1. Introduction

The main purposes of this paper are (i) to consider how a prime p acts on a fixed unital tracial -probability space, (ii) to study operator-algebraic, and operator-theoretic information of crossed product -algebras obtained from -dynamical systems obtained from (i), and (iii) to investigate free probability on free product of crossed product -algebras of (ii). Also, we apply our main results to the cases where a fixed unital tracial -probability space is generated by a countable discrete group.

1.1. Preview and Motivation

The relations between primes and operator algebras have been studied in various different approaches. For instance, we studied how primes act on certain types of von Neumann algebras, especially, generated by p-adic, and Adelic measure spaces (e.g., ^{[3, 6]}). Meanwhile, in ^[4] and ^[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 certain Krein-space representations (e.g., ^[8]). Also, in ^[7], we considered free-probabilistic structures on a Hecke algebra for a fixed prime p. For more about number-theoretic motivation, see e.g., ^{[9, 10, 13]}.

In ^[1], we considered free-distributional data of certain operators in a C^*-algebra , induced by a (pure-algebraic) -algebra consisting of all measurable functions over under Hilbert-space representation, for a prime For all we defined C^*-probability spaces , where are kind of “filtered" linear functionals implying the number-theoretic data on in terms of the usual p-adic integration on for a prime From the system

of C^*-probability spaces, the free product C^*-probability space

is constructed, and the free probability on this C^*-probability space, called the Adelic probability space, is studied.

In ^[2], we established weighted-semicircular elements, and corresponding semicircular elements in certain Banach -probability spaces induced by a C^*-probability space

Especially, the weighted-semicircularity of ^[2] is dictated by free-distributional data on obtained in ^[1].

Motivated by the main results of ^[1] and ^[2], we here study how our p-adic-analytic structures affect the original free-probabilistic data on arbitrarily fixed unital tracial -probability spaces. To do that, we establish suitable W^*-dynamical systems induced by the σ-algebra of . From such dynamical systems, we construct corresponding crossed product -algebras, and investigate -probability on them. In particular, we focus on how p-adic-analytic data (containing number-theoretic information) affect the original free-distributional data.

Remark and Emphasis Even though our proceeding works are highly motivated by those of ^[1] and ^[2], the aims, approaches, tools, theories and results are totally independent from each other.

1.2. Overview

In Sections 2, we briey introduce backgrounds and a motivation of our works. In particular, we introduce p-adic number fields for primes and consider p-adic calculus on based on p-adic calculus, we construct free-probabilistic models on the *-algebras consisting of all measurable functions on for primes Free products of *-probability spaces is observed, too.

In Section 3, we understand the σ-algebra as a semigroup equipped with the set-intersection, for a fixed prime and then act this semigroup on an arbitrarily taken unital tracial -probability space . A -dynamical system is well-established by the semigroup acting on and the corresponding crossed product -algebra is considered. By determining a suitable linear functional on such a crossed product -algebra, free-probabilistic informations are investigated.

In Section 4, certain “sub"-free-probabilistic structures of the -probability spaces of Section 3 are established and studied.

In Section 5, from the -probability spaces of Section 4, we construct free product -probability spaces, and study operator-algebraic, and operator-theoretic (especially, spectral-theoretic) properties of such free product -probability spaces.

As an application of our main results of Sections 4 and 5, we restrict our interests to the cases where a fixed unital tracial -probability space is induced by a countable discrete group, in Section 6. For example, we concentrate on the case where a fixed group is a free group with multi generators.

2. Preliminaries

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

2.1. Free Probability

Readers can check fundamental analytic-and-combinatorial free probability from ^[12] and ^[14] (and the cited papers therein). Free probability is understood as the noncommutative 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 ^{[1, 2, 3, 4, 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 C^*-probability spaces in the sense of ^[12] and ^[14], without detailed introduction. However, rough introduction would be given whenever they are needed in text.

2.2. Calculus on

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 σ-algebra . Recall also that, is a well-defined ring algebraically.

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 , where

is the unit disk of , consisting of all p-adic integers.

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

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

and hence, the pair orms a well-determined -probability space. Remark that it is a “commutative" -probability space (and hence, it is well-covered by (noncommutative) free probability theory).

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, and

by the measure-theoretic information in (2.2.1), for all

Now, let be the vector space of all -measurable functions on , i.e.,

(2.2.5)

So, if and only if

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

Then it forms a *-algebra over . Indeed, the vector space of (2.2.5) is an algebra under the usual functional addition, and multiplication. Also, this algebra has the adjoint,

where having their conjugates in .

Let Then one can define the p-adic integral of f by

(2.2.6)

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

satisfying the following result.

Proposition 2.1. (See ^[1]) Let and let Then there exist such that

and

where is in the sense of (2.2.7).

By (2.2.8), one obtains that if

then

where are in the sense of (2.2.8), for all for all , whenever of (2.2.7) is nonempty in

2.3. 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 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." Remark that is a commutative -algebra, but, for our later purposes, we construct free-probabilistic settings on .

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

(2.3.1)

with their boundaries

Define a linear functional by

(2.3.2)

Then, by (2.3.2), one obtains that

since

with help of (2.2.8), for all

Moreover, by the commutativity on ,

and hence, this linear functional of (2.3.2) is a trace on .

Definition 2.1. The free probability space is called the p-adic free probability space, for where is the linear functional (2.3.2) on .

Let be in the sense of (2.3.1) 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

(2.3.3)

Inductive to (2.3.3), we obtain the following result.

Proposition 2.2. (See ^[1]) Let for Then

(2.3.4)

and hence,

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

(2.3.5)

where means the Kronecker delta, and hence,

So, we obtain the following computations.

Proposition 2.3. Let for Then

(2.3.6)

and hence,

where

Proof. The proof of (2.3.6) is done by (2.3.5).

Thus, one can get that, for any

where is in the sense of (2.2.8)

(2.3.7)

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

Also, if then

(2.3.8)

where

because

In (2.3.8), it is clear that, if is empty, then

where is the empty set in

Thus, one can get that there exist such that

(2.3.9)

where

by (2.3.8) and (2.2.10), for all

By (2.3.9), we obtain the following general result under induction.

Theorem 2.4. Let and let for for Let

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

(2.3.10)

and

Proof. The proof of (2.3.10) is done by induction on (2.3.9). For more details, see ^[1] and ^[2].

Of course, if is empty in then the formula (2.3.10) vanishes. By (2.3.10), we obtain that, if

then

(2.3.11)

where are in the sense of (2.2.10), for all

The above joint free-moment formula (2.3.11) provides a universal tool to compute the free-distributional data of free random variables in our p-adic free probability space

2.4. Free Product W^*-Probability Spaces

Let be arbitrary -probability spaces, consisting of C^*-algebras , and corresponding linear functional , for where is an arbitrary countable (finite or infinite) index set. The free product -algebra

is the -algebra generated by the noncommutative reduced words in having a new linear functional

The -algebra A is understood as a Banach space,

(2.4.1)

with

where

for all and where the direct product , and the tensor product are topological on Banach spaces.

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

then one can understand as an equivalent Banach-space vector

contained in a direct summand, of (2.4.1).

We denote this relation by

(2.4.2)

Of course, the left-hand side means the -algebra element of while, the right-hand side means the Banach-space vector of in the sense of (2.4.1).

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

(2.4.3)

for all

Notation and Remark Let be a free reduced word in as above. The power in (2.4.3) means the k-th power of as a new element of which is regarded as a vector in

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

in Note that only if in the index set then the above free non-reduced word forms a free reduced word in

For example, let be a free reduced word with

as a vector,

Then

in but

i.e.,

as free reduced words in

So, in the text below, if we use the term “” for a fixed free reduced word , then it is in the sense of (2.4.3). In the following text, we will not use the concept “.” However, we want to emphasize the difference between and in the free product algebra at this moment.

Similar to and , one can understand the adjoints and of a fixed free reduced word in i.e.,

and

in

So, the free product linear functional on satisfies that, whenever is a reduced free word in satisfying (2.4.2), then

(2.4.4)

by (2.4.3), for all Sometimes, by abusing (2.4.3), one can / may write

whenever is a “free reduced word” in for all

Now, let

We say that a is a free sum in if all summands of b are contained in “mutually-distinct” direct summands of a Banach space A of (2.4.1), as free reduced words. Then, similar to the above observation, one can realize that

(2.4.5)

satisfying

for all

Here, remark that each summand of (2.4.5) satisfies (2.4.4).

Notation and Remark Similar to the free-reduced-word case, if b is a free sum in the sense of (2.4.5), then one can consider

where the summands are forms of free “non-reduced” words in A. In the following text, we will not use the concept “” in . But, as before, we emphasize the difference between and , for a fixed free sum b of .

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

3. Act on W^*-Probability Spaces

In this section, we regard the -algebra of a p-adic number field as a monoid equipped with its binary operation, the set-intersection And we act this monoid on arbitrarily given -probability spaces. In other words, we construct a monoidal -dynamical systems. And then, corresponding crossed product -algebras are established. Free probability on such -algebras is considered.

3.1. Act on W^*-Probability Spaces

Let be a von Neumann algebra, i.e., it is a unital -subalgebra of the operator algebra consisting of all bounded operators on a Hilbert space Assume further that we fix a linear functional tr on satisfying

for all and

where means the identity operator of , which is the identity operator of . i.e., the linear functional is a unital trace on . So, the pair forms a unital tracial -probability space in the sense of ^[11] and ^[14].

For a prime let be the -Hilbert space over the p-adic number field , i.e.,

(3.1.1)

It has its inner product ,

and the corresponding -norm

Remark that, for any this vector is expressed by

where is a finite or an infinite (limit of finite) sum(s) under -topology.

Let be the Hilbert space where a fixed von Neumann algebra acts, and let be the -Hilbert space (3.1.1). Then one has the tensor product Hilbert space ,

(3.1.2)

for

Understand now the -algebra of as a semigroup equipped with its binary operation, the set-intersection i.e.,

Each element of is acting on by

(3.1.3)

for all

i.e., there exists a well-defined semigroup-action of the semigroup such that

(3.1.4)

acting on , by (3.1.3).

This action of the semigroup acting on , is called the characteristic(-functional) action. By the very definition (3.1.4) of , we denote the images of simply by , for all

With help of the action of (3.1.4), one can construct a semigroup-action of acting on the fixed von Neumann algebra as follows

(3.1.5)

where is the tensor product Hilbert space (3.1.2). Indeed, one can understand as on and m as on Then the image of (3.1.5) is regarded as a process sending

(3.16)

for for all

Theorem 3.1. Let be in the sense of (3.1.5). Then the triple forms a well-defined semigroup -dynamical system in

Proof. It suffices to show that is indeed a well-defined semigroup action of acting on . By the definition (3.1.5), satisfying (3.1.6), one has that

since in

for all for all i.e.,

for all

Therefore, the morphism is a well-defined semigroup-action of acting on in

The above theorem shows that there is a well-defined semigroup -dynamical system in

Definition 3.1. The semigroup -dynamical system is called the p-adic -dynamical system in

Let be the p-adic -dynamical system in Then one can construct the corresponding crossed product -algebra,

(3.1.7)

by the -subalgebra of generated by and satisfying the -relation:

for all for al and

i.e., the -relation is

(3.1.8)

and

for all and for all

Definition 3.2. The crossed product -algebra of (3.1.7), with the -relation (3.1.8), is called the p-adic dynamical -algebra induced by the p-adic -dynamical system .

Now, let be the von Neumann algebra ,

(3.1.9)

which is the -algebra acting on . i.e., all elements of are understood as the multiplication operators on , i.e., if is an arbitrary element of of (3.1.9), then it is acting on by

where is the multiplication operator on ,

As usual, we denote multiplication operators of simply by

Note that all elements of are expressed by the finite or infinite (limits of finite) sums

under -topology.

Let's define the tensor product -algebra of a given von Neumann algebra , and the von Neumann algebra , where means the tensor product of von Neumann algebras.

Define now a -subalgebra by the conditional tensor product -algebra of and .

(3.1.10)

generated by satisfying the -relation:

(3.1.11)

and

(3.1.12)

for all and under linearity and product topology.

Then we obtain the following isomorphism theorem.

Theorem 3.2. Let be the p-adic dynamical -algebra (3.1.7) induced by the p-adic -dynamical system and let be the conditional tensor product -algebra (3.1.10) of and , satisfying (3.1.11). Then

(3.1.12)

where “” means “being *-isomorphic.”

Proof. Let be the conditional tensor product -algebra (3.1.10), satisfying the -relation (3.1.11), and let be the p-adic dynamical -algebra (3.1.7) induced by the p-adic dynamical system satisfying the -relation (3.1.8). Define a morphism

by the linear transformation satisfying

(3.1.13)

for all for all and

By the very definition (3.1.13) of the linear morphism , one can verify it is bounded and injective. Moreover, for any generating elements of we can always find the elements in So, under topology, the morphism is surjective, too. i.e., is a bijective bounded linear transformation.

Observe that

by (3.1.11)

by (3.1.8)

for all and

And hence, for any one has

Therefore, this bijective linear transformation is multiplicative, equivalently, it is a bounded algebra-isomorphism.

Also, it satisfies that

by (3.1.11)

in , for all for and

And hence, for any

Thus, the bounded algebra-isomorphism is a *-isomorphism.

So, two von Neumann algebras and are *-isomorphic from each other since there exists a *-isomorphism of (3.1.13).

By the structure theorem (3.1.12), one can understand the conditional tensor product -algebra

and the p-adic dynamical -algebra

as the same von Neumann algebra. In the rest of this paper, we use and alternatively, and we call these identified von Neumann algebras, a p-adic dynamical -algebra.

3.2. Free Probability on p-Adic Dynamical W^*-Algebra

In this section, we fix the p-adic dynamical -algebra

induced by a p-dynamical -dynamical system which is also understood as the conditional tensor product -algebra

where is the -von Neumann algebra

As we assumed at the beginning of Section 9, let be a fixed unital trace on i.e., the pair forms a unital tracial -probability space. Define now a linear functional on the -Neumann algebra by

(3.2.1)

So, for any the linear functional of (3.2.1) satisfies that

(3.2.2)

for some for all

by (2.2.8).

Since the morphism of (3.2.1) is a well-defined linear functional, the pair is a -probability space.

Remark 3.1. Check the similarity-but-difference between our (pure-algebraic) *-probability spaces , and the -probability spaces .

By (3.2.2), one obtains that

for all

Define now a bounded linear transformation from our p-adic dynamical -algebra to the -von Neumann algebra

by

(3.2.3)

Proposition 3.3. The linear morphisms of (3.2.3) are conditional expectations from onto for all

Proof. Since by (3.1.12), the linear transformation is bounded by the very definition (3.2.3).

(3.2.4)

in

Now, let

and

Then

where mean the iterated Cartesian product (k-times), for all for all sets

i.e.,

(3.2.5)

for all and

Also, satisfies that, for any (as above),

by (3.1.8) or (3.1.11)

i.e.,

(3.2.6)

Therefore, this surjective bounded linear transformation of (3.2.3) is a conditional expectation from onto , by (3.2.4), (3.2.5) and (3.2.6).

So, the pair of our p-adic dynamical -algebra , and the conditional expectation of (3.2.3) forms an amalgamated -valued -probability space with amalgamation over (e.g., ^[12]).

Remember that the von Neumann algebra forms a well-determined -probability space , where is in the sense of (3.2.1). One can define a linear functional on by

(3.2.9)

where is the conditional expectation (3.2.3) from onto , and is the linear functional (3.2.1) on , for

Clearly, by the linearity of and that of , the morphism of (3.2.9) is a well-defined linear functional on inducing a corresponding -probability space

Definition 3.3. The -probability spaces of a p-adic dynamical -algebra and the linear functional of (3.2.9) is called the p-adic dynamical -probability spaces.

3.3. On p-Adic Dynamical W^*-Probability Spaces

Let and let

be a p-adic dynamical -algebra, and let be the linear functional (3.2.9) on , where is the linear functional (3.2.1) on , and is the conditional expectation (3.2.3). Let be the corresponding p-adic dynamical -probability space.

In this section, we study free-distributional data on in terms of by computing free moments, and free cumulants of arbitrarily fixed free random variables of In particular, we concentrate on studying free distributions of generating elements of , for and

Lemma 3.4. Let for and Then there exist such that

(3.3.1)

and

Proof. The proof of (3.3.1) is done by straightforward computations by the definition (3.2.9) of Indeed,

by (2.2.8).

From now on, for convenience, we will use the following notations;

Notation Denote simply by in for all and

Now, observe that, if for for then

(3.3.2)

by (3.3.1), where

where are in the sense of (2.2.8).

Proposition 3.5. Let and for for Then

(3.3.3)

where

where are in the sense of (2.2.8).

Proof. The free-moment formula (3.3.3) is proven by (3.3.2), under the -relation (3.1.8) on

The above joint free-moment formula (3.3.3) provides the free-distributional data of generating elements of our p-adic dynamical -probability space

Based on (3.3.3), one can obtain that, for any

for for

by the Möbius inversion of ^[12]

(3.3.4)

by (3.3.2) and (3.3.3), where is the free cumulant with respect to (e.g., ^[12]).

Now, let be free random variables in with and for and let be the n-tuples in , for Then

(3.3.5)

with

where

by (3.3.4).

In particular, by (3.3.5), we obtain the following freeness condition on the p-adic dynamical -algebra

Theorem 3.6. Let be free random variables in the p-adic dynamical -probability space , with and , for . If

(3.3.6)

(3.3.7)

and if and are free in then and are free in

Proof. Let be a “mixed” n-tuple in , for Then

(3.3.8)

with

for all for all where

by (3.3.5).

Under the conditions (3.3.6) and (3.3.7), one has that

for all for all

So, the formula (3.3.8) satisfies that

where is the free cumulant in terms of on the -probability space .

by the assumption that and are free in .

Indeed, since all mixed free cumulants of and vanish, under conditions (3.3.6) and (3.3.7), all mixed free cumulants of and vanish, equivalently, and are free on

The above theorem shows that, under the assumptions (3.3.6) and (3.3.7), the freeness on implies the freeness on

4. Certain p-Adic Dynamical Subalgebras of

Let p be a fixed prime in and let be a unital tracial -probability space. And let

be a p-adic dynamical -algebra, inducing the p-adic dynamical -probability space

In this section, we are interested in certain sub-structures of the p-adic dynamical -algebra Define a -subalgebra of by the -algebra generated by

i.e.,

(4.1)

Since are the mutually-orthogonal projections in , satisfying

(4.2)

and

it is not difficult to obtain the following structure theorem of the -subalgebra of .

Proposition 4.1. Let be the -subalgebra (4.1) of . Then

(4.3)

in , where “” means the direct product of -algebras.

Proof. The structure theorem (4.3) is obtained by the relation (4.2) of the generators of .

Let be the -subalgebra (4.1) of . Then one can construct the -subalgebra of our p-adic dynamical -algebra by

(4.4)

where is the conditional tensor product under the -relation.

Since the -algebra of (4.4) is understood as the -subalgebra of too.

Theorem 4.2. Let be the -subalgebra (4.4) of our p-adic dynamical -algebra Then

(4.5)

Proof. Observe that

where is in the sense of (4.1) in

by (4.3)

By (4.5), one can understand the -subalgebra is a certain diagonal-like subalgebra of

Definition 4.1. Let and let be the p-adic dynamical -algebra induced by the p-adic -dynamical system and let be the -subalgebra (4.4) of This -algebra is called the p(-adic)-dynamical diagonal -subalgebra of

Recall that has its well-defined linear functional of (3.2.9). So, one can naturally restrict this linear functional on . For convenience, we denote such a restricted linear functional again by Thus, a -probability space is well-determined.

Definition 4.2. Let be the p-dynamical diagonal -subalgebra of a p-adic dynamical -algebra for a prime Then the corresponding -probability space is said to be the p-dynamical diagonal -probability space.

Let be a p-dynamical diagonal -probability space, and let

(4.6)

for all

Let be in the sense of (4.6). Then, for any

(4.7)

by the -relation on

Therefore, one can get the following free-distributional data of free random variables of (4.6), for all

Proposition 4.3. Let be a free random variable in a p-dynamical diagonal -probability space for Then

(4.8)

for all Moreover, if for then

(4.9)

with

Proof. The free-moment formula (4.8) is obtained by (4.7) and (3.3.3). So, it suffices to prove the joint free-moment formula (4.9).

Let and Then

by the -relation on

i.e.,

(4.10)

in

Therefore, by (4.10), one has

Therefore, the joint free-moment formula (4.9) holds.

The above free-moment formulas (4.8) and (4.9) characterize the free-moment of operators in the sense of (4.6) in a p-dynamical diagonal -probability space So, we obtain the following generalized free-distributional data.

Theorem 4.4. Let be a free random variable in a p-dynamical diagonal -probability space where are in the sense of (4.6), for for all for Then

(4.11)

for all for all

Proof. Let be given as above in Then, for any for one can get that

since for all or by (4.5)

and hence, we obtain that

Therefore, the free-moment formula (4.11) holds.

The above joint free-moment formula (4.11) characterizes the free-distributional data of elements of a p-dynamical diagonal -probability space under linearity.

Recall that the Euler totient function is an arithmetic function,

defined by

(4.12)

for all

It is well-known that if _ is the Euler totient function (4.12), then

(4.13)

where means “divides ,” or “ is a divisor of .”

For instance,

(4.14)

by (4.12) and (4.13), for all

Now, observe that, if is in the sense of (4.6), and if

then

by (4.9) and (4.11)

(4.15)

for all Similar to (4.15), one can get that

for all by (4.8).

Motivated by (4.15) and (4.15)’, we define a new linear functional on by

(4.16)

Then the pair is a well-determined -probability space, where is in the sense of (4.16).

Definition 4.3. The -probability space of a p-dynamical diagonal -algebra , and the linear functional of (4.16) is called the p-dynamical -probability space of

By the very construction, our p-dynamical -probability space has the following free-distributional information.

Theorem 4.5. Let be a free random variable in a p-dynamical -probability space , where are in the sense of (4.6) for for all for and is in the sense of (4.16). Then

(4.17)

for all for all

Proof. Let be given as above in the p-dynamical -probability space . Then, for any for one can get that

since , for all or by (4.5)

and hence, we obtain that

Therefore, the free-moment formula (4.17) holds.

By the above free-moment formula (4.17), we obtain the following corollary.

Corollary 4.6. Let be in the sense of (4.6) in the p-dynamical -probability Then

(4.18)

and

for all for all

Proof. The proof of (4.18) is done by (4.17). Also, the free-moment formulas of (4.18) can be proven by (4.15) and (4.15)’; under the definition (4.16).

5. Free Product W^*-Probability Spaces of

For a prime one can have a corresponding p-dynamical diagonal -algebra from a -dynamical system

where is a fixed unital tracial -probability space. By defining a linear functional on one obtains the p-dynamical -probability space

For a fixed -probability space assume we have family of -probability spaces for all i.e., in the similar manner with Section 3, suppose we have -probability spaces

(5.1)

where are in the sense of (3.1.5) for every prime in

Let be the family (5.1) of prime-dynamical -probability spaces 's. Now, we consider the free product -probability space ,

i.e.,

(5.2)

Definition 5.1. The free product -probability space of (5.2) is called the Adelic-dynamical -probability space over .

By the very definition (5.2) of the Adelic-dynamical -probability space over , we obtain the following structure theorem.

Theorem 5.1. Let be our Adelic-dynamical -probability space over . Then is -isomorphic to the conditional tensor product -algebra,

(5.3)

where are the -subalgebras (4.1) of for all and

satisfying that

Proof. Let S be the Adelic-dynamical -probability space (5.2) over a fixed - probability space Then, by the very construction (5.2), one obtains that

in terms of the linear functional where satisfying,

for all

By the structure theorem (5.3), we understand the free product -algebra and its -isomorphic -algebra,

(5.4)

as the same -algebra.

Notation and Assumption For convenience, we denote the conditional-tensor factor, which is a free product -algebra,

(5.5)

i.e.,

where is in the sense of (5.4), and is in the sense of (5.5).

By the structure theorem (5.3) of the Adelic-dynamical -probability space over , we understand and as the same -algebras, and denote them by .

Let be the Adelic-dynamical -probability space over , and let

(5.6)

where

with for

We are interested in the cases where a free random variable T of (5.6) is a free reduced word in , equivalently, we are interested in the case where the prime-sequence

from T of (5.6) is alternating (See (5.6)’, and Section 2.4).

By Section 2.4, for a fixed free reduced word T of (5.6), we have

(5.7)

for all meanwhile,

as a free “non-reduced” word in . Remark that only if in then the above free non-reduced word of forms a free reduced word in .

Here, we are interested in the free-distributional data of the free reduced word of (5.6) in the Adelic-dynamical -probability space over .

Theorem 5.2. Let be a free reduced word (5.6) in the Adelicdynamical -probability space over a fixed -probability space , where

for for Then there exist such that

(5.8)

and

for all

Note that the quantity is free from the choice of the powers of , for all

Proof. Let T be a free reduced word of in the sense of (5.6). Then, by (5.7), one obtains that

for all So,

by (4.17) and (4.18)

for all

The above free-moment formula (5.8) shows that, under certain rational-scalar-multiples, the free-distributional data of are determined by the free-distributional data on a fixed -probability space , where is free reduced word in the sense of (5.6) whose factors satisfy (5.6)’.

Now, let

(5.9)

with

for for

Suppose S of (5.9) is a free sum in . Equivalently, assume that the primes

from the operator S of (5.9), satisfying (5.9)’, are mutually-distinct from each other in Since the primes are mutually-distinct from each other, all summands of are contained in the mutually-distinct free blocks of (5.9)’, for all and hence, they are free from each other in .

Now, recall that

(5.10)

for all by Section 2.4. Meanwhile,

where the summands

are free non-reduced words in , for all

for all (See Section 2.4.)

We are interested in the free-distributional data of for a free sum of (5.9) in .

Theorem 5.3. Let be a free sum (5.9) in the Adelic-dynamical -probability space , satisfying (5.9)’. Then

(5.11)

for all

Proof. By (5.10), if S of (5.9) is a free word in , then one has that

in , for all Thus,

for all

6. Adelic-Dynamical W^*-Probability Spaces over Group W^*-Probability Spaces

In this section, let's apply the main results of Sections 4 and 5 to certain special cases. In particular, we are interested in the cases where fixed unital tracial -probability spaces are induced by countable discrete groups. In the following, all groups are automatically assumed to be countable discrete groups.

Let be a group. Then one can construct the corresponding group Hilbert space,

(6.1)

equipped with its orthonormal basis ,

satisfying

where means the -inner product of the -space of (6.1).

Then the group acts on via an action , satisfying that

(6.2)

For convenience, we denote by , for all Then these images of the action satisfy

(6.3)

and

where mean the inverses of

So, one can realize that

(6.4)

for all by (6.2) and (6.3), where is the group-identity of , and hence is the identity operator on the Hilbert space of (6.1).

The above relation (6.4) guarantees that are unitary operators on , satisfying

where mean the inverses of on .

So, we call the action of acting on , the left-regular unitary representation. Also, the Hilbert-space representation of a group is also called the left-regular unitary representation of .

Define now a -subalgebra of the operator algebra by

(6.5)

where mean the -topology closures of subsets of .

We call the -subalgebra of (6.5), the group -algebra of (in ). By the very definition (6.5) of if , then

where means a finite, or an infinite (limits of finite) sum(s under -topology).

Thus, on the group -algebra, we define the canonical trace by

(6.6)

So, one can understand the canonical tracial values mean the coefficients of the -terms of elements of

Remark that

and hence, the linear functional of (6.6) is a unital trace.

Definition 6.1. Let be the group -algebra (6.5) of in and let be the canonical trace (6.6) on . Then the unital tracial -probability space is said to be the group -probability space of .

Let be a given group, and let be the corresponding group -probability space of . Then, by our general cases in Sections 3, 4 and 5, one can construct the -dynamical systems

and the corresponding crossed product -algebras

for all

Thus, one obtains the p-dynamical diagonal subalgebras

where are the p-adic diagonal -subalgebra of for all

So, as in Section 5, we have the system

(6.7)

of -probability spaces 's.

From the family of (6.7), we can have the corresponding Adelic-dynamical -probability space ,

(6.8)

i.e.,

Let be a generating element of the Adelic-dynamical -probability space , for and where are in the sense of (6.2). Then one obtains that

i.e.,

(6.9)

by (4.17) and (5.8), for all and for all where is the group-identity of .

Thus, one can obtain the following generalized result of (6.9).

Theorem 6.1. Let be free random variables of the Adelic-dynamical -probability space of (6.8), for a fixed group , for for for Assume that

(6.10)

Let be the N-tuple in primes obtained from (6.10). If this N-tuple is an alternating sequence, equivalently, if

and if

for all then we have that

(6.11)

for all

Proof. By (6.10), we have

in the Adelic-dynamical -probability space of a group , for Assume that the prime-sequence is an alternating N-tuple in and suppose

(6.12)

for all

By the alternating property of the operator

(6.13)

forms a free reduced word in , satisfying that

(6.14)

for all

Thus, by (6.14), one can get that

by (6.14)

by (6.12)

Remark here that the existence of for in the assumption (6.12), is guaranteed by the very definition of the semigroup-action of the semigroup and the structure theorem (5.3), which can be re-expressed by (5.4). i.e.,

as a -homomorphic image, and hence, there does exist

such that Therefore, the condition (6.12) in the above theorem is meaningful.

Let be a free reduced word in in the sense of (6.13) above. Notice the difference between and , as we discussed in Section 2.4, i.e.,

meanwhile,

as a “non-reduced” word, for all Thus, one can verify that

whenever in

By the free-moment formula (6.11), one obtains the following corollary.

Corollary 6.2. Let in the Adelic-dynamical -probability space of a group , for and If in for some then

(6.15)

for all

Note that, in the above corollary case, one has

and hence, we automatically obtain that

(6.16)

by (6.15), for all

Indeed, the formula (6.16) holds, because is chosen in a single free block of .

However, one may take

for for where

Then, even though are identical to , the free reduced word

satisfy that

(6.17)

by (6.11).

Corollary 6.3. Let in , for all for for a fixed and and let in . If the corresponding prime-sequence satisfies

then

(6.18)

Proof. The proof of (6.18) is done by (6.11) and (6.17).

Now, let's concentrate on free blocks

of our Adelic-dynamical -probability space of a group .

Definition 6.2. Let be a free block of the Adelic-dynamical -probability space of a group , for a fixed We say that this fixed free block is -independent in if

(6.19)

for all for all

By the very definition (6.19), one can realize that if a free block of the Adelic-dynamical -probability space of a group is -independent, then the semigroup-action not only induces a -isomorphism from to for all but also forms the identity morphism from to for all Therefore, one can characterize the -independence (6.19) of a free block of as follows.

Theorem 6.4. Let be a free block of the Adelic-dynamical -probability space of a group , for a fixed Then is -independent in the sense of (6.19), if and only if

(6.20)

in .

Proof. By the discussions in the very above paragraphs, if a free block is -independent, then

Conversely, if a free block of is -isomorphic to , then the semigroup-action of the sub-semigroup generated by of the semigroup satisfies (6.19). i.e., for all the action forms the identity -isomorphism from

Therefore, the above -isomorphic relation holds, then is -independent in the Adelic-dynamical -probability space .

Then above characterization (6.20) shows that a free block of is -independent, if and only if

equivalently, the conditionality on tensor product can be ignored.

Theorem 6.5. Let be the Adelic-dynamical -probability space of a group , and let be some free blocks of , for for Let

where and If are -independent in the sense of (6.19), for all in , and if

then we obtain

(6.21)

for all

Proof. By the assumption that the prime-sequence is an alternating sequence in the operator is a free reduced word in , satisfying that

So, for any one has that

However, by the -independence (6.19) of , for all one can refine that

for all by (6.20).

Therefore, we have that

for all

So, by (6.15), (6.16) and (6.21), one can get the following corollary.

Corollary 6.6. Let in , for where and Assume that the free block is -independent in . Then

(6.22)

for all

Proof. Let , where is -independent in . Then

for all because is contained in a single free block in . Thus,

as we discussed in (6.16).

By the -independence (6.19) of ,

by (6.20).

So, one obtains that

for all

Therefore, the free-moment formula (6.22) holds.

We finish this section with a special case where a given group is the free group with n-generators. Recall that the free group is a non-abelian group generated by its n-many generators , with no relators (or relations) on it, i.e.,

for all In particular, we are interested in the cases where We cannot help emphasizing the importance of free groups not only in algebra, and combinatorics, but also in operator algebra theory (e.g., see ^[11] and ^[14]).

In the rest of this section, let be the free group with n-generators , for and

the p-adic dynamical -probability spaces, for all and let

be the corresponding Adelic-dynamical -probability space.

We concentrate on the generating operators

(6.23)

for all for all

Lemma 6.7. Let be in the sense of (6.23) in the Adelic-dynamical -probability space of the given free group , for for If is -independent in , then

(6.24)

Proof. By (6.21), or (6.22), one can get that

under the -independence (6.19) of in , for all where means the group-identity of .

By the very construction of the free group , if is a generator of , then and hence, for all Therefore, the vanishing free-moment formula (6.24) holds.

More generally, one can get the following result.

Lemma 6.8. Let be in the sense of (6.23) in , for for for all for Suppose the free blocks are -independent in , for all Let

If

then we have

(6.25)

Proof. By assumption, the operator T is a free reduced word in . So,

by (6.24), under the -independence, for all

Observe that:

Lemma 6.9. Let be in the sense of (6.23) in the Adelicdynamical -probability space of the given free group , for Assume that the free block is -independent in . Let W be an operator of ,which is either

for some Then

(6.26)

for all

Proof. Let in a fixed free block of the Adelic-dynamical -probability space , for Then the adjoint is also contained in , and hence,

for all Thus, for any

for all

Observe that

in inside , for all by the -independence of in . Similarly, we have

i.e., one obtains that

(6.27)

By (6.27), let's concentrate on For any

(6.28)

and similarly,

by (6.27), under the -independence of in .

So, by (6.28), we have that if is either , or for some then

by (6.27).

Thus, if either , or then

for all for all Therefore, the free-distributional data (6.26) is obtained.

Motivated by (6.26), we obtain the following generalized result.

Proposition 6.10. Let be in the sense of (6.23) in the Adelic-dynamical -probability space of the given free group , for and Assume that the free block is -independent in . Let

and let

and

If in then

(6.29)

for all

Proof. As we discussed in the proof of (6.26), if T is given as above, then

and hence, are contained in the same free block in , for all Under the -independence of , if

then

satisfying

Therefore, one has that

for all

By the above three lemmas and proposition, we obtain the following free-distributional data on the Adelic-dynamical -probability space of the given free group .

Theorem 6.11. Let be the Adelic-dynamical -probability space of a given free group , for and let be in the sense of (6.23) in , contained in the free block , for Assume that this free block is-independent in . Let

Also, let

and

Then we obtain that

(6.30)

for all for all

Proof. Let be in the sense of (6.23) in the -independent free block of our Adelic-dynamical -probability space of the free group , for Let

and let

As we discussed in the proof of (6.26), indeed, the operator is contained in the free block of .

Assume first say Then

and hence,

by (6.24) and (6.25).

Similarly, one can verify that if then

Suppose now that Then, by (6.29), one obtains that

The above free-moment formula (6.30) characterizes the free distributions of individual generating elements of the Adelic-dynamical -algebra of the free group , for all for all whenever a free block is -independent in for

By the above free-distributional data (6.30), we obtain the following result.

Theorem 6.12. Let be in the sense of (6.23) in the Adelicdynamical -probability space of a free group , for , for for Suppose the corresponding prime-sequence

is an alternating sequence in and assume that the free blocks are -independent in , for all Let

where

for all

Then we obtain that

(6.31)

for all

Proof. Under hypothesis, the factors of are contained in the free blocks of , for all i.e.,

By the assumption that is an alternating sequence in the operator is a free reduced word in , satisfying that

Thus, one has that

with

for all by (6.30), under the -independence (6.19) of in , for all

Therefore, we have that

for all

Therefore, the free-moment formula (6.31) holds true.

The above theorem characterizes the free distributions of generating elements 's of in terms of the joint free-moment formula (6.31).

References

[1]	I. Cho, Free Product C*-Algebras Induced by *-Algebras over p-Adic Number Fields for Primes p, (2016). Submitted to J. Numb. Theo. Anal. Acad.
	In article
[2]	I. Cho, Semicircular Elements Induced by p-Adic Number Fields, (2016). Submitted to Adv. Oper. Theo.
	In article
[3]	I. Cho, On Dynamical Systems Induced by p-Adic Number Fields, Opuscula Math., 35, no. 4, (2015). 445-484.
	In article		View Article
[4]	I. Cho, Free-Distributional Data of Arithmetic Functions and Corresponding Generating Functions Determined by Gaps of Primes, Compl. Anal. Oper. Theo., vol 8, issue 2, (2014). 537-570.
	In article		View Article
[5]	I. Cho, Classification of Arithmetic Functions and Corresponding Free-Moment L-Functions, Bull. Korean Math. Soc., 52, no. 3, (2015). 717-734.
	In article		View Article
[6]	I. Cho, Dynamical Systems on Arithmetic Functions Determined by Prims, Banach J. Math. Anal., 9, no. 1, (2015). 173-215.
	In article		View Article
[7]	I. Cho, and T. Gillespie, Free Probability on the Hecke Algebra, Compl. Anal. Oper. Theo., (2014).
	In article
[8]	I. Cho, and P. E. T. Jorgensen, Krein-Space Operators Induced by Dirichlet Characters, Special Issues: Contemp. Math.: Commutative and Noncommutative Harmonic Analysis and Applications, Amer. Math. Soc., (2014) 3-33.
	In article
[9]	T. Gillespie, Superposition of Zeroes of Automorphic L-Functions and Functoriality, Univ. of Iowa, (2010). PhD Thesis.
	In article
[10]	T. Gillespie, Prime Number Theorems for Rankin-Selberg L-Functions over Number Fields, Sci. China Math., 54, no. 1, (2011). 35-46.
	In article		View Article
[11]	F. Radulescu, Random Matrices, Amalgamated Free Products and Subfactors of the W*-Algebra of a Free Group of Nonsingular Index, Invent. Math., 115, (1994). 347-389.
	In article		View Article
[12]	R. Speicher, Combinatorial Theory of the Free Product with Amalgamation and Operator-Valued Free Probability Theory, Amer. Math. Soc. Mem., vol 132, no. 627, (1998).
	In article
[13]	V. S. Vladimirov, I. V. Volovich, and E. I. Zelenov, p-Adic Analysis and Mathematical Physics, Ser. Soviet & East European Math., vol 1, (1994). World Scientific.
	In article
[14]	D. Voiculescu, K. Dykemma, and A. Nica, Free Random Variables, CRM Monograph Series, vol 1., (1992).
	In article