Abstract

In this paper, we study “semicircular-like” elements in free product Banach *-algebras induced by Haar-measurable functions over p-adic number fields , 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.

Keywords: free probability p-Adic number fields Hilbert-space representations weighted-semicircular elements semicircular elements weight ratios

1. 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 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.

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 on a Hecke algebra for a fixed prime p.

In ², we considered certain free random variables in a *-algebra consisting of all measurable functions over for primes and its Hilbert-space representation. Under representation, corresponding -algebras of are constructed, and free probability on is studied there. In particular, for all we define -probability spaces where are kind of sectionized linear functional implying the number-theoretic free-distributional data on in terms of the usual p-adic integration on Moreover, from the system

of -probability spaces, we constructed, and studied free probability on the free product -probability space,

called the Adelic -probability space.

Motivated by the main results of ², in the paper ⁴, by using the free-probabilistic information from a single -probability space we established weighted-semicircular elements in a free product Banach -probability space generated by and realized that these operators generate semicircular elements.

In Sections 2, we briefly introduce backgrounds and a motivation of our proceeding works.

Our free-probabilistic models on is established and considered in Sections 3. And then, in Section 4, we construct suitable Hilbert-space representations of the free-probabilistic models of Section 3, preserving the free-distributional data implying number-theoretic information. Under representation, corresponding -algebras are constructed from

In Section 5, we consider -subalgebraic structures in generated by mutually-orthogonal -many projections, sectionizing or filterizing and study free probability on them, for primes

In Sections 6, we construct certain Banach *-probability spaces by defining so-called the radial operators on And we realize that the generating operators of are weighted-semicircular under suitable linear functionals on for every prime These weighted-semicircular elements generate semicircular elements in

In Section 7, we enlarge our weighted-semicircularity and semicircularity under free product Banach *-probability space of '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.

2. Preliminaries

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

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}). In particular, we will use combinatorial approach of Speicher (e.g., ¹²).

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 -probability spaces in the sense of ¹² and ¹³, without detailed introduction. However, rough introduction would be given whenever they are needed in text.

2.2. Calculus on

For more about p-adic, or Adelic analysis, see ¹³. Let be the p-adic number fields for equipped with the non-Archimedean p-norms (on ), where is the set of all primes in the natural numbers (or the positive integers) This Banach space is also understood as a measure space

equipped with the left-and-right additive invariant Haar measure on the -algebra Recall also that, is a well-defined ring algebraically. If then

for some i.e.,

If and hence, if in then is said to be a p-adic integer of The subset of consisting of all p-adic integers, is called the unit disk of Indeed, every p-adic integer satisfies i.e.,

Remark that and all -measurable subsets are either finite-or-infinite unions, or finite intersections of the subsets formed by

As a topological space, the p-adic number field contains its basis elements

(2.2.1)

satisfying the basis property,

and the chain property,

and the measure-theoretic data,

for all

By understanding as a measure space, one can establish a *-algebra over as a *-algebra consisting of all -measurable functions f,

where the sum means a finite sum, and are the usual characteristic functions of Of course, the adjoint of is defined to be

where mean the conjugates of for all

On one can naturally define a linear functional

(2.2.2)

and hence, the pair forms a well-determined *-probability space.

Remark 2.1. Remark that our *-probabilistic structure is a “commutative” measure-theoretic structure which is not under usual free-probability-theory point of view. However, free probability theory naturally covers measure theory on commutative algebras, and hence, we use the terminology, “*-probability space,” under enlarged sense in this text.

Define now subsets of by

(2.2.3)

We call such -measurable subsets the k-th boundaries of the basis elements of (2.2.1), which are also -measurable subsets, for all By the basis property in (2.2.1), one obtains that

(2.2.4)

where means the disjoint union. Also, by measure-theoretic data in (2.2.1), one has

for all

Now, let be as above, and let Then, by definition, is expressed by

(2.2.5)

where P means the finite sum, and are the usual characteristic functions of

If is expressed by (2.2.5), then the adjoint is determined to be

where having their conjugates in

Let Then

(2.2.6)

by (2.2.2).

Note that, by (2.2.4), if then there exists a subset of such that

(2.2.7)

Thus, by (2.2.6) and (2.2.7), one obtains the following proposition.

Proposition 2.1. (See ²) Let and let Then there exist such that

(2.2.8)

and

where is in the sense of (2.2.7).

3. Free Probability on

Throughout this section, fix a prime and let be the corresponding p-adic number field, and let be the -algebra consisting of all measurable functions on In this section, let's establish a suitable free-probabilistic model on the -algebra Remark again that free probability provides a universal tool to study free distributions on “noncommutative” algebras, and hence, it covers the cases where given algebras are “commutative.” Even though is a commutative -algebra, for our purposes, we understand our p-adic-analytic settings on under free-probability language and terminology (See Remark 2.1).

Let be the basis elements (2.2.1) of the topology for i.e.,

(3.1)

with their boundaries

Define a linear functional by the p-adic integration (2.2.2), i.e.,

(3.2)

Then, by (3.2), one obtains that

since

with help of (2.2.7) and (2.2.8), for all

Definition 3.1. The free probability space is called the p-adic free probability space, for where is the linear functional (3.2) on

Let be in the sense of (3.1) in and for all Then

by (2.2.3), where means the maximum in

Say in Then in by (2.2.3). Therefore, in So, if in then

Thus, one can verify that

(3.3)

Inductive to (3.3), we obtain the following result.

Proposition 3.1. (See ²) Let for Then

(3.4)

and hence,

Now, let be the k-th boundary of in for all Then, for one obtains that

(3.5)

where means the Kronecker delta, and hence,

So, we obtain the following computations.

Proposition 3.2. Let for Then

(3.6)

and hence,

where

Proof. The proof of (3.6) is done by (3.5).

Thus, one can get that, for any

where is in the sense of (2.2.8)

(3.7)

where are in the sense of (2.2.8), for all

Also, if then

(3.8)

where

because

In (3.8), it is clear that, if is empty, then

where is the empty set in

Thus, one can get that there exist such that

(3.9)

where

by (3.8) and (2.2.10), for all

By (3.9), we obtain the following general result under induction.

Theorem 3.3. Let and let or for Let

where are in the sense of (2.2.7), for Then there exist such that

(3.10)

and

Proof. The proof of (3.10) is done by induction on (3.9). See ² for details.

4. Representations of

Fix a prime Let be the p-adic free probability space. Now, we construct a representation of By understanding as a measure space, construct the -space,

(4.1)

over consisting of all square-integrable -measurable functions on Then this -space is a well-defined Hilbert space equipped with its inner product

(4.2)

Naturally, is the -norm completion in where

where is the inner product (4.2) on

Definition 4.1. We call the Hilbert space of (4.1), the p-adic Hilbert space.

By the very construction (4.1) of the p-adic Hilbert space our -algebra acts on via an algebra-action

(4.3)

for all i.e., the morphism of (4.3) is an action of acting on the Hilbert space i.e., for any the image is a multiplication operator on with its symbol contained in the operator algebra of all (bounded linear) operators on

Notation Denote by for all Also, for convenience, denote simply by or all For instance,

and

for all where are in the sense of (3.1), and are the corresponding boundaries of in for all

It is not difficult to check that

and

Therefore, one obtains that:

Proposition 4.1. The pair is a well-determined Hilbert-space representation of

Definition 4.2. The Hilbert-space representation is said to be the p-adic (Hilbert-space) representation of

Depending on the p-adic representation of one can construct the -algebra in the operator algebra Here, is the operator algebra consisting of all (bounded linear) operators on equipped with the operator-norm,

for all where means the -norm (4.2)’ on

Definition 4.3. Let be the operator-norm closure of in the operator algebra i.e.,

(4.4)

where mean the operator-norm closures of subsets of Then the -algebra is called the p-adic -algebra of

5. Free Probability on

Throughout this section, let's fix a prime Let be the corresponding p-adic free probability space, and let be the p-adic representation of inducing the corresponding p-adic -algebra of (4.4). In this section, we consider suitable free-probabilistic models on In particular, we are interested in a system of linear functionals on determined by the j-th boundaries of

Define a linear functional by a linear morphism,

(5.1)

for all where is the inner product (4.2) on the p-adic Hilbert space of (4.1).

First, remark that, if then

where is finite or infinite (limit of finite) sum(s), under -topology of

Definition 5.1. Let and let be the linear functional (5.1) on the p-adic -algebra Then the -probability space is said to be the j-th (p-adic) -probability space.

So, one can get the system

of -probability spaces for a fixed -algebra

Now, fix and take the corresponding j-th -probability space For and an element one has that

(5.2)

for some in

Proposition 5.1. Let and for a fixed Then there exists such that

(5.3)

and

Proof. Remark that the element is a projection in in the sense that:

So,

Thus, by (5.2), we obtain (5.3).

The above proposition characterizes the free distributions of in the j-th -probability space for

More precisely, we obtain the following theorem.

Theorem 5.2. Let and for a fixed for for Then there exists such that

(5.4)

and

for all

Proof. Let be -measurable subsets of for and let

Then, one has that

satisfying

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 in the j-th -probability space for

As corollaries of (5.4), we obtain the following results.

Corollary 5.3. Let be in the sense of (3.1), and the k-th boundaries of in for all Then

(5.5)

and

for all for

6. Semigroup -Subalgebras of

Let be the p-adic -algebra for as in Section 5. Take operators

(6.1)

for all for

As we have seen, these operators are projections on the p-adic Hilbert space in i.e.,

for all We now restrict our interests to these projections of (6.1).

Definition 6.1. Fix Let be the -subalgebra

(6.2)

where are projections (6.1), for all We call this -subalgebra the p-adic boundary (-)subalgebra of

We originally defined such p-adic boundary subalgebras in ⁴.

Proposition 6.1. (See ⁴) Let be the p-adic boundary subalgebra (6.2) of the p-adic -algebra Then

(6.3)

in

By the structure theorem (6.3) of it acts like a diagonal subalgebra inside Since p-adic boundary subalgebras are -subalgebras of one can naturally get the -probability spaces

i.e., we have a family

(6.4)

where the linear functional are restrict linear functional of on for all and

7. Weighted-Semicircularity

Let be the p-adic -algebra, and let be the boundary subalgebra (8.2) of satisfying the structure theorem (6.3):

where are projections of (6.1) on for all

Then we have

(7.0.1)

Now, let be the Euler totient function, which is an arithmetic function

defined by

(7.0.2)

for all where mean the cardinalities of sets X, and gcd means the greatest common divisor.

It is well-known that

for all where means “ divides ” or “ is a divisor of ”

Thus, one has

(7.0.3)

by (7.0.2).

So, one can get that

(7.0.4)

by (7.0.1) and (7.0.3), for for all

Now, Define new linear functionals

by linear morphisms satisfying that

(7.0.5)

for all and inducing new -probability spaces

(7.0.6)

as in (6.4), where are in the sense of (7.0.5).

Proposition 7.1. Let be a -probability space (7.0.6), and let be generating projections of for all Then

(7.0.7)

Proof. The free-moment formula (7.0.7) is proven by (7.0.5), because are projections in

Let be an arbitrary topological *-probability space (-probability space, or -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 on

Definition 7.1. Let a be a self-adjoint operator in It is said to be even in if all odd free moments of a vanish, i.e.,

(7.1.1)

Let a be a “self-adjoint” operator of satisfying the even-ness (7.1.1). Then it is said to be semicircular in if

(7.1.2)

where are the n-th Catalan number,

for all

It is well-known that, if is the free cumulant on in terms of (in the sense of ¹²), then a self-adjoint operator a is semicircular in if and only if

(7.1.3)

for all (e.g., see ^{11, 14}, and cited papers therein). The above characterization (7.1.3) is obtained by the Möbius inversion of ¹².

Thus, the semicircular operators a of 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.

Definition 7.2. Let be a self-adjoint operator. It is said to be weighted-semicircular in with its weight (in short, -semicircular), if there exists such that

(7.1.4)

for all where is the free cumulant in terms of 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 is t0-semicircular, if and only if (i) it is even in and (ii) there exists such that

(7.1.5)

for all where mean the m-th Catalan numbers, for all

7.2. Tensor Product Banach *-Algebra

Let be a -probability space (7.0.6), for Throughout this section, we fix in and the corresponding -probability space

Define now a Banach-space operators (or bounded linear transformations) and “acting on the -algebra ” by linear morphisms satisfying,

(7.2.1)

on for all

Definition 7.3. The Banach-space operators and on in the sense of (7.2.1) are called the p-creation, respectively, the p-annihilation on for Define a new Banach-space operator by

(7.2.2)

We call it the p-radial operator on

Let be the p-radial operator of (7.2.2) acting on Construct a Banach algebra by

(7.2.3)

where means the operator space consisting of all bounded linear transformations on equipped with its operator-norm defined by

where

giving the -norm topology for (which is the subspace topology of that for ), where means the Hilbert-space -norm on the p-adic Hilbert space for all

On define the adjoint (*) by

(7.2.4)

where with their conjugates It is indeed a well-defined adjoint on (See ⁴).

Then, equipped with the adjoint (7.2.4), this Banach algebra of (7.2.3) forms a Banach *-algebra.

Definition 7.4. Let be a Banach *-algebra (7.2.3) in for We call the p-radial (Banach-*-) algebra on

Let be the p-radial algebra on Construct now the tensor product *-algebra by

(7.2.5)

where means the tensor product of Banach *-algebras.

Take now a generating element for some where are in the sense of (6.1) in and

with axiomatization: the identity operator of on satisfying

for all

Define now a bounded linear morphism

by a linear transformation satisfying that:

(7.2.6)

for all where is the minimal integer greater than or equal to for all

By the cyclicity of the tensor factor of and by the fact that: all generating elements of are mutually orthogonal projections, the above morphism is well-defined as a linear transformation.

Now, consider how our p-radial operator acts on First observe that if and are the p-creation, respectively, the p-annihilation on then

(7.2.7)

for all

Lemma 7.2. Let be the p-creation, respectively, the p-annihilation on Then

(7.2.8)

where is the identity operator on

Proof. The formula (7.2.8) holds by (6.3) and (7.2.7).

Also, the formula (7.2.8) shows that the Banach-space operators and are acting commutatively on Therefore, one can obtain that

(7.2.9)

with identity:

for all where

By considering the formulas (7.2.8) and (7.2.9) together, one obtains the following proposition.

Proposition 7.3. Let be the p-radial operator on Then

(7.2.10)

(7.2.11)

for all

Proof. 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.

7.3. Weighted-Semicircular Elements in

Fix and let be the tensor product Banach *-algebra in the sense of (7.2.5), and let be the linear transformation of (7.2.6). Throughout this section, fix an element

(7.3.1)

where are projections (6.1), generating Observe that

(7.3.2)

for all for all By (7.3.2), one can realize that the operators of (7.3.1) generate for all

Consider that, if is in the sense of (7.3.1), for then

(7.3.3)

by (7.2.6) and (7.3.2), for all

Now, for any fixed define a linear functional on by

(7.3.4)

where is in the sense of (7.0.5).

By the linearity of both and the morphism of (7.3.4) is a well-defined linear functional on So, the pair forms a Banach *-probability space in the sense of ¹² and ¹⁴.

By (7.3.3) and (7.3.4), if is in the sense of (7.3.1), then

(7.3.5)

for all

Theorem 7.4. Let for a fixed Then is -semicircular in More precisely, one obtains that

(7.3.6)

for all Equivalently, if means a free cumulant in terms of the linear functional of (9.3.5) on then

(7.3.7)

for all

Proof. The formula (7.3.6) is proven by the straightforward computations from (7.3.5), with help of (7.2.10) and (7.2.11). Also, the formula (7.4.7) is obtained by the Möbius inversion of ¹² from (7.3.6). See ⁴ for more details.

8. Semicircularity on

For all let and be in the sense of (7.2.5), respectively, (7.3.4). Then, one has the corresponding Banach *-probability spaces,

(8.0.1)

for all

Let be the generating elements (7.3.1) of for where are in the sense of (8.0.1). Then a generating element of is -semicircular in by (7.3.6) and (7.3.7). i.e.,

(8.0.2)

and

for all for all where

for all

Let be arbitrary topological *-probability spaces (e.g., C^*-probability spaces, or W^*-probability spaces, or Banach *-probability spaces, etc.) consisting of topological *-algebras (e.g., C^*-algebras, or W^*-algebras, or Banach *-algebras, etc.), and corresponding bounded linear functional for where is an arbitrary countable (finite or infinite) index set.

The free product topological *-probability space of is a new topological *-probability space, consisting of the free product topological *-algebra

generated by the noncommutative reduced words, called free reduced words, in (under product topology), having a new linear functional

where satisfies that: if is a free reduced word in then

for all for all (e.g., see ^{11, 12, 14}).

We denote the above relations together by

The topological *-algebra is understood as a Banach space,

(8.1.1)

with

where

for all and where and are the direct product, respectively, the tensor product of Banach spaces (e.g., see ^{12, 14}).

In particular, if an element is of the form of free reduced word,

with

then one can understand a as an equivalent Banach-space vector

contained in the minimal direct summand (which is a closed subspace) of A in (8.1.1).

We denote this relation by

(8.1.2)

Of course, the left-hand side a of (8.1.2) means the operator in A, while, the right-hand side means the Banach-space vector in the direct summand of the Banach space A of (8.1.1). Note that, under same argument, one may understand a as an operator in the *-subalgebra

where means the tensor product of topological *-algebras.

Notation 8.1. In the rest of this paper, we will call the above *-subalgebra of A, the minimal free summand of A containing a given free reduced word

Remark that, if a is a free reduced word in A, then

(8.1.3)

in the minimal free summand of A containing a in the sense of Notation 8.1, for all Remark also that even though the above relation (8.1.3) holds inside the minimal free summand, it does not hold fully in A, in general (in particular, whenever k > 1).

Notation and Remark 8.1. (From below, NR 8.1) Let be a free reduced word in A, as above. The power in (8.1.3) means the k-th power of a as an element of the minimal free summand of A containing a.

To avoid the confusion, we will use the notation as a construction of new free “non-reduced” word,

in A.

For example, let be a free reduced word with

which is equivalent to

Then

in the minimal free summand of A containing a, but

i.e.,

is a free reduced word in A.

So, in the text below, if we use the term “” for a fixed free reduced word a, then it is in the sense of (8.1.3) contained in the minimal free summand of A containing a; meanwhile, if we use the term “” then it means a free (non-reduced) word in A.

Note that only if a is a free reduced word with its length-1 in A, and hence, its minimal free summand of A is identical to the free block in A, for all

Similar to and one can understand the adjoints and of a fixed free reduced word a in A as follows;

in the minimal free summand of A containing a, but

in A.

The free product linear functional on A satisfies that, whenever a is a reduced free word in A of (8.1.2), then

(8.1.4)

in the minimal free summand of A containing a, by (8.1.3), for all Sometimes, by abusing (8.1.3), one can / may write

for all

Note that, in general,

However, the equality holds, if a is a free reduced word with its length-1.

Now, let

We say that such an element b is a free sum in A, if all summands of b are contained in “mutually-distinct” minimal free summands of A containing them as free reduced words. Then, similar to the above terminology, one can realize

(8.1.5)

in the direct sum of the minimal free summands of A containing for all (also, called the minimal free summand of A containing the free sum b). So, it satisfies

(8.1.6)

for all Here, each summand of (8.1.6) satisfies (8.1.4).

Notation and Remark 8.2. (From below, NR 8.2) Similar to NR 8.1, if b is a free sum in the sense of (8.1.5), then one can consider

where the summands of are free (non-reduced) words in A.

For more about (free-probabilistic) free product algebras, and corresponding free probability spaces, see ^{11, 12, 14} and cited papers therein.

8.2. Free Product C^*-Probability Space

In this section, we will use the same concepts and notations introduced in Section 8.1.

By (8.0.1), we have the family

of Banach *-probability spaces.

Thus, one can define the free product Banach *-probability space,

(8.2.1)

as in Section 8.1.

Definition 8.1. The Banach *-probability space of (8.2.1) is called the radial-Adelic (Banach-*-) probability space. If we understand as a Banach *-algebra, we call it the radial-Adelic (Banach *-) algebra.

Let be the radial-Adelic probability space (8.2.1). Then, we obtain a subset

of consisting of -semicircular elements in the free blocks of Remark here that, by the choice of in the family, all are taken from the mutually-distinct free blocks of It means that all elements are mutually-free from each other in

Recall that a subset of an arbitrary *-probability space is said to be a free family, if, for all pairs of “distinct” elements and of the corresponding operators and are free in (e.g., ^{12, 14}).

Definition 8.2. Let be a free family in a *-probability space This family S is said to be a free semicircular family, if every element of S is semicircular, for all Similarly, the family S is called a free weighted-semicircular family, if all elements of S are weighted-semicircular, for all

So, by the very construction (8.2.1) of our radial-Adelic probability space we obtain the following fact.

Theorem 8.1. Let be the radial-Adelic probability space (8.2.1). And let be the generating operators of

(8.2.2) A family is a weighted-semicircular free family in for a fixed

(8.2.3) A family is a weighted-semicircular free family in for a fixed

(8.2.4) A family is a weighted-semicircular free family in

Proof. First, let's take a family by fixing a prime p. Then one can understand this family is taken from

which forms a free block of because

by (8.2.1). Therefore, the subset

is a free family in for the fixed prime So, the family is also a free family in

Now, assume we have a family in for a fixed integer Then, this family is a subset of

because

by (8.2.1). Thus, the subset

forms a free family in for the fixed integer Therefore, the family is a free family in

Finally, consider the family We re-write this family by

Each family is a free family in for each Note that the family forms a free family, and hence, the sub-families is a free family in too.

Therefore, the family

is a free family in

Recall that all elements are -semicircular in for all and and hence, they are weighted-semicircular in Therefore, the statements (8.2.2), (8.2.3) and (8.2.4) hold.

8.3. Semicircular Elements Induced by

Let be the radial Adelic probability space (8.2.1). Define the elements of by

(8.3.1)

Then, by the self-adjointness of the operators of (8.3.1) are self-adjoint in

Also, one obtains the following free-cumulant computation; if is the free cumulant on with respect to the linear functional on the radial-Adelic algebra then

(8.3.2)

by the bimodule-map property of free cumulant (e.g., ¹²), for all where are the free cumulants on the free blocks of in terms of the linear functional on for all

By (8.3.2), we obtain the following result.

Theorem 8.2. Let be free random variables (8.3.1) of the radial-Adelic probability space for Then are semicircular elements, and the families

(8.3.3)

and

form free semicircular families in

Proof. Consider that

by (8.3.2)

by the -semicircularity of in

(8.3.4)

for all

By the free-cumulant computation (8.3.4), the self-adjoint operators are semicircular in by (6.1.3), for all

Thus, the families and of (8.3.3) form free families in by (8.2.2), (8.2.3) and (8.2.4), respectively, because all elements are simply scalar multiples of contained in mutually-distinct free blocks of for all Therefore, by (8.3.4), the families of (8.3.3) are free semicircular families in

9. Operators of Generated by

Let be the radial-Adelic probability space in the sense of (8.2.1), and let

be the free semicircular family in (8.3.3), where are the free blocks of for all Throughout this section, fix and we consider certain free-distributional data of operators generated by the free semicircular family

(9.1)

in for

and

Theorem 9.1. Let be an operator (9.1) generated by the free semicircular family of (8.3.3) in the radial-Adelic probability space If either the prime-sequence or the integer-sequence is an alternating sequence, then

(9.2)

in the minimal free summand of containing

Under the same hypothesis, if either in or in then

(9.3)

in for all

Proof. Let be in the sense of (9.1) in Assume either the prime-sequence or the integer-sequence is an alternating sequence. Then this operator T is a free reduced word in and hence,

in the minimal free summand of containing by (8.1.3), for all

Therefore, one has that

in the free block, by the semicircularity of

Now, let T be given as above in Suppose either or is alternating in respectively, in and assume further that

Then this operator is not only a free reduced word in but also are free reduced words in satisfying

in for all Therefore, we have

for all by the semicircularity of

Now, remark that

in by the self-adjointness of our semicircular elements for all So,

for all moreover, it forms a free reduced word in too, because either

is alternating in respectively, in Also, since either

the operators are free reduced words in too, for all

And hence, similar to the above case

for all

Therefore, the free-distributional data (9.3) holds.

The formulas (9.2) and (9.3) characterize the free distributions of free reduced words where either in or in

Let and be arbitrary topological *-probability spaces, and let and b be “self-adjoint.” These operators and are said to be identically free-distributed, if

(9.4)

Motivated by (9.4), one can obtain the following universalized result.

Theorem 9.2. Let be an arbitrary topological *-probability space, and let be a free semicircular family in A, for some Let

where and are mutually distinct in Then

(9.5)

in the minimal free summand of A containing M; for all

Also, if M is as above in and if in then

(9.6)

Proof. Since the family is assumed to be a free semicircular family, all elements are not only free from each other in but also semicircular in By the definitions of the operator is a free reduced word in satisfying the equivalences,

in the minimal free summand of A containing M.

Therefore, by the identically-free-distributedness (9.4), one can obtain the free-distributional data (9.5) in the minimal free summand of A containing where are the *-subalgebras of A generated by for all

Similar to (9.5), we obtain the free-distributional data (9.6) by (9.3), because form free reduced words in 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 generated by our free semicircular family 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.

10. Operators of Generated by

In this section, we use same concepts and notations introduced in Notation 8.1, NR 8.1, and NR 8.2. Throughout this section, we fix and let

and

Let be the weighted-semicircular free family (8.2.4) of i.e.,

(10.1)

consisting of -semicircular elements

which are free from each other in the radial-Adelic probability space for all

Define operators of by the operators, induced by the free family of (10,1), by

(10.2)

in

Theorem 10.1. Let be an operator of (10.2), generated by the free weighted-semicircular family of (10.1), in the radial-Adelic probability space Assume that either the prime-sequence or the integer-sequence is alternating in respectively, in Then, in the minimal free summand of containing it, we have

(10.3)

for all

Proof. Let be in the sense of (10.2) in and assume either

is an alternating sequence in respectively, in

Then this operator X is a free reduced word in So, it satisfies that

(10.4)

in the minimal free summand of containing for all

By the equivalence (10.4), one has that

for all by the -semicircularity of for all

Therefore, the free-moment formula (10.3) holds in the minimal free summand of containing

Remark that, by the equivalence (10.4), one can check the self-adjointness of the operator in the minimal free summand of containing it, i.e.,

by the self-adjointness of for all This self-adjointness guarantees that the free-moment formula (10.3) characterizes the free distribution of in the minimal free summand of containing it. (Remark that the above self-adjointness does not hold in It holds “in the minimal free summand.”)

Theorem 10.2. Let be an operator (10.2) in the radial-Adelic probability space Assume that either or is alternating in respectively, in If either

then we have

(10.5)

for all

Proof. Let be in the sense of (10.2) in and suppose that either

or

Then the operator forms a free reduced word in Also, suppose that either

Then the powers of X form free reduced words “in ” too, for all Thus, one can get that

(10.6)

for all So, the first equality of the formula (10.5) holds by (10.6) Note that X is not self-adjoint in because

Thus, similar to (10.6), one obtains that

(10.7)

for all

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

11. 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 space and the Banach *-probability space itself.

Let and be subsets of defined by

(11.1)

and

Then the family of (11.1) is a free weighted-semicircular family (by (8.2.4)), and the family of (11.1) is a free semicircular family (by (8.3.3)) in

Let be a finite sequence in Then the subset of induced by the sequence is defined to be

For example, if in then in

Similarly, for a finite sequence in one can determine the corresponding subset of for all

Let be a finite sequence in and let

for

If we fix

and

where means the size or the length of finite sequences, then one can define the corresponding operators and of by

(11.2)

in generated by Q, respectively, by of (11.1).

The construction (11.2) shows that if in then and are the operators in the sense of (10.2), respectively (9.1). While, if then

in the free block which is the very minimal free summand of 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

Definition 11.1. Let and be in the sense of (11.2) in for a fixed If (or, ) is a free reduced word in or equivalently, if either W or J is alternating, then the operator (or, ) is called a weighted-semicircular(resp., a semicircular) reduced word of generated by (resp., by )).

Moreover, if “N > 1," and if either in or in then the corresponding weighted-semicircular reduced word (resp., the corresponding semicircular reduced word ) is said to be a power-w(eighted)-s(emicircular) reduced word (resp., a power-s(emicircular) reduced word) of generated by (resp., by ). Remark that power-w-s reduced words and power-s reduced words are defined only if the free reduced words have their length

If is a power-w-s reduced word of then satisfies

(11.3)

in by (10.3) and (10.5), and if is a power-s reduced word of then (11.4)

(11.4)

by (9.2) and (9.3), for all

Definition 11.2. Let and be a weighted-semicircular reduced word (not necessarily power-w-s), and a semicircular reduced word (not necessarily power-s) in respectively. Define a sequence by

(11.5)

with axiomatization:

We call this sequence of (11.5), the weight-ratio of and in

Observe that, by the definition (11.5) (with axiomatization: ), if and are a power-w-s reduced word, respectively, a power-s reduced word in then

by (11.3) and (11.4)

where

(11.6)

for all and then

by (11.6), i.e.,

(11.7)

where is in the sense of (11.6).

Proposition 11.1. Let be the weight-ratio (11.5) of a power-w-s reduced word and a power-s reduced word in the sense of (11.2). Then

(11.8)

where is in the sense of (11.6).

Proof. The proof of (11.8) is done by (11.7), under our axiomatization:

Note that if is a weighted-semicircular reduced word (which is not necessarily power-w-s), and if is a semicircular reduced word (which is not necessarily power-s) in then the above relation (11.8) for does not hold in general.

We will denote the k-th entry of the weight-ratio of and by

(11.9)

So, if and are a power-w-s reduced word, respectively, a power-s reduced word in then

by (11.7) and (11.9), for all

By definition, the k-th entry (11.9) of the weight-ratio shows how the k-th free moment of and that of are different free-distributionally. In other words, the sequence indicates how the free distribution of and that of can be distinguished.

Suppose one takes be a finite sequence of “even” numbers, i.e., for all In such a case, one may / can say the N-tuple is even, for convenience.

If is a power-w-s reduced word, and hence, is a power-s reduced word in and if is even, then

(11.10)

by (11.8), since

Corollary 11.2. Let and be a power-w-s reduced word, respectively, a power-s reduced word in and let be the weight-ratio of and If is even, then

(11.11)

Let be the weight-ratio (11.5). By understanding it as a R-sequence, one can consider convergence, or divergence of the sequence. Note that every entry of is a nonnegative real number.

If is the weight-ratio of a power-w-s reduced word and a power-s reduced word and if we denote the first entry

then one obtains that

by (11.8). Thus, one can have the following cases; (i) if then

(ii) if then

(iii) if then

and (iv) if then

Proposition 11.3. Let in and let and be a power-w-s reduced word, and a power-s reduced word in respectively. Let be the weightratio of and and let

(11.12) If then converges to 1.

(11.13) If then diverges to .

(11.14) If then converges to

Proof. Under hypothesis, let in Then, by (11.8) and (11.11), one has that

So, by the discussions (i), (ii), (iii) and (iv) in the very above paragraph, we obtain the convergence conditions (11.12), (11.13) and (11.14) of our weight-ratio

Perhaps, by the convergence conditions (11.12), (11.13) and (11.14), one may /can say our power-w-s reduced word is

to a power-s reduced word in the radial-Adelic probability space 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

satisfies

then the free distribution of is close to the free distribution of and the close-ness may / can be measured by the limit 0 of the weight-ratio (which means the free distribution of (determined by weighted-semicircularity) is “very” much close, or 0-much close to that of (determined by semicircularity)). However, such conclusions, or estimations are too rough. So, we provide a following new way to determine the close-ness, motivated by (11.11).

Definition 11.3. Let be the weight-ratio of a weighted-semicircular reduced word and a semicircular reduced word (which are not necessarily power-w-s, respectively, power-s) of As a R-sequence, let

and suppose

If then is said to be close to with its ratio (in short, is -close to ) in Otherwise, we say is not close to in

By definition, a weighted-semicircular reduced word is -close to a semicircular reduced word in if and only if the free distribution of is close to that of and the close-ness can be measured by the nonnegative quantity

in

Let and and let

(11.15)

and

be the corresponding weighted-semicircular reduced word, respectively, semicircular reduced word in If two operators X and M are in the sense of (11.15), then

in the free block in for all (See Section 8.1).

These operators X and M of (11.15) have their free distributions,

(11.16)

respectively,

by (11.15)’.

By the free distributions (11.16), the operators X and M of (11.15) have their weight-ratio,

(11.17)

under axiomatization:

Thus one can obtain the following close-ness of X to M.

Theorem 11.4. Let and in the radial-Adelic probability space where is a -semicircular element in and is a semicircular element in for and for a fixed

(11.18) If in then X is 1-close to M in

(11.19) If in then there exists such that (i) is the minimal quantity satisfying and (ii) X is -close to M in

(11.20) If in then X is not close to M in

Proof. Suppose is fixed, and let X and M be given as above in Then the weight-ratio R of X and M satisfies

(11.21)

by (11.17).

Assume first that in Then the above weight-ratio R of (11.21) satisfies that

So, one obtains that there exists such that (i) it is the minimal quantity satisfying and (ii)

because for all The existence of such quantity is guaranteed, because for any natural number there always exist such that is even in

It shows that if in Z; then is 1-close to M in and hence, the statement (11.18) holds.

Now, suppose in Then the weight-ratio R of (11.21) satisfies that

because in where means the absolute value of in

So, by the similar manner with the proof of (11.18), there exists such that (i) is the minimal quantity satisfying and (ii)

since the weight-ratio R forms a monotonically decreasing sequence in

It shows that, if in then there exists a unique such that X is -close to M in where is the minimal quantity satisfying Therefore, the statement (11.19) holds true.

Finally, assume that in Then the weight-ratio

of (11.21) contains a sub-sequence

which is a monotonically “strictly” increasing sequence in where is the minimal quantity in satisfying

Since the weight-ratio R is a monotonically increasing sequence, one can get that

by the monotonically-strictly-increasing-ness of the sub-sequence of which implies that

So, X is not close to M in Thus, the statement (11.20) holds.

The above theorem, expressed by the close-ness conditions (11.18), (11.19) and (11.20), characterizes how much our weighted-semicircular element s close to a semicircular element in for

Corollary 11.5. Let and and let be the -semicircular element, and the semicircular element in

(11.22) If in then is 1-close to in

(11.23) If in then is -close to in

(11.24) If in then is not close to in

Proof. The proofs of (11.22), (11.23) and (11.24) are done by those of (11.18), (11.19) and (11.20), respectively, by putting n = 1. Remark that, in this case, the weight-ratio satisfies that

The above corollary, expressed by (11.22), (11.23) and (11.24), illustrates the close-ness of our weighted-semicircular elements and the corresponding semicircular elements

More generally, one can obtain the following corollary in terms of identically-free-distributedness.

Corollary 11.6. Let be our weighted-semicircular reduced word (11.2) in for and let s be an arbitrary semicircular element in a topological *-probability space Define a sequence of X and by

(11.25)

under axiomatization:

Also, define a -sequence

where are in the sense of (11.25). Then

(11.26)

where is the minimal quantity satisfying in Therefore, up to (11.25), we may / can say that X is -close to for the fixed where are in the sense of (11.26).

Proof. If is a -sequence (11.25), where X is the weighted-semicircular reduced word and is the n-th power of an arbitrary semicircular element s in a topological *-probability space And let

Since and our free semicircular reduced word are identically free-distributed, the sequence of (11.25) is identical to our weight-ratio as -sequences.

Therefore, one obtains the formula (11.26), by (11.18), (11.19) and (11.20).

The above corollary shows how our weighted-semicircularity (on ) differs (or, is close) from (resp., to) the semicircular law, and the difference (or the close-ness) of them is measured by (11.26). Thus, by abusing notation, for example, one can/ may say that: a weighted-semicircular reduced word of is -close to the n-th power of any arbitrary semicircular element whenever in for all

The following proposition precisely determine the minimal quantity obtained in our close-ness conditions (11.19) and (11.26).

Proposition 11.7. Let be our -semicircular reduced word in the radial-Adelic probability space and assume that in and hence, is -close to any semicircular element s in a topological *-probability space where is the minimal quantity satisfying Then

(11.27)

for all It means that, if in then X is either -close, or -close to

Proof. Suppose n is even. Then for all and hence, 1 is the minimal quantity making Assume now that n is odd. Then if and only if is even in So, the smallest even quantity is in Therefore, 2 is the minimal quantity making So, the above equality (11.27) holds.

The equality (11.27) completely characterizes the minimal quantity making for all

Corollary 11.8. Let X be a weighted-semicircular reduced word in for for a fixed Assume that in

(11.28) If then X is -close to in

(11.29) If then X is -close to in

where is the semicircular element in

Proof. The statements (11.28) and (11.29) are the direct consequences of (11.19) and (11.27).

Now, assume that N > 1 in ; and let

(11.30)

and

be power-w-s, respectively, power-s reduced words in Then, the weight-ratio satisfies

by (11.8). So, if we write

(11.31)

then

as a -sequence.

Lemma 11.9. Let and be a power-w-s, respectively, a power-s reduced words (11.30) in and let be the weight-ratio of and If there exists at least one such that is odd in then is the zero sequence, i.e.,

Proof. Let and be given as above in with in Assume now that there exists at least one such that Then the quantity of (11.31) is identical to 0. since So, the corresponding weight-ratio satisfies that

By the above lemma, one can obtain the following close-ness condition.

Theorem 11.10. Let and be a power-w-s, respectively, a power-s reduced words (11.30) in If there exists at least one such that is odd in then is 0-close to in i.e.,

(11.32)

Proof. Under hypothesis, the weight-ratio

becomes the zero -sequence by the above lemma, where is in the sense of (11.31). So,

by the above lemma. Equivalently, it shows that is 0-close to in

By the above close-ness condition (11.32), we now focus on the cases where equivalently, is even in the sense that all entries are even in for all If is even, then the quantity

and hence, the weight-ratio

is a -sequence without zero entries.

Definition 11.4. Let be a power-w-s reduced word, and hence, the corresponding power-s reduced word of (11.30) in where in Assume that is even. Then we call such operators and an even power-w-s reduced word, respectively, an even power-s reduced word in

We obtain the following additional close-ness condition of and of (11.30) in the radial-Adelic probability space

Theorem 11.11. Let and be an “even” power-w-s reduced word, respectively an “even” power-s reduced word in in the sense of (11.30), and let

(11.33) If in then is -close to in

(11.34) If in then is not close to in

Proof. Let be the weight-ratio of and Then, by (11.10), one obtains

by the even-ness of So, it is not difficult to check that: if in then is the constant sequence having all its entries 1, i.e.,

Thus, one can get that

So, if then is 1-close to in

Suppose now that in Then the sequence is a monotonically strictly decreasing sequence. So,

Therefore, if then is -close to in

Therefore, the close-ness condition (11.33) holds.

Assume now that in Then, the weight-ratio is a monotonically strictly increasing sequence, implying that

It shows that is not close to in Therefore, the close-ness condition (11.34) holds true.

In the rest of this section, we refine the above main results (11.33) and (11.34) more in detail.

Let and be an even power-w-s reduced word, respectively an even power-s reduced word in in the sense of (11.30), and let

be the weight-ratio of and where

(11.35)

Now, let be the integer-sequence of and with the corresponding subset of Now, decompose by

(11.36)

where

and

where and are the maximal sub-sequences of inducing respectively

For example, let with in Then

with

Depending on and obtained from (11.36), let's decide the sub-sequences and of the prime-sequence under the rule: if and only if there exists for all for all Similarly, determine the sub-sequences and of , under similar manner.

Then the first entry of (11.35) in the weight-ratio of and can be re-written as a -quantity,

(11.37)

with

as a positive real number in

By the close-ness conditions (11.33) and (11.34), and by (11.37), we obtain the following refined main result.

Theorem 11.12. Let be an even power-w-s reduced word, and the corresponding even power-s reduced word in as in (11.30), and let be the weight-ratio of and where is in the sense of (11.35) in Suppose the quantity is factorized by

where

are in the sense of (11.37), where and are the sub-sequences in the sense of (11.36). If

(11.38)

Then is -closed to in

If the above condition (11.38) does not hold, then is not close to in Proof. Suppose in as in (11.37). Then, by the very constructions,

Assume now the condition (11.38) holds in Then the positive -quantity satisfies

Therefore, by (11.33), is -close to in

By (11.34), if the condition (11.38) does not hold, then is not close to in

So, by (11.32), (11.33), (11.34), and (11.38), one can summarize the close-ness of power-w-s reduced words as follows.

Corollary 11.13. Let and be a power-w-s reduced word, respectively, a power-s reduced word in and let

with

(I) If there exists at least one such that is odd in then is 0-close to in .

(II) Let and be an even power-w-s, respectively an even power-s reduced words in and let be decomposed by (11.37). If in then is -close to in Otherwise, is not close to in

The above corollary provides the close-ness of power-w-s reduced words and power-s reduced words in

By the above close-ness, and by the identically-free-distributedness, one obtains the following corollary, too.

Corollary 11.14. Let be a power-w-s reduced word of (11.30) in the radial-Adelic probability space and let S be a free reduced word in an arbitrary topological *-probability space in where is an alternating N-tuple of “free,” semicircular elements of and where in A, for N > 1 in Let

be a -sequence. Then there exists a nonnegative number in such that

Moreover, we have the following cases:

(11.39) If there exists at least one such that is odd in then

(11.40) If X is an even power-w-s reduced word in then in and

Proof. The proofs of (11.39) and (11.40) are done by the identically-free-distributedness of our power-s reduced word of (11.30) and the free reduced word S, with help of (11.32), (11.33), (11.34) and (11.38).

The above corollary shows the close-ness between free distributions of our power-w-s reduced words of the radial-Adelic probability space generated by the free weighted-semicircular family and free distributions of certain free reduced words generated by an arbitrary free semicircular family.

References