Mathematical Physics II 3039434950, 9783039434954

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Mathematical Physics II Edited by

Enrico De Micheli Printed Edition of the Special Issue Published in Mathematics

www.mdpi.com/journal/mathematics

Mathematical Physics II

Mathematical Physics II

Editor Enrico De Micheli

MDPI • Basel • Beijing • Wuhan • Barcelona • Belgrade • Manchester • Tokyo • Cluj • Tianjin

Editor Enrico De Micheli Consiglio Nazionale delle Ricerche Italy

Editorial Office MDPI St. Alban-Anlage 66 4052 Basel, Switzerland

This is a reprint of articles from the Special Issue published online in the open access journal Mathematics (ISSN 2227-7390) (available at: https://www.mdpi.com/journal/mathematics/special issues/Mathematical Physic II).

For citation purposes, cite each article independently as indicated on the article page online and as indicated below: LastName, A.A.; LastName, B.B.; LastName, C.C. Article Title. Journal Name Year, Article Number, Page Range.

ISBN 978-3-03943-495-4 (Hbk) ISBN 978-3-03943-496-1 (PDF)

c 2020 by the authors. Articles in this book are Open Access and distributed under the Creative  Commons Attribution (CC BY) license, which allows users to download, copy and build upon published articles, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications. The book as a whole is distributed by MDPI under the terms and conditions of the Creative Commons license CC BY-NC-ND.

Contents About the Editor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Preface to ”Mathematical Physics II” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

Ilwoo Cho and Palle Jorgensen Primes in Intervals and Semicircular Elements Induced by p-Adic Number Fields Qp over Primes p Reprinted from: Mathematics 2019, 7, 199, doi:10.3390/math7020199 . . . . . . . . . . . . . . . . .

1

Jin Liang and Chengwei Zhang Study on Non-Commutativity Measure of Quantum Discord Reprinted from: Mathematics 2019, 7, 543, doi:10.3390/math7060543 . . . . . . . . . . . . . . . . . 37 Tongshuai Liu and Huanhe Dong ¨ The Prolongation Structure of the Modified Nonlinear Schrodinger Equation and Its Initial-Boundary Value Problem on the Half Line via the Riemann-Hilbert Approach Reprinted from: Mathematics 2019, 7, 170, doi:10.3390/math7020170 . . . . . . . . . . . . . . . . . 45 Enrico De Micheli On the Connection between Spherical Laplace Transform and Non-Euclidean Fourier Analysis Reprinted from: Mathematics 2020, 8, 287, doi:10.3390/math8020287 . . . . . . . . . . . . . . . . . 63 Salem Ben Sa¨ıd, Sara al-Blooshi, Maryam al-Kaabi, Aisha al-Mehrzi and Fatima al-Saeedi A Deformed Wave Equation and Huygens’ Principle Reprinted from: Mathematics 2019, 8, 10, doi:10.3390/math8010010 . . . . . . . . . . . . . . . . . . 93 Mutaz Mohammad On the Gibbs Effect Based on the Quasi-Affine Dual Tight Framelets System Generated Using the Mixed Oblique Extension Principle Reprinted from: Mathematics 2019, 7, 952, doi:10.3390/math7100952 . . . . . . . . . . . . . . . . . 105 Ping Chen, Suizheng Qiu, Shichao Liu, Yi Zhou, Yong Xin, Shixin Gao, Xi Qiu and Huaiyu Lu Preliminary Analysis of a Fully Ceramic Microencapsulated Fuel Thermal–Mechanical Performance Reprinted from: Mathematics 2019, 7, 448, doi:10.3390/math7050448 . . . . . . . . . . . . . . . . . 119 Bing Dai, Ying Chen, Guoyan Zhao, Weizhang Liang and Hao Wu A Numerical Study on the Crack Development Behavior of Rock-Like Material Containing Two Intersecting Flaws Reprinted from: Mathematics 2019, 7, 1223, doi:10.3390/math7121223 . . . . . . . . . . . . . . . . 133 Neda Moayyeri, Sadjad Gharehbaghi and Vagelis Plevris Cost-Based Optimum Design of Reinforced Concrete Retaining Walls Considering Different Methods of Bearing Capacity Computation Reprinted from: Mathematics 2019, 7, 1232, doi:10.3390/math7121232 . . . . . . . . . . . . . . . . 149

v

About the Editor Enrico De Micheli is currently Senior Researcher at the National Research Council (CNR) of Italy and teaches Quantum Information Theory and Computation at the Information Technology Institute (ISICT) of the University of Genova (Italy). He received his degree in Theoretical Physics from the University of Genova in 1986. He worked as research fellow at the Institute for Scientific Research (IRST) of Trento. He has been visiting scientist at the University of British Columbia in Vancouver (Canada). He is reviewer of the AMS and Editor/Referee of several mathematical journals and has published some 60 papers in peer-reviewed journals. His research interests include Special Functions of Mathematical Physics, Potential Theory, Quantum Mechanics, Thermal Quantum Field Theory, Approximation Theory, Inverse Problems, and Computational Physics.

vii

Preface to ”Mathematical Physics II” The mysterious charm of Mathematical Physics is beautifully represented in the celebrated 1960 paper of Eugene Wigner ”The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” It is indeed hard for us to understand the astonishing appropriateness of the language of mathematics for the formulation of the laws of physics and its capability to do predictions, appropriateness that emerged immediately at the beginning of the scientific thought and was splendidly depicted by Galileo: ”The grand book, the Universe, is written in the language of Mathematics.” Paraphrasing the words of Bertrand Russell, in this marriage the supreme beauty, cold and austere, of Mathematics complements the supreme beauty, warm and engaging, of Physics. This book, which consists of nine articles, gives a flavor of this beauty and covers various topics related to physics and engineering. A brief outline of these topics is given hereafter. The study of free probability in certain probability spaces induced by functions on p-adic number fields is here a very interesting example of the application of p-adic mathematical methods for modeling physical phenomena.

Within the quantum information processing framework is

presented the analysis of a non-commutative measure of quantum discord in the two-qubit case.

The Riemann–Hilbert problem plays relevant but different roles in two papers.

In one

paper, the Riemann–Hilbert problem, formulated with respect to the spectral parameter, and the ¨ prolongation structure theory are used to analyze the modified nonlinear Schrodinger equation. In the other case, the Riemann–Hilbert structure emerges from the holomorphic extension of certain Legendre expansions, leading thus to an explicit connection between spherical Laplace transform and non-Euclidean Fourier transform. Remaining in the spectral analysis field, the study of a deformed wave equation, with the Laplacian being replaced by a differential-difference Dunkl operator, shows its relation with a generalized Fourier transform and the non-existence of the related Huygens principle. The Gibbs phenomenon is then the subject of a paper where the primary tool of analysis is the representation of suitable functions in terms of dual tight framelets. Finally, numerical analysis and optimization methods are the main mathematical devices used to study equations that are relevant in the study of material properties, such as thermomechanical performances, flaw dynamics, and bearing capacity of structures. In conclusion, as Editor of this Special Issue, I wish to thank the authors of the articles for their valuable contributions, the referees for their precious reviews, and Ms. Julie Shi and Ms. Grace Wang of MDPI for their kind assistance. Enrico De Micheli Editor

ix

mathematics Article

Primes in Intervals and Semicircular Elements Induced by p-Adic Number Fields Q p over Primes p Ilwoo Cho 1, * and Palle Jorgensen 2 1 2

*

Department of Mathematics & Statistics, Saint Ambrose University, 421 Ambrose Hall, 518 W. Locust St., Davenport, IA 52803, USA Department of Mathematics, University of Iowa, 14 McLean Hall, Iowa City, IA 52242, USA; [email protected] Correspondence: [email protected]

Received: 11 December 2018; Accepted: 15 February 2019; Published: 19 February 2019

Abstract: In this paper, we study free probability on (weighted-)semicircular elements in a certain Banach ∗-probability space (LS, τ 0 ) induced by measurable functions on p-adic number fields Q p over primes p. In particular, we are interested in the cases where such free-probabilistic information is affected by primes in given closed intervals of the set R of real numbers by defining suitable “truncated” linear functionals on LS. Keywords: free probability; primes; p-adic number fields; Banach ∗-probability spaces; weighted-semicircular elements; semicircular elements; truncated linear functionals MSC: 05E15; 11G15; 11R47; 11R56; 46L10; 46L54; 47L30; 47L55

1. Introduction In [1,2], we constructed-and-studied weighted-semicircular elements and semicircular elements induced by p-adic number fields Q p , for all p ∈ P , where P is the set of all primes in the set N of all natural numbers. In this paper, we consider certain “truncated” free-probabilistic information of the weighted-semicircular laws and the semicircular law of [1]. In particular, we are interested in free distributions of certain free reduced words in our (weighted-)semicircular elements under conditions dictated by the primes p in a “suitable” closed interval [t1 , t2 ] of the set R of real numbers. Our results illustrate how the original (weighted-)semicircular law(s) of [1] is (resp., are) distorted by truncations on P . 1.1. Preview and Motivation Relations between primes and operators have been widely studied not only in mathematical fields (e.g., [3–6]), but also in other scientific fields (e.g., [7]). For instance, we studied how primes act on certain von Neumann algebras generated by p -adic and Adelic measure spaces in [8,9]. Meanwhile, in [10], primes are regarded as linear functionals acting on arithmetic functions, understood as Krein-space operators under the representation of [11]. Furthermore, in [12,13], free-probabilistic structures on Hecke algebras   H GL2 (Q p ) are studied for p ∈ P . These series of works are motivated by number-theoretic results (e.g., [4,5,7]). In [2], we constructed weighted-semicircular elements { Q p,j } j∈Z and corresponding semicircular elements {Θ p,j } j∈Z in a certain Banach ∗-algebra LS p induced from the ∗-algebra M p consisting of measurable functions on a p-adic number field Q p , for p ∈ P . In [1], the free product Banach ∗-probability   space LS, τ 0 of the measure spaces {LS p ( j)} p∈P ,j∈Z of [2] were constructed over both primes and integers, and weighted-semicircular elements { Q p,j } p∈P , j∈Z and semicircular elements {Θ p,j } p∈P , j∈Z were studied in LS, as free generators. Mathematics 2019, 7, 199; doi:10.3390/math7020199

1

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Mathematics 2019, 7, 199

In this paper, we are interested in the cases where the free product linear functional τ 0 of [1] on the Banach ∗-algebra LS is truncated in P . The distorted free-distributional data from such truncations   are considered. The main results characterize how the original free distributions on LS, τ 0 are affected by the given truncations on P . 1.2. Overview We briefly introduce the backgrounds of our works in Section 2. In the short Sections 3–8, we construct the Banach ∗-probability space (LS, τ 0 ) and study weighted-semicircular elements Q p,j and corresponding semicircular elements Θ p,j in (LS, τ 0 ), for all p ∈ P , j ∈ Z.   In Section 9, we define a free-probabilistic sub-structure LS = LS, τ 0 of the Banach ∗-probability   space LS, τ 0 , having possible non-zero free distributions, and study free-probabilistic properties of LS. Then, truncated linear functionals of τ 0 on LS and truncated free-probabilistic information on LS are studied. The main results illustrate how our truncations distort the original free distributions on LS (and hence, on LS). In Section 10, we study free sums X of LS having their free distribution, the (weighted-)semicircular law(s), under truncation. Note that, in general, if free sums X have more than one summand as operators, then X cannot be (weighted-)semicircular in LS. However, certain truncations make them be. In Section 11, we investigate a type of truncation (compared with those of Sections 9 and 10). In particular, certain truncations inducing so-called prime-neighborhoods are considered. The unions of such prime-neighborhoods provide corresponding distorted free probability on LS (different from that of Sections 9 and 10). 2. Preliminaries In this section, we briefly introduce the backgrounds of our proceeding works. 2.1. Free Probability Readers can review free probability theory from [14,15] (and the cited papers therein). Free probability is understood as the noncommutative operator-algebraic version of classical measure theory and statistics. The classical independence is replaced by the freeness, by replacing measures on sets with linear functionals on noncommutative (∗-)algebras. It has various applications not only in pure mathematics (e.g., [16–20]), but also in related topics (e.g., see [2,8–11]). Here, we will use the combinatorial free probability theory of Speicher (e.g., see [14]). In the text, without introducing detailed definitions and combinatorial backgrounds, free moments and free cumulants of operators will be computed. Furthermore, the free product of ∗-probability spaces in the sense of [14,15] is considered without detailed introduction. Note now that one of our main objects, the ∗-algebra M p of Section 3, are commutative, and hence, (traditional, or usual “noncommutative”) free probability theory is not needed for studying functional analysis or operator algebra theory on M p , because the freeness on this commutative structure is trivial. However, we are not interested in the free-probability-depending operator-algebraic structures of commutative algebras, but in statistical data of certain elements to establish (weighted-)semicircular elements. Such data are well explained by the free-probability-theoretic terminology and language. Therefore, as in [2], we use “free-probabilistic models” on M p to construct and study our (weighted-)semicircularity by using concepts, tools, and techniques from free probability theory “non-traditionally.” Note also that, in Section 8, we construct “traditional” free-probabilistic structures, as in [1], from our “non-traditional” free-probabilistic structures of Sections 3–7 (like the free group factors; see, e.g., [15,19]).

2

Mathematics 2019, 7, 199

2.2. Analysis of Q p For more about p-adic and Adelic analysis, see [7]. Let p ∈ P , and let Q p be the p-adic number field. Under the p-adic addition and the p-adic multiplication of [7], the set Q p forms a field algebraically. It is equipped with the non-Archimedean norm |.| p , which is the inherited p-norm on the set Q of all rational numbers defined by:     | x | p = pk ba  = p1k , p

whenever x = pk ba in Q, where k, a ∈ Z, and b ∈ Z \ {0}. For instance,   8   = 23 × 1  = 13 = 1 , 3 2 3 8 2 2

and:

and:

  8   =  3−1 × 8  = 3 3 3 8   = 3 q

1 q0

1 3−1

= 3,

= 1, whenever q ∈ P \ {2, 3}.

The p-adic number field Q p is the maximal p-norm closure in Q. Therefore, under norm topology, it forms a Banach space (e.g., [7]). Let us understand the Banach field Q p as a measure space,   Q p = Q p , σ(Q p ), μ p , where σ (Q p ) is the σ-algebra of Q p consisting of all μ p -measurable subsets, where μ p is a left-and-right additive invariant Haar measure on Q p satisfying: μ p (Z p ) = 1, where Z p is the unit disk of Q p , consisting of all p-adic integers x satisfying | x | p Moreover, if we define: Uk = pk Z p = { pk x ∈ rQ p : x ∈ Z p },

≤

1. (1)

for all k ∈ Z (with U0 = Z p ), then these μ p -measurable subsets Uk ’s of (1) satisfy:

Q p = ∪ Uk , k ∈Z

and: μ p (Uk ) =

1 = μ p ( x + Uk ) , ∀ x ∈ Q p , pk

(2)

and:

· · · ⊂ U2 ⊂ U1 ⊂ U0 = Z p ⊂ U−1 ⊂ U−2 ⊂ · · ·. In fact, the family {Uk }k∈Z forms a basis of the Banach topology for Q p (e.g., [7]). Define now subsets ∂k of Q p by: ∂k = Uk \ Uk+1 , f or allk ∈ Z.

(3)

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

Q p = ∂k , k ∈Z

and: μ p (∂k ) = μ p (Uk ) − μ p (Uk+1 ) = and: 3

1 1 − k +1 , pk p

(4)

Mathematics 2019, 7, 199



if k1 = k2 , otherwise,

∂k1 ∅

∂k1 ∩ ∂k2 =

for all k, k1 , k2 ∈ Z, where is the disjoint union and ∅ is the empty set. Now, let M p be the algebra,   M p = C {χS : S ∈ σ(Q p )} , 

(5)



where χS are the usual characteristic functions of S ∈ σ Q p . Then the algebra M p of (5) forms a well-defined ∗-algebra over C, with its adjoint, ∗  ∑

S∈σ( G p )

de f

=

tS χS

∑

S∈σ( G p )

tS χS ,

where tS ∈ C, having their conjugates tS in C. Let ∑ tS χS ∈ M p . Then, one can define the p-adic integral by: S∈σ( G p )

⎛

Qp

⎞

⎝

∑

S∈σ (Q p )

tS χS ⎠ dμ p =

∑

S∈σ(Q p )

t S μ p ( S ).

(6)

Note that, by (4), if S ∈ σ(Q p ), then there exists a subset ΛS of Z, such that: ΛS = { j ∈ Z : S ∩ ∂ j = ∅},

(7)

satisfying:

 Qp

χS dμ p =



∑ χS∩∂ j dμ p

Qp

j∈ΛS

  = ∑ μ p S ∩ ∂j j∈ΛS

by (6)

≤

∑ μp

j∈ΛS

  ∂j =

∑



j∈ΛS

1 1 − j +1 pj p

 ,

(8)

by (4), for all S ∈ σ(Q p ), where ΛS is in the sense of (7). Proposition 1. Let S ∈ σ (Q p ), and let χS ∈ M p . Then, there exist r j ∈ R, such that: 0 ≤ r j ≤ 1inR, f orallj ∈ ΛS , and:

 Qp

Proof. The existence of r j = (9) is obtained by (8).

μ p (S∩∂ j ) μ p (∂ j )

χS dμ p = ∑ r j j∈ΛS



1 pj

−

1 p j +1

(9)  .

, for all j ∈ Z, is guaranteed by (7) and (8). The p-adic integral in

3. Free-Probabilistic Model on M p Throughout this section, fix a prime p ∈ P , and let Q p be the corresponding p-adic number field and M p be the ∗-algebra (5) consisting of μ p -measurable functions on Q p . Here, we establish a suitable (non-traditional) free-probabilistic model on M p implying p-adic analytic data. 4

Mathematics 2019, 7, 199

Let Uk be the basis elements (1) of the topology for Q p with their boundaries ∂k of (3), i.e., Uk = pk Z p , f orallk ∈ Z,

(10)

and: ∂k = Uk \ Uk+1 , for all k ∈ Z. Define a linear functional ϕ p : M p → C by the p-adic integration (6),

ϕp ( f ) =

Qp

Then, by (9) and (11), one obtains:   ϕ p χU j =

1 pj

f dμ p , f orall f ∈ M p .

  and ϕ p χ∂ j =

1 pj

−

(11)

1 , p j +1

for all j ∈ Z.   Definition 1. We call the pair M p , ϕ p the p-adic (non-traditional) free probability space for p ∈ P , where ϕ p is the linear functional (11) on M p .   Remark 1. As we discussed in Section 2.1, we study the measure-theoretic structure M p , ϕ p as a   free-probabilistic model on M p for our purposes. Therefore, without loss of generality, we regard M p , ϕ p   as a non-traditional free-probabilistic structure. In this sense, we call M p , ϕ p the p-adic free probability   space for p. The readers can understand M p , ϕ p as the pair of a commutative ∗-algebra M p and a linear functional ϕ p , having as its name the p-adic free probability space. Let ∂k be the kth boundary Uk \ Uk+1 of Uk in Q p , for all k ∈ Z. Then, for k1 , k2 ∈ Z, one obtains that: χ∂k χ ∂k = χ∂k 2

1

by (4), and hence,

  ϕ p χ ∂k χ∂k 2

1

1

∩∂k2

= δk1 ,k2 χ∂k , 1

  = δk1 ,k2 ϕ p χ∂k 1   = δk1 ,k2 1k1 − k11+1 , p

(12)

p

where δ is the Kronecker delta. Proposition 2. Let ( j1 , ..., jN ) ∈ Z N , for N ∈ N. Then: N

Π χ∂ j = δ( j1 ,...,jN ) χ∂ j in M p ,

l =1

and hence,

 ϕp



N

Π χ∂ j

l =1

1

l

l



= δ( j1 ,...,jN )

where:

 δ( j1 ,...,jN ) =

N −1

1 1 − j +1 p j1 p1 

Π δjl ,jl +1

l =1



 ,

(13)

 δjN ,j1 .

Proof. The proof of (13) is done by induction on (12).   Thus, one can get that, for any S ∈ σ Q p ,  ϕ p (χS ) = ϕ p

5



∑ χS∩∂ j

j∈ΛS

(14)

Mathematics 2019, 7, 199

where ΛS is in the sense of (7).     = ∑ ϕ p χS∩∂ j = ∑ μ p S ∩ ∂ j j∈ΛS   j∈ΛS = ∑ r j p1j − p j1+1 ,

(15)

j∈ΛS

by (13), where 0 ≤ r j ≤ 1 are in the sense of (9) for all j ∈ ΛS .   Furthermore, if S1 , S2 ∈ σ Q p , then: 

=

χ S1 χ S2

= =

 ∑ χ S1 ∩ ∂ k

k ∈ Λ S1

∑

(k,j)∈ΛS1 ×ΛS2

∑

j∈ΛS1 ,S2

∑ χ S2 ∩ ∂ j

j ∈ Λ S2

δk,j χ(S1 ∩S2 )∩∂ j

(16)

χ(S1 ∩S2 )∩∂ j ,

where: ΛS1 ,S2 = ΛS1 ∩ ΛS2 .   Proposition 3. Let Sl ∈ σ(Q p ), and let χSl ∈ M p , ϕ p , for l = 1, ..., N, for N ∈ N. Let: N

ΛS1 ,...,S N = ∩ ΛSl in Z, l =1

where ΛSl are in the sense of (7), for l = 1, ..., N. Then, there exist r j ∈ R, such that: 0 ≤ r j ≤ 1 in R, for j ∈ ΛS1 ,...,S N , 

and: ϕp

N



Π χ Sl

l =1

=



∑

j∈ΛS1 ,...,S N

rj

1 1 − j +1 pj p

 .

(17)

Proof. The proof of (17) is done by induction on (16) with the help of (15).   4. Representations of M p , ϕ p   Fix a prime p in P , and let M p , ϕ p be the p-adic free probability space. By understanding Q p as a measure space, construct the L2 -space H p ,     de f H p = L2 Q p , σ(Q p ), μ p = L2 Q p ,

(18)

over C. Then, this L2 -space H p of (18) is a well-defined Hilbert space equipped with its inner product 2 ,

de f

h1 , h2 2 =

Qp

h1 h2∗ dμ p ,

(19)

for all h1 , h2 ∈ H p . Definition 2. We call the Hilbert space H p of (18), the p-adic Hilbert space. By the definition (18) of the p-adic Hilbert space H p , our ∗-algebra M p acts on H p , via an algebra-action α p , α p ( f ) (h) = f h, f orallh ∈ H p , (20) for all f ∈ M p . 6

Mathematics 2019, 7, 199

p

p

Notation: Denote α p ( f ) of (20) by α f , for all f ∈ M p . Furthermore, for convenience, denote αχS   p simply by αS , for all S ∈ σ Q p .  By (20), the linear morphism α p is indeed a well-determined ∗-algebra-action of M p acting on H p p (equivalently, every α f is a ∗-homomorphism from M p into the operator algebra B( H p ) of all bounded operators on H p , for all f ∈ M p ), since: p

αf

1 f2

(h) = f 1 f 2 h = f 1 ( f 2 h)   p p p = f 1 α f 2 ( h ) = α f α f 2 ( h ), 1

for all h ∈ H p , implying that:

p

αf for all f 1 , f 2 ∈ M p ; and:   p α f ( h1 ), h2

p

1 f2

p

= α f α f2 ,

(21)

1

 = f h1 , h2 2 = Q p f h1 h2∗ dμ p   = Q p h1 f h2∗ dμ p = Q p h1 (h2 f ∗ )∗ dμ p    p = Q p h1 ( f ∗ h2 )∗ dμ p = h1 , α f ∗ (h2 ) ,

2

2

for all h1 , h2 ∈ H p , for all f ∈ M p , implying that: 

 p ∗

αf

= α f ∗ , f orall f ∈ M p ,

(22)

where 2 is the inner product (19) on H p . Proposition 4. The linear morphism α p of (20) is a well-defined ∗-algebra-action of M p acting on H p . Equivalently, the pair ( H p , α p ) is a Hilbert-space representation of M p . Proof. The proof is done by (21) and (22).   Definition 3. The Hilbert-space representation H p , α p is said to be the p-adic representation of M p . Depending on the p-adic representation ( H p , α p ) of M p , one can construct the C ∗ -subalgebra M p of the operator algebra B( H p ). Definition 4. Define the C ∗ -subalgebra M p of the operator algebra B( H p ) by:     de f p Mp = α p Mp = C α f : f ∈ Mp ,

(23)

where X mean the operator-norm closures of subsets X of B( H p ). Then, this C ∗ -algebra M p is called the p-adic   C ∗ -algebra of the p-adic free probability space M p , ϕ p . 5. Free-Probabilistic Models on M p

  Throughout this section, let us fix a prime p ∈ P , and let M p , ϕ p be the corresponding p-adic   free probability space. Let H p , α p be the p-adic representation of M p , and let M p be the p-adic   C ∗ -algebra (23) of M p , ϕ p . We here construct suitable free-probabilistic models on M p . In particular, we are interested in a p system { ϕ j } j∈Z of linear functionals on M p , determined by the jth boundaries {∂ j } j∈Z of Q p . p

Define a linear functional ϕ j : M p → C by a linear morphism, p

de f

ϕ j ( a) =

    a χ∂ j , χ∂ j , 2

7

(24)

Mathematics 2019, 7, 199

for all a ∈ M p , for all j ∈ Z, where 2 is the inner product (19) on the p-adic Hilbert space H p of (18). Remark that if a ∈ M p , then: a=

∑

S ∈ σ (Q p )

p

tS αS , in M p

(with tS ∈ C), where ∑ is a finite or infinite (i.e., limit of finite) sum(s) under the C ∗ -topology for M p . p Thus, the linear functionals ϕ j of (24) are well defined on M p , for all j ∈ Z, i.e., for any fixed j ∈ Z, one has that:           p   ϕ j ( a) =  ∑ tS χS∩∂ j , χ∂ j  2  S ∈ σ (Q p )        (25) =  ∑ tS μ p χS∩∂ j   S ∈ σ (Q p )           ≤ μ p ∂ j  ∑ tS  ≤ p1j − p j1+1  a ,  S ∈ σ (Q p )  where:    a = sup  a(h)2 : h ∈ H p with h2 = 1 is the C ∗ -norm on M p (inherited by the operator norm on the operator algebra B( H p )), and .2 is the Hilbert-space norm,   f 2 = f , f 2 , ∀ f ∈ H p , induced by the inner product 2 of (19). Therefore, for any fixed integer j ∈ Z, the corresponding p linear functional ϕ j of (24) is bounded on M p . Definition 5. Let j ∈ Z, and let ϕ j be the linear functional (24) on the p-adic C ∗ -algebra M p . Then, the pair   p M p , ϕ j is said to be the jth p-adic (non-traditional) C ∗ -probability space. p

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

    p Fix j ∈ Z, and take the corresponding jth p-adic C ∗ -probability space M p , ϕ j . For S ∈ σ Q p p

and a generating operator αS of M p , one has that: p

p

ϕ j (αS )

    p = α S ( χ ∂ j ), χ ∂ j = χ S ∩ ∂ j , χ ∂ j 2 2   = Q p χS∩∂ j χ∗∂ j dμ p = Q p χS∩∂ j χ∂ j dμ p

by (19)

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



χS∩∂ j dμ p = Q p  rS p1j − p j1+1 ,

  μ p S ∩ ∂j

(26)

(27)

    p p p Proposition 5. Let S ∈ σ Q p and αS = αχS ∈ M p , ϕ j , for a fixed j ∈ Z. Then, there exists rS ∈ R, such that: 0 ≤ rS ≤ 1 in R, 8

Mathematics 2019, 7, 199

and: p

ϕj



  p n

αS



= rS

1 1 − j +1 pj p

 , f or all n ∈ N.

(28)

p

Proof. Remark that the generating operator αS is a projection in M p , in the sense that:  ∗  2 p p p αS = αS = αS , in M p , so,



 p n

αS

p

= αS , for all n ∈ N.

Thus, for any n ∈ N, we have:  n   p p p p = ϕ j (αS ) = rS p1j − ϕj αS

1 p j +1

 ,

for some 0 ≤ rS ≤ 1 in R, by (27). As a corollary of (28), one obtains the following corollary. Corollary 1. Let ∂k be the kth boundaries (10) of Q p , for all k ∈ Z. Then: p

ϕj



n 

p

α∂

k



= δj,k

1 1 − j +1 pj p

 (29)

for all n ∈ N, for all j ∈ Z. Proof. The formula (29) is shown by (28). 6. Semigroup C ∗ -Subalgebras S p of M p Let M p be the p-adic C ∗ -algebra (23) for an arbitrarily-fixed p ∈ P . Take operators: p

Pp,j = α∂ ∈ M p , j

(30)

where ∂ j are the jth boundaries (10) of Q p , for the fixed prime p, for all j ∈ Z. Then, these operators Pp,j of (30) are projections on the p-adic Hilbert space H p in M p , i.e., ∗ = P = P2 , Pp,j p,j p,j

for all j ∈ Z. We now restrict our interest to these projections Pp,j of (30). Definition 6. Fix p ∈ P . Let S p be the C ∗ -subalgebra:     S p = C ∗ { Pp,j } j∈Z = C { Pp,j } j∈Z o f M p ,

(31)

where Pp,j are projections (30), for all j ∈ Z. We call this C ∗ -subalgebra S p the p-adic boundary (C ∗ -)subalgebra of M p . The p-adic boundary subalgebra S p of the p-adic C ∗ -algebra M p satisfies the following structure theorem. Proposition 6. Let S p be the p-adic boundary subalgebra (31) of the p-adic C ∗ -algebra M p . Then:   ∗-iso ∗-iso S p = ⊕ C · Pp,j = C⊕|Z| , j∈Z

in M p . 9

(32)

Mathematics 2019, 7, 199

Proof. The proof of (32) is done by the mutual orthogonality of the projections { Pp,j } j∈Z in M p . Indeed, one has: p

p

p

Pp,j1 Pp,j2 = α∂ α∂ = α∂ j1

j2

j1 ∩ ∂ j2

= δj1 ,j2 Pp,j1 ,

in S p , for all j1 , j2 ∈ Z. p

Define now linear functionals ϕ j (for a fixed prime p) by: ( p)

ϕj

p

= ϕ j |S p onS p ,

(33)

p

where ϕ j in the right-hand side of (33) are the linear functionals (24) on M p , for all j ∈ Z. 7. Weighted-Semicircular Elements Let M p be the p-adic C ∗ -algebra, and let S p be the p-adic boundary subalgebra (31) of M p , satisfying the structure theorem (32). Fix p ∈ P . Recall that the generating projections Pp,j of S p satisfy: ( p)

ϕj



 1 1 Pp,j = j − j+1 , ∀ j ∈ Z, p p

(34)

by (33) (also see (28) and (29)). Now, let φ be the Euler totient function, an arithmetic function: φ : N → C, defined by: φ(n) = |{k ∈ N : k ≤ n, gcd(n, k) = 1}| ,

(35)

for all n ∈ N, where | X | mean the cardinalities of sets X and gcd is the greatest common divisor. It is well known that:    , 1 − 1q Π φ(n) = n q∈P , q|n

for all n ∈ N, where “q | n” means “q divides n.” For instance,   1 φ( p) = p − 1 = p 1 − , ∀p ∈ P. p Thus: ( p)

ϕj



Pp,j



= =



1 pj

p p j +1

− 

1 p j +1

1−



1 p

=



=

1 pj



1−

1 p

(36)



φ( p) , p j +1

by (34), (35), and (36), for all Pp,j ∈ S p . More generally, ( p)

ϕj



   φ( p) Pp,k = δj,k , ∀ p ∈ P , k ∈ Z. p j +1 p

(37)

Now, for a fixed prime p, define new linear functionals τj on S p , by linear morphisms satisfying that: 1 p ( p) ϕ , onS p , (38) τj = φ( p) j p

for all j ∈ Z, where ϕ j are in the sense of (33).

10

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Then, one obtains new (non-traditional) C ∗ -probabilistic structures,   p {S p ( j) = S p , τj : p ∈ P , j ∈ Z},

(39)

p

where τj are in the sense of (38). p

Proposition 7. Let S p ( j) = (S p , τj ) be in the sense of (39), and let Pp,k be generating operators of S p ( j), for p ∈ P , j ∈ Z. Then:   δj,k p n τj Pp,k = j+1 , f oralln ∈ N. (40) p Proof. The formula (40) is proven by (37) and (38). Indeed, since Pp,k are projections in S p ( j),       p p n = τj Pp,k = δj,k p j1+1 , τj Pp,k for all n ∈ N, for all p ∈ P , and j, k ∈ Z. 7.1. Semicircular and Weighted-Semicircular Elements Let ( A, ϕ) be an arbitrary (traditional or non-traditional) topological ∗-probability space (C ∗ -probability space, or W ∗ -probability space, or Banach ∗-probability space, etc.), consisting of a (noncommutative, resp., commutative) topological ∗-algebra A (C ∗ -algebra, resp., W ∗ -algebra, resp., Banach ∗-algebra, etc.), and a (bounded or unbounded) linear functional ϕ on A. Definition 7. Let a be a self-adjoint element in ( A, ϕ). It is said to be even in ( A, ϕ), if all odd free moments of a vanish, i.e.,   ϕ a2n−1 = 0, f oralln ∈ N. (41) Let a be a “self-adjoint,” and “even” element of ( A, ϕ) satisfying (41). Then, it is said to be semicircular in ( A, ϕ), if: ϕ( a2n ) = cn , f oralln ∈ N, (42) where ck are the kth Catalan number, ck =

 1 k +1

2k k



=

1 (2k )! k +1 (k!)2

=

(2k)! , k!(k+1)!

for all k ∈ N0 = N ∪ {0}. It is well known that, if k n (...) is the free cumulant on A in terms of a linear functional ϕ (in the sense of [14]), then a self-adjoint element a is semicircular in ( A, ϕ), if and only if: ⎛

⎞



k n ⎝ a, a, ......, a⎠ =   

1 0

if n = 2 otherwise,

(43)

n-times

for all n ∈ N (e.g., see [14]). The above equivalent free-distributional data (43) of the semicircularity (42) are obtained by the Möbius inversion of [14]. Motivated by (43), one can define the weighted-semicircularity. Definition 8. Let a ∈ ( A, ϕ) be a self-adjoint element. It is said to be weighted-semicircular in ( A, ϕ) with its weight t0 (in short, t0 -semicircular), if there exists t0 ∈ C× = C \ {0}, such that: ⎛

⎞



k n ⎝ a, a, ...., a⎠ =    n-times

11

t0 0

if n = 2 otherwise,

(44)

Mathematics 2019, 7, 199

for all n ∈ N, where k n (...) is the free cumulant on A in terms of ϕ. By the definition (44) and by the Möbius inversion of [14], one obtains the following free-moment characterization (45) of the weighted-semicircularity (44): A self-adjoint element a is t0 -semicircular in ( A, ϕ), if and only if there exists t0 ∈ C× , such that: n

ϕ( an ) = ωn t02 c n , 2



where: ωn =

1 0

if n is even if n is odd,

(45)

for all n ∈ N, where cm are the mth Catalan numbers for all m ∈ N0 . Thus, from below, we use the weighted-semicircularity (44) and its characterization (45) alternatively. 7.2. Tensor Product Banach ∗-Algebra LS p   p Let S p (k) = S p , τk be a (non-traditional) C ∗ -probability space (39), for p ∈ P , k ∈ Z. Define bounded linear transformations c p and a p “acting on the p-adic boundary subalgebra S p of M p ,” by linear morphisms satisfying,   c p Pp,j = Pp,j+1 , and:

  a p Pp,j = Pp,j−1 ,

(46)

on S p , for all j ∈ Z. By (46), these linear transformations c p and a p are bounded under the operator-norm induced by the C ∗ -norm on S p . Therefore, the linear transformations c p and a p are regarded as Banach-space operators “acting on S p ,” by regarding S p as a Banach space (under its C ∗ -norm). i.e., c p and a p are   elements of the operator space B S p consisting of all bounded operators on the Banach space S p . Definition 9. The Banach-space operators c p and a p of (46) are called the p-creation, respectively,   the p-annihilation on S p , for p ∈ P . Define a new Banach-space operator l p ∈ B S p by: l p = c p + a p onS p .

(47)

We call it the p-radial operator on S p . Let l p be the p-radial operator c p + a p of (47) on S p . Construct a closed subspace L p of B(S p ) by: L p = C[l p ] ⊂ B(S p ),

(48)

where Y means the operator-norm-topology closure of every subset Y of the operator space B(S p ). By the definition (48), L p is not only a closed subspace, but also a well-defined Banach algebra embedded in the vector space B(S p ). On this Banach algebra L p , define the adjoint (∗) by: ∞

∞

k =0

k =0

∑ sk l kp ∈ L p −→ ∑ sk l kp ∈ L p ,

(49)

where sk ∈ C with their conjugates sk ∈ C. Then, equipped with the adjoint (49), this Banach algebra L p of (48) forms a Banach ∗-algebra inside B(S p ). Definition 10. Let L p be a Banach ∗-algebra (48) in the operator space B(S p ) for p ∈ P . We call it the p-radial (Banach-∗-)algebra on S p . 12

Mathematics 2019, 7, 199

Let L p be the p-radial algebra (48) on S p . Construct now the tensor product Banach ∗-algebra LS p by: (50) LS p = L p ⊗C S p , where ⊗C is the tensor product of Banach ∗-algebras (Remark that S p is a C ∗ -algebra and L p is a Banach ∗-algebra; and hence, the tensor product Banach ∗-algebra LS p of (50) is well-defined.). Take now a generating element l kp ⊗ Pp,j , for some k ∈ N0 = N ∪ {0}, and j ∈ Z, where Pp,j is in the sense of (30) in S p , with axiomatization: l 0p = 1S p , the identity operator on S p , in B(S p ), satisfying:

  1S p Pp,j = Pp,j , for all Pp,j ∈ S p ,

for all j ∈ Z. By (50) and (32), the elements l kp ⊗ Pp,j indeed generate LS p under linearity, because: 

l p ⊗ Pp,j

k

= l kp ⊗ Pp,j ,

for all k ∈ N0 , and j ∈ Z, for p ∈ P , and their self-adjointness. We now focus on such generating operators of LS p . Define a linear morphism: E p : LS p → S p by a linear transformation satisfying that:  j +1  k +1   p l kp ( Pp,j ), E p l kp ⊗ Pp,j = [ 2k ] + 1 for all k ∈ N0 , j ∈ Z, where

  k 2

(51)

is the minimal integer greater than or equal to 2k , for all k ∈ N0 ; for example, 3 2

=2=

  4 2

.

By the cyclicity (48) of the tensor factor L p of LS p , and by the structure theorem (32) of the other tensor factor S p of LS p , the above morphism E p of (51) is a well-defined bounded surjective linear transformation. Now, consider how our p-radial operator l p of (47) works on S p . Observe first that: if c p and a p are the p-creation, respectively, the p-annihilation on S p , then:     c p a p Pp,j = Pp,j = a p c p Pp,j , for all j ∈ Z, p ∈ P , and hence: c p a p = 1S p = a p c p onS p .

(52)

Lemma 1. Let c p , a p be the p-creation, respectively, the p-annihilation on S p . Then:  n  n cnp anp = c p a p = 1S p = a p c p = a p c p , and:

n

n

c p1 anp2 = anp2 c p1 onS p , for all n, n1 , n2 ∈ N0 . Proof. The formula (53) holds by (52). By (53), one can get that: 13

(53)

Mathematics 2019, 7, 199

n  l np = c p + a p = ∑nk=0



n k

ckp anp−k on S p ,

with identities; c0p = 1S p = a0p , for all n ∈ N, where:



n k

(54)



=

n! , k!(n−k )!

∀ k ≤ n ∈ N0 .

Thus, one obtains the following proposition. Proposition 8. Let l p ∈ L p be the p-radial operator on S p . Then: −1 does not contain 1S p − term, and l 2m p

 l 2m p contains its 1S p − term,

2m m

(55)



· 1S p ,

(56)

for all m ∈ N. Proof. The proofs of (55) and (56) are done by straightforward computations by (53) and (54). See [1] for more details. 7.3. Weighted-Semicircular Elements Q p,j in LS p Fix p ∈ P , and let LS p = L p ⊗C S p be the tensor product Banach ∗-algebra (50) and E p be the linear transformation (51) from LS p onto S p . Throughout this section, fix a generating element: Q p,j = l p ⊗ Pp,j o f LS p ,

(57)

for j ∈ Z, where Pp,j is a projection (30) generating S p . Observe that:  n Qnp,j = l p ⊗ Pp,j = l np ⊗ Pp,j ,

(58)

for all n ∈ N, for all j ∈ Z. If Q p,j ∈ LS p is in the sense of (57) for j ∈ Z, then:  j +1  n +1       p l np Pp,j , E p Qnp,j = E p l np ⊗ Pp,j =  n  2 +1

(59)

by (51) and (58), for all n ∈ N. 0 on LS by: Now, for a fixed j ∈ Z, define a linear functional τp,j p p

0 = τj ◦ E p onLS p , τp,j p

where τj =

1 φ( p)

( p)

ϕj

(60)

is the linear functional (38) on S p . p

0 of (60) is a bounded linear By the bounded-linearity of both τj and E p , the morphism τp,j functional on LS p . By (59) and (60), if Q p,j is in the sense of (57), then:

  0 Qnp,j τp,j

=

n +1   ( p j +1 ) p l np ( Pp,j ) , n +1 τj [2]

14

(61)

Mathematics 2019, 7, 199

for all n ∈ N.   0 Theorem 1. Let Q p,j = l p ⊗ Pp,j ∈ LS p , τp,j , for a fixed j ∈ Z. Then, Q p,j is p2( j+1) -semicircular in   0 LS p , τp,j . More precisely,    n 2 0 Qnp,j = ωn p2( j+1) c n , τp,j (62) 2

for all n ∈ N, where ωn are in the sense of (45). Equivalently, if 0 of (61) on LS , then: of the linear functional τp,j p ⎛ 0,p,j kn

⎞

⎜ ⎟ ⎝ Q p,j , Q p,j , ..., Q p,j ⎠ =   



0,p,j k n (...)

p 2( j +1) 0

is the free cumulant on LS p in terms

if n = 2 otherwise,

(63)

n-times

for all n ∈ N. Proof. The free-moment formula (62) is obtained by (55), (56) and (61). The free-cumulant formula (63) is obtained by (62) under the Möbius inversion of [14]. See [1] for details. 8. Semicircularity on LS For all p ∈ P , j ∈ Z, let:

  0 LS p ( j) = LS p , τp,j

(64)

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

Definition 11. We call the pairs LS p ( j) of (64) the jtextth p-adic filters, for all p ∈ P , j ∈ Z. Let Q p,k = l p ⊗ Pp,k be the kth generating elements of the jth p-adic filter LS p ( j) of (64), for all k ∈ Z, for fixed p ∈ P , j ∈ Z. Then, the generating elements { Q p,k }k∈Z of the jth p-adic filter LS p ( j) satisfy that:    δj,k p2( j+1) if n = 2 0,p,j Q p,k , ..., Q p,k = kn 0 otherwise, and:

    n  2 0 Qnp,k = δj,k ωn p2( j+1) c n , τp,j 2

(65)

for all p ∈ P , j ∈ Z, for all n ∈ N, by (62) and (63), where:  1 if n is even ωn = 0 if n is odd, for all n ∈ N. For the family:

!

  " 0 : p ∈ P, j ∈ Z LS p ( j) = LS p , τp,j

of jth p-adic filters of (64), one can define the free product Banach ∗-probability space, denote

LS =



LS, τ 0

 de f =

as in [14,15], with: 15



p∈P , j∈Z

LS p ( j).

(66)

Mathematics 2019, 7, 199

LS =



p∈P , j∈Z

LS p , and τ 0 =

 τ0 . p∈P , j∈Z p,j

  Note that the pair LS = LS, τ 0 of (66) is a well-defined “traditional or noncommutative” Banach ∗-probability space. For more about the (free-probabilistic) free product of free probability spaces, see [14,15].   Definition 12. The Banach ∗-probability space LS = LS, τ 0 of (66) is called the free Adelic filtration. Let LS be the free Adelic filtration (66). Then, by (65), one can take a subset:   Q = Q p,j = l p ⊗ Pp,j ∈ LS p ( j) p∈P , j∈Z of LS, consisting of “jth ” generating elements Q p,j of the “jth ” p-adic filters LS p ( j), which are the free blocks of LS, for all j ∈ Z, for all p ∈ P . Lemma 2. Let Q be the above family in the free Adelic filtration LS. Then, all elements Q p,j of Q are p2( j+1) -semicircular in the free Adelic filtration LS. Proof. Since all self-adjoint elements Q p,j of the family Q are chosen from mutually-distinct free blocks LS p ( j) of LS, they are p2( j+1) -semicircular in LS p ( j). Indeed, since every element Q p,j ∈ Q is from a free block LS p ( j), the powers Qnp,j are free reduced words with their lengths-N in LS p ( j) in LS. Therefore, each element Q p,j ∈ Q satisfies that:     0 τ 0 Qnp,j = τp,j Qnp,j = ωn pn( j+1) c n , 2

equivalently,   k0n Q p,j , ..., Q p,j

0,p,j

= kn 

=



Q p,j , ..., Q p,j

p 2( j +1) 0



if n = 2 otherwise,

for all n ∈ N, by (62) and (63), where k0n (...) is the free cumulant on LS in terms of τ 0 . Therefore, all elements Q p,j ∈ Q are p2( j+1) -semicircular in LS, for all p ∈ P , j ∈ Z. Furthermore, since all p2( j+1) -semicircular elements Q p,j ∈ Q are taken from the mutually-distinct free blocks LS p ( j) of LS, they are mutually free from each other in the free Adelic filtration LS of (66), for all p ∈ P , j ∈ Z. Recall that a subset S = { at }t∈Δ of an arbitrary (topological or pure-algebraic) ∗-probability space ( A, ϕ) is said to be a free family, if, for any pair (t1 , t2 ) ∈ Δ2 of t1 = t2 in a countable (finite or infinite) index set Δ, the corresponding free random variables at1 and at2 are free in ( A, ϕ) (e.g., [7,14]). Definition 13. Let S = { at }t∈Δ be a free family in an arbitrary topological ∗-probability space ( A, ϕ). This family S is said to be a free (weighted-)semicircular family, if it is not only a free family, but also a set consisting of all (weighted-)semicircular elements at in ( A, ϕ), for all t ∈ Δ. Therefore, by the construction (66) of the free Adelic filtration LS, we obtain the following result. Theorem 2. Let LS be the free Adelic filtration (66), and let:

Q = { Q p,j ∈ LS p ( j)} p∈P , j∈Z ⊂ LS,

(67)

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

16

Mathematics 2019, 7, 199

Proof. Let Q be a subset (67) in LS. Then, all elements Q p,j of Q are p2( j+1) -semicircular in LS by the above lemma, for all p ∈ P , j ∈ Z. Furthermore, all elements Q p,j of Q are mutually free from each other in LS, because they are contained in the mutually-distinct free blocks LS p ( j) of LS, for all p ∈ P , j ∈ Z. Therefore, the family Q of (67) is a free weighted-semicircular family in LS. Now, take elements:

1 Q p,j , ∀ p ∈ P , j ∈ Z, p j +1

de f

Θ p,j =

(68)

in LS, where Q p,j ∈ Q, where Q is the free weighted-semicircular family (67) in the free Adelic filtration LS. Then, by the self-adjointness of Q p,j , these operators Θ p,j of (68) are self-adjoint in LS, as well, because: p j+1 ∈ Q ⊂ R in C, satisfying p j+1 = p j+1 , for all p ∈ P , j ∈ Z. Furthermore, one obtains the following free-cumulant computation; if k0n (...) is the free cumulant on LS in terms of τ 0 , then:     0,p,j 1 1 = kn k0n Θ p,j , ..., Θ p,j j+1 Q p,j , ..., p j+1 Q p,j p  n (69)  0,p,j  1 = kn Q p,j , ..., Q p,j , p j +1 0,p,j

by the bimodule-map property of the free cumulant (e.g., see [14]), for all n ∈ N, where k n (...) are the 0 of (60) on LS , free cumulants (63) on the free blocks LS p ( j) in terms of the linear functionals τp,j p for all p ∈ P , j ∈ Z. Theorem 3. Let Θ p,j = Then, the family:

1 Q p j+1 p,j

be free random variables (68) of the free Adelic filtration LS, for Q p,j ∈ Q.   Θ = Θ p,j ∈ LS p ( j) : p ∈ P , j ∈ Z

(70)

forms a free semicircular family in LS. Proof. Consider that:

n     0,p,j  kn Q p,j , ..., Q p,j by (69) k0n Θ p,j , ..., Θ p,j = p j1+1 ⎧  2  0,p,j  ⎪ 1 ⎪ Q p,j , Q p,j if n = 2 ⎪ ⎨ p j +1 k 2 = n  ⎪  ⎪ 0,p,j  ⎪ 1 ⎩ k = 0 otherwise, Q , ..., Q p j +1

n

p,j

p,j

by the p2( j+1) -semicircularity of Q p,j ∈ Q in LS:

=

⎧  ⎨ ⎩ 0

1

p j +1

2 

p j +1

2

= 1 if n = 2

(71)

otherwise,

for all n ∈ N. By the free-cumulant computation (71), these self-adjoint free random variables Θ p,j ∈ LS p ( j) are semicircular in LS by (43), for all p ∈ P , j ∈ Z. Furthermore, the family Θ of (70) forms a free family in LS, because all elements Θ p,j are the scalar-multiples of Q p,j ∈ Q, contained in mutually-distinct free blocks LS p ( j) of LS, for all j ∈ Z, p ∈ P. Therefore, this family Θ of (70) is a free semicircular family in LS.

17

Mathematics 2019, 7, 199

Now, define a Banach ∗-subalgebra LS of LS by: de f

LS = C[Q]inLS,

(72)

where Q is the free weighted-semicircular family (67) and Y means the Banach-topology closures of subsets Y of LS. Then, one can obtain the following structure theorem for the Banach ∗-algebra LS of (72) in LS. Theorem 4. Let LS be the Banach ∗-subalgebra (72) of the free Adelic filtration LS generated by the free weighted-semicircular family Q of (67). Then:

LS = C [Θ]inLS,

(73)

where Θ is the free semicircular family (70) and where “=” means “being identically same as sets.” Moreover, ∗-iso

LS =



p∈P , j∈Z

  ∗-iso C { Q p,j } = C

'



p∈P , j∈Z

( { Q p,j } ,

(74)

∗-iso

in LS, where “ = ” means “being Banach-∗-isomorphic,” and:   C { Q p,j } are Banach ∗-subalgebras of LS p ( j), for all p ∈ P , j ∈ Z, in LS. Here, () in the first ∗-isomorphic relation of (74) is the (free-probability-theoretic) free product of [14,15], and () in the second ∗-isomorphic relation of (74) is the (pure-algebraic) free product (generating noncommutative algebraic free words in the family Q). Proof. Let LS be the Banach ∗-subalgebra (72) of LS. Then, all generating operators Q p,j ∈ Q of LS are contained in mutually-distinct free blocks LS p ( j) of LS, and hence, the Banach ∗-subalgebras   C { Q p,j } of LS are contained in the free blocks LS p ( j), for all p ∈ P , j ∈ Z. Therefore, as embedded sub-structures of LS, they are free from each other. Equivalently, ∗-iso

LS =



p∈P , j∈Z

  C { Q p,j } inLS,

(75)

by (66).   Since every free block C { Q p,j } of the Banach ∗-algebra LS of (75) is generated by a single self-adjoint (weighted-semicircular) element, every operator T of LS is a limit of linear combinations of free words in the free family Q of (67), which form noncommutative free “reduced” words (in the sense of [14,15]), as operators in LS of (75). Note that every (pure-algebraic) free word in Q has a unique free reduced word in LS, under operator-multiplication on LS (and hence, on LS). Therefore, the ∗-isomorphic relation (75) guarantees that: '

∗-iso

LS = C



p∈P , j∈Z

( { Q p,j } ,

(76)

where the free product () in (76) is pure-algebraic. Remark that, indeed, the relation (76) holds well, because all weighted-semicircular elements of Q are self-adjoint; if: N

n

T = Π Q pl ,j ∈ LS l =1

l l

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

18

Mathematics 2019, 7, 199

N

n

T ∗ = Π Q p N −l +1 ,j l =1

N − l +1 N − l +1

∈ LS

is a free word of LS in Q, as well. Therefore, by (75) and (76), the structure theorem (74) holds true. Note now that Q p,j ∈ Q satisfy: Q p,j = p j+1 Θ p,j , for all p ∈ P , j ∈ Z, where Θ p,j are semicircular elements in the family Θ of (70). Therefore, the free blocks of (75) satisfy that:       C { Q p,j } = C { p j+1 Θ p,j } = C {Θ p,j } , (77) for all p ∈ P , j ∈ Z. Thus, one can get that:

∗-iso

LS =



p∈P , j∈Z

  C {Θ p,j } ,

(78)

by (75) and (77). With similar arguments of (75), we have:

LS = C[Θ], set − theoretically,

(79)

by (78). Therefore, the identity (73) holds true by (79). As a sub-structure, one can restrict the linear functional τ 0 of (66) on LS to that on LS, i.e., one can obtain the Banach ∗-probability space, 

denote

LS, τ 0 =

 τ 0 |LC .

(80)

Definition 14. Let (LS, τ 0 ) be the Banach ∗-probability space (80). Then, we call (LS, τ 0 ) the semicircular (free Adelic sub-)filtration of LS. Note that, by (66), all elements of the semicircular filtration (LS, τ 0 ) provide “possible” non-vanishing free distributions in the free Adelic filtration LS. Especially, all free reduced words of LS in the generator set { Q p,j } p∈P,j∈Z have non-zero free distributions only if they are contained in (LS, τ 0 ). Therefore, studying free-distributional data on (LS, τ 0 ) is to study possible non-zero free-distributional data on LS. 9. Truncated Linear Functionals on LS In number theory, one of the most interesting, but difficult topics is to find a number of primes or a density of primes contained in closed intervals [t1 , t2 ] of the real numbers R (e.g., [3,6,21,22]). Since the theory is deep, we will not discuss more about it here. Hhowever, motivated by the theory, we consider certain “suitable” truncated linear functionals on our semicircular filtration (LS, τ 0 ) of (80) in the free Adelic filtration LS of (66). Notation: From below, we will use the following notations to distinguish their structural differences; denote LS = the Banach ∗-subalgebra (72) of LS,   denote LS0 = the semicircular filtration LS, τ 0 of (80). 

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Mathematics 2019, 7, 199

9.1. Linear Functionals {τ(t) }t∈R on LS Let LS0 be the semicircular filtration (LS, τ 0 ) of the free Adelic filtration LS. Furthermore, let Q and Θ be the free weighted-semicircular family (67), respectively, the free semicircular family (70) of LS, generating LS by (73) and (74). We here truncate τ 0 on LS for a fixed real number t ∈ R. First, recall and remark that: τ0 =



p∈P , j∈Z

0 on LS, τp,j

by (66) and (80). Therefore, one can sectionize τ 0 over P , as follows; τ 0 =  τp0 on LS, p∈P

with: 0 onLS p , f orall p ∈ P , τp0 =  τp,j j∈Z

where:

de f

LS p =  C[{Θ p,j }] ⊂ LS ⊂ LS, j∈Z

(81)

(82)

for each p ∈ P , under (74). From below, we understand the Banach ∗-subalgebras LS p of LS as free-probabilistic sub-structures, denote

LS( p) =



 LS p , τp0 , f orall p ∈ P .

(83)

Lemma 3. Let LS pl be in the sense of (82) in the semicircular filtration LS0 , for l = 1, 2. Then, LS p1 and LS p2 are free in LS0 , if and only if p1 = p2 in P . Proof. The proof is directly done by (81) and (82). Indeed,   LS =  C {Θ p,j } p∈P , j∈Z



= 

p∈P

   C {Θ p,j }



j∈Z

=  LS p , p∈P

by (80) and (82). Therefore, LS p1 and LS p2 are free in LS0 , if and only if p1 = p2 in P . Fix now t ∈ R, and define a new linear functional τ(t) on LS by:

de f

τ(t) =

⎧  τp0 ⎪ ⎪ ⎪ ⎨ p≤t ⎪ ⎪ ⎪ ⎩ O

on  LS p ⊂ LS p≤t

 on LS \

  LS p ,

(84)

p≤t

where τp0 are the linear functionals (81) on the Banach ∗-subalgebras LS p of (82) in LS0 , for all p ∈ P , and O means the zero linear functional on LS, satisfying that: O( T ) = 0, for all T ∈ LS. For convenience, if there is no confusion, we simply write the definition (84) as: denote

τ(t) =

20

 τp0 .

p≤t

(85)

Mathematics 2019, 7, 199

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

τ(t) =

τ(t) of (84) O

if t ≥ 2 if t < 2,

(86)

for all t ∈ R. In the following text, τ(t) mean the linear functionals in (86), satisfying (84) whenever t ≥ 2, for all t ∈ R. In fact, we are not interested in the cases where t < 2. For example, τ √3 = O, τ(2.1003) = τ20 , andτ(6) = τ20  τ30  τ50 , (

2

)

on LS, under (85), etc. Theorem 5. Let Q p,j ∈ Q and Θ p,j ∈ Θ in the semicircular filtration LS0 , for p ∈ P , j ∈ Z, and let t ∈ R and τ(t) , the corresponding linear functional (86) on LS. Then:    if t ≥ p ω n p 2( j +1) c n 2 τ(t) Qnp,j = 0 if t < p, and: τ(t)



Θnp,j





=

ωn c n 2 0

if t ≥ p if t < p,

(87)

for all n ∈ N. Proof. By the p2( j+1) -semicircularity of Q p,j ∈ Q, the semicircularity of Θ p,j ∈ Θ in the semicircular filtration LS0 , and by the definition (86), if t ≥ p in R, then:       0 Qnp,j = τp0 Qnp,j = τp,j τ(t) Qnp,j

= ωn p2( j+1) c n2 ,

and:

  τ(t) Θnp,j

    0 Θnp,j = τp0 Θnp,j = τp,j

= ωn c n2 , by (62), (71), and (81), for all n ∈ N. If t < p, then: τ(t) = Therefore, in such cases,



2≤q1 = (N \ {1}) ∪ { ∞ }, then either (I) or (II) holds true, where: ∗-iso (I) L( Fn ) = L( F∞ ), for all n ∈ N∞ >1 , ∗-iso

(II) L( Fn1 ) = L( Fn2 ), if and only if n1 = n2 ∈ N∞ >1 , ∗-iso

where “ = ” means “being W ∗ -isomorphic.” Depending on the author’s knowledge, he does not know which one is true at this moment. We here have similar troubles. Under the similar difficulties, we are not sure at this moment that LSt and LS(t) (or LSt and LS) are ∗-isomorphic or not (and hence, free-isomorphic or not). However, definitely, LSt is free-homomorphic “into” LS(t) in the semicircular filtration LS0 , by the above theorem. The above free-homomorphic relation (90) lets us understand all “non-zero” free distributions of free reduced words of LS(t) as those of LSt , for all t ∈ R, by the injectivity of a free-homomorphism Φt of (91). Corollary 2. All free reduced words T of the semicircular t-filtration LS(t) in Q ∪ Θ, having non-zero free distributions, are contained in the Banach ∗-probability space LSt of (89), whenever t is suitable. The converse holds true, as well. Proof. The proof of this characterization is done by (87), (89), and (90). In particular, the injectivity of the free-homomorphism Φt of (91) guarantees that this characterization holds. Therefore, whenever we consider a non-zero free-distribution having free reduced words T of semicircular t-filtrations LS(t) , they are regarded as free random variables of the Banach ∗-probability spaces LSt of (89), for all suitable t ∈ R. 9.2. Truncated Linear Functionals τt1