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.
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.
In this section, we briefly mention about backgrounds of our proceeding works.
2.1. Free ProbabilityReaders 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.
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).
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.
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
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
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
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
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 ElementsFix 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.
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 SpaceIn 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.
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
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.
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
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.
[1] | I. Cho, Representations and Corresponding Operators Induced by Hecke Algebras, Compl. Anal. Oper. Theo., 10, no. 7, (2016). 1453-1499. | ||
In article | View Article | ||
[2] | I. Cho, Free W*-Dynamical Systems from p-Adic Number Fields and the Euler Totient Functions, Mathematics, 3, (2015). 1095-1138. | ||
In article | View 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 Semicircular Families in Free Product Banach *-Algebras Induced by p-Adic Number Fields over Prims p, Compl. Anal. Oper. Theo., 11, no. 3, (2017). 507-565. | ||
In article | View Article | ||
[5] | I. Cho, Free Distributional Data of Arithmetic Functions and Corresponding Generating Functions, Compl. Anal. Oper. Theo., vol. 8, issue:2, (2014). 537-570. | ||
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., vol. 9, issue 7, (2015). 1491-1531. | ||
In article | View Article | ||
[8] | I. Cho, and P. E. T. Jorgensen, Krein-Space Operators Induced by Dirichlet Char-acters, Special Issues: Contemp. Math.: Commutative and Noncommutative Harmonic Analysis and Applications, Amer. Math. Soc., (2014). 3-33. | ||
In article | View Article | ||
[9] | T. Gillespie, Superposition of Zeroes of Automorphic L-Functions and Functoriality, Univ. of Iowa, PhD Thesis. (2010). | ||
In article | View 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 C*-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 | View Article | ||
[13] | V. S. Vladimirov, I. V. Volovich, and E. I. Zelenov, p-Adic Analysis and Math-ematical Physics, Ser. Soviet & East European Math., vol 1, (1994). World Scientific. | ||
In article | View Article | ||
[14] | D. Voiculescu, K. Dykemma, and A. Nica, Free Random Variables, CRM Monograph Series, vol 1., (1992). Published by Amer. Math. Soc.. | ||
In article | View Article | ||
This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit
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[1] | I. Cho, Representations and Corresponding Operators Induced by Hecke Algebras, Compl. Anal. Oper. Theo., 10, no. 7, (2016). 1453-1499. | ||
In article | View Article | ||
[2] | I. Cho, Free W*-Dynamical Systems from p-Adic Number Fields and the Euler Totient Functions, Mathematics, 3, (2015). 1095-1138. | ||
In article | View 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 Semicircular Families in Free Product Banach *-Algebras Induced by p-Adic Number Fields over Prims p, Compl. Anal. Oper. Theo., 11, no. 3, (2017). 507-565. | ||
In article | View Article | ||
[5] | I. Cho, Free Distributional Data of Arithmetic Functions and Corresponding Generating Functions, Compl. Anal. Oper. Theo., vol. 8, issue:2, (2014). 537-570. | ||
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., vol. 9, issue 7, (2015). 1491-1531. | ||
In article | View Article | ||
[8] | I. Cho, and P. E. T. Jorgensen, Krein-Space Operators Induced by Dirichlet Char-acters, Special Issues: Contemp. Math.: Commutative and Noncommutative Harmonic Analysis and Applications, Amer. Math. Soc., (2014). 3-33. | ||
In article | View Article | ||
[9] | T. Gillespie, Superposition of Zeroes of Automorphic L-Functions and Functoriality, Univ. of Iowa, PhD Thesis. (2010). | ||
In article | View 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 C*-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 | View Article | ||
[13] | V. S. Vladimirov, I. V. Volovich, and E. I. Zelenov, p-Adic Analysis and Math-ematical Physics, Ser. Soviet & East European Math., vol 1, (1994). World Scientific. | ||
In article | View Article | ||
[14] | D. Voiculescu, K. Dykemma, and A. Nica, Free Random Variables, CRM Monograph Series, vol 1., (1992). Published by Amer. Math. Soc.. | ||
In article | View Article | ||