THE CHARACTERIZATIONS OF A SEMICIRCLE LAW BY

THE CERTAIN FREENESS IN A C^{∗}− PROBABILITY SPACE

Osamu Hiwatashi^{∗}, Masaru Nagisa^{∗} and Hiroaki Yoshida^{†}

∗ Department of Mathematics and Informatics Faculty of Science Chiba University

1-33, Yayoi-Cho, Inage-Ku, Chiba 263 Japan

† Department of Information Sciences Faculty of Science Ochanomizu University

2-1-1, Otsuka, Bunkyou, Tokyo 112 Japan

Abstract. In usual probability theory, various characterizations of the Gaussian law have been obtained. For instance, independence of the sample mean and the sample variance of independently identically distributed random variables characterizes the Gaussian law and the property of remaining independent under rotations characterizes the Gaussian random variables. In this paper, we consider the free anologue of such a kind of characterizations replacing independence by freeness. We show that freeness of the certain pair of the linear form and the quadratic form in freely identically distributed noncommutative random variables, which covers the case for the sample mean and the sample variance, characterizes the semicircle law. Moreover we give the alternative proof for Nica’s result that the property of remaining free under rotations characterizes a semicircular system. Our proof is more direct and straightforward one.

0 Introduction In [Vo1], D. Voiculescu began studying operator algebra free products from the

probabilistic point of view. The idea is to look at free products as an analogue of tensor products and to develop a corresponding noncommutative probabilitic framework where freeness is given a treatment similar to independence. The analogy between freeness and independence is that around freeness, several concepts can be developed similar to those around independence, free random variables, the central limit theorem for free random variables, the addition of free random variables, process with free increments etc..

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Especially the central limit theorem for free random variables holds with limit distribution equal to a semicircle law. And semicircle laws play in many respects the role of the Gaussian laws, when independence is replaced by freeness in a noncommutative probability space, consequently, semicircle laws are in some sense the free analogue of the Gaussian ones. The explanations for occurrence of the semicircle law, both in the free central limit theorem and Wigner’s works (see [Wi]) on asymptotics of large random matrices, are found in [Vo3].

In usual probability theory, various characterizations of the Gaussian laws have been obtained, for instance see [KLR]. Thus it is natural to consider the free analogue of such characterization problems of the semicircle law in noncommutative case too. In [BV], H. Bercovici and D. Voiculescu, however, showed that the free analogue of the Cramér theorem on the decomposition of a semicircle law is failure. Nevertheless, A. Nica showed that the property of remaining free under rotations characterizes a semicircular system in [Ni], which is the free analogue of the wellknown fact in usual probability theory, that the property of remaining independent under rotations characterizes the Gaussian random variables, see [Fe], Section III.4.

In this paper, we establish some characterizations of the semicircle law by the freeness of noncommutative random variables replacing independence by freeness. We have three sections in this paper. Section 1 is devoted to preliminary materials for noncommutative probability spaces and free random variables.

In section 2, we consider the characterization of the semicircle law by freeness of the certain pair of the linear from and the quadratic form in freely identically distributed noncommutative random variables, which covers the free analogue of the well-known fact in usual probability theory that the independence of the sample mean and the sample variance of independently identically distributed random variables characterizes the Gaussian laws [KS]. We show the semicircleness of the distribution by calculating the higher moments with induction because our distribution has a compact support.

In section 3, we give the alternative proof for the above Nica’s work. He showed this characterization as an application of the multidimensional R−transform introduced in his paper [Ni], that is the multivariate version of Voiculescu’s R−transform [Vo2] for a noncommutative random variable, the free analogue of the cumulants generating series. However, we show it by using the same technique that we use in section 2 and our poof is rather direct and straightforward one.

1 Noncommutative probability spaces and a semicircular system

This section contains preliminaries concerning with noncommutative probability spaces and free random variables. Recall that a usual probability space is (Ω, Σ, μ), where Ω is a base space, Σ is a σ− algebra and μ is a probability measure (i.e. positive and satisfying μ(Ω) = 1). A random variable is a mesurable function f : Ω → ℂ, and if f is integrable then its expectation E(f) is given by

(1.1)  E(f) = ∫_{ω∈Ω} fdμ(ω).

We can consider a noncommutative probability space in a purely algebraic frame as an analogue of the above usual probability space.

Definition 1.1. A noncommutative probability space is (A, ϕ), where A is a unital algebra and ϕ : A → ℂ is a linear functional with ϕ(1) = 1. We say that (A, ϕ) is a C^{∗}− probability space when, in addition, A is a C^{∗}− algebra and ϕ is a state, and it is a W^{∗}− probability space when A is a von Neumann algebra and ϕ is a normal state.

One can define independence in a noncommutative probability space, generalizing the usual definition. Note that independence of usual random variables was actually express in terms of the subalgebra of L^{∞}(Ω), so it makes sense to speak of independence of subalgebras in a noncommutative context.

Definition 1.2. Let (A, ϕ) be a noncommutative probability space, and A_{i} ⊂ A be subalgebra (i ∈ I), for the index set I. We say that the family (A_{i})_{i∈I} is independent if

(i) for every i_{1}, i_{2} ∈ I (i_{1} ≠ i_{2}),

x_{1}x_{2} = x_{2}x_{1} for x_{1} ∈ A_{i_{1}} and x_{2} ∈ A_{i_{2}},

(ii)

ϕ(x_{1}x_{2} ⋯ x_{n}) = ϕ(x_{1})ϕ(x_{2}) ⋯ ϕ(x_{n})
whenever x_{j} ∈ A_{i_{j}} and i_{1}, i_{2}, ⋯, i_{n} are distinct.

Note that if 1 ∈ A_{i} (ii) is equivalent to (ii)′. (ii)′

ϕ(x_{1}x_{2} ⋯ x_{n}) = 0
whenever x_{j} ∈ A_{i_{j}} and i_{1}, i_{2}, ⋯, i_{n} are distinct and ϕ(x_{j}) = 0 for all j.

This form of independence is based on the tensor product of algebras. We will replace this concept by a more noncommutative one which takes instead of the tensor product the reduced free product. This concept is due to D. Vioculescu and explained below.

Definition 1.3. Let (A, ϕ) be a noncommutative probability space and A_{i} be subalgebra of A containing the identity element of A, 1 ∈ A_{i} ⊂ A, for i ∈ I. We say that the family (A_{i})_{i∈I} is free if

ϕ(x_{1}x_{2} ⋯ x_{n}) = 0

whenever x_{j} ∈ A_{i_{j}} and i_{1} ≠ i_{2} ≠ ⋯ ≠ i_{n} and ϕ(x_{j}) = 0 for all j.

A family of subsets X_{i} ⊂ A (resp. elements x_{i} ∈ A) will be called free if the family of subalgebras A_{i} generated by {1} ∪ X_{i} (resp. {1, x_{i}}) is free.

Freeness is related to independence only by analogy, since independence is about mutually commuting noncommutative subalgebras and freeness is highly noncommutative. The definition of freeness allows us to calculate all moments, the expectation of a product element. This calculation is done according the following recursive procedure.

Let i_{1} ≠ i_{2} ≠ ⋯ ≠ i_{n} and x_{j} ∈ A_{i_{j}} be given, where the family of subalgebras (A_{i}) is free. We put

(1.2)  x_{j}^{∘} = x_{j} − ϕ(x_{j})1

then x_{j}^{∘} ∈ A_{i_{j}} and ϕ(x_{j}^{∘}) = 0. It follows that

(1.3) \begin{aligned}ϕ(x_{1}x_{2} ⋯ x_{n}) & = ϕ((x_{1}^{∘} + ϕ(x_{1})1)(x_{2}^{∘} + ϕ(x_{2})1) ⋯ (x_{n}^{∘} + ϕ(x_{n})1)) \\ & = ∑_{π,σ} ϕ(x_{π(1)})ϕ(x_{π(2)}) ⋯ ϕ(x_{π(m)})ϕ(x_{σ(1)}^{∘}x_{σ(2)}^{∘} ⋯ x_{σ(n−m)}^{∘}), \end{aligned}

where the sum runs over all partitions of {1, 2, ⋯, n} into two ordered sets, {π(1), π(2), ⋯, π(m)} and {σ(1), σ(2), ⋯, σ(n − m)} with m ≥ 1. From the definition, the term for m = 0 vanishes. After combining neighboring elements of the same algebra, the terms x_{σ(1)}^{∘}x_{σ(2)}^{∘} ⋯ x_{σ(n−m)}^{∘} can be written as a product y_{1}y_{2} ⋯ y_{n′} with y_{j} ∈ A_{l_{j}} and l_{1} ≠ l_{2} ≠ ⋯ ≠ l_{n′} . Now we can repeat this procedure and because of n′ < n, this procedure will be stopped after finitely many steps.

In general, the calculation of a moment becomes much more combinatorially complicated for longer one. We will, however, have the next useful property of the calculation procedure which will be sometimes used in this paper.

Lemma 1.4. Let (A_{i})_{i∈I} be a free family of subalgebras in a noncommutative probability space (A, ϕ). Consider a product x_{1}x_{2} ⋯ x_{n} with x_{j} ∈ A_{i_{j}} for j = 1, 2, ⋯, n. If there is one algebra which appears only once, i.e. these exists the number k with i_{k} ≠ i_{j} for all k ≠ j, then we have

(1.4)  ϕ(x_{1}x_{2} ⋯ x_{n}) = ϕ(x_{k})ϕ(x_{1} ⋯ x_{k−1}x_{k+1} ⋯ x_{n}).

This lemma follows from the definition of freeness. Of course, if x_{1} ⋯ x_{k−1}x_{k+1} ⋯ x_{n} fulfils again the requirements of this lemma, we may repeart this procedure.

We call an expectation of the form ϕ(x^{n}) the simple moment of x of order n.

Definition 1.5. Let (A, ϕ) be a noncommutative probability space. A random variable is an element x ∈ A. The distribution of x is the linear functional μ_{x} on ℂ[X] (the algebra of complex polynomials in the variable X), defined by

(1.5)  μ_{x}(P (X)) = ϕ(P (x)), for all P ∈ ℂ[X].

Note that the distribution of a random variable x ∈ A is nothing more than a way of discribing the simple moments.

Remark 1.6. In a C^{∗}− probability space (A, ϕ), if x is a self-adjoint element of A then the distribution of x, μ_{x}, extends to a compactly supported measure on ℝ, namely there exists a unique measure, dμ_{x} on ℝ such that

(1.6)  ∫ P (t)dμ_{x}(t) = ϕ(P (x)).

An important distribution in a noncommutative probability space is the semicircle law, which plays in free probability theory a role analogous to one of Gaussian law in usual probability theory.

Definition 1.7. The semicircle law centered at m ∈ ℝ and of radius r > 0 is distribution γ_{m,r} : ℂ[X] → ℂ definied by

(1.7)  γ_{m,r}(P (X)) = \frac{2}{πr^{2}} ∫_{m−r}^{m+r} P (t)\sqrt{r^{2} − (t − m)^{2}}dt.

Suppose (A, ϕ) is a noncommutative probability space, an element x ∈ A will be called semicircular centered at m ∈ ℝ and of radius r > 0 if its distribution μ_{x} is γ_{m,r}.

It is known that the semicircle law play in many respects the role of the Gaussian one, when independence is replaced by freeness. And similary to the Gaussian case, the semicircle law is determined by its moments of orders one and two; γ_{m,r}(X) = m

and γ_{m,r}(X^{2}) = m^{2} + \frac{r^{2}}{4} .

Remark 1.8. Concerning with the moments of a centered (m = 0) semicircular element x of radius r, it is easily done via integration by parts and induction that

(1.8)  \begin{aligned}ϕ(x^{k}) & = ∫_{−r}^{r} t^{k}\sqrt{r^{2} − t^{2}}dt \\ & = \{ \begin{matrix}0 & if k = 2m + 1, \\ \frac{(2m)!}{m!(m + 1)!} (\frac{r^{2}}{4} )^{m} & if k = 2m, for m ≥ 0. \end{matrix} \end{aligned}

The number C_{m} = \frac{(2m)!}{m!(m + 1)!} is known as the Catalan number, which will appear in various fields of combinatorics. Moreover, the second moment of the centered semicircular element of radius r is given by \frac{r^{2}}{4} so the relations (1.8) can be read as

(1.9)  \{ \begin{matrix}ϕ(x^{2m+1}) = 0, \\ ϕ(x^{2m}) = C_{m}ϕ(x^{2})^{m}, & for m ≥ 0. \end{matrix}

As we noted before, the distribution of a self-adjoint element in a C^{∗}− probability space extends to a compactly supported probability measure on ℝ and if it is absolutely continuous with respect to the Lebesgue measure and has the continuous probability density function, then it can be completely determined by its moments. Hence we can assert that, for a self-adjoint element x in a C^{∗}− probability space (A, ϕ), x is a centered semicircular element if and only if the moments of x satisfy the relations (1.9).

Definition 1.9. In a C^{∗}− probability space (A, ϕ), a family (x_{i})_{i=1,2,··· ,n} of random variables is called freely identically distributed if it is a free family of the self-adjoint elements and each x_{i} has the same distribution. We call a family (x_{i})_{i=1,2,··· ,n} of freely identically distributed random variables semicircular system if the identical distribution of each x_{i} is given by the centered semicircle law γ_{0,r} for some r > 0.

Speaking in the terms of its moments, they satisfy the following relations:

(1.10)  \{ \begin{matrix}ϕ(x_{i}^{2m+1}) = 0, \\ ϕ(x_{i}^{2m}) = C_{m}ϕ(x_{i}^{2})^{m} & for 1 ≤ i ≤ n and m ≥ 0, \\ ϕ(x_{i}^{2}) = ϕ(x_{j}^{2}) > 0 & for 1 ≤ i, j ≤ n, \end{matrix}

where C_{m} are the Catalan numbers.

2 Freeness of the linear form and the quadratic form

In this section, we will give the characterization of a semicircular system by the freeness of the linear form and the quadratic form in noncommutative random variables.

In commutative case, for a given family of independently identically distributed random variables, the most important linear form and quadratic form are the sample mean and the sample variance, respectively.

It is known that if the identical distribution is Gaussian then the sample mean and the sample variance are independent. Conversely, independence of the sample mean and the sample variance implies the Gaussianity of the identical distribution. That is, independence of the sample mean and the sample variance characterizes the Gaussian law [KS].

Having the above fact in mind, we establish the noncommutative version of such a characterization replacing independence by freeness.

We shall begin with the key result which is found in [VDN], Proposition 5.1.2.

Proposition 2.1. Let (x_{i})_{i=1,2,··· ,n} be a semicircular system in a C^{∗}− probability space (A, ϕ) and let T = (t_{ij}) ∈ O(n) be an n × n real orthogonal matrix. We put

(2.1)  y_{i} = ∑_{j=1}^{n} t_{ij}x_{j}, (i = 1, 2, ⋯, n).

Then the family (y_{i})_{i=1,2,··· ,n} is also a semicircular system.

In Proposition 5.1.2 in [VDN], the traciality of the state ϕ was imposed, but in order to have the above statement, it is no matter to regard the C^{∗}− algebra A as the C^{∗}− free product ∏ ∗_{i=1}^{n}A_{i} of the algebras A_{i} generated by {1, x_{i}}. Since each A_{i} is abelian and the state on ∏ ∗_{i=1}^{n}A_{i} is determined by its restriction on each A_{i}, see Proposition 2.5.5 in [VDN], so the traciality of the state is automatically derived in this case.

We consider the certain pair of a linear form and a quadratic form which covers the case of the sample mean and the sample variance.

Proposition 2.2. Let (x_{i})_{i=1,2,··· ,n} be a semicircular system in a C^{∗}− probability space (A, ϕ). Let A = (a_{ij}) ∈ M_{n}(ℝ) be an n × n real matrix and b = t(b_{1}, b_{2}, ⋯, b_{n}) ∈ ℝ^{n} a non-zero n− dimensional real vector satisfying with

(2.2)  Ab = 0 and ^{t}bA = 0.

Then we have that the linear form ℓ = ∑_{i=1}^{n} b_{i}x_{i} and the quadratic form q = ∑_{i,j=1}^{n} a_{ij}x_{i}x_{j} are free.

Proof. Without loss of generality, we may asuume that ||b|| = (∑_{i=1}^{n} b_{i}^{2})^{1/2} = 1. Take the orthonormal system of ℝ^{n}, {c_{1}, c_{2}, ⋯, c_{n−1}, c_{n} = b} and denote c_{i} = t(c^{i}_{1}, c^{i}_{2}, ⋯, c^{i}_{n}). We make the orthogonal matrix S by Sc_{i} = e_{i}, where e_{i} is

the i-th canonical unit vector ^{t}(0, ⋯, 1^{⌣}^{i}, ⋯, 0). By the assumption (2.2), we have SA^{t}SSb = 0 and ^{t}b^{t}SSA^{t}S = 0, which implies that the matrix SA^{t}S can be written of the form

(2.3)  SA^{t}S = ( \begin{matrix}& & & 0 \\ & α_{ij} & & ⋮ \\ & & & 0 \\ 0 & … & 0 & 0 \end{matrix} ) = ∑_{i,j=1}^{n−1} α_{ij}E_{ij},

where E_{ij} means the (i, j)−th matrix unit and α_{ij} is a constant. Thus we obtain that

(2.4)  A = ∑_{i,j=1}^{n−1} α_{ij}c_{i}^{t}c_{j}

and the quadratic form q turns out

(2.5)  q = ∑_{i,j=1}^{n−1} α_{ij}y_{i}y_{j},

where y_{i} = ∑_{k=1}^{n} c^{i}_{k}x_{k}, i = 1, 2, ⋯, n. From Proposition 2.1, the elements y_{1}, y_{2}, ⋯, y_{n−1} and ℓ = b_{1}x_{1} + b_{2}x_{2} + ⋯ + b_{n}x_{n} are free. Hence we obtain that ℓ is free from the C^{∗}− subalgebra generated by {y_{1}, y_{2}, ⋯, y_{n−1}}, namely ℓ and q = ∑_{i,j=1}^{n−1} α_{ij}y_{i}y_{j} = ∑_{i,j=1}^{n} a_{ij}x_{i}x_{j} are free. Because we know that, in general, if x, y, z are free in (A, ϕ) then x is free from the algebra C^{∗}({y, z}) ⊂ A, see Proposition 2.5.5 in [VDN].

□

Theorem 2.3. Let x_{1}, x_{2}, ⋯, x_{n} be freely identically distributed random variables with zero expectations, ϕ(x_{i}) = 0, in a C^{∗}−probability space (A, ϕ). Let A = (a_{ij}) ∈ M_{n}(ℝ) be an n × n non-negative definite real matrix and b = t(b_{1}, b_{2}, ⋯, b_{n}) ∈ ℝ^{n} be an n− dimensional non-negative vector satisfying with the conditions

(2.6)  Ab = 0 and ∑_{i=1}^{n} b_{i}a_{ii} ≠ 0.

We put the linear form ℓ = ∑_{i=1}^{n} b_{i}x_{i} and the quadratic form q = ∑_{i,j=1}^{n} a_{ij}x_{i}x_{j}. Then the forms ℓ and q are free if and only if the family (x_{i})_{i=1,2,··· ,n} is a semicircular system.

Proof. First, we show that ϕ(x_{1}^{2m+1}) = 0 for m ≥ 0 by induction on m.

For m = 0, it is nothing but our assumption. For m ≥ 1, we consider the expectation ϕ(ℓq^{m}). Using the freeness of ℓ and q, we have ϕ(ℓq^{m}) = ϕ(ℓ)ϕ(q^{m}) = 0. On the other hand, we have the following expansion:

(2.7) \begin{aligned}ϕ(ℓq^{m}) & = ϕ((∑_{i=1}^{n} b_{i}x_{i})( ∑_{i,j=1}^{n} a_{ij}x_{i}x_{j})^{m}) \\ & = ∑_{i=1}^{n} b_{i}a_{ii}^{m}ϕ(x_{i}^{2m+1}) \\ & + lin\{ϕ(x_{1}^{ι_{1,1}})ϕ(x_{1}^{ι_{1,2}}) ⋯ ϕ(x_{1}^{ι_{1,k(1)}})ϕ(x_{2}^{ι_{2,1}})ϕ(x_{2}^{ι_{2,2}}) ⋯ \\ & ⋯ ϕ(x_{2}^{ι_{2,k(2)}}) ⋯ ϕ(x_{n}^{ι_{n,1}})ϕ(x_{n}^{ι_{n,2}}) ⋯ ϕ(x_{n}^{ι_{n,k(n)}}) ; \\ & ∑_{j=1}^{n} ∑_{l=1}^{k(j)} ι_{j,l} = 2m + 1, #\{j : k(j) > 0\} ≥ 2\}. \end{aligned}

Here the second term of the above expression means that the linear combination of the products of the simple moments. And this linear combination is homogeneous of dgree 2m + 1 in the sense that ∑_{j=1}^{n} ∑_{l=1}^{k(j)} ι_{j,l} = 2m + 1. If all ι_{j,l} are even integers then the summation ∑ ι_{j,l} should be even. Nevertheless ∑ ι_{j,l} is odd number so we can find a moment of some odd order among the factors of each summand, and its order is strictly smaller than 2m + 1. Thus the induction hypothesis implies that the term of this linear combination must be 0. Hence we have that

(2.8)  ϕ(ℓq^{m}) = ∑_{i=1}^{n} b_{i}a_{ii}^{m}ϕ(x_{i}^{2m+1}) = 0

and then

(2.9)  (∑_{i=1}^{n} b_{i}a_{ii}^{m})ϕ(x_{1}^{2m+1}) = 0

since all x_{i}’s have the same distribution. From the assumption, it follows that ∑_{i=1}^{n} b_{i}a_{ii}^{m} > 0, which implies that ϕ(x_{1}^{2m+1}) = 0.

Next using induction on m, we shall show that ϕ(x_{1}^{2m}) can be expressed as

(2.10)  ϕ(x_{1}^{2m}) = α_{m}ϕ(x_{1}^{2})^{m},

where α_{m} is the constant which depends only on the matrix A and the vector b. For m = 1, it is clear that α_{1} = 1. For m ≥ 2, we assume that the constants α_{1}, α_{2}, ⋯, α_{m−1} have been determined by the matrix A and the vector b, as the induction hypothesis. Now we consider the expectation ϕ(ℓ^{2}q^{m−1}) and expand it in two different ways. The first way is as follows:

(2.11)

ϕ(ℓ^{2}q^{m−1}) = ϕ(ℓ^{2})ϕ(q^{m−1})  by the freeness of ℓ and q

= ϕ( ∑_{i,j=1}^{n} b_{i}b_{j}x_{i}x_{j})ϕ(( ∑_{i,j=1}^{n} a_{ij}x_{i}x_{j})^{m−1}).

The freeness of (x_{i}) implies that ϕ(x_{i}x_{j}) = 0 if i ≠ j. Thus we have, on the first factor of (2.11),

(2.12)  ϕ( ∑_{i,j=1}^{n} b_{i}b_{j}x_{i}x_{j}) = ϕ(∑_{i=1}^{n} b_{i}^{2}x_{i}^{2}) = ∑_{i=1}^{n} b_{i}^{2}ϕ(x_{i}^{2}) = (∑_{i=1}^{n} b_{i}^{2})ϕ(x_{1}^{2}),

because x_{i}’s have the same distribution. The second factor of (2.11) can be expanded into 2(m−1)-homogeneous linear combination of the products of the simple moments. And only the summands constituted from the simple moments of even orders could appear because the moments of odd orders will vanish as we have shown just above. Using the induction hypothesis and the assumption that x_{i}’s have the same distribution, the second factor can be written as

(2.13)  ϕ(( ∑_{i,j=1}^{n} a_{jk}x_{i}x_{j})^{m−1}) = F_{1}(A, b, α_{1}, α_{2}, ⋯, α_{m−1})ϕ(x_{1}^{2})^{m−1},

where F_{1} is the constant determined by the matrix A, the vector b and the constants α_{1}, α_{2}, ⋯, α_{m−1}. Combinding the equalities (2.11), (2.12) and (2.13), we have

(2.14)  ϕ(ℓ^{2}q^{m−1}) = (∑_{i=1}^{n} b_{i}^{2})F_{1}(A, b, α_{1}, α_{2}, ⋯, α_{m−1})ϕ(x_{1}^{2})^{m}.

The other way for the expansion is as follows:

(2.15) \begin{aligned}ϕ(ℓ^{2}q^{m−1}) & = ϕ(( ∑_{i,j=1}^{n} b_{i}b_{j}x_{i}x_{j})( ∑_{k,l=1}^{n} a_{kl}x_{k}x_{l})^{m−1}) \\ & = (∑_{i=1}^{n} b_{i}^{2}a_{ii}^{m−1})ϕ(x_{1}^{2m}) \\ & + lin\{ϕ(x_{1}^{ι_{1,1}})ϕ(x_{1}^{ι_{1,2}}) ⋯ ϕ(x_{1}^{ι_{1,k(1)}})ϕ(x_{2}^{ι_{2,1}})ϕ(x_{2}^{ι_{2,2}}) ⋯ \\ & ⋯ ϕ(x_{2}^{ι_{2,k(2)}}) ⋯ ϕ(x_{n}^{ι_{n,1}})ϕ(x_{n}^{ι_{n,2}}) ⋯ ϕ(x_{n}^{ι_{n,k(n)}}) ; \\ & ∑_{j=1}^{n} ∑_{l=1}^{k(j)} ι_{j,l} = 2m, #\{j : k(j) > 0\} ≥ 2\} \end{aligned}

As we used in (2.7), the second term of the above expression means that the linear combination of the products of the simple moments. Moreover this linear combination is homogeneous of dgree 2m in the sense that ∑_{j=1}^{n} ∑_{l=1}^{k(j)} ι_{j,l} = 2m. Then only the products of the moments of even orders, which are strictly smaller than 2m, could appear because we know that the moments of odd orders will vanish. Again use the induction hypothesis and the assumption that x_{i}’s have the same distribution, this linear combination can be written as

(2.16)  F_{2}(A, b, α_{1}, α_{2}, ⋯, α_{m−1})ϕ(x_{1}^{2})^{m},

where F_{2} means the constant determined by the matrix A, the vector b and α_{1}, α_{2}, ⋯, α_{m−1}. Consequently, we obtain that

(2.17) ϕ(ℓ^{2}q^{m−1}) = (∑_{i=1}^{n} b_{i}^{2}a_{ii}^{m−1})ϕ(x_{1}^{2m}) + F_{2}(A, b, α_{1}, α_{2}, ⋯, α_{m−1})ϕ(x_{1}^{2})^{m}.

From the equalities (2.14) and (2.17), it follows that

(2.18)  \begin{matrix}ϕ(x_{1}^{2m}) = (∑_{i=1}^{n} b_{i}^{2}a_{ii}^{m−1})^{−1}\{( & ∑_{i=1}^{n} b_{i}^{2})F_{1}(A, b, α_{1}, α_{2}, ⋯, α_{m−1}) \\ − & F_{2}(A, b, α_{1}, α_{2}, ⋯, α_{m−1})\}ϕ(x_{1}^{2})^{m}, \end{matrix}

where the assumption implies that ∑_{i=1}^{n} b_{i}^{2}a_{ii}^{m−1} > 0. Hence we obtain that α_{m} is determined by the matix A and the vector b, which ends induction.

By the way, since the matrix A and the vector b satisfy the conditions of Proposition 2.2, if we set (x_{1}, x_{2}, ⋯, x_{n}) be a semicircular system then ℓ and q become a free pair. So the constants α_{i}’s are the universal constants determined only by the freeness of ℓ and q. This implies that α_{m} is the Catalan number C_{m} = \frac{(2m)!}{(m + 1)!m!}.

□

Corollary 2.4. Let x_{1}, x_{2}, ⋯, x_{n} be freely identically distributed random variables in a C^{∗}− probability space (A, ϕ). We put x̄ = \frac{1}{n} ∑_{i=1}^{n} x_{i} (the sample mean) and v = \frac{1}{n} ∑_{i=1}^{n}(x_{i} − x̄)^{2} (the sample variance) are free if and only if (x_{i}) is the semicircular system.

Proof. As similar to the commutative case, v is given by putting the coefficient matrix A = (a_{ij}) as a_{ij} = \frac{1}{n}δ_{ij} − \frac{1}{n^{2}} , where δ_{ij} is the Kronecker’s delta and x̄ is given by the vector b = (\frac{1}{n}, \frac{1}{n}, …, \frac{1}{n}). Then the matrix A and the vector b satisfy the requirements of Theorem 2.3.

□

3 Remaining free under rotations and semicircle laws

In this section, we give the alternative proof for Nica’s work [Ni], that the property of remaining free under rotations characterizes semicircular elements. This is the free analogue of the well-known fact in usual probability theory that property of remaining independent under rotations characterizes Gaussian random varibles, see for instance Section III.4 in [Fe].

In [Ni], A. Nica introduced the multidimensional R−transform (free analogue of the cumulants generating series) and investigated its behavior under linear transformations of the coodinates. Using these results, he proved the above characterization. We will, however, apply the same method which we have used in the previous section, that is a direct calculation of the higher moments of noncommutative random variables, in order to give the alternative proof.

We shall begin with the definition of the some property for a matrix which will be required for an orthogonal transformation of random variables.

Definition 3.1. Let A = (a_{ij}) ∈ M_{n}(ℝ) be an n × n real matrix. For the pair of integers (i, j) ∈ {1, 2, ⋯, n} × {1, 2, ⋯, n}, we consider the set K_{(i,j)} = {k | 1 ≤ k ≤ n, a_{ki}a_{kj} ≠ 0}. If the set K_{(i,j)} has at least two elements, i.e. #K_{(i,j)} ≥ 2, then we write i ∼^{A} j or simply write i ∼ j if no confusion.

Moreover, we write i ≈^{A} j if there exist numbers j_{1}, j_{2}, ⋯, j_{l} such that i = j_{1}, j_{1} ∼ j_{2}, ⋯,j_{l−1} ∼ j_{l}, j_{l} = j.

Definition 3.2. A matrix A is called indecomposable if i ≈^{A} j for any i and j.

Proposition 3.3. Let A ∈ M_{n}(ℝ) be an orthogonal matrix. Then A is indecomposable if and only if C_{1}AC_{2} does not have the form

(3.1)  ( \begin{matrix}A_{1} & 0 \\ 0 & A_{2} \end{matrix} )

for any permutation matrices C_{1}, C_{2}, where A_{1}, A_{2} are orthogonal matrices and their dimensions are strictly smaller than n.

Proof The part of “if” is trivial. Now, we assume that A is not indecomposable. Then without loss of generality, we may assume that 1 ≈^{A} 2 ≈^{A} ⋯ ≈^{A} k and i ≉^{A} j for every 1 ≤ i ≤ k , k + 1 ≤ j ≤ n.

Note first that, in general, if row vectors (b_{1}, b_{2}, ⋯, b_{n}) and (c_{1}, c_{2}, ⋯, c_{n}) in ℝ^{n}satisfy

(3.2)  b_{1}c_{1} + b_{2}c_{2} + ⋯ + b_{n}c_{n} = 0

and

(3.3)  #{k : b_{k}c_{k} ≠ 0} ≤ 1

then b_{l} ≠ 0 implies c_{l} = 0. This means that a_{ki} ≠ 0 implies a_{kj} = 0 if i ≁ j.

Exchanging column vectors of A, we may assume that there exist 1 = i_{0} ≤ i_{1} ≤ ⋯ ≤ i_{k} ≤ n such that

(3.4)  \begin{matrix}a_{st} & ≠ 0,  i_{s−1} ≤ t ≤ i_{s} \\ a_{st} & = 0,  i_{s} < t ≤ n \end{matrix}

for 1 ≤ s ≤ k. In the case 1 ≤ i ≤ k < j ≤ n, i ≉^{A} j. So we have i ≁ j. By the above note and our assumption, we have

(3.5)  a_{st} = 0,

if k < s ≤ n , 1 ≤ t ≤ i_{k} or 1 ≤ s ≤ k, i_{k} < t ≤ n. The linear independence of row (resp. column) vectors of A implies k ≤ i_{k} (resp. k ≥ i_{k}). So we get k = i_{k}. That is, A is decomposed to a direct sum of two orthogonal matrices.

□

Remark 3.4. The condition that A is indecomposable coincides with the stronger condition which can be found in Theorem 5.3′ in [Ni].

Theorem 3.5. Let A = (a_{ij}) ∈ O(n) be an n × n - real orthogonal matrix and indecomposable in the sense of Defnition 3.2. Let x_{1}, x_{2}, ⋯, x_{n} be non-scalar selfadjoint elements of a C^{∗}− probability space (A, ϕ) with zero expectation, ϕ(x_{i}) = 0. If the family (x_{1}, x_{2}, ⋯, x_{n}) is free and the family (y_{1}, y_{2}, ⋯, y_{n}) is still free, where y_{i} = ∑_{j=1}^{n} a_{ij}x_{j}, then the family of noncommutative random variables (x_{1}, x_{2}, ⋯, x_{n}) is a semicircular system.

Proof. It is enough to show the relations (1.10).

First we shall show that ϕ(x_{i}^{2m+1}) = 0 by induction on m. For m = 0, it is nothing but our assumption. For m ≥ 1, all we have to prove is that ϕ(x_{i_{0}}^{2m+1}) = ϕ(x_{j_{0}}^{2m+1}) only for a pair (i_{0}, j_{0}) with i_{0} ∼ j_{0}, because the matrix A is indecomposable in the sense of Definition 3.2.

By the definition, given a pair (i_{0}, j_{0}) with i_{0} ∼ j_{0}, we can find the integers k_{1} and k_{2} such that a_{k_{1}i_{0}}a_{k_{1}j_{0}} ≠ 0 and a_{k_{2}i_{0}}a_{k_{2}j_{0}} ≠ 0. Here we consider the n expectations, ϕ(y_{k_{1}}^{2m−1}y_{k_{2}}y_{ℓ}) for ℓ = 1, 2, ⋯, n.

If ℓ ≠ k_{2} then,using the freeness of y_{k}’s and applying the Lemma 1.4, it follows that

(3.6)  ϕ(y_{k_{1}}^{2m−1}y_{k_{2}}y_{ℓ}) = ϕ(y_{k_{2}})ϕ(y_{k_{1}}^{2m−1}y_{ℓ}) = 0

since ϕ(y_{k_{2}}) = 0. And if ℓ = k_{2} then we have

(3.7)  ϕ(y_{k_{1}}^{2m−1}y_{k_{2}}y_{ℓ}) = ϕ(y_{k_{1}}^{2m−1})ϕ(y_{k_{2}}^{2}).

Apply the same method as we used in the proof of Thorem 2.3, to the simple moment ϕ(y_{k_{1}}^{2m−1}). That is, we expand ϕ(y_{k_{1}}^{2m−1}) to the (2m − 1) homogeneous linear combination of the products of the simple moments of x_{i}’s and the induction hypothesis makes each summand vanish, i.e.

(3.8)  ϕ(y_{k_{1}}^{2m−1}) = 0.

Consequently, we obtain that

(3.9)  ϕ(y_{k_{1}}^{2m−1}y_{k_{2}}y_{ℓ}) = 0 for all ℓ = 1, 2, ⋯, n.

On the other hand, ϕ(y_{k_{1}}^{2m−1}y_{k_{2}}y_{ℓ}) can be expanded as follows:

(3.10) \begin{aligned}ϕ(y_{k_{1}}^{2m−1}y_{k_{2}}y_{ℓ}) & = ϕ((∑_{j=1}^{n} a_{k_{1}j}x_{j})^{2m−1}(∑_{j=1}^{n} a_{k_{2}j}x_{j})(∑_{j=1}^{n} a_{ℓj}x_{j})) \\ & = ϕ(∑_{j=1}^{n} a_{k_{1}j}^{2m−1}a_{k_{2}j}a_{ℓj}x_{j}^{2m+1}) \\ & + lin\{ϕ(x_{1}^{ι_{1,1}})ϕ(x_{1}^{ι_{1,2}}) ⋯ ϕ(x_{1}^{ι_{1,k(1)}}) ⋯ \\ & ⋯ ϕ(x_{n}^{ι_{n,1}})ϕ(x_{n}^{ι_{n,2}}) ⋯ ϕ(x_{n}^{ι_{n,k(n)}}) ; \\ & ∑_{j=1}^{n} ∑_{l=1}^{k(j)} ι_{j,l} = 2m + 1, #\{j : k(j) > 0\} ≥ 2\} \\ & = ∑_{j=1}^{n} a_{k_{1}j}^{2m−1}a_{k_{2}j}a_{ℓj}ϕ(x_{j}^{2m+1}). \end{aligned}

Of course, the part of the linear combination of the products of the simple moments will vanish by the induction hypothesis and only the highest simple moments can survive. Thus we have the linear equations,

(3.11)  ∑_{j=1}^{n} a_{k_{1}j}^{2m−1}a_{k_{2}j}a_{ℓj}ϕ(x_{j}^{2m+1}) = 0  (ℓ = 1, 2, ⋯, n),

which can be written in the matrix form as

(3.12)  A ( a_{k_{1}1}^{2m−1}a_{k_{2}1}ϕ(x_{1}^{2m+1}) ; a_{k_{1}2}^{2m−1}a_{k_{2}2}ϕ(x_{2}^{2m+1}) ; ⋮ ; a_{k_{1}n}^{2m−1}a_{k_{2}n}ϕ(x_{n}^{2m+1}) ) = ( 0 0 ⋮ 0 ) .

The invertibility of the matrix A implies that

a_{k_{1}j}^{2m−1}a_{k_{2}j}ϕ(x_{j}^{2m+1}) = 0 (j = 1, 2, ⋯, n).

Since the integers k_{1} and k_{2} have been chosen such that a_{k_{1}i_{0}}a_{k_{1}j_{0}} ≠ 0 and a_{k_{2}i_{0}}a_{k_{2}j_{0}} ≠ 0, it follows that ϕ(x_{i_{0}}^{2m+1}) = ϕ(x_{j_{0}}^{2m+1}) = 0.

Next we shall show the sameness of the variances that

(3.13)  ϕ(x_{1}^{2}) = ϕ(x_{2}^{2}) = ⋯ = ϕ(x_{n}^{2}).

Because of the indecomposablity of the matrix A, it is enough to show that ϕ(x_{i_{0}}^{2}) = ϕ(x_{j_{0}}^{2}) only for a pair (i_{0}, j_{0}) of integers with i_{0} ∼ j_{0}. From the definition of i_{0} ∼ j_{0}, we can take the integer k_{1} such that a_{k_{1}i_{0}}a_{k_{1}j_{0}} ≠ 0. Here we consider the n expectations, ϕ(y_{k_{1}}y_{ℓ}), ℓ = 1, 2, ⋯, n. It is direct consequence of the freeness of families (x_{i}) and (y_{j}) that

(3.14)  ϕ(y_{k_{1}}y_{ℓ}) = ∑_{i=1}^{n} a_{k_{1}i}a_{ℓi}ϕ(x_{i}^{2}) = \{ \begin{matrix}0 & if ℓ ≠ k_{1} \\ ϕ(y_{k_{1}}^{2}) & if ℓ = k_{1} \end{matrix} ,

which can be written in the matrix form that

(3.15)  A ( a_{k_{1}1}ϕ(x_{1}^{2}) ; ⋮ ; a_{k_{1}k_{1}}ϕ(x_{k_{1}}^{2}) ; ⋮ ; a_{k_{1}n}ϕ(x_{n}^{2}) ) = ( 0 ; ⋮ ; ϕ(y_{k_{1}}^{2}) ; ⋮ ; 0 ) ← the k_{1}th row.

The orthogonality of the matrix A imples that

(3.16)  ( a_{k_{1}1}ϕ(x_{1}^{2}) ; a_{k_{1}2}ϕ(x_{2}^{2}) ; ⋮ ; a_{k_{1}n}ϕ(x_{n}^{2}) ) = ( a_{k_{1}1} ; a_{k_{1}2} ; ⋮ ; a_{k_{1}n} ) ϕ(y_{k_{1}}^{2}),

because A^{−1} is given by^{t}A. As the integer k_{1} has been chosen such that a_{k_{1}i_{0}}a_{k_{1}j_{0}} ≠ 0, it is clear that a_{k_{1}i_{0}} ≠ 0 and a_{k_{1}j_{0}} ≠ 0. Hence we obtain that ϕ(x_{i_{0}}^{2}) = ϕ(x_{j_{0}}^{2}).

Finally, we shall show with induction that ϕ(x_{i}^{2m}) can be written in the form

(3.17)  ϕ(x_{i}^{2m}) = α_{m}^{(i)}ϕ(x_{1}^{2})^{m},

where α_{m}^{(i)} is the constant determined by the matrix A.

For m = 1, α_{1}^{(i)} = 1 (i = 1, 2, ⋯, n) as we have shown just above. For m ≥ 2, we assume that the constants α_{k}^{(i)} (i = 1, 2, ⋯, n, k = 1, 2, ⋯, m − 1) have been determined by the matrix A. Here take a pair (i_{0}, j_{0}) of integers with i_{0} ∼ j_{0} and find integers k_{1} and k_{2} such that a_{k_{1}i_{0}}a_{k_{1}j_{0}} ≠ 0 and a_{k_{2}i_{0}}a_{k_{2}j_{0}} ≠ 0. Then consider the n expectations, ϕ(y_{k_{1}}^{2m−2}y_{k_{2}}y_{ℓ}), ℓ = 1, 2, ⋯, n. By freeness of (y_{i}), we have

(3.18)  ϕ(y_{k_{1}}^{2m−2}y_{k_{2}}^{2}) = \{ \begin{matrix}0 & if ℓ ≠ k_{2} \\ ϕ(y_{k_{1}}^{2m−2})ϕ(y_{k_{2}}^{2}) & if ℓ = k_{2} \end{matrix} .

Then ϕ(y_{k_{1}}^{2m−2}) can be expanded as the (2m − 2) homogeneous linear combination of products of the simple moments, in which only ones of even orders will survive. Hence the induction hypothesis implies that

(3.19)  ϕ(y_{k_{1}}^{2m−2}) = F (A, α_{1}^{(1)}, α_{2}^{(1)}, ⋯, α_{m−1}^{(1)}, α_{1}^{(2)}, ⋯, α_{m−1}^{(n)})ϕ(x_{1}^{2})^{m−1},

where F means the constant determined by the matrix A and the constants α_{1}^{(1)}, α_{2}^{(1)}, ⋯, α_{m−1}^{(1)}. On the second factor of (3.18), we have

(3.20)  \begin{aligned}ϕ(y_{k_{2}}^{2}) & = ϕ( ∑_{i,j=1}^{n} a_{k_{2}i}a_{k_{2}j}x_{i}x_{j}) \\ & = ϕ(∑_{i=1}^{n} a_{k_{2}i}^{2}x_{i}^{2}) = (∑_{i=1}^{n} a_{k_{2}i}^{2}) ϕ(x_{1}^{2}), \end{aligned}

by the knowledge that all ϕ(x_{i}^{2}) are equal. Combining (3.18), (3.19) and (3.20), we obtain that

(3.21)  ϕ(y_{k_{1}}^{2m−2}y_{k_{2}}^{2}) = F (A)(∑_{i=1}^{n} a_{k_{2}i}^{2})ϕ(x_{1}^{2})^{m},

which means that ϕ(y_{k_{1}}^{2m−2}y_{k_{2}}^{2}) can be expressed as the product of the constant F (A) determined by the matrix A and the power of the variance, ϕ(x_{1}^{2})^{m}.

On the other hand, the n expectations ϕ(y_{k_{1}}^{2m−2}y_{k_{2}}y_{ℓ}) could be expaned as follows:

(3.22) \begin{aligned}ϕ(y_{k_{1}}^{2m−2}y_{k_{2}}y_{ℓ}) & = ϕ((∑_{j=1}^{n} a_{k_{1}j}x_{j})^{2m−2}(∑_{j=1}^{n} a_{k_{2}j}x_{j})(∑_{j=1}^{n} a_{ℓj}x_{j})) \\ & = ϕ(∑_{j=1}^{n} a_{k_{1}j}^{2m−2}a_{k_{2}j}a_{ℓj}x_{j}^{2m}) \\ & + lin\{ϕ(x_{1}^{ι_{1,1}})ϕ(x_{1}^{ι_{1,2}}) ⋯ ϕ(x_{1}^{ι_{1,k(1)}}) ⋯ \\ & ⋯ ϕ(x_{n}^{ι_{n,1}})ϕ(x_{n}^{ι_{n,2}}) ⋯ ϕ(x_{n}^{ι_{n,k(n)}}) ; \\ & ∑_{j=1}^{n} ∑_{l=1}^{k(j)} ι_{j,l} = 2m, #\{j : k(j) > 0\} ≥ 2\}. \end{aligned}

Again we apply the similar argument as we used in the proof of Theorem 2.3. That is, the part of the linear combination in the above equality is 2m - homogeneous and only the products of moments of even orders, which are strictly smaller than 2m, could appear. Now the induction hypothesis implies that this linear combination can be written as the product of the constant determined by the matrix A and the power of the variance ϕ(x_{1}^{2})^{m}, i.e.

(3.23)  ϕ(y_{k_{1}}^{2m−2}y_{k_{2}}y_{ℓ}) = ∑_{j=1}^{n} a_{k_{1}j}^{2m−2}a_{k_{2}j}a_{ℓj}ϕ(x_{j}^{2m}) + F_{ℓ}(A)ϕ(x_{1}^{2})^{m},

where F_{ℓ}(A) means some constant determined by the matrix A.

Having in mind the relations (3.18), (3.21) and (3.23), it turns out that

(3.24)  ∑_{j=1}^{n} a_{k_{1}j}^{2m−2}a_{k_{2}j}a_{ℓj}ϕ(x_{j}^{2m}) = G_{ℓ}(A)ϕ(x_{1}^{2})^{m} for ℓ = 1, 2, ⋯, n,

where we denote G_{ℓ}(A) the constants determined by the matrix A as

(3.25)  G_{ℓ}(A) = \{ \begin{matrix}−F_{ℓ}(A) & if ℓ ≠ k_{2}, \\ F (A)(∑_{i=1}^{n} a_{k_{2}i}^{2}) − F_{k_{2}}(A) & if ℓ = k_{2}. \end{matrix}

In the matrix form, we obtain that

(3.26)  A ( a_{k_{1}1}^{2m−2}a_{k_{2}1}ϕ(x_{1}^{2m}) ; a_{k_{1}2}^{2m−2}a_{k_{2}2}ϕ(x_{2}^{2m}) ; ⋮ ; a_{k_{1}n}^{2m−2}a_{k_{2}n}ϕ(x_{n}^{2m}) ) = ( G_{1}(A) ; G_{2}(A) ; ⋮ ; G_{n}(A) ) ϕ(x_{1}^{2})^{m}.

The invertibility of the matrix A ( in this case A^{−1} is given by ^{t}A) implies that

(3.27)  a_{k_{1}j}^{2m−2}a_{k_{2}j}ϕ(x_{j}^{2m}) = (∑_{i=1}^{n} a_{ji}G_{i}(A))ϕ(x_{1}^{2})^{m} for j = 1, 2, ⋯, n.

Since we chose the integers k_{1} and k_{2} such that a_{k_{1}i_{0}}a_{k_{1}j_{0}} ≠ 0 and a_{k_{2}i_{0}}a_{k_{2}j_{0}} ≠ 0, it follows that ϕ(x_{i_{0}}^{2m}) and ϕ(x_{j_{0}}^{2m}) can be written as ϕ(x_{i_{0}}^{2m}) = α_{m}^{(i_{0})}ϕ(x_{1}^{2})^{m} and ϕ(x_{j_{0}}^{2m}) = α_{m}^{(j_{0})}ϕ(x_{1}^{2})^{m}, where α_{m}^{(i_{0})}, α_{m}^{(j_{0})} are the constants determined by the matrix A. Since the pair (i_{0}, j_{0}) has been taken arbitrary, this ends induction.

By the way, if we set (x_{1}, x_{2}, ⋯, x_{n}) be a semicircular system then the family (y_{1}, y_{2}, ⋯, y_{n}) is still free system by Proposition 2.1. The scalar α_{m}^{(i)} must coincide with the Catalan number C_{m} = \frac{(2m)!}{(m + 1)!m!}, since the constants α_{m}^{(i)}’s are the universal constants determined only by the freeness.

□

If x is a centered semicircular element of some radius then the random variable λx, any scalar multiple of x, is still a centered semicircular element. That is, the relations (1.9) are invariant under dilations. It should be noted that, in the proof of Theorem 3.5, we have shown the relations (1.9) directly for the pair of random variables x_{i_{0}} and x_{j_{0}} with i_{0} ∼ j_{0}.

Moreover our assumption that each x_{i} has zero expectation ϕ(x_{i}) = 0 is not essential requirement. Because freeness will be kept in translations. Thus we can give the following slight extended characterization of the above theorem as corollary.

Corollary 3.6. Let A = (a_{ij}) ∈ O(n) be an n × n - real orthogonal matrix and we put B = (b_{ij}) as

(3.28)  B = ( \begin{matrix}λ_{1} & & & & & 0 \\ & λ_{2} \\ & & . & . & . \\ 0 & & & & & λ_{n} \end{matrix} ) A ( \begin{matrix}μ_{1} & & & & & 0 \\ & μ_{2} \\ & & . & . & . \\ 0 & & & & & μ_{n} \end{matrix} ) ,

where λ_{1}λ_{2} ⋯ λ_{n} ≠ 0 and μ_{1}μ_{2} ⋯ μ_{n} ≠ 0. Let x_{1}, x_{2}, ⋯, x_{n} be non-scalar selfadjoint elements of a C^{∗}− probability space (A, ϕ). We assume that the family (x_{1}, x_{2}, ⋯, x_{n}) is free and the family (y_{1}, y_{2}, ⋯, y_{n}) is still free, where y_{i} = ∑_{j=1}^{n} b_{ij}x_{j}. If i ≠ j and i ∼^{B} j then we have that x_{i} and x_{j} are semicircular elements.

Finally we comment that we can have futhermore characterizations of a semicircle law which are related to quadratic forms or the free analogue of noncentral chisquare distributions. They will be discussed in the sequent paper [HKNY].

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