Difference between Semicircular-like Laws Induced by p-Adic Number Fields and the Semicircular Law

Abstract In this paper, we study “semicircular-like” elements in free product Banach *-algebras induced by Haar-measurable functions over p-adic number fields p, for primes p. And we investigate how the free distributions of operators generated by our mutually-free weighted-semicircular elements are close enough to (or far from) those of free reduced words generated by arbitrary mutually-free semicircular elements.


Introduction
The main purposes of this paper are (i) to establish weighted-semicircular elements and corresponding semicircular elements in a certain Banach *-probability space induced by measurable functions over p-adic number fields , p  for primes p, (ii) to consider free-distributional data of weighted-semicircular elements, and those of operators generated by free, weighted-semicircular elements under free product, and (iii) to investigate how the free-distributional data of the operators generated by weighted-semicircular elements are close to those of operators generated by semicircular elements. In particular, the main results of the topic (iii) illustrate how the weights of our weighted-semicircular elements distort (or affect) the semicircular law, and such distortions are measured by so-called the weight-ratios of weighted-semicircular elements.

Preview and Motivation
We have considered how primes act on operator algebras. The relations between primes and operators have been studied in various different approaches. For instance, we studied how primes act on certain von Neumann algebras generated by p-adic, and Adelic measure spaces (e.g., [6]). Meanwhile, in [5], primes are regarded as linear functionals acting on arithmetic functions. In such a case, one can understand arithmetic functions as Krein-space operators (for fixed primes), under Krein-space representations (e.g., [8]). Also, in [1], [3], [4] and [7], we considered free-probabilistic structures In [2], we considered certain free random variables in a *-algebra ,  of * C -probability spaces, we constructed, and studied free probability on the free product * C -probability space,

LS
In Section 7, we enlarge our weighted-semicircularity and semicircularity under free product Banach *-probability space LS of p LS 's. And, free distributions of operators generated by free weighted-semicircular family, and those of operators generated by free semicircular family are computed, in Sections 8,9 and 10. In Section 11, the close-ness (or difference) between free distributions of self-adjoint operators generated by semicircular elements, and those of operators generated by our weighted-semicircular elements is considered by defining so-called the weight-ratios. Such weight-ratios measure how our weighted-semicircularity is close to semicircularity.

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

Free Probability
Readers can check fundamental analytic-and-combinatorial free probability from [12] and [13] (and the cited papers therein). Free probability is understood as the non-commutative operator-algebraic version of classical probability theory (covering commutative cases). The classical independence is replaced by the freeness, by replacing measures to linear functionals. It has various applications not only in pure mathematics (e.g., [11]), but also in related scientific topics (for example, see [3,4,6,8]).
Especially, in the text, without introducing detailed definitions and combinatorial backgrounds, free moments and free cumulants of operators will be computed. Also, we use free product of * C -probability spaces in the sense of [12] and [13], without detailed introduction. However, rough introduction would be given whenever they are needed in text.
 for all . k ∈  Then, for 1 2 , , k k ∈  one obtains that , where δ means the Kronecker delta, and hence, So, we obtain the following computations.
Thus, one can get that, for any where 0 1 j r ≤ ≤ are in the sense of (2.2.8), for all .
Proof. The proof of (3.10) is done by induction on (3.9). See [2] for details.

Representations of ( )
Remark that the element S α is a projection in , p M in the sense that: Thus, by (5.2), we obtain (5.3). The above proposition characterizes the free distributions More precisely, we obtain the following theorem. ,..., Then, one has that 1 in , Therefore, by (5.3), the formula (5.4) holds.
The above joint free-moment formula (5.4) characterizes the free-distributions of finitely many projections 1 ,..., . j ∈  As corollaries of (5.4), we obtain the following results. Corollary 5.3. Let k U be in the sense of (3.1), and , and ( ) , for all , j ∈  for . p ∈ As we have seen, these operators , . j ∈  We now restrict our interests to these projections , p j P of (6.1).
We originally defined such p-adic boundary subalgebras p S in [4].
Now, let φ be the Euler totient function, which is an for all , n ∈  where X mean the cardinalities of sets X, and gcd means the greatest common divisor.
It is well-known that Thus, one has ( ) for all , p ∈ and , , j k ∈  inducing new * C -probability spaces , .
Proof. The free-moment formula (7.0.7) is proven by

Semicircular and Weighted-Semicircular Elements
A ϕ be an arbitrary topological *-probability space ( * C -probability space, or * W -probability space, or Banach *-probability space, etc.) equipped with a topological *-algebra A (C_-algebra, resp., W_-algebra, resp., Banach *-algebra), and a bounded linear functional ϕ on . where n c are the n-th Catalan number, is the free cumulant on A in terms of ϕ (in the sense of [12]), then a self-adjoint operator a is semicircular in ( )
Thus, the semicircular operators a of ( ) , A ϕ can be re-defined by the self-adjoint operators satisfying the free-cumulant characterization (7.1.3).
Motivated by (7.1.3), one can define so-called the weighted-semicircular elements.
It is said to be weighted-semicircular in ( ) is the free cumulant in terms of ϕ in the sense of [12]. By the definition (7.1.4), and by the Möbius inversion, we obtained the following free-moment characterization of (7.1.4) in [4]: A self-adjoint operator a in a *-probability space ( ) where ⊗  means the tensor product of Banach *-algebras. Take now a generating element , , with identity: By considering the formulas (7.2.8) and (7.2.9) together, one obtains the following proposition. , The proofs of (7.2.10) and (7.2.11) are done by straightforward computations under (7.2.8) and (7.2.9). See [4] for more details.
We denote the above relations together by with , for all 1,..., , for all , n ∈  and where ⊕ and ⊗ are the direct product, respectively, the tensor product of Banach spaces (e.g., see [12,14]).
In particular, if an element a A ∈ is of the form of free reduced word, To avoid the confusion, we will use the notation ( ) ,  (non-reduced word) (reduced word) a a a a a a a a a   a a a a a

a a a a a a = is a free reduced word in A.
So, in the text below, if we use the term " k a " for a fixed free reduced word a, then it is in the sense of (8.
The free product linear functional ϕ on A satisfies that, whenever a is a reduced free word in A of (8. where the summands of ( ) For more about (free-probabilistic) free product algebras, and corresponding free probability spaces, see [11,12,14] and cited papers therein.

Free Product C * -Probability Space (L S , τ 0 )
In this section, we will use the same concepts and notations introduced in Section 8.  A ϕ (e.g., [12,14]).

, ,
A ϕ for all In the proof of (9.5) and (9.6), we use the identically free-distributedness of the semicircular law. Remark 9.1. Our main results (9.2) and (9.3) show that the free distributions of operators (9.1) in LS generated by our free semicircular family Θ of (8.3.3) are determined universally, up to the identically-free-distributedness of semicircularity, by (9.5) and (9.6). It demonstrates that we lost some interesting "local" free-probabilistic information came from our number-theoretic settings from p-adic, and Adelic analysis. So, we escape from such a universalization in Section 10 below.

Operators of LS Generated by 
In this section, we use same concepts and notations introduced in Notation 8.1, NR 8.1, and NR 8  in .

LS
for all . k ∈  So, the first equality of the formula (10.5) holds by (10.6) Note that X is not self-adjoint in , for all . k ∈  The formula (10.7) proves the second equality of (10.5). Therefore, by (10.6) and (10.7), the free-distributional data (10.5) holds.
The free-distributional data (10.5) not only gives free-probabilistic information of the operators (10.2), but also provides tools to compute the (joint) free distributions of operators (10.2) in the radial-Adelic probability space .

How Close Our Weighted-Semicircularity is to Semicircularity?
In Sections 9 and 10, we studied free-distributional data of the free reduced words generated by our semicircular elements, and those generated by our weighted-semicircular elements. In particular, we considered such data both in minimal free summands of the radial-Adelic probability

LS
Then the family  of (11.1) is a free weighted-semicircular family (by (8.2.4)), and the family Θ of (11.1) is a free semicircular family (by (8.3.3)) in .  (11.2) in LS, generated by Q, respectively, by Θ of (11.1).

LS
LS which is the very minimal free summand of LS containing them. So, the operators formed by (11.2) are not only the operators in Sections 9 and 10, but also our weighted-semicircular, and semicircular elements in Sections 7 and 8 in .   ,

LS
with axiomatization: : , is in the sense of (11.6). : , is in the sense of (11.6).
Proof. The proof of (11.8) is done by (  LS In other words, we may / can have three characterizations for close-ness determined by (11.12), (11.13) and (11.14). For instance, by (11.14), one may / can conclude that: if the nonnegative real quantity ( ) by (11.17). Assume first that 1 j = − in .
 Then the above weight-ratio R of (11.21) satisfies that  LS We obtain the following additional close-ness condition of N X and N M of (11.30) in the radial-Adelic probability space . LS Theorem 11.11. Let N X and N M be an "even" power-ws reduced word, respectively an "even" power-s reduced word in LS in the sense of (11.30), and let ( )