-homomorphisms contained in the homomorphism semigroup Hom(LS), by (99). Thus, "some" generating operators of A, σ 1n (e,k) = σ n (e,k) ∈ σ(LS) ⊂ ∑(LS) preserve the free probability on LS. However, not "all" generators preserve the free probability on LS. For instance, a generating operator σ * n (e,k) ∈ ∑(LS) \ σ(LS) is a * -homomorphism, but not a free-homomorphism on LS, whenever n = 0 in N 0 , for any (e, k) ∈ N ± 0 . Thus, it is interesting to check how the pi-shift-operator semigroup ∑(LS) deform the original free-distributional data on LS.
If Θ p,j ∈ X is a semicircular element in LS, then for all l ∈ N.
Thus, this operator W 2 (Θ p,j ) has the zero free distribution in LS. For example, the statement (iii) in Theorem 17 holds true.
The above theorem illustrates how our semicircular law on LS is distorted by the action of the pi-shift-operator semigroup ∑(LS). It explains how the pi-shift-operator algebra A distorts the free probability on the semicircular Adelic filterization LS.

Discussion
As we have seen, there exists an interesting type of Banach-space operators acting on the semicircular elements induced by {Q p } p∈P , deforming their free distributions, the semicircular law. In particular, the operator-theoretic properties of our pi-shift operators are provided in Theorems 9-16; and some deformations of the semicircular law induced by {Q p } p∈P are characterized in Theorem 17.
It is interesting that the pi-shift operators actint on our semicircular elements are like the classical, or generalized Toeplitz operators on Hilbert spaces. In future, one may find connections, or differences between pi-shift-like operators on semicircular elements (as newly-introduced Banach-space operators), and Toeplitz operators (as well-known Hilbert-space operators).
Funding: This research received no external funding.