R-cyclic families of matrices in free probability

Alexandru Nica ^{∗} Department of Pure Mathematics

University of Waterloo Waterloo, Ontario, N2L 3G1, Canada (e-mail: anica@math.uwaterloo.ca)

Dimitri Shlyakhtenko ^{†} Department of Mathematics U.C.L.A. Los Angeles, CA 90095-1555, USA (email: shlyakht@math.ucla.edu)

Roland Speicher Department of Mathematics and Statistics Queen’s University Kingston, Ontario K7L 3N6, Canada (email: speicher@mast.queensu.ca)

Abstract

We introduce the concept of “R-cyclic family” of matrices with entries in a noncommutative probability space; the definition consists in asking that only the “cyclic” non-crossing cumulants of the entries of the matrices are allowed to be non-zero.

Let A_{1}, …, A_{s} be an R-cyclic family of d×d matrices over a non-commutative probability space (A, φ). We prove a convolution-type formula for the explicit computation of the joint distribution of A_{1}, …, A_{s} (considered in M_{d}(A) with the natural state), in terms of the joint distribution (considered in the original space (A, φ)) of the entries of the s matrices. Several important situations of families of matrices with tractable joint distributions arise by application of this formula.

Moreover, let A_{1}, …, A_{s} be a family of d × d matrices over a non-commutative probability space (A, φ), let D ⊂ M_{d}(A) denote the algebra of scalar diagonal matrices, and let C be the subalgebra of M_{d}(A) generated by {A_{1}, …, A_{s}} ∪ D. We prove that the R-cyclicity of A_{1}, …, A_{s} is equivalent to a property of C – namely that C is free from M_{d}(C), with amalgamation over D.

∗Research supported by a grant of NSERC, Canada.

†Partially supported by an NSF Postdoctoral Fellowship.

Introduction

In the influential paper [21], Voiculescu introduced the concepts of circular and semicircular systems, and used them to obtain results about the fundamental groups of the von Neumann algebras associated to free groups. There are three main properties of the circulars and semicirculars which are essential for the arguments in [21]:

(a) the compression of a semicircular system by a projection free from the system is again a semicircular system;

(b) in the polar decomposition of a circular element, the polar part is free from the positive part;

(c) one can obtain semicircular systems consisting of matrices over a non-commutative probability space, if the entries of these matrices are themselves chosen (in an appropriate way) to be circular/semicircular and free.

Each of (a), (b), (c) points to a direction of investigation in the combinatorics of free probability.

Concerning (a) and (b), the things are now pretty well understood. For (a), we know a general formula describing the distribution of the compression by a free projection (see [9]), or even more generally for what happens when we perform a compression by a free matrix unit (see [17], [8]). For (b), the relevant class of elements to be studied is the one of “R-diagonal elements”, introduced in [10], and which turns out to have a lot of good properties (see e.g. [5], or [12]–[14]).

With (c) the situation is not that clear. If we look at the case of only one matrix, then the problem is to give effective methods for computing the distribution of the matrix, by starting from the joint distribution of its entries. Of course, the distribution of the matrix is always completely determined by the joint distribution of its entries; the issue is here about the word “effective”. It is unlikely that one can give a nice formula which would work in full generality. The problem is more like this: to what kind of matrices can one generalize the nice facts known about matrices of free circular/semicircular elements? We look for a situation which is general enough to contain interesting examples, but also particular enough so that a nice formula does exist.

In this paper we propose the concept of R-cyclic matrix (or more generally, of R-cyclic family of matrices), which we believe is a good framework for studying the direction (c).

The definition is in terms of the joint R-transform of the entries of the matrix – where the R-transform is the free probabilistic counterpart for the characteristic function of the joint distribution. The coefficients of the R-transform are called non-crossing cumulants. The definition of an R-cyclic matrix goes by asking that only the cyclic non-crossing cumulants of the entries survive; see Definition 2.2 in Section 2 below, and see Sections 2.3-2.6 for examples.

If A is an R-cyclic matrix, then all the information about the distribution of A is stored in the family of cyclic cumulants of its entries. These cyclic cumulants can be in turn nicely stored in one formal power series f (in d non-commuting variables, where d × d is the size of A); the series f is called “the determining series” of A. Our problem is then to find an effective method for computing the distribution of the R-cyclic matrix A, in terms of its determining series f. In the Section 2 of the paper we show that this problem can be treated by using a convolution-type formula:

(I) R_{A}(z) = \frac{1}{d}(f ⋆ H_{d})( \underbrace{z, …, z}_{d} ),

where: R_{A} is the R-transform of A; H_{d} is a certain universal series in d indeterminates; and ⋆ is a convolution-type operation introduced in [9], which appears to play an important role in combinatorial free probability (see review in Section 1 below). The formula (I) can be extended to the case of an R-cyclic family of matrices (see Definition 2.9 and Theorem 2.10 in Section 2), and can be used to obtain various situations when one gets a family of matrices with computable joint distribution. Some applications are presented in the Section 3 of the paper.

Section 4 is about operations with matrices in an R-cyclic family. It is trivial from the definition that if A_{1}, …, A_{s} is an R-cyclic family (of d × d matrices over a non-commutative probability space (A, φ)), then one can add to A_{1}, …, A_{s}:

(a) a linear combination of A_{1}, …, A_{s}, or

(b) any scalar diagonal matrix, and the enlarged family is still R-cyclic. In Lemma 4.2 we show that a similar statement is true when one adds to A_{1}, …, A_{s} a product of some of the matrices in the family; this comes as a fairly easy application of a formula for non-crossing cumulants with products for entries, which was found in [6].

The considerations of Section 4 show that the property of a family of matrices A_{1}, …, A_{s} ∈ M_{d}(A) of being R-cyclic is really a property of the algebra C generated together by A_{1}, …, A_{s} and the set of scalar diagonal matrices. The rest of the paper is devoted to identifying what this property of C exactly is. The result turns out to be the following (Theorem 8.2):

(II)  \{ \begin{matrix}the family \\ A_{1}, …, A_{s} \\ is R-cyclic \end{matrix} \} ⇔ \{ \begin{matrix}C is free from M_{d}(C), \\ with amalgamation over \\ scalar diagonal matrices \end{matrix} \} ,

where A_{1}, …, A_{s} and C are as above, and where the algebra M_{d}(C) of scalar d × d matrices is identified as a subalgebra of M_{d}(A) in the natural way.

In the paper [14] we had shown that an element a ∈ A is R-diagonal if and only if the matrix [ \begin{matrix}0 & a \\ a^{∗} & 0 \end{matrix} ] ∈ M_{2}(A) is free from M_{2}(C), with amalgamation over scalar diagonal matrices. But it is easy to see, directly from the definitions, that a is R-diagonal if and only if the matrix [ \begin{matrix}0 & a \\ a^{∗} & 0 \end{matrix} ] is R-cyclic. Hence the above equivalence (II) can be viewed as an ample generalization of the named result from [14].

The equivalence in (II) is obtained by studying non-crossing operator-valued cumulants, in the sense of [19]; a few basic facts about operator-valued cumulants are reviewed in Section 5, and the proof of (II) is shown in Section 8. In between 5 and 8 we have two short sections where we derive some explicit formulas (used in Section 8) for operator-valued cumulants with respect to the algebra M_{d}(C) (in Section 6), and with respect to the algebra of scalar diagonal matrices (in Section 7).

1. Basic concepts for the combinatorics of free probability

As a preparation for the theorems proved in Section 2, we review here a few basic concepts and facts used in combinatorial free probability. We use the framework of a noncommutative probability space, by which we will simply understand a pair (A, φ) where A is a complex unital algebra (“the algebra of random variables”) and φ : A → C (“the expectation”) is a linear functional, normalized by φ(1) = 1. We assume that the reader has some familiarity with the concept of freeness for families of elements in (A, φ) (see e.g. [22], Chapter 2).

In the combinatorial study of freeness, an important role is played by the concepts of moment series and R-transform of a family of non-commuting random variables. The definition of the first of these two concepts is straightforward: if (A, φ) is a non-commutative probability space, and if a_{1}, …, a_{s} are in A, then the numbers of the form:

φ( a_{r_{1}} ⋯ a_{r_{n}} ), n ≥ 1, 1 ≤ r_{1}, …, r_{n} ≤ s,  (1.1)

are called the joint moments of a_{1}, …, a_{s}; the moment series of a_{1}, …, a_{s} is the power series in s non-commuting indeterminates z_{1}, …, z_{s} which has the joint moments as coefficients. That is:

M_{a_{1},...,a_{s}}(z_{1}, …, z_{s}) := ∑_{n=1}^{∞} ∑_{r_{1},...,r_{n}=1}^{s} φ(a_{r_{1}} ⋯ a_{r_{n}})z_{r_{1}} ⋯ z_{r_{n}}.  (1.2)

The (less straightforward) definition of the R-transform can be placed within the framework of a certain convolution operation on formal power series which will be used in Section 2, and is reviewed next (in Sections 1.1-1.2, followed by the definition of the R-transform in Section 1.3).

1.1 Non-crossing partitions. Let π = {B_{1}, …, B_{k}} be a partition of {1, …, n} – i.e. B_{1}, …, B_{k} are pairwisely disjoint non-void sets (called the blocks of π), and B_{1} ∪ ⋯ ∪ B_{k} = {1, …, n}. We say that π is non-crossing if for every 1 ≤ i < j < k < l ≤ n such that i is in the same block with k and j is in the same block with l, it necessarily follows that all of i, j, k, l are in the same block of π. The set of non-crossing partitions of {1, …, n} will be denoted by NC(n).

For π, ρ ∈ NC(n), we write “π ≤ ρ” if each block of ρ is a union of blocks of π. Then “≤” is a partial order relation on NC(n), called the refinement order. It turns out that (NC(n), ≤) is in fact a lattice, i.e. every two partitions in NC(n) have a lowest upper bound and a greatest lower bound with respect to ≤.

For π ∈ NC(n) we will denote by perm_{π} the permutation of {1, …, n} which has the blocks of π as cycles, in such a way that if B = {k_{1} < ⋯ < k_{p−1} < k_{p}} is a block of π then we have

perm_{π}(k_{1}) = k_{2}, …, perm_{π}(k_{p−1}) = k_{p}, perm_{π}(k_{p}) = k_{1}.

( For example, if π = \{ \{1, 2, 5\}, \{3, 4\} \} ∈ NC(5), then perm_{π} = ( \begin{matrix}1 & 2 & 3 & 4 & 5 \\ 2 & 5 & 4 & 3 & 1 \end{matrix} ). ) The set {perm_{π} | π ∈ NC(n)} has a nice interpretation in terms of the geometry of the Cayley graph of the symmetric group (see [1]), and can be a useful instrument in considerations about the lattice NC(n).

Unlike the lattice of all partitions of {1, 2, …, n}, NC(n) is anti-isomorphic to itself. We will in fact make extensive use of a canonical anti-isomorphism Kr : NC(n) → NC(n), introduced in [7] and called the Kreweras complementation map. The map Kr can be conveniently described by using the permutations associated to non-crossing partitions, via the following formula:

perm_{π} ∘ perm_{Kr(π)} = γ_{n}, ∀ π ∈ NC(n),  (1.3)

where γ_{n} is the forward cycle on {1, …, n} (γ_{n}(1) = 2, …, γ_{n}(n − 1) = n, γ_{n}(n) = 1).

1.2 The operation of boxed convolution. Let s be a positive integer. We denote by Θ_{s} the set of all series of the form appearing in Equation (1.2):

Θ_{s} = \{f \begin{matrix}| & f(z_{1}, …, z_{s}) = ∑_{n=1}^{∞} ∑_{r_{1},...r_{n}=1}^{s} α_{r_{1},...,r_{n}}z_{r_{1}} ⋯ z_{r_{n}} \\ | & where α_{r_{1},...,r_{n}} ∈ C (n ≥ 1, 1 ≤ r_{1}, …, r_{n} ≤ s) \end{matrix} \} .  (1.4)

For a series f as in (1.4), we will use the notation

[ coef (r_{1}, …, r_{n}) ](f)  (1.5)

to denote the coefficient α_{r_{1},...,r_{n}} of z_{r_{1}} ⋯ z_{r_{n}} in f.

The operation of boxed convolution, ⋆, is an associative binary operation on the set Θ_{s}. Its definition is inspired from the combinatorial theory of convolution in a lattice, as developed by Rota and his collaborators (see e.g. [2]; the lattices of relevance for the definition of ⋆ are those of non-crossing partitions, NC(n) for n ≥ 1).

In order to state the definition of ⋆, it is convenient to first expand the notations for coefficients introduced in (1.5). If n ≥ 1, 1 ≤ r_{1}, …, r_{n} ≤ s, and if B = {k_{1} < k_{2} < ⋯ < k_{p}} is a non-void subset of {1, …, n}, then by “(r_{1}, …, r_{n})|B” we will understand the p-tuple (r_{k_{1}}, r_{k_{2}}, …, r_{k_{p}}) (for example (r_{1}, r_{2}, r_{3}, r_{4}, r_{5})|{2, 3, 5} = (r_{2}, r_{3}, r_{5})). Then for a series f ∈ Θ_{s} we introduce the following “generalized coefficients”:

[coef (r_{1}, …, r_{n}); π](f) := ∏_{B block of π} [ coef (r_{1}, …, r_{n})|B ](f),  (1.6)

for every n ≥ 1, 1 ≤ r_{1}, …, r_{n} ≤ s, and for every π ∈ NC(n). (For example if n = 4 and π = {{1, 3}, {2}, {4}}, then

[coef (r_{1}, r_{2}, r_{3}, r_{4}); π](f) = [coef (r_{1}, r_{3})](f) · [coef (r_{2})](f) · [coef (r_{4})](f),

for any 1 ≤ r_{1}, r_{2}, r_{3}, r_{4} ≤ s.)

By using the notation introduced in (1.6), the boxed convolution f ⋆ g of two series f, g ∈ Θ_{s} is described by the formula:

[coef (r_{1}, …, r_{n})](f ⋆ g) :=  (1.7)

∑_{π∈NC(n)}[coef (r_{1}, …, r_{n}); π](f) · [coef (r_{1}, …, r_{n}); Kr(π)](g),

holding for every n ≥ 1 and 1 ≤ r_{1}, …, r_{n} ≤ s, and where Kr(π) is the Kreweras complement of the partition π ∈ NC(n).

It can be shown that ⋆ is associative and unital, where the unit is the series Δ(z_{1}, …, z_{s}) := z_{1} + ⋯ + z_{s}. A series f ∈ Θ_{s} is invertible with respect to ⋆ if and only if its coefficients of degree 1, [coef (r)](f), 1 ≤ r ≤ s, are all different from 0 (see [9], Section 3).

1.3 R-transform and free cumulants. Let a_{1}, …, a_{s} be an s-tuple of elements in a non-commutative probability space (A, φ). The R-transform of the s-tuple, R_{a_{1},...,a_{s}}, is a series in the set Θ_{s} of Equation (1.4). A succinct way of introducing R_{a_{1},...,a_{s}} goes by using the boxed convolution ⋆ and a special series Möb_{s} ∈ Θ_{s}, called the Möbius series.

Möb_{s} is defined as the inverse under ⋆ of the “zeta series in s indeterminates”,

Zeta_{s}(z_{1}, …, z_{s}) := ∑_{n=1}^{∞} ∑_{r_{1},...,r_{n}=1}^{s} z_{r_{1}} ⋯ z_{r_{n}}.  (1.8)

It is not hard to determine the coefficients of Möb_{s} explicitly:

Möb_{s}(z_{1}, …, z_{s}) = ∑_{n=1}^{∞} ∑_{r_{1},...,r_{n}=1}^{s}(−1)^{n+1} \frac{(2n − 2)!}{(n − 1)!n!} z_{r_{1}} ⋯ z_{r_{n}}  (1.9)

(see e.g. [9], Remark 3.8).

Now, if (A, φ) is a non-commutative probability space, and if a_{1}, …, a_{s} ∈ A, then we define:

R_{a_{1},...,a_{s}} := M_{a_{1},...,a_{s}} ⋆ Möb_{s},  (1.10)

where M_{a_{1},...,a_{s}} is the moment series from Equation (1.2). It is clear that R_{a_{1},...,a_{s}} contains the same information about a_{1}, …, a_{s} as the moment series, since Equation (1.10) can be re-written equivalently as

M_{a_{1},...,a_{s}} = R_{a_{1},...,a_{s}} ⋆ Zeta_{s}.  (1.11)

Following [18], it is customary to denote the coefficient of z_{r_{1}} ⋯ z_{r_{n}} in R_{a_{1},...,a_{s}} by:

k_{n}(a_{r_{1}}, …, a_{r_{n}}).  (1.12)

More generally, given n ≥ 1, 1 ≤ r_{1}, …, r_{n} ≤ s, and a partition π ∈ NC(n), we use the notation

k_{π}(a_{r_{1}}, …, a_{r_{n}})  (1.13)

for the “generalized coefficient” [coef (r_{1}, …, r_{n}); π](R_{a_{1},...,a_{s}}) defined as in Equation (1.6). These generalized coefficients are called the non-crossing cumulants of the s-tuple a_{1}, …, a_{s}. It is worth keeping in mind that for any n ≥ 1 and π ∈ NC(n), it makes sense to view k_{π} as a multilinear map from A^{n} to C (see [18]).

1.4 R-transform and freeness. The R-transform and the boxed convolution turn out to have very pleasant properties in connection to the addition and multiplication of free n-tuples – see [20], [9]. Even more importantly, R-transforms (or equivalently, non-crossing cumulants) can be used to provide a neat characterization of freeness. To be precise: let a′_{1}, …, a′_{m}, a′′_{1}, …, a′′_{n} be elements of the non-commutative probability space (A, φ); then the freeness of the families {a′_{1}, …, a′_{m}} and {a′′_{1}, …, a′′_{n}} is equivalent to the equation

R_{a′_{1},...,a′_{m},a′′_{1},...,a′′_{n}}(z′_{1}, …, z′_{m}, z′′_{1}, …, z′′_{n}) =  (1.14)
= R_{a′_{1},...,a′_{m}}(z′_{1}, …, z′_{m}) + R_{a′′_{1},...,a′′_{n}}(z′′_{1}, …, z′′_{n}).

It is obvious how Equation (1.14) extends by induction to the case of s (instead of just two) families of elements. Note that in the case of s families having one element each, we obtain the following: the elements a_{1}, …, a_{s} ∈ A form a free family if and only if we have that

R_{a_{1},...,a_{s}}(z_{1}, …, z_{s}) = R_{a_{1}}(z_{1}) + ⋯ + R_{a_{s}}(z_{s}).  (1.15)

1.5 Extended boxed convolution. Let s and d be positive integers. Consider the set Θ_{sd} of power series in sd non-commuting indeterminates z_{1,1}, …, z_{r,i}, …, z_{s,d}. The same formula as in Equation (1.7) above can be used to define a “convolution operation”, denoted in what follows by ˜ ⋆, which gives a right action of Θ_{d} on Θ_{sd}. More precisely, if f ∈ Θ_{sd} and g ∈ Θ_{d} then we define f ˜ ⋆ g ∈ Θ_{sd} by the following formula:

[coef ( (r_{1}, i_{1}), …, (r_{n}, i_{n}) )](f ˜ ⋆ g) :=  (1.16)

∑_{π∈NC(n)}[coef ( (r_{1}, i_{1}), …, (r_{n}, i_{n}) ); π](f) · [coef (i_{1}, …, i_{n}); Kr(π)](g),

holding for every n ≥ 1 and for every 1 ≤ r_{1}, …, r_{n} ≤ s, 1 ≤ i_{1}, …, i_{n} ≤ d. Some trivial adjustments of the considerations made in Section 4 of [9] for ⋆ show that ˜ ⋆ is indeed a right action of Θ_{d} on Θ_{sd}, in the sense that the equation

(f ˜ ⋆ g) ˜ ⋆ h = f ˜ ⋆ (g ⋆ h)  (1.17)

holds for every f ∈ Θ_{sd} and g, h ∈ Θ_{d}.

Let us also record the fact that:

f ˜ ⋆ Zeta_{d} = f ⋆ Zeta_{sd}, ∀ f ∈ Θ_{sd}  (1.18)

(where on the right-hand side of (1.18), ⋆ denotes the boxed convolution operation on Θ_{sd}). This relation is obvious if one takes into account the fact that any Zeta series has all the coefficients equal to 1.

From (1.17) and (1.18) it is immediate that one also has:

f ˜ ⋆ Möb_{d} = f ⋆ Möb_{sd}, ∀ f ∈ Θ_{sd}.  (1.19)

Note that, as a consequence, we can write the relation

M_{a_{1,1},...,a_{r,i},...,a_{s,d}} ˜ ⋆ Möb_{d} = R_{a_{1,1},...,a_{r,i},...,a_{s,d}},  (1.20)

holding for any family {a_{r,i} | 1 ≤ r ≤ s, 1 ≤ i ≤ d} of elements in some non-commutative probability space (A, φ).

1.6 Dilations and scalar multiples of power series. Let s be a positive integer, let f be a series in Θ_{s}, and let α be a complex number. We denote by f ∘ D_{α} the series in Θ_{s} which is defined by the equation: “(f ∘ D_{α})(z_{1}, …, z_{s}) = f(αz_{1}, …, αz_{s})”, or more rigorously by the fact that:

[coef(r_{1}, …, r_{n})](f ∘ D_{α}) = α^{n} · [coef(r_{1}, …, r_{n})](f), ∀ n ≥ 1, 1 ≤ r_{1}, …, r_{n} ≤ s.

The formulas relating ⋆ with dilation and with scalar multiplication which are proved in [9] can be easily extended to the case of ˜ ⋆. Concerning dilation we have:

(f ∘ D_{α}) ˜ ⋆ g = f ˜ ⋆ (g ∘ D_{α}) = (f ˜ ⋆ g) ∘ D_{α},  (1.21)

for every f ∈ Θ_{ds}, g ∈ Θ_{d}, α ∈ C. Concerning scalar multiplication we have the formula:

(αf) ˜ ⋆ (αg) = α( (f ˜ ⋆ g) ∘ D_{α}), ∀ f ∈ Θ_{ds}, g ∈ Θ_{d}, α ∈ C.  (1.22)

It is sometimes convenient to use Equation (1.22) in the form:

(αf) ˜ ⋆ g = α( f ˜ ⋆ (\frac{1}{α}g ∘ D_{α}) ),  (1.23)

holding for f ∈ Θ_{ds}, g ∈ Θ_{d}, and α ∈ C \ {0}.

1.7 The special series H_{d}. Let d be a positive integer. In this paper we also encounter the “geometric series in d separate indeterminates”,

G_{d}(z_{1}, …, z_{d}) = ∑_{n=1}^{∞} ∑_{i=1}^{d} z_{i}^{n} ( = \frac{z_{1}}{1 − z_{1}} + ⋯ + \frac{z_{d}}{1 − z_{d}} ),  (1.24)

and a series derived from G_{d} which can be described as follows:

H_{d} := G_{d} ⋆ (d · Möb_{d} ∘ D_{1/d}).  (1.25)

To give an idea of how H_{d} looks like, here is its truncation to order three:

H_{d}(z_{1}, …, z_{d}) = ∑_{i=1}^{d} z_{i} + ∑_{i_{1},i_{2}=1}^{d}(δ_{i_{1},i_{2}} − \frac{1}{d})z_{i_{1}}z_{i_{2}}

+ ∑_{i_{1},i_{2},i_{3}=1}^{d} (δ_{i_{1},i_{2},i_{3}} − \frac{1}{d}(δ_{i_{1},i_{2}} + δ_{i_{1},i_{3}} + δ_{i_{2},i_{3}}) + \frac{2}{d^{2}} )z_{i_{1}}z_{i_{2}}z_{i_{3}} + ⋯

Note that a direct application of Equation (1.23) (in the particular case when ˜ ⋆ is ⋆ on Θ_{d}, and α = 1/d) gives the alternative formula:

H_{d} = d · ( (\frac{1}{d}G_{d}) ⋆ Möb_{d} ).  (1.26)

Furthermore, the latter equation has the following interpretation. Let tr_{d} denote the normalized trace on the algebra M_{d}(C), and consider the matrices P_{1}, …, P_{d} ∈ M_{d}(C) where P_{i} has its (i, i)-entry equal to 1 and all the other entries equal to 0. Then, obviously:

M_{P_{1},...,P_{d}} = \frac{1}{d}G_{d}

(moment series considered in the non-commutative probability space (M_{d}(C), tr_{d}) ); hence:

(\frac{1}{d}G_{d}) ⋆ Möb_{d} = M_{P_{1},...,P_{d}} ⋆ Möb_{d} = R_{P_{1},...,P_{d}},

and the formula (1.26) for H_{d} takes the form

H_{d} = d · R_{P_{1},...,P_{d}}.  (1.27)

An application of Equation (1.27) is that for every n ≥ 2, every k ∈ {1, …, n}, and every fixed indices i_{1}, …, i_{k−1}, i_{k+1}, …, i_{n} ∈ {1, … d}, we have:

∑_{i=1}^{d}[coef(i_{1}, …, i_{k−1}, i, i_{k+1}, …, i_{n}](H_{d}) = 0.  (1.28)

Indeed, the sum on the left-hand side of (1.28) is equal to:

d · ∑_{i=1}^{d} k_{n}(P_{i_{1}}, …, P_{i_{k−1}}, P_{i}, P_{i_{k+1}}, …, P_{i_{n}}) (by (1.27))

= k_{n}(P_{i_{1}}, …, P_{i_{k−1}}, I, P_{i_{k+1}}, …, P_{i_{n}}) (by the multilinearity of k_{n}), and the latter quantity equals 0 by (1.14) and the fact that the identity matrix I is free from {P_{1}, …, P_{d}} in (M_{d}(C), tr_{d}).

2. R-cyclic matrices and their R-transforms

2.1 Notation. Let (A, φ) be a non-commutative probability space, and let d be a positive integer. Consider the algebra M_{d}(A) of d × d matrices over A. We denote by φ_{d} the linear functional on M_{d}(A) defined by the formula:

φ_{d}( [a_{i,j}]_{i,j=1}^{d} ) = \frac{1}{d} ∑_{i=1}^{d} φ(a_{i,i}).  (2.1)

Then (M_{d}(A), φ) is a non-commutative probability space, too.

2.2 Definition. Let (A, φ) and d be as above. A matrix A = [a_{i,j}]_{i,j=1}^{d} ∈ M_{d}(A) is said to be R-cyclic if the following condition holds:

k_{n}(a_{i_{1},j_{1}}, …, a_{i_{n},j_{n}}) = 0

for every n ≥ 1 and every 1 ≤ i_{1}, j_{1}, …, i_{n}, j_{n} ≤ d for which it is not true that j_{1} = i_{2}, …, j_{n−1} = i_{n}, j_{n} = i_{1}.

If the matrix A is R-cyclic, then the series:

f(z_{1}, …, z_{d}) := ∑_{n=1}^{∞} ∑_{i_{1},...,i_{n}=1}^{d} k_{n}(a_{i_{n},i_{1}}, a_{i_{1},i_{2}}, …, a_{i_{n−1},i_{n}})z_{i_{1}}z_{i_{2}} ⋯ z_{i_{n}}  (2.2)

is called the determining series of A.

2.3 Example. Consider a diagonal matrix,

A := [ \begin{matrix}a_{1} & & & & 0 \\ & . & . & . \\ 0 & & & & a_{d} \end{matrix} ] ∈ M_{d}(A),

where (A, φ) and d are as above. An application of Equation (1.15) shows that A is Rcyclic if and only if the elements a_{1}, …, a_{d} form a free family; if this is the case, then the determining series of A coincides with the joint R-transform R_{a_{1},...,a_{d}}.

For more elaborate examples we will use the framework of a ∗-probability space, which is also the one most frequently encountered in applications. A ∗-probability space is a noncommutative probability space (A, φ) where A is a ∗-algebra, and φ has the property that φ(a^{∗}) = \overline{φ(a)}, ∀ a ∈ A.

2.4 Example. Let (A, φ) be a ∗-probability space, and let {e_{i,j} | 1 ≤ i, j ≤ d} be a family of elements of A which satisfy the following relations: e_{i,j}^{∗} = e_{j,i} for all 1 ≤ i, j ≤ d, e_{i,j}e_{k,l} = δ_{j,k}e_{i,l} for all 1 ≤ i, j, k, l ≤ d, and ∑_{i=1}^{d} e_{i,i} = I. We will assume in addition that φ(e_{i,j}) = 0 whenever i ≠ j, and that φ(e_{1,1}) = ⋯ = φ(e_{d,d}) = 1/d. We denote by (C, ψ) the compression of (A, φ) by e_{1,1}, i.e:

C := e_{1,1}Ae_{1,1}, ψ := d · φ|C.

Let now a be a selfadjoint element of A, which is free from {e_{i,j} | 1 ≤ i, j ≤ d}. We compress a by the matrix unit formed by the e_{i,j}’s, and we move the compressions under the projection e_{1,1}; that is, we consider the family of elements:

c_{i,j} := e_{1,i}ae_{j,1} ∈ C, 1 ≤ i, j ≤ d.

One can compute explicitly the free cumulants of the family {c_{i,j} | 1 ≤ i, j ≤ d}, and obtain that for every n ≥ 1 and 1 ≤ i_{1}, j_{1}, …, i_{n}, j_{n} ≤ d:

k_{n}(c_{i_{1},j_{1}}, …, c_{i_{n},j_{n}}) = \{ \begin{matrix}d^{−(n−1)}k_{n}(a, …, a) & if j_{1} = i_{2}, …, j_{n−1} = i_{n}, j_{n} = i_{1} \\ 0 & otherwise \end{matrix}

(see Theorem 8.14 or Theorem 17.3 in the notes [11]). In other words, the matrix C = [c_{i,j}]_{i,j=1}^{d} ∈ M_{d}(C) is R-cyclic, with determining series:

f(z_{1}, …, z_{d}) = ∑_{n=1}^{∞} ∑_{i_{1},...,i_{n}=1}^{d} d^{−(n−1)}k_{n}(a, …, a)z_{i_{1}} ⋯ z_{i_{n}}

= d · ∑_{n=1}^{∞} k_{n}(a, …, a) · (\frac{z_{1} + ⋯ + z_{d}}{d} )^{n}

= d · R_{a}( \frac{z_{1} + ⋯ + z_{d}}{d} ),

where R_{a} is the R-transform of a, in the original space (A, φ).

2.5 Example. Let (A, φ) be a ∗-probability space. Let a ∈ A be an R-diagonal element, by which we mean that the joint R-transform of a and a^{∗} is of the form

R_{a,a^{∗}}(z_{1}, z_{2}) = ∑_{n=1}^{∞} α_{n}( (z_{1}z_{2})^{n} + (z_{2}z_{1})^{n} )

for a sequence of real coefficients (α_{n})_{n=1}^{∞} (see [10]). The series f(z) := ∑_{n=1}^{∞} α_{n}z^{n} is called the determining series of a.

Now consider the non-commutative probability space (M_{2}(A), φ_{2}) defined as in Section 2.1, and the selfadjoint matrix:

A = [ \begin{matrix}0 & a \\ a^{∗} & 0 \end{matrix} ] ∈ M_{2}(A).

One immediately checks that A is R-cyclic (and in fact that also conversely, the R-cyclicity of A implies the R-diagonality of a). Moreover, the determining series of A (as defined in Section 2.2) coincides with the determining series of the R-diagonal element a. A number of results known about R-diagonal elements can be incorporated in the theory of R-cyclic matrices by using this trick.

2.6 Example. The situation discussed in the Example 2.5 can be generalized to the one of a selfadjoint matrix with free R-diagonal entries. More precisely, let (A, φ) be a ∗-probability space, let d be a positive integer, and suppose that the elements {a_{i,j} | 1 ≤ i, j ≤ d} of A have the following properties:

(i) a_{i,j}^{∗} = a_{j,i}, ∀ 1 ≤ i, j ≤ d;

(ii) a_{i,j} is R-diagonal whenever i ≠ j;

(iii) the d(d + 1)/2 families: {a_{i,i}} for 1 ≤ i ≤ d, together with {a_{i,j}, a_{j,i}} for 1 ≤ i < j ≤ d, are free in (A, φ). Then the matrix A := [a_{i,j}]_{i,j=1}^{d} ∈ M_{d}(A) is R-cyclic. Indeed, the freeness condition (iii) combined with the R-diagonality of a_{i,j} for i ≠ j implies that the only free cumulants made with the entries of A which could possibly be non-zero are:

\{ \begin{matrix}k_{n}(a_{i,i}, …, a_{i,i}) & with n ≥ 1, 1 ≤ i ≤ d, and \\ k_{n}(a_{i,j}, a_{j,i}, …, a_{i,j}, a_{j,i}) & with n ≥ 1 even, 1 ≤ i, j ≤ d, i ≠ j; \end{matrix}

all these cumulants fall within the pattern allowed by the definition of R-cyclicity.

2.7 Remarks. 1) Variations of the Example 2.6 can be fabricated, such that nonselfadjoint matrices are obtained. For this purpose, it is more natural to use the concept of R-cyclic family of matrices, given in Definition 2.9 below, and the d×d matrix which appears should be considered together with its adjoint. We mention that a particularly intriguing construction of this type – upper triangular d × d matrix with circular ∗-distribution – was studied recently in [3].

2) In the Example 2.6 one can take the a_{i,j}’s to be circular/semicircular, thus obtaining a matrix A as considered in [21]. Recall that a ∈ A is said to be semicircular of radius r if a = a^{∗} and if

φ(a^{n}) = \frac{2}{πr^{2}} ∫_{−r}^{r} t^{n}\sqrt{r^{2} − t^{2}} dt, ∀ n ≥ 1;

and that c ∈ A is said to be circular of radius r if it is of the form c = (a + ib)/\sqrt{2}, where each of a, b is semicircular of radius r, and a is free from b. It can be shown (see e.g. [22], Chapter 3) that if a ∈ A is semicircular of radius r, then k_{2}(a, a) = r^{2}/4 and k_{n}(a, a, …, a) = 0 for n ≠ 2. As an easy consequence (see e.g. [10]), a circular element c of radius r is R-diagonal with R_{c,c^{∗}}(z_{1}, z_{2}) = (r^{2}/4) · (z_{1}z_{2} + z_{2}z_{1}). Thus an example of R-cyclic matrix A = [a_{i,j}]_{i,j=1}^{d} is provided by the case when every a_{i,i} is semicircular (of some radius r_{i,i}), every a_{i,j} with i ≠ j is circular (of some radius r_{i,j}), and the conditions (i), (iii) of Example 2.6 are satisfied.

The following theorem indicates how the distribution of an R-cyclic matrix (considered in the non-commutative probability space (M_{d}(A), φ_{d}) ) can be obtained from the determining series of the matrix.

2.8 Theorem. Suppose that A is an R-cyclic matrix, and let f denote the determining series of A. Then we have the formulas:

M_{A}(z) = \frac{1}{d}(f ⋆ G_{d})( \underbrace{z, …, z}_{d times} ),  (2.3)

and

R_{A}(z) = \frac{1}{d}(f ⋆ H_{d})( \underbrace{z, …, z}_{d times} ),  (2.4)

where the series G_{d} and H_{d} are as defined in Section 1.7.

Before starting on the proof of Theorem 2.8, it is convenient to observe that the discussion about R-cyclicity can be generalized without much effort to the situation of a family of matrices, as follows.

2.9 Definition. Let (A, φ) be a non-commutative probability space, and let d be a positive integer. Let A_{1} = [a_{i,j}^{(1)}]_{i,j=1}^{d}, …, A_{s} = [a_{i,j}^{(s)}]_{i,j=1}^{d} be matrices in M_{d}(A). We say that the family A_{1}, …, A_{s} is R-cyclic if the following condition holds:

k_{n}(a_{i_{1},j_{1}}^{(r_{1})}, …, a_{i_{n},j_{n}}^{(r_{n})}) = 0,

for every n ≥ 1, every 1 ≤ r_{1}, …, r_{n} ≤ s, and every 1 ≤ i_{1}, j_{1}, …, i_{n}, j_{n} ≤ d for which it is not true that j_{1} = i_{2}, …, j_{n−1} = i_{n}, j_{n} = i_{1}.

If the family A_{1}, …, A_{s} is R-cyclic, then the power series in ds indeterminates:

f(z_{1,1}, …, z_{s,d}) :=  (2.5)

∑_{n=1}^{∞} ∑_{i_{1},...,i_{n}=1}^{d} ∑_{r_{1},...,r_{n}=1}^{s} k_{n}( a_{i_{n},i_{1}}^{(r_{1})}, a_{i_{1},i_{2}}^{(r_{2})}, …, a_{i_{n−1},i_{n}}^{(r_{n})} ) · z_{r_{1},i_{1}}z_{r_{2},i_{2}} ⋯ z_{r_{n},i_{n}}

is called the determining series of the family.

2.10 Theorem. Suppose that A_{1}, …, A_{s} is an R-cyclic family of matrices, with determining series f. Then we have the formulas:

M_{A_{1},...,A_{s}}(z_{1}, …, z_{s}) = \frac{1}{d}(f ˜ ⋆ G_{d})( \underbrace{z_{1}, …, z_{1},}_{d times} …, \underbrace{z_{s}, …, z_{s}}_{d times} ),  (2.6)

and

R_{A_{1},...,A_{s}}(z_{1}, …, z_{s}) = \frac{1}{d}(f ˜ ⋆ H_{d})( \underbrace{z_{1}, …, z_{1},}_{d times} …, \underbrace{z_{s}, …, z_{s}}_{d times} ),  (2.7)

where the operation ˜ ⋆ is as described in Section 1.5, and where the series G_{d} and H_{d} are as defined in Section 1.7.

In the proof of Theorem 2.10 we will use the following lemma:

2.11 Lemma. Consider the framework of Theorem 2.10. Let n be a positive integer, let π be in NC(n), and consider some indices 1 ≤ r_{1}, …, r_{n} ≤ s, 1 ≤ i_{1}, …, i_{n} ≤ d. Then we have the equality:

k_{π}( a_{i_{n},i_{1}}^{(r_{1})}, a_{i_{1},i_{2}}^{(r_{2})}, …, a_{i_{n−1},i_{n}}^{(r_{n})} ) =  (2.8)
[coef( (r_{1}, i_{1}), …, (r_{n}, i_{n}) ); π ](f) · [coef( i_{1}, …, i_{n} ); Kr(π) ](G_{d}).

Proof. We will work with the permutations associated to π and to Kr(π) (as discussed in Section 1.1). We will use cyclic notations modulo n for indices – i.e, “i_{k+1}” will mean “i_{1}” if k = n and “i_{k−1}” will mean “i_{n}” if k = 1.

Since every coefficient G_{d} is equal either to 0 or to 1, the generalized coefficient of G_{d} appearing on the right-hand side of (2.8) also is 0 or 1. So we have two cases.

Case 1: [coef( i_{1}, …, i_{n} ); Kr(π) ](G_{d}) = 1.

By writing explicitly what the generalized coefficient of G_{d} is, we find that:

\{ \begin{matrix}1 ≤ k, l ≤ n, \\ k, l in the same block of Kr(π) \end{matrix} \} ⟹ i_{k} = i_{l}.  (2.9)

Under this assumption, we have to show that:

k_{π}( a_{i_{n},i_{1}}^{(r_{1})}, a_{i_{1},i_{2}}^{(r_{2})}, …, a_{i_{n−1},i_{n}}^{(r_{n})} ) = [coef( (r_{1}, i_{1}), …, (r_{n}, i_{n}) ); π ](f).  (2.10)

Each of the two sides of (2.10) is a product of factors indexed by the blocks of π; we will prove (2.10) by showing that actually for any given block B of π, the factor corresponding to B on the left-hand side of (2.10) is equal to the factor corresponding to B on the right-hand side of (2.10).

So let us fix a block B = {k_{1} < k_{2} < ⋯ < k_{p}} of π. The factor corresponding to B on the left-hand side of (2.10) is:

k_{p}( a_{i_{k_{1}−1},i_{k_{1}}}^{(r_{k_{1}})} , a_{i_{k_{2}−1},i_{k_{2}}}^{(r_{k_{2}})} , …, a_{i_{kp−1},i_{kp}}^{(r_{kp})} )  (2.11)

(where recall that if k_{1} = 1, then we use i_{n} for “i_{k_{1}−1}”); the factor corresponding to B on the right-hand side of (2.10) is [coef( (r_{k_{1}}, i_{k_{1}}), …, (r_{k_{p}}, i_{k_{p}}) ) ](f), i.e:

k_{p}( a_{i_{kp},i_{k_{1}}}^{(r_{k_{1}})} , a_{i_{k_{1}},i_{k_{2}}}^{(r_{k_{2}})} , …, a_{i_{k_{p−1}},i_{kp}}^{(r_{kp})} ).  (2.12)

But now, let us notice that k_{1} and k_{2}−1 belong to the same block of Kr(π), and same for k_{2} and k_{3} −1, …, same for k_{p} and k_{1} −1. This is easily seen by looking at the permutations associated to π and Kr(π): we have that

perm_{π}(k_{1}) = k_{2}, …, perm_{π}(k_{p−1}) = k_{p}, perm_{π}(k_{p}) = k_{1},

so from Eqn.(1.3) we get that:

perm_{Kr(π)}(k_{2} − 1) = k_{1}, …, perm_{Kr(π)}(k_{p} − 1) = k_{p−1}, perm_{Kr(π)}(k_{1} − 1) = k_{p}.

As a consequence of this remark and of the implication stated in (2.9), we see that the expressions appearing in (2.11) and (2.12) are actually identical.

Case 2: [coef( i_{1}, …, i_{n} ); Kr(π) ](G_{d}) = 0.

In this case we know that (2.9) does not hold, and we have to show that the left-hand side of (2.8) is equal to 0.

It is immediate that, under the current assumption, we can find 1 ≤ k, l ≤ n such that:

perm_{Kr(π)}(l) = k, and i_{k} ≠ i_{l}.  (2.13)

Indeed, if it were true that i_{k} = i_{l} whenever perm_{Kr(π)}(l) = k, then by moving along the cycles of perm_{Kr(π)} we would find that (2.9) holds.

By taking into account the relation between perm_{π} and perm_{Kr(π)}, we see that for k, l as in (2.13) we also have that perm_{π}(k) = l + 1. Hence k and l + 1 belong to the same block B of π; and moreover, if the block B is written as B = {k_{1} < k_{2} < ⋯ < k_{p}}, then there exists an index j, 1 ≤ j ≤ p such that k = k_{j} and l + 1 = k_{j+1} (with the convention that if k = k_{p}, then l + 1 = k_{1}). But then the fact that i_{k} ≠ i_{l} reads: i_{k_{j}} ≠ i_{k_{j+1}−1}, which in turn implies that

k_{p}( a_{i_{k_{1}−1},i_{k_{1}}}^{(r_{k_{1}})} , a_{i_{k_{2}−1},i_{k_{2}}}^{(r_{k_{2}})} , …, a_{i_{kp−1},i_{kp}}^{(r_{kp})} ) = 0

(by the definition of R-cyclicity). Since the latter expression is the factor corresponding to B in the product defining k_{π}( a_{i_{n},i_{1}}^{(r_{1})}, a_{i_{1},i_{2}}^{(r_{2})}, …, a_{i_{n−1},i_{n}}^{(r_{n})} ), we conclude that the left-hand side of (2.8) is indeed equal to 0. QED

Proof of Theorem 2.10. Let n be a positive integer, and consider some indices 1 ≤ r_{1}, …, r_{n} ≤ s, 1 ≤ i_{1}, …, i_{n} ≤ d. By summing over π ∈ NC(n) in the Equation (2.8) of Lemma 2.11, and by taking into account the properties of non-crossing cumulants and of boxed convolution, we get:

φ(a_{i_{n},i_{1}}^{(r_{1})}a_{i_{1},i_{2}}^{(r_{2})} ⋯ a_{i_{n−1},i_{n}}^{(r_{n})}) = [coef( (r_{1}, i_{1}), …, (r_{n}, i_{n}) ) ](f ˜ ⋆ G_{d}).  (2.14)

For every 1 ≤ i ≤ d, let us denote by P_{i} ∈ M_{d}(A) the matrix which has I (the unit of A) on the (i, i)-entry, and has all the other entries equal to 0. It is immediately verified that

φ_{d}(A_{r_{1}}P_{i_{1}} ⋯ A_{r_{n}}P_{i_{n}}) = \frac{1}{d}φ(a_{i_{n},i_{1}}^{(r_{1})}a_{i_{1},i_{2}}^{(r_{2})} ⋯ a_{i_{n−1},i_{n}}^{(r_{n})}).

By combining this equation with (2.14), we get an equality of power series in ds variables, which is stated as follows:

M_{A_{1}P_{1},...,A_{r}P_{i},...,A_{s}P_{d}} = \frac{1}{d}(f ˜ ⋆ G_{d}).  (2.15)

The Equation (2.6) is an immediate consequence of (2.15), since we have for every n ≥ 1 and 1 ≤ r_{1}, …, r_{n} ≤ s:

φ_{d}(A_{r_{1}} ⋯ A_{r_{n}}) = ∑_{i_{1},...,i_{n}=1}^{d} φ_{d}(A_{r_{1}}P_{i_{1}} ⋯ A_{r_{n}}P_{i_{n}})

= ∑_{i_{1},...,i_{n}=1}^{d}[coef( (r_{1}, i_{1}), …, (r_{n}, i_{n}) ) ](M_{A_{1}P_{1},...,A_{s}P_{d}})

= \frac{1}{d} ∑_{i_{1},...,i_{n}=1}^{d}[coef( (r_{1}, i_{1}), …, (r_{n}, i_{n}) ) ](f ˜ ⋆ G_{d});

the latter quantity is easily seen to be the coefficient of z_{r_{1}} ⋯ z_{r_{n}} in the series:

\frac{1}{d}(f ˜ ⋆ G_{d})( \underbrace{z_{1}, …, z_{1},}_{d times} …, \underbrace{z_{s}, …, z_{s}}_{d times} ),

hence (2.6) follows.

On the other hand let us ˜ ⋆-convolve with Möb_{d} on the right, on both sides of (2.15). On the left-hand side we get M_{A_{1}P_{1},...,A_{s}P_{d}} ˜ ⋆ Möb_{d}, which is equal to R_{A_{1}P_{1},...,A_{s}P_{d}} (see Equation (1.20) in Section 1.5). On the right-hand side we get:

( \frac{1}{d}(f ˜ ⋆ G_{d}) ) ˜ ⋆ Möb_{d} = \frac{1}{d}( f ˜ ⋆ G_{d} ˜ ⋆ (dMöb_{d} ∘ D_{1/d}) ) (by Eqn.(1.23) )
= \frac{1}{d}(f ˜ ⋆ H_{d}) (by the definition of H_{d} in Section 1.7).

So we obtain the equation:

R_{A_{1}P_{1},...,A_{r}P_{i},...,A_{s}P_{d}} = \frac{1}{d}(f ˜ ⋆ H_{d}),  (2.16)

out of which (2.7) is obtained in the same way as (2.6) was obtained from (2.15). QED

2.12 Remark. The proof of Theorem 2.10 obtains the Equations (2.15) and (2.16), stronger than what was originally stated, and which show better the significance of the series f ˜ ⋆ G_{d} and f ˜ ⋆ H_{d}.

3. Applications of Theorem 2.10.

We will concentrate on applications to a family A_{1}, …, A_{s} of selfadjoint d × d matrices over a ∗-probability space (A, φ). By keeping in mind the motivating example from [21], it is of particular interest to put into evidence situations where the family A_{1}, …, A_{s} is free in (M_{d}(A), φ_{d}), and where the individual R-transform of each of A_{1}, …, A_{s} is determined explicitly. It seems that some important situations of this kind appear as a consequence of a “partial summation condition”, described in the next proposition.

3.1 Proposition. Let (A, φ) be a ∗-probability space, let d, s be positive integers, and let A_{1} = [a_{i,j}^{(1)}]_{i,j=1}^{d}, …, A_{s} = [a_{i,j}^{(s)}]_{i,j=1}^{d} form an R-cyclic family of selfadjoint matrices in M_{d}(A). We denote the determining series of A_{1}, …, A_{s} by f. Suppose that for every n ≥ 1 and every 1 ≤ r_{1}, …, r_{n} ≤ s, 1 ≤ i_{1}, …, i_{n} ≤ d, the sum:

∑_{i_{1},...,i_{n−1}=1}^{d} [coef( (r_{1}, i_{1}), …, (r_{n−1}, i_{n−1}), (r_{n}, i_{n}) )](f) =: λ_{r_{1},...,r_{n}}  (3.1)

does not depend on i_{n} (even though the sum is only over i_{1}, …, i_{n−1}). Then:

R_{A_{1},...,A_{s}}(z_{1}, …, z_{s}) = ∑_{n=1}^{∞} ∑_{r_{1},...,r_{n}=1}^{s} λ_{r_{1},...,r_{n}}z_{r_{1}} ⋯ z_{r_{n}}.  (3.2)

Proof. The Equation (3.2) is equivalent to the fact that for every n ≥ 1 and every 1 ≤ r_{1}, …, r_{n} ≤ s we have:

k_{n}(A_{r_{1}}, …, A_{r_{n}}) = λ_{r_{1},...,r_{n}}.  (3.3)

We fix n and r_{1}, …, r_{n} about which we show that (3.3) is true. The case when n = 1 is trivial, so we will assume that n ≥ 2.

The Equation (2.7) of Theorem 2.10 gives us the formula:

k_{n}(A_{r_{1}}, …, A_{r_{n}}) =

\frac{1}{d} ∑_{i_{1},...,i_{n}=1}^{d} ∑_{π∈NC(n)} [coef( (r_{1}, i_{1}), …, (r_{n}, i_{n}) ); π](f) · [coef(i_{1}, …, i_{n}); Kr(π)](H_{d}).

We will write this in the form:

k_{n}(A_{r_{1}}, …, A_{r_{n}}) = ∑_{π∈NC(n)} T_{π},  (3.4)

where for every π ∈ NC(n) we set:

T_{π} := \frac{1}{d} ∑_{i_{1},...,i_{n}=1}^{d} [coef( (r_{1}, i_{1}), …, (r_{n}, i_{n}) ); π](f) · [coef(i_{1}, …, i_{n}); Kr(π)](H_{d}). (3.5)

We first consider the quantity T_{π} defined in (3.5) in the special case when π = 1_{n}, the partition of {1, …, n} which has only one block. In this case Kr(π) is the partition into n blocks of one element; since all the coefficients of degree 1 of H_{d} are equal to 1, it follows that

[coef(i_{1}, …, i_{n}); Kr(1_{n})](H_{d}) = 1, ∀ i_{1}, …, i_{n} ∈ {1, …, d}.

We hence get:

T_{1_{n}} = \frac{1}{d} ∑_{i_{1},...,i_{n}=1}^{d} [coef( (r_{1}, i_{1}), …, (r_{n}, i_{n}) )](f).

The partial summation property of the series f (given in Eqn.(3.1)) implies that the latter sum is equal to λ_{r_{1},...,r_{n}}. Thus, in view of (3.4), the proof will be over if we can show that T_{π} = 0 for every π ≠ 1_{n} in NC(n).

So for the remaining of the proof we fix a partition π ≠ 1_{n} in NC(n). Moreover, we will also fix a block B_{o} of π which is an interval, B_{o} = [p, q] ∩ Z with 1 ≤ p ≤ q ≤ n (every non-crossing partition has such a block). The considerations below, leading to the conclusion that T_{π} = 0, will be made by looking at the case when B_{o} has more than one element; the case when |B_{o}| = 1 (which is similar, and easier) is left as an exercise to the reader. We denote by “Rest” the set of blocks of π which are different from B_{o}.

Let us now look at at the Kreweras complement Kr(π). It is immediate that {p}, {p + 1}, …, {q − 1} are one-element blocks of Kr(π). We denote by B′_{o} the block of Kr(π) which contains q; observe that B′_{o} has more than one element – indeed, it is clear that p − 1 also belongs to B′_{o} (where if p = 1, then “p − 1” means “n”; even in this case we have that p − 1 ≠ q, since it was assumed that π ≠ 1_{n}). Moreover, let us denote by Rest′ the set of blocks of Kr(π) (if any) which remain after {p}, {p + 1}, …, {q − 1} and B′_{o} are deleted.

For any i_{1}, …, i_{n} ∈ {1, …, d} we have:

[coef( (r_{1}, i_{1}), …, (r_{n}, i_{n}) ); π](f) · [coef(i_{1}, …, i_{n}); Kr(π)](H_{d}) =
[coef( (r_{p}, i_{p}), …, (r_{q}, i_{q}) )](f) · [coef(i_{1}, …, i_{n})|B′_{o}](H_{d})·  (3.6)

· ∏_{B∈Rest}[coef( (r_{1}, i_{1}), …, (r_{n}, i_{n}) )|B](f) · ∏_{B′∈Rest′}[coef(i_{1}, …, i_{n})|B′](H_{d})

(we took into account that the factors [coef(i_{p})](H_{d}), …, [coef(i_{q−1})](H_{d}), which should also appear on the right-hand side of (3.6), are all equal to 1). The indices i_{p}, …, i_{q−1} appear only in the factor “[coef( (r_{p}, i_{p}), …, (r_{q}, i_{q}) )](f)” of (3.6). Thus, if in (3.6) we sum over i_{p}, …, i_{q−1}, and make use of the partial summation property from (3.1), then we get:

λ_{r_{p},...,r_{q}} · [coef(i_{1}, …, i_{n})|B′_{o}](H_{d})·  (3.7)

· ∏_{B∈Rest}[coef( (r_{1}, i_{1}), …, (r_{n}, i_{n}) )|B](f) · ∏_{B′∈Rest′}[coef(i_{1}, …, i_{n})|B′](H_{d})

(expression depending on some arbitrary indices i_{1}, …, i_{p−1}, i_{q}, …, i_{n}, chosen from {1, …, d}).

Next, in (3.7) we sum over the index i_{q}. The only factor in (3.7) which involves i_{q} is “[coef(i_{1}, …, i_{n})|B′_{o}](H_{d})”, so as a result of this new summation we get:

λ_{r_{p},...,r_{q}} · { ∑_{i_{q}=1}^{d}[coef(i_{1}, …, i_{n})|B′_{o}](H_{d}) }·

· ∏_{B∈Rest}[coef( (r_{1}, i_{1}), …, (r_{n}, i_{n}) )|B](f) · ∏_{B′∈Rest′}[coef(i_{1}, …, i_{n})|B′](H_{d}).

But, as an immediate consequence of the remark concluding Section 1.7, we have that ∑_{i_{q}=1}^{d}[coef(i_{1}, …, i_{n})|B′_{o}](H_{d}) = 0.

The conclusion that we draw from the preceding three paragraphs is the following: for any choice of the indices i_{1}, …, i_{p−1}, i_{q+1}, …, i_{n} ∈ {1, …, d}, we have that

∑_{i_{p},...,i_{q}=1}^{d} [coef( (r_{1}, i_{1}), …, (r_{n}, i_{n}) ); π](f) · [coef(i_{1}, …, i_{n}); Kr(π)](H_{d}) = 0.

It only remains that we sum over i_{1}, …, i_{p−1}, i_{q+1}, …, i_{n} in the latter equation, to obtain the desired fact that T_{π} = 0. QED

3.2 Corollary. Let (A, φ) be a ∗-probability space, let d, s be positive integers, and let A_{1} = [a_{i,j}^{(1)}]_{i,j=1}^{d}, …, A_{s} = [a_{i,j}^{(s)}]_{i,j=1}^{d} form an R-cyclic family of selfadjoint matrices in M_{d}(A). Suppose that the s families of entries {a_{i,j}^{(r)} | 1 ≤ i, j ≤ d}, with 1 ≤ r ≤ s, are free in (A, φ). Moreover, for every 1 ≤ r ≤ s let f_{r} ∈ Θ_{d} be the determining series of A_{r}. We assume that for every n ≥ 1 and for every 1 ≤ r ≤ s, 1 ≤ i ≤ d, the sum:

∑_{i_{1},...,i_{n−1}=1}^{d} [coef(i_{1}, …, i_{n−1}, i)](f_{r}) =: λ_{n}^{(r)}  (3.8)

does not depend on the choice of i (but only on n and r). Then the matrices A_{1}, …, A_{s} are free in (M_{d}(A), φ_{d}), and have R-transforms

R_{A_{r}}(z) = ∑_{n=1}^{∞} λ_{n}^{(r)}z^{n}, 1 ≤ r ≤ s.  (3.9)

Proof. Let f denote the determining series of the whole R-cyclic family A_{1}, …, A_{s}. The condition of freeness between the families of entries of A_{1}, …, A_{s} implies the formula:

f(z_{1,1}, …, z_{r,i}, …, z_{s,d}) = ∑_{r=1}^{s} f_{r}(z_{r,1}, …, z_{r,i}, …, z_{r,d}),

where f_{r} is (as in the statement of the corollary) the determining series for just the R-cyclic matrix A_{r}. It is immediate that f satisfies the partial summation condition described in Equation (3.1) of Proposition 3.1, where we set:

λ_{r_{1},...,r_{n}} = \{ \begin{matrix}λ_{n}^{(r)} & if r_{1} = ⋯ = r_{n} = r \\ 0 & otherwise. \end{matrix}

Thus the Proposition 3.1 can be applied, and gives us:

R_{A_{1},...,A_{s}}(z_{1}, …, z_{s}) = ∑_{r=1}^{s} ∑_{n=1}^{∞} λ_{n}^{(r)}z_{r}^{n},

which (by virtue of Equation (1.15) in Section 1) is equivalent to saying that A_{1}, …, A_{s} are free and and have the indicated individual R-transforms. QED

The Corollary 3.2 can be in turn particularized to the situation of a family of matrices with free R-diagonal entries (on the line of Example 2.6). The precise spelling of this particular case goes as follows.

3.3 Corollary. Let (A, φ) be a ∗-probability space, let d, s be positive integers, and suppose that the elements {a_{i,j}^{(r)} | 1 ≤ i, j ≤ d, 1 ≤ r ≤ s} of A have the following properties:

(i) For every 1 ≤ i ≤ d and 1 ≤ r ≤ s, the element a_{i,i}^{(r)} is selfadjoint. We denote the R-transform of a_{i,i}^{(r)} as ∑_{n=1}^{∞} α_{i,i;n}^{(r)}z^{n}.

(ii) For every 1 ≤ i, j ≤ d such that i ≠ j, and for every 1 ≤ r ≤ s, the element a_{i,j}^{(r)} is Rdiagonal and has (a_{i,j}^{(r)})^{∗} = a_{j,i}^{(r)}. We denote the determining series of a_{i,j}^{(r)} as ∑_{n=1}^{∞} α_{i,j;2n}^{(r)}z^{n}; we also set α_{i,j;2n−1}^{(r)} := 0, ∀ n ≥ 1.

(iii) The sd(d + 1)/2 families: {a_{i,i}^{(r)}} for 1 ≤ i ≤ d, 1 ≤ r ≤ s, together with {a_{i,j}^{(r)}, a_{j,i}^{(r)}} for 1 ≤ i < j ≤ d, 1 ≤ r ≤ s are free in (A, φ).

Suppose moreover that for every n ≥ 1 and every 1 ≤ r ≤ s, 1 ≤ i ≤ d, the sum:

∑_{j=1}^{d} α_{i,j;n}^{(r)} =: λ_{n}^{(r)}  (3.10)

does not actually depend on i. Then the matrices A_{1} = [a_{i,j}^{(1)}]_{i,j=1}^{d}, …, A_{s} = [a_{i,j}^{(s)}]_{i,j=1}^{d} are free in (M_{d}(A), φ_{d}), and have R-transforms

R_{A_{r}}(z) = ∑_{n=1}^{∞} λ_{n}^{(r)}z^{n}, 1 ≤ r ≤ s.

3.4 Remark. The summation conditions (3.10) become extremely simple when the elements a_{i,i}^{(r)} are semicircular, and the elements a_{i,j}^{(r)} with i ≠ j are circular. Indeed, in this case we have that α_{i,j;n}^{(r)} = 0 whenever n ≠ 2, and that α_{i,j;2}^{(r)} is one quarter of the squared radius of the circular/semicircular element a_{i,j}^{(r)} (compare to Remark 2.7.2). Thus in this case if we denote the radius of a_{i,j}^{(r)} by γ_{i,j}^{(r)}, then (3.10) amounts to asking that for every 1 ≤ r ≤ s the matrix of squared radii [γ_{i,j}^{(r)}]_{i,j=1}^{d} has constant sums along its columns:

∑_{j=1}^{d}(γ_{1,j}^{(r)})^{2} = ⋯ = ∑_{j=1}^{d}(γ_{d,j}^{(r)})^{2} =: γ_{r}^{2}.

The conclusion of Corollary 3.3 becomes that the matrices A_{1} = [a_{i,j}^{(1)}]_{i,j=1}^{d}, …, A_{s} = [a_{i,j}^{(s)}]_{i,j=1}^{d} are free, and that A_{r} is semicircular of radius γ_{r}, for 1 ≤ r ≤ s. This particular case of Corollary 3.3 is very close to Proposition 2.9 of [21], and can also be obtained by the methods used there (approximations in distribution by large Gaussian random matrices).

Another particularization of Proposition 3.1 covers a situation when the matrices A_{1}, …, A_{s} are not free, and which is motivated by results about free compressions (see Sections 8 and 17 of [11]; the case of only one matrix appeared in Example 2.4 above).

3.5 Corollary. Let (A, φ) be a ∗-probability space, let d, s be positive integers, and let A_{1} = [a_{i,j}^{(1)}]_{i,j=1}^{d}, …, A_{s} = [a_{i,j}^{(s)}]_{i,j=1}^{d} form an R-cyclic family of selfadjoint matrices in M_{d}(A). Suppose that the cyclic cumulants of the entries of these matrices depend only on the superscript indices:

k_{n}(a_{i_{n},i_{1}}^{(r_{1})}, a_{i_{1},i_{2}}^{(r_{2})}, …, a_{i_{n−1},i_{n}}^{(r_{n})}) =: α_{r_{1},...,r_{n}},  (3.11)

for every n ≥ 1 and every 1 ≤ r_{1}, …, r_{n} ≤ s, 1 ≤ i_{1}, …, i_{n} ≤ d. Then:

R_{A_{1},...,A_{s}}(z_{1}, …, z_{s}) = ∑_{n=1}^{∞} ∑_{r_{1},...,r_{n}=1}^{s} d^{n−1}α_{r_{1},...,r_{n}}z_{r_{1}} ⋯ z_{r_{n}}.  (3.12)

Proof. If f denotes the determining series of A_{1}, …, A_{s}, then the coefficients of f are:

[coef( (r_{1}, i_{1}), …, (r_{n}, i_{n}) )](f) =: α_{r_{1},...,r_{n}},

∀ n ≥ 1, ∀ 1 ≤ r_{1}, …, r_{n} ≤ s, 1 ≤ i_{1}, …, i_{n} ≤ d. It is obvious that the partial summation condition of Equation (3.1) holds, where λ_{r_{1},...,r_{n}} := d^{n−1}α_{r_{1},...,r_{n}}. QED

4. Algebras generated by R-cyclic families

4.1 Remark. Let (A, φ) be a non-commutative probability space, let d be a positive integer, and let A_{1}, …, A_{s} be an R-cyclic family of matrices in M_{d}(A). Directly from the definition of R-cyclicity, and by using some basic properties of the non-crossing cumulants, it is easy to observe several “operations” that can be performed on the family A_{1}, …, A_{s} without affecting its R-cyclicity. For instance, it is trivial that re-ordering the s matrices does not affect R-cyclicity, and same about the operation of deleting one of the matrices from the family. Another operation which clearly does not affect the R-cyclicity of A_{1}, …, A_{s} consists in arbitrarily re-scaling the entries of the matrices (multiply the (i, j)-entry of A_{r} by some constant λ_{i,j}^{(r)}, for every 1 ≤ i, j ≤ d, 1 ≤ r ≤ s). Let us also observe that:

(a) If we enlarge A_{1}, …, A_{s} with a matrix A ∈ span{A_{1}, …, A_{s}}, then the enlarged family A_{1}, …, A_{s}, A is still R-cyclic. This is a direct consequence of the multilinearity of the cumulant functionals k_{n} : A^{n} → C, n ≥ 1.

(b) If we enlarge A_{1}, …, A_{s} with a scalar diagonal matrix D (which has the diagonal entries of the form λ_{i}I, 1 ≤ i ≤ d, and the off-diagonal entries equal to 0), then the enlarged family A_{1}, …, A_{s}, D is still R-cyclic. This is a consequence of the fact that a non-crossing cumulant of n ≥ 2 variables is 0 if at least one of its entries is in CI (same kind of argument as in the last phrase of Section 1).

In connection to (b) of Remark 4.1, note that we could not use there a scalar matrix which is not diagonal – indeed, the R-cyclicity condition asks in particular that every offdiagonal entry of every matrix in the family lies in the kernel of the state φ.

Now, in the framework of the same R-cyclic family A_{1}, …, A_{s} as above, where we assume that s ≥ 2, let us also observe that:

4.2 Lemma. If A_{s+1} := A_{1}A_{2}, then the enlarged family A_{1}, …, A_{s}, A_{s+1} is still Rcyclic.

Proof. We will use a formula for free cumulants with products as entries, as developed in [6]. In fact we can set the proof by induction, in such a way that we only use a particular case of this formula, which had already appeared in [19]. The particular case in question says that for any 1 ≤ m < n and any x_{1}, …, x_{n} in A we have:

k_{n−1}(x_{1}, …, x_{m−1}, x_{m}x_{m+1}, x_{m+2}, …, x_{n})

= k_{n}(x_{1}, …, x_{n}) + k_{m}(x_{1}, …, x_{m}) · k_{n−m}(x_{m+1}, …, x_{n})

+ ∑_{j=2}^{m} k_{m−j+1}(x_{j}, …, x_{m}) · k_{n−m+j−1}(x_{1}, …, x_{j−1}, x_{m+1}, …, x_{n})  (4.1)

+ ∑_{j=m+1}^{n−1} k_{j−m}(x_{m+1}, …, x_{j}) · k_{n−j+m}(x_{1}, …, x_{m}, x_{j+1}, …, x_{n}).

Now, let us return to the matrices A_{1}, …, A_{s+1} appearing in the statement of the lemma. For 1 ≤ r ≤ s + 1 and 1 ≤ i, j ≤ d we denote by a_{i,j}^{(r)} the (i, j)-entry of A_{r}. The hypothesis that A_{s+1} = A_{1}A_{2} thus says that

a_{i,j}^{(s+1)} = ∑_{k=1}^{d} a_{i,k}^{(1)}a_{k,j}^{(2)}, ∀ 1 ≤ i, j ≤ d.  (4.2)

We will prove by induction on l ≥ 0 the following statement:

St(l) \{ \begin{matrix}For every n ≥ 1, r_{1}, …, r_{n} ∈ \{1, …, s + 1\} and i_{1}, j_{1}, …, i_{n}, j_{n} ∈ \{1, …, d\} \\ such that \{m | 1 ≤ m ≤ n, r_{m} = s + 1\} has l elements \\ and for which it is not true that j_{1} = i_{2}, …, j_{n−1} = i_{n}, j_{n} = i_{1} \\ we have that k_{n}(a_{i_{1},j_{1}}^{(r_{1})}, …, a_{i_{n},j_{n}}^{(r_{n})}) = 0. \end{matrix}

If l = 0, the statement St(l) amounts precisely to the hypothesis that the family A_{1}, …, A_{s} is R-cyclic. For the rest of the proof we fix an l ≥ 1, for which we assume that the statements St(0), …, St(l − 1) are true, and for which we prove that the statement St(l) is also true.

Consider n ≥ 1, r_{1}, …, r_{n} ∈ {1, …, s + 1} and i_{1}, j_{1}, …, i_{n}, j_{n} ∈ {1, …, d} such that {m | 1 ≤ m ≤ n, r_{m} = s + 1} has l elements, and for which it is not true that j_{1} = i_{2}, …, j_{n−1} = i_{n}, j_{n} = i_{1}. Moreover, let us fix an index m, 1 ≤ m ≤ n, such that r_{m} = s + 1. Our goal is to show that k_{n}(a_{i_{1},j_{1}}^{(r_{1})}, …, a_{i_{n},j_{n}}^{(r_{n})}) = 0, but in view of (4.2) and of the multilinearity of k_{n} it suffices to verify that:

k_{n}(a_{i_{1},j_{1}}^{(r_{1})}, …, a_{i_{m−1},j_{m−1}}^{(r_{m−1})}, a_{i_{m},k}^{(1)}a_{k,j_{m}}^{(2)}, a_{i_{m+1},j_{m+1}}^{(r_{m+1})}, …, a_{i_{n},j_{n}}^{(r_{n})}) = 0, ∀ 1 ≤ k ≤ d.  (4.3)

Finally, let us also fix an index k ∈ {1, …, d} about which we will show that (4.3) holds. This in fact will be an immediate application of the formula (4.1). Indeed, let us pick an index p ∈ {1, …, n} such that j_{p} ≠ i_{p+1}; for the sake of clarity of the presentation we will assume that we know the relative position of p and m – say for instance that p < m − 1 (all the cases are treated similarly). We apply the formula (4.1) to the cumulant (4.3), and obtain a sum of n + 1 terms T_{1}, T_{2}, …, T_{n+1} where each of these terms is either a cumulant or a product of two cumulants:

k_{n}(a_{i_{1},j_{1}}^{(r_{1})}, …, a_{i_{p},j_{p}}^{(r_{p})}, a_{i_{p+1},j_{p+1}}^{(r_{p+1})}, …, a_{i_{m},k}^{(1)}a_{k,j_{m}}^{(2)}, …, a_{i_{n},j_{n}}^{(r_{n})})  (4.4)
= T_{1} + T_{2} + ⋯ + T_{n+1}.

The list of superscript indices on the left-hand side of (4.4) is r_{1}, …, r_{m−1}, 1, 2, r_{m+1}, …, r_{n}, containing l − 1 occurrences of s + 1. So the induction hypothesis will apply and will give us that T_{1} = ⋯ = T_{n+1} = 0 on the right-hand side of (4.4), provided that we make sure that each of T_{1}, …, T_{n+1} still violates the cyclicity condition of the subscript indices. The violation of cyclicity for subscript indices is trivial for all of T_{1}, …, T_{n+1} with one exception, because in general the neighboring indices j_{p} ≠ i_{p+1} will not be separated. The exception is for the term:

k_{m−p}(a_{i_{p+1},j_{p+1}}^{(r_{p+1})}, …, a_{i_{m},k}^{(1)}) · k_{n+1−m+p}(a_{i_{1},j_{1}}^{(r_{1})}, …, a_{i_{p},j_{p}}^{(r_{p})}, a_{k,j_{m}}^{(2)}, …, a_{i_{n},j_{n}}^{(r_{n})});

but here the cyclicity condition of the subscript indices is still violated, since we must have that either k ≠ i_{p+1} or that j_{p} ≠ k. QED

By combining the various “R-cyclicity preserving operations” which were observed in the Remark 4.1 and Lemma 4.2, we arrive to the following statement (which in some sense collects these observations together):

4.3 Theorem. Let (A, φ) be a non-commutative probability space, let d be a positive integer, and let A_{1}, …, A_{s} be an R-cyclic family of matrices in M_{d}(A). We denote by D the algebra of scalar diagonal matrices in M_{d}(A), and by C the subalgebra of M_{d}(A) which is generated by {A_{1}, …, A_{s}} ∪ D. Then every finite family of matrices from C is R-cyclic.

This theorem will be put into a better perspective by the result in Section 8.

5. Review of operator-valued cumulants

Let B be a unital algebra over C. By a B-probability space we understand a pair (M, E), where:

– M is an algebra containing B as a unital subalgebra (by which we mean that B is identified as a unital subalgebra of M, in some well-determined way);

– E : M → B is a linear map with the properties that E(b) = b for every b ∈ B, and E(b_{1}xb_{2}) = b_{1}E(x)b_{2} for every b_{1}, b_{2} ∈ B, x ∈ M.

If (M, E) is a B-probability space and if x_{1}, …, x_{s} ∈ M, then the expressions of the form:

E(b_{0}x_{r_{1}}b_{1} ⋯ x_{r_{n}}b_{n}), with n ≥ 1, 1 ≤ r_{1}, …, r_{n} ≤ s, b_{0}, b_{1}, …, b_{n} ∈ B

are called joint B-moments of the family x_{1}, …, x_{s}. Moreover, if (M̃, Ẽ) also is a Bprobability space and if x̃_{1}, …, x̃_{s} ∈ M̃, we will say that the families x_{1}, …, x_{s} and x̃_{1}, …, x̃_{s} have identical B-distributions if

E(b_{0}x_{r_{1}}b_{1} ⋯ x_{r_{n}}b_{n}) = Ẽ(b_{0}x̃_{r_{1}}b_{1} ⋯ x̃_{r_{n}}b_{n})  (5.1)

for every n ≥ 1, 1 ≤ r_{1}, …, r_{n} ≤ s, and b_{0}, b_{1}, …, b_{n} ∈ B.

While the joint B-moments generalize the joint moments appearing in Eqn.(1.1) of Section 1, it is in general inconvenient to introduce a concept of “B-moment series” analogous to the one defined by Eqn.(1.2). Similarly, rather than introducing B-valued R-transforms, it is more convenient to just consider the B-valued generalization for the concept of noncrossing cumulant. Following the development of [19], this can be done as described in Proposition 5.2 below.

5.1 Notations. Let π, ρ be partitions in NC(p) and NC(q) respectively, where p, q ≥ 1. Let k be in {0, 1, …, q}. By ins(π ↦ ρ; k) we will denote the non-crossing partition in NC(p + q) which is obtained by “inserting π between the elements k and k + 1 of ρ”. Formally this means that the set {k + 1, …, k + p} is a union of blocks of ins(π ↦ ρ; k), and that:

(i) the restriction of ins(π ↦ ρ; k) to {k + 1, …, k + p} is naturally identified to π; (ii) the restriction of ins(π ↦ ρ; k) to {1, 2, …, p + q} \ {k + 1, …, k + p} is naturally

identified to ρ.

For example, if π = { {1}, {2, 3} } ∈ NC(3) and ρ = { {1, 2} } ∈ NC(2), then: ins(π ↦ ρ; 0) = { {1}, {2, 3}, {4, 5} }; ins(π ↦ ρ; 1) = { {1, 5}, {2}, {3, 4} }; ins(π ↦ ρ; 2) = { {1, 2}, {3}, {4, 5} }.

5.2 Proposition (see [19], Section 3.2). Let (M, E) be a B-probability space. There exists a family of functionals {k_{π}^{(B)} | π ∈ ∪_{n=1}^{∞}NC(n)} uniquely determined by the following properties:

(1) For π ∈ NC(n), k_{π}^{(B)} is a multilinear functional from M^{n} to B.

(2) If π ∈ NC(p), ρ ∈ NC(q), k ∈ {0, 1, …, q}, and if σ := ins(π ↦ ρ; k) ∈ NC(p + q), then for every x_{1}, …, x_{p+q} ∈ M we have:

\{ \begin{matrix}k_{σ}^{(B)}(x_{1}, …, x_{p+q}) = k_{ρ}^{(B)}(x_{1}, …, x_{k}b, x_{k+p+1}, …, x_{p+q}) \\ where b := k_{π}^{(B)}(x_{k+1}, …, x_{k+p}). \end{matrix}  (5.2)

(3) For every n ≥ 1 and x_{1}, …, x_{n} ∈ M we have:

∑_{π∈NC(n)} k_{π}^{(B)}(x_{1}, x_{2}, …, x_{n}) = E(x_{1}x_{2} ⋯ x_{n}).  (5.3)

5.3 Remarks and Notations. 1) In the condition (1) of Proposition 5.2, by “multilinear” we understand C-multilinear. The functionals k_{π}^{(B)} turn out to actually have B-multilinearity properties, namely that:

k_{π}^{(B)}(bx_{1}, x_{2}, …, x_{n}) = b · k_{π}^{(B)}(x_{1}, x_{2}, …, x_{n}),
k_{π}^{(B)}(x_{1}, x_{2}, …, x_{n}b) = k_{π}^{(B)}(x_{1}, x_{2}, …, x_{n}) · b,

also that

k_{π}^{(B)}(x_{1}, …, x_{i}b, x_{i+1}, …, x_{n}) = k_{π}^{(B)}(x_{1}, …, x_{i}, bx_{i+1}, …, x_{n})

for every π ∈ NC(n), x_{1}, …, x_{n} ∈ M, b ∈ B and 1 ≤ i ≤ n − 1. The C-multilinearity stated in (1) of Proposition 5.2 is however more convenient when using the uniqueness part of the proposition.

2) For every n ≥ 1, we will denote by k_{n}^{(B)} : M^{n} → B the functional k_{1_{n}}^{(B)}, where 1_{n} is the partition of {1, …, n} into only one block.

The knowledge of the functionals {k_{n}^{(B)} | n ≥ 1} really determines the whole family {k_{π}^{(B)} | π ∈ ∪_{n=1}^{∞}NC(n)}, via the Equation (5.2) and the observation that the only noncrossing partitions that are irreducible for the operation of insertion are those of the form 1_{n}. So in a certain sense the functionals k_{π}^{(B)} with π not of the form 1_{n} are just some derived objects; but nevertheless, the k_{π}^{(B)}’s are important for stating the essential condition (3) of Proposition 5.2, which can be viewed as a B-valued analogue for Eqn.(1.11) in Section 1.

3) Let (M, E) and (M̃, Ẽ) be B-probability spaces, and consider the families of elements x_{1}, …, x_{s} ∈ M, x̃_{1}, …, x̃_{s} ∈ M̃. We say that the families x_{1}, …, x_{s} and x̃_{1}, …, x̃_{s} have identical B-cumulants if:

k_{n}^{(B)}(x_{r_{1}}b_{1}, …, x_{r_{n−1}}b_{n−1}, x_{r_{n}}) = k_{n}^{(B)}(x̃_{r_{1}}b_{1}, …, x̃_{r_{n−1}}b_{n−1}, x̃_{r_{n}}),  (5.4)

for every n ≥ 1, 1 ≤ r_{1}, …, r_{n} ≤ s, and b_{1}, …, b_{n−1} ∈ B.

If x_{1}, …, x_{s} and x̃_{1}, …, x̃_{s} have identical B-cumulants, then the Equations (5.4) actually hold with “k_{π}^{(B)}” instead of k_{n}^{(B)}; this is immediate from (5.2), by an induction argument. Another induction argument and the use of Equation (5.3) show that x_{1}, …, x_{s} and x̃_{1}, …, x̃_{s} have identical B-cumulants if and only if the two families are identically B-distributed in the sense of Equation (5.1). Hence, similarly to the scalar case reviewed in Section 1, the B-cumulants offer an alternative to working with B-moments.

It is useful to record the following generalization (in Proposition 5.5) of the uniqueness part of Proposition 5.2. In all the considerations of this paper, by “B-bimodule” we will understand a left-and-right B-module, where the left and the right action of B commute with each other.

5.4 Definition. Let X be a B-bimodule, and suppose that for every n ≥ 1 and π ∈ NC(n) we have a C-multilinear functional f_{π} : X^{n} → B. We say that the family of functionals {f_{π} | π ∈ ∪_{n=1}^{∞}NC(n)} has the insertion property if the following holds: if σ = ins(π ↦ ρ; k) with π ∈ NC(p), ρ ∈ NC(q), k ∈ {0, 1, …, q}, and if x_{1}, …, x_{p+q} ∈ X, then:

\{ \begin{matrix}f_{σ}(x_{1}, …, x_{p+q}) = f_{ρ}(x_{1}, …, x_{k} · b, x_{k+p+1}, …, x_{p+q}) \\ where b := f_{π}(x_{k+1}, …, x_{k+p}) ∈ B. \end{matrix}  (5.5)

5.5 Proposition. Let X be a B-bimodule, and suppose that for every n ≥ 1 and π ∈ NC(n) we have two C-multilinear functionals f_{π}, g_{π} : X^{n} → B. If both the families {f_{π} | π ∈ ∪_{n=1}^{∞}NC(n)} and {g_{π} | π ∈ ∪_{n=1}^{∞}NC(n)} have the insertion property, and if:

∑_{π∈NC(n)} f_{π}(x_{1}, …, x_{n}) = ∑_{π∈NC(n)} g_{π}(x_{1}, …, x_{n}),  (5.6)

for every n ≥ 1 and x_{1}, …, x_{n} ∈ X, then we must have that f_{π} = g_{π} for all π ∈ ∪_{n=1}^{∞}NC(n).

The proof of Proposition 5.5 is done by induction on n (where π ∈ NC(n)), and is an immediate adaptation of arguments in [19], Section 3.2.

The main use of B-cumulants is as tool for studying freeness with amalgamation over B. Recall that this is defined as follows (cf. e.g. [22], Section 3.8).

5.6 Definition. Let (M, E) be a B-probability space and let M_{1}, …, M_{s} be subalgebras of M such that M_{1}, …, M_{s} ⊃ B. We say that M_{1}, …, M_{s} are free with amalgamation over B if for every n ≥ 1 and every r_{1}, …, r_{n} ∈ {1, …, s} such that r_{1} ≠ r_{2}, r_{2} ≠ r_{3}, …, r_{n−1} ≠ r_{n} we have:

\{ \begin{matrix}x_{1} ∈ M_{r_{1}}, x_{2} ∈ M_{r_{2}}, …, x_{n} ∈ M_{r_{n}} \\ E(x_{1}) = E(x_{2}) = ⋯ = E(x_{n}) = 0 \end{matrix} \} ⇒ E(x_{1}x_{2} ⋯ x_{n}) = 0.  (5.7)

5.7 Remark. The important characterization of freeness described in Remark 1.4 can be generalized to the B-valued framework. More precisely: if (M, E) and B ⊂ M_{1}, …, M_{s} ⊂ M are as above, then the freeness of M_{1}, …, M_{s} with amalgamation over B is equivalent to the following condition:

k_{n}^{(B)}(x_{1}, …, x_{n}) = 0
whenever x_{1} ∈ M_{r_{1}}, …, x_{n} ∈ M_{r_{n}}
are such that ∃ 1 ≤ k < l ≤ n with r_{k} ≠ r_{l}.

See [19], Section 3.3.

5.8 Notations. For the remaining of this section we will suppose that besides the algebra B (which was fixed from the beginning of the section) we have also fixed:

– a unital subalgebra D ⊂ B;

– a linear map τ : B → D with the properties that τ(d) = d for every d ∈ D, and that τ(d_{1}bd_{2}) = d_{1}τ(b)d_{2} for every d_{1}, d_{2} ∈ D, b ∈ B.

We will assume moreover that τ is faithful (or non-degenerate) in the sense that if b ∈ B has the property that τ(bb′) = 0 for all b′ ∈ B, then b = 0.

In the Notations 5.8, observe that any B-probability space (M, E) induces a D-probability space (M, E_{D}), where we set E_{D} := τ ∘ E.

5.9 Proposition. Let (M, E) and (M̃, Ẽ) be B-probability spaces, and consider the corresponding D-probability spaces (M, E_{D}) and (M̃, Ẽ_{D}). Suppose that C ⊂ M and C̃ ⊂ M̃ are subalgebras which contain D, and suppose that each of C and C̃ is free from B with amalgamation over D (in its corresponding space). Let x_{1}, …, x_{s} be in C, and let x̃_{1}, …, x̃_{s} be in C̃. If the families x_{1}, …, x_{s} are identically D-distributed, then the two families are also identically B-distributed.

Proof. We have to show that:

E_{B}(b_{0}x_{r_{1}}b_{1} ⋯ x_{r_{n}}b_{n}) = Ẽ_{B}(b_{0}x̃_{r_{1}}b_{1} ⋯ x̃_{r_{n}}b_{n}),

for every n ≥ 1, 1 ≤ r_{1}, …, r_{n} ≤ s, and b_{0}, b_{1}, …, b_{n} ∈ B. In view of the faithfulness of τ : B → D, this will follow if we can show that:

τ( E_{B}(b_{0}x_{r_{1}}b_{1} ⋯ x_{r_{n}}b_{n})b′ ) = τ(Ẽ_{B}(b_{0}x̃_{r_{1}}b_{1} ⋯ x̃_{r_{n}}b_{n})b′ ),  (5.8)

(for every n, r_{1}, …, r_{n}, b_{0}, b_{1}, …, b_{n} as before, and for every b′ ∈ B). By absorbing b′ into E_{B} and into Ẽ_{B}, and by taking into account that τ ∘ E_{B} = E_{D}, τ ∘ Ẽ_{B} = Ẽ_{D}, we reduce (5.8) to:

E_{D}(b_{0}x_{r_{1}}b_{1} ⋯ x_{r_{n}}b_{n}) = Ẽ_{D}(b_{0}x̃_{r_{1}}b_{1} ⋯ x̃_{r_{n}}b_{n}),  (5.9)

for every n ≥ 1, 1 ≤ r_{1}, …, r_{n} ≤ s, and b_{0}, b_{1}, …, b_{n} ∈ B. Finally, (5.9) follows from the definition of freeness with amalgamation plus an induction argument, by using the hypotheses that x_{1}, …, x_{s} and x̃_{1}, …, x̃_{s} have identical D-distributions, and that C, C̃ are free from B with amalgamation over D. QED

6. Cumulants with respect to the algebra of d-by-d scalar matrices

6.1 Notations. In this section we fix a positive integer d, and we consider the algebra B := M_{d}(C). If (A, φ) is any non-commutative probability space, then the algebra M_{d}(A) gets a structure of B-probability space, where we view B as a subalgebra of M_{d}(A) via the natural identification:

[λ_{i,j}]_{i,j=1}^{d} = [λ_{i,j}I]_{i,j=1}^{d}  (6.1)

(with I = the unit of A). The expectation E_{B} : M_{d}(A) → B is defined by the formula:

E_{B}( [a_{i,j}]_{i,j=1}^{d} ) := [ φ(a_{i,j}) ]_{i,j=1}^{d}.  (6.2)

Thus we are in the situation when we can consider B-valued cumulants for families of matrices in M_{d}(A).

The goal of the section is to give an explicit formula for the B-valued cumulant of a family of matrices, in terms of the scalar cumulants of the entries of these matrices.

6.2 Theorem. In the framework considered above let A_{1}, …, A_{n} be matrices in M_{d}(A), where A_{m} = [a_{i,j}^{(m)}]_{i,j=1}^{d} for 1 ≤ m ≤ n. Then for every 1 ≤ i, j ≤ d, the (i, j)-entry λ_{i,j} of the B-valued cumulant k_{n}^{(B)}(A_{1}, …, A_{n}) is given by the formula:

λ_{i,j} = ∑_{i_{1},...,i_{n−1}=1}^{d} k_{n}(a_{i,i_{1}}^{(1)}, a_{i_{1},i_{2}}^{(2)}, …, a_{i_{n−2},i_{n−1}}^{(n−1)}, a_{i_{n−1},j}^{(n)}).  (6.3)

Proof. For every n ≥ 1 and π ∈ NC(n), we define a multilinear functional f_{π} : (M_{d}(A))^{n} → B, by the following formula:

(i, j) − entry of f_{π}(A_{1}, …, A_{n}) :=  (6.4)

∑_{i_{1},...,i_{n−1}=1}^{d} k_{π}(a_{i,i_{1}}^{(1)}, a_{i_{1},i_{2}}^{(2)}, …, a_{i_{n−2},i_{n−1}}^{(n−1)}, a_{i_{n−1},j}^{(n)}),

for every A_{1}, …, A_{n} ∈ M_{d}(A) and every 1 ≤ i, j ≤ d (and where a_{k,l}^{(m)} stands for the (k, l)- entry of the matrix A_{m}). We will verify that the family of functionals {f_{π} | π ∈ ∪_{n=1}^{∞}NC(n)} satisfies the conditions (2) and (3) from Proposition 5.2, which determine uniquely the Bvalued cumulant functionals. Once this is done, the equality f_{π} = k_{π}^{(B)} applied to the partition π = 1_{n} (of {1, …, n} into only one block) will give the statement of the theorem.

We start with the verification of condition (3) (about summation). Given n ≥ 1 and A_{1}, …, A_{n} ∈ M_{d}(A), we look at:

∑_{π∈NC(n)} f_{π}(A_{1}, …, A_{n}).  (6.5)

For every 1 ≤ i, j ≤ d, the (i, j)-entry of the matrix appearing in (6.5) is equal to:

∑_{π∈NC(n)} ∑_{i_{1},...,i_{n−1}=1}^{d} k_{π}(a_{i,i_{1}}^{(1)}, a_{i_{1},i_{2}}^{(2)}, …, a_{i_{n−2},i_{n−1}}^{(n−1)}, a_{i_{n−1},j}^{(n)})

= ∑_{i_{1},...,i_{n−1}=1}^{d} ( ∑_{π∈NC(n)} k_{π}(a_{i,i_{1}}^{(1)}, a_{i_{1},i_{2}}^{(2)}, …, a_{i_{n−2},i_{n−1}}^{(n−1)}, a_{i_{n−1},j}^{(n)}) )

= ∑_{i_{1},...,i_{n−1}=1}^{d} φ(a_{i,i_{1}}^{(1)}a_{i_{1},i_{2}}^{(2)} ⋯ a_{i_{n−2},i_{n−1}}^{(n−1)}a_{i_{n−1},j}^{(n)})

(by the relation between scalar cumulants and moments). It is clear that the latter quantity is equal to φ of the (i, j)-entry of A_{1}A_{2} ⋯ A_{n}. Hence the matrix in (6.5) is equal to E_{B}(A_{1}A_{2} ⋯ A_{n}) (as desired).

We now move to the verification of condition (2) (about insertion). Suppose that σ = ins(π ↦ ρ; k), where π ∈ NC(p), ρ ∈ NC(q), 0 ≤ k ≤ q, and where the Notations 5.1 are used. Given matrices A_{1}, …, A_{p+q} ∈ M_{d}(A), we want to verify that:

f_{σ}(A_{1}, …, A_{p+q}) = f_{ρ}(A_{1}, …, A_{k−1}, A_{k}B, A_{k+p+1}, …, A_{p+q}),  (6.6)

where

B := f_{π}(A_{k+1}, …, A_{k+p}).  (6.7)

We fix i and j in {1, …, d}, and we work on the (i, j)-entry of the left-hand side of (6.6). By the definition of f_{σ} this equals:

∑_{i_{1},...,i_{p+q−1}=1}^{d} k_{σ}(a_{i,i_{1}}^{(1)}, a_{i_{1},i_{2}}^{(2)}, …, a_{i_{p+q−2},i_{p+q−1}}^{(p+q−1)}, a_{i_{p+q−1},j}^{(p+q)}),

so by using the insertion property for scalar cumulants, we can re-write it as:

∑_{i_{1},...,i_{p+q−1}=1}^{d} k_{π}(a_{i_{k},i_{k+1}}^{(k+1)}, …, a_{i_{k+p−1},i_{k+p}}^{(k+p)})·  (6.8)
·k_{ρ}(a_{i,i_{1}}^{(1)}, …, a_{i_{k−1},i_{k}}^{(k)}, a_{i_{k+p},i_{k+p+1}}^{(k+p+1)}, …, a_{i_{p+q−2},i_{p+q−1}}^{(p+q−1)}, a_{i_{p+q−1},j}^{(p+q)}).

Now, let us denote:

B =: [β_{i,j}]_{i,j=1}^{d} ∈ B, BA_{k} =: [x_{i,j}]_{i,j=1}^{d} ∈ M_{d}(A),  (6.9)

where B is the matrix defined by (6.7). If in the summation of (6.8) we first sum over the indices i_{k+1}, …, i_{k+p−1}, we arrive to:

∑_{i_{1},...,i_{k},i_{k+p},...,i_{p+q−1}=1}^{d} ( ∑_{i_{k+1},...,i_{k+p−1}=1}^{d} k_{π}(a_{i_{k},i_{k+1}}^{(k+1)}, …, a_{i_{k+p−1},i_{k+p}}^{(k+p)}))·

·k_{ρ}(a_{i,i_{1}}^{(1)}, …, a_{i_{k−1},i_{k}}^{(k)}, a_{i_{k+p},i_{k+p+1}}^{(k+p+1)}, …, a_{i_{p+q−2},i_{p+q−1}}^{(p+q−1)}, a_{i_{p+q−1},j}^{(p+q)})

= ∑_{i_{1},...,i_{k},i_{k+p},...,i_{p+q−1}=1}^{d} β_{i_{k},i_{k+p}} · k_{ρ}(a_{i,i_{1}}^{(1)}, …, a_{i_{k−1},i_{k}}^{(k)}, a_{i_{k+p},i_{k+p+1}}^{(k+p+1)}, …, a_{i_{p+q−2},i_{p+q−1}}^{(p+q−1)}, a_{i_{p+q−1},j}^{(p+q)})

(by taking into account the definition of f_{π}, the Equation (6.7), and the notation in (6.9))

= ∑_{i_{1},...,i_{k},i_{k+p},...,i_{p+q−1}=1}^{d} k_{ρ}(a_{i,i_{1}}^{(1)}, …, a_{i_{k−1},i_{k}}^{(k)}β_{i_{k},i_{k+p}}, a_{i_{k+p},i_{k+p+1}}^{(k+p+1)}, …, a_{i_{p+q−2},i_{p+q−1}}^{(p+q−1)}, a_{i_{p+q−1},j}^{(p+q)})

= ∑_{i_{1},...,i_{k−1},i_{k+p},...,i_{p+q−1}=1}^{d} k_{ρ}(a_{i,i_{1}}^{(1)}, …, a_{i_{k−2},i_{k−1}}^{(k−1)}, x_{i_{k−1},i_{k+p}}, a_{i_{k+p},i_{k+p+1}}^{(k+p+1)}, …, a_{i_{p+q−1},j}^{(p+q)})

(by summing over i_{k} and by taking into account the definition of the x_{i,j}’s in (6.9)). The last expression is exactly the (i, j)-entry of the matrix on the right-hand side of (6.6), and this concludes the proof. QED

6.3 Remark. We actually arrived to prove a stronger formula than originally announced in Theorem 6.2, namely that

(i, j) − entry of k_{π}^{(B)}(A_{1}, …, A_{n}) =  (6.10)

∑_{i_{1},...,i_{n−1}=1}^{d} k_{π}(a_{i,i_{1}}^{(1)}, a_{i_{1},i_{2}}^{(2)}, …, a_{i_{n−2},i_{n−1}}^{(n−1)}, a_{i_{n−1},j}^{(n)}),

for every non-crossing partition π ∈ NC(n) and every A_{1}, …, A_{n} ∈ M_{d}(A) (and where a_{k,l}^{(m)} denotes the (k, l)-entry of the matrix A_{m}).

6.4 Remark. For A_{1}, …, A_{n} ∈ M_{d}(A) as above, one sometimes denotes by A_{1} ⊙ A_{2} ⊙ ⋯ ⊙ A_{n} the matrix in M_{d}(A ⊗ A ⊗ ⋯ ⊗ A) which has the (i, j)-entry equal to:

∑_{i_{1},...,i_{n−1}=1}^{d} a_{i,i_{1}}^{(1)} ⊗ a_{i_{1},i_{2}}^{(2)} ⊗ ⋯ ⊗ a_{i_{n−1},j}^{(n)}, 1 ≤ i, j ≤ d.

The operation ⊙ is for instance used in considerations on tensor products of operator spaces (see e.g. Section 8.1 of [4], or Section 3 of [16]).

The statement of Theorem 6.2 can be given a nice form if we use ⊙, as follows: instead of viewing the scalar-valued cumulant k_{n} as a multilinear map from A^{n} to C, let us view it as a linear map from the n-fold tensor product A ⊗ ⋯ ⊗ A into C. When we go to d × d matrices, k_{n} then induces a linear application k̃_{n} from M_{d}(A ⊗ ⋯ ⊗ A) to M_{d}(C), hence to B; this is given by the formula

k̃_{n}( [x_{i,j}]_{i,j=1}^{d} ) := [k_{n}(x_{i,j})]_{i,j=1}^{d}, ∀ [x_{i,j}]_{i,j=1}^{d} ∈ M_{d}(A ⊗ ⋯ ⊗ A).

It is immediate that with these notations, the statement of Theorem 6.2 takes the form:

k_{n}^{(B)}(A_{1}, …, A_{n}) = k̃_{n}(A_{1} ⊙ ⋯ ⊙ A_{n}), ∀ A_{1}, …, A_{n} ∈ M_{d}(A).  (6.11)

7. Cumulants with respect to the algebra of scalar diagonal matrices

7.1 Notations. The framework for this section is similar to the one of Section 6, but where instead of the algebra B = M_{d}(C) we consider the algebra D of scalar diagonal d × d matrices. In other words D = span{P_{1}, …, P_{d}}, where P_{i} denotes the matrix which has its (i, i)-entry equal to 1 and all the other entries equal to 0.

If (A, φ) is any non-commutative probability space, then the algebra M_{d}(A) gets a natural structure of D-probability space, where we view D as a subalgebra of M_{d}(A) via the natural identification:

[ \begin{matrix}λ_{1} & & & & 0 \\ & . & . & . \\ 0 & & & & λ_{d} \end{matrix} ] = [ \begin{matrix}λ_{1}I & & & & 0 \\ & . & . & . \\ 0 & & & & λ_{d}I \end{matrix} ]  (7.1)

(with I = the unit of A). The expectation E_{D} : M_{d}(A) → D is defined by the formula:

E_{D}( [a_{i,j}]_{i,j=1}^{d} ) := [ \begin{matrix}φ(a_{1,1}) & & & & 0 \\ & . & . & . \\ 0 & & & & φ(a_{d,d}) \end{matrix} ] .  (7.2)

Thus we are in the situation when we can consider D-valued cumulants for families of matrices in M_{d}(A).

Following the same line as in the preceding section, we consider the problem of expressing the D-cumulants of a family of matrices from M_{d}(A) in terms of the scalar cumulants of the entries of these matrices. It does not seem that there exists a nice formula holding in general, but it is still possible to get one in the case of R-cyclic families. In fact we will consider a class larger than the one of R-cyclic families, as described in the next theorem.

7.2 Theorem. In the framework considered above, let A_{1}, …, A_{s} be a family of matrices in M_{d}(A), where A_{r} = [a_{i,j}^{(r)}]_{i,j=1}^{d} for 1 ≤ r ≤ s. Suppose that for every n ≥ 1, 1 ≤ r_{1}, …, r_{n} ≤ s, 1 ≤ i_{1}, …, i_{n}, j ≤ d we have:

j ≠ i_{n} ⇒ k_{n}(a_{j,i_{1}}^{(r_{1})}, a_{i_{1},i_{2}}^{(r_{2})}, …, a_{i_{n−1},i_{n}}^{(r_{n})}) = 0.  (7.3)

Then the D-valued cumulants of the family A_{1}, …, A_{s} are described by the following formula:

k_{n}^{(D)}(A_{r_{1}}Λ_{1}, …, A_{r_{n−1}}Λ_{n−1}, A_{r_{n}}) =  (7.4)

∑_{i_{1},...,i_{n}=1}^{d} λ_{i_{1}}^{(1)} ⋯ λ_{i_{n−1}}^{(n−1)} · k_{n}(a_{i_{n},i_{1}}^{(r_{1})}, a_{i_{1},i_{2}}^{(r_{2})}, …, a_{i_{n−1},i_{n}}^{(r_{n})})P_{i_{n}},

holding for n ≥ 2, r_{1}, …, r_{n} ∈ {1, …, s}, and where

Λ_{k} := [ \begin{matrix}λ_{1}^{(k)} & & & & 0 \\ & . & . & . \\ 0 & & & & λ_{d}^{(k)} \end{matrix} ] ∈ D, 1 ≤ k ≤ n − 1.

Proof. Let X be the free D-bimodule with s generators X_{1}, …, X_{s}. As a vector space over C, X has dimension d^{2}s, and has a natural basis given by the elements P_{i}X_{r}P_{j}, with 1 ≤ i, j ≤ d and 1 ≤ r ≤ s.

For every n ≥ 1 and π ∈ NC(n) we consider the C-multilinear functionals f_{π} and g_{π} from X^{n} to D, determined as follows (by their action on the natural basis of X^{n}):

f_{π}(P_{i_{1}}X_{r_{1}}P_{j_{1}}, …, P_{i_{n}}X_{r_{n}}P_{j_{n}}) = k_{π}^{(D)}(P_{i_{1}}A_{r_{1}}P_{j_{1}}, …, P_{i_{n}}A_{r_{n}}P_{j_{n}})  (7.5)

and

g_{π}(P_{i_{1}}X_{r_{1}}P_{j_{1}}, …, P_{i_{n}}X_{r_{n}}P_{j_{n}}) =  (7.6)
δ_{i_{1},j_{n}}δ_{i_{2},j_{1}} ⋯ δ_{i_{n},j_{n−1}} · k_{π}(a_{j_{n},j_{1}}^{(r_{1})}, a_{j_{1},j_{2}}^{(r_{2})}, …, a_{j_{n−1},j_{n}}^{(r_{n})})P_{j_{n}},

for n ≥ 1 and 1 ≤ r_{1}, …, r_{n} ≤ s, 1 ≤ i_{1}, …, i_{n}, j_{1}, …, j_{n} ≤ d. An immediate linearity argument shows that:

f_{π}(Γ_{1}X_{r_{1}}Λ_{1}, …, Γ_{n}X_{r_{n}}Λ_{n}) = k_{π}^{(D)}(Γ_{1}A_{r_{1}}Λ_{1}, …, Γ_{n}A_{r_{n}}Λ_{n}),  (7.7)

and

g_{π}(Γ_{1}X_{r_{1}}Λ_{1}, …, Γ_{n}X_{r_{n}}Λ_{n}) =  (7.8)

∑_{j_{1},...,j_{n}=1}^{d} γ_{j_{n}}^{(1)} · λ_{j_{1}}^{(1)}γ_{j_{1}}^{(2)} ⋯ λ_{j_{n−1}}^{(n−1)}γ_{j_{n−1}}^{(n)} · λ_{j_{n}}^{(n)} · k_{π}(a_{j_{n},j_{1}}^{(r_{1})}, a_{j_{1},j_{2}}^{(r_{2})}, …, a_{j_{n−1},j_{n}}^{(r_{n})})P_{j_{n}},

for every n ≥ 1, 1 ≤ r_{1}, …, r_{n} ≤ s and Γ_{1}, Λ_{1}, …, Γ_{n}, Λ_{n} ∈ D, where:

Γ_{k} := [ \begin{matrix}γ_{1}^{(k)} & & & & 0 \\ & . & . & . \\ 0 & & & & γ_{d}^{(k)} \end{matrix} ] , Λ_{k} := [ \begin{matrix}λ_{1}^{(k)} & & & & 0 \\ & . & . & . \\ 0 & & & & λ_{d}^{(k)} \end{matrix} ] , 1 ≤ k ≤ n.

Now, both the families of functionals {f_{π} | π ∈ ∪_{n=1}^{∞}NC(n)} and {g_{π} | π ∈ ∪_{n=1}^{∞}NC(n)} satisfy the insertion property considered in Definition 5.4. For the f_{π}’s this is an immediate consequence of the corresponding property for the D-valued cumulant functionals {k_{π}^{(D)} | π ∈ ∪_{n=1}^{∞}NC(n)}. For the g_{π}’s the insertion property follows from a calculation very similar in nature to the one shown in the proof of Theorem 6.2, and which, due to its routine character, will be left to the reader. (The reader who will have the patience to go through this calculation will notice that it effectively makes use of the implication (7.3) stated in the hypothesis of the current theorem.)

We next show that f_{π} = g_{π} for every π ∈ ∪_{n=1}^{∞}NC(n). Proposition 5.5 combined with a linearity argument shows that all we need to check is the equality:

∑_{π∈NC(n)} g_{π}(P_{i_{1}}X_{r_{1}}P_{j_{1}}, …, P_{i_{n}}X_{r_{n}}P_{j_{n}}) = ∑_{π∈NC(n)} f_{π}(P_{i_{1}}X_{r_{1}}P_{j_{1}}, …, P_{i_{n}}X_{r_{n}}P_{j_{n}})

(for some fixed n ≥ 1, 1 ≤ r_{1}, …, r_{n} ≤ s, 1 ≤ i_{1}, j_{1}, …, i_{n}, j_{n} ≤ d). And indeed, we compute:

∑_{π∈NC(n)} g_{π}(P_{i_{1}}X_{r_{1}}P_{j_{1}}, …, P_{i_{n}}X_{r_{n}}P_{j_{n}}) =

δ_{i_{1},j_{n}}δ_{i_{2},j_{1}} ⋯ δ_{i_{n},j_{n−1}} · ( ∑_{π∈NC(n)} k_{π}(a_{j_{n},j_{1}}^{(r_{1})}, a_{j_{1},j_{2}}^{(r_{2})}, …, a_{j_{n−1},j_{n}}^{(r_{n})}) )P_{j_{n}}

= δ_{i_{1},j_{n}}δ_{i_{2},j_{1}} ⋯ δ_{i_{n},j_{n−1}} · φ(a_{j_{n},j_{1}}^{(r_{1})}a_{j_{1},j_{2}}^{(r_{2})} ⋯, a_{j_{n−1},j_{n}}^{(r_{n})})P_{j_{n}}

= E_{D}(P_{i_{1}}A_{r_{1}}P_{j_{1}}P_{i_{2}}A_{r_{2}}P_{j_{2}} ⋯ P_{i_{n}}A_{r_{n}}P_{j_{n}})

= ∑_{π∈NC(n)} k_{π}^{(D)}(P_{i_{1}}A_{r_{1}}P_{j_{1}}, …, P_{i_{n}}A_{r_{n}}P_{j_{n}})

= ∑_{π∈NC(n)} f_{π}(P_{i_{1}}X_{r_{1}}P_{j_{1}}, …, P_{i_{n}}X_{r_{n}}P_{j_{n}}).

But if f_{π} = g_{π}, then one can equate the right-hand sides of the Equations (7.7) and (7.8). By doing this for π = 1_{n} (the partition of {1, …, n} into only one block), and by appropriately choosing Γ_{1}, Λ_{1}, …, Γ_{n}, Λ_{n} ∈ D, one obtains the Equation (7.4) from the conclusion of the theorem. QED

7.3 Remark. In the framework of the Notations 7.1, let A_{1}, …, A_{s} be an R-cyclic family of matrices in M_{d}(A), where A_{r} = [a_{i,j}^{(r)}]_{i,j=1}^{d} for 1 ≤ r ≤ s. Then Theorem 7.2 gives us an interpretation for the cyclic cumulants of the entries of A_{1}, …, A_{s} (i.e., for the coefficients of the determining series of the family A_{1}, …, A_{s}). More precisely, for every n ≥ 1, 1 ≤ r_{1}, …, r_{n} ≤ s and 1 ≤ i_{1}, …, i_{n} ≤ d, we have that:

k_{n}( a_{i_{n},i_{1}}^{(r_{1})}, a_{i_{1},i_{2}}^{(r_{2})}, …, a_{i_{n−2},i_{n−1}}^{(r_{n−1})}, a_{i_{n−1},i_{n}}^{(r_{n})} ) =  (7.9)
(i_{n}, i_{n}) − entry of k_{n}^{(D)}(A_{r_{1}}P_{i_{1}}, …, A_{r_{n−1}}P_{i_{n−1}}, A_{r_{n}}).

7.4 Remark. In analogy to Remark 6.4, one can also reformulate the result of Theorem 7.2 by using the ⊙-product. Let us denote by k̃_{n}^{D} the counterpart of k̃_{n} (from Remark 6.4) which is suitable for working with D. That is, k̃_{n}^{D} is the linear application from M_{d}(A ⊗ ⋯ ⊗ A) to D given by the formula:

k̃_{n}^{D}( [x_{i,j}]_{i,j=1}^{d} ) := [δ_{i,j}k_{n}(x_{i,j})]_{i,j=1}^{d}, ∀ [x_{i,j}]_{i,j=1}^{d} ∈ M_{d}(A ⊗ ⋯ ⊗ A).

It is immediate that with these notations, the statement of Theorem 7.2 takes the following form (where we have set the matrices Λ_{1}, …, Λ_{n−1} from Equation (7.4) to be equal to the unit of D):

k_{n}^{(D)}(A_{r_{1}}, …, A_{r_{n}}) = k̃_{n}^{D}(A_{r_{1}} ⊙ ⋯ ⊙ A_{r_{n}}), ∀ 1 ≤ r_{1}, …, r_{n} ≤ s.  (7.10)

It is a natural question if the same kind of formula is true when we consider other algebras of scalar d × d matrices (instead of B and D, as we have in the Equations (6.11) and (7.10)). Let us consider the case of the smallest possible such algebra, namely C (corresponding to scalar multiples of the identity d × d matrix). Again we have a linear application k̃_{n}^{C} : M_{d}(A ⊗ ⋯ ⊗ A) → C, given by the formula:

k̃_{n}^{C}( [x_{i,j}]_{i,j=1}^{d} ) := \frac{1}{d} ∑_{i=1}^{d} k_{n}(x_{i,i}), ∀ [x_{i,j}]_{i,j=1}^{d} ∈ M_{d}(A ⊗ ⋯ ⊗ A).

The question becomes: under what conditions on the matrices A_{1}, …, A_{s} ∈ M_{d}(A) can we infer that:

k_{n}(A_{r_{1}}, …, A_{r_{n}}) = k̃_{n}^{C}(A_{r_{1}} ⊙ ⋯ ⊙ A_{r_{n}}),  (7.11)

for every 1 ≤ r_{1}, …, r_{n} ≤ s? It turns out (see [15]) that (7.11) can be guaranteed if we know that k_{n}^{(D)}(A_{r_{1}}, …, A_{r_{n}}), which is a priori an element in D, is actually an element in C. In the context of R-cyclic matrices, this amounts precisely to the situation discussed in Proposition 3.1; indeed, the “partial summation property” stated in Equation (3.1) asks that k_{n}^{(D)}(A_{r_{1}}, …, A_{r_{n}}) is a scalar multiple of the identity d × d matrix, while on the other hand the conclusion of Proposition 3.1 (as appearing e.g. in Equation (3.3)) is tantamount to (7.11).

8. Characterization of R-cyclicity as freeness with amalgamation

In this section we combine the frameworks used in the Sections 6 and 7. That is, for a fixed integer d ≥ 1 we will consider both the algebra B = M_{d}(C) and its subalgebra D consisting of diagonal matrices. For every 1 ≤ i, j ≤ d we will denote by V_{i,j} ∈ B the matrix which has 1 on the (i, j)-entry and 0 on all the other entries. (Note that the matrices denoted up to now by “P_{i}” have become “V_{i,i}”, for 1 ≤ i ≤ d.)

If (A, φ) is any non-commutative probability space, then M_{d}(A) is at the same time a B-probability space and a D-probability space, where the identifications D ⊂ B ⊂ M_{d}(A) and the expectations E_{B} : M_{d}(A) → B, E_{D} : M_{d}(A) → D are as described in the Sections 6 and 7. Note that the restriction of E_{D} to B is faithful; this implies that the discussion concluding the Section 5 (and in particular the Proposition 5.9) can be applied in this framework.

8.1 Lemma. In the framework considered above, let C_{1}, C_{2}, …, C_{n} ∈ M_{d}(A) form an R-cyclic family, where n ≥ 2. Suppose that:

(i) for m ∈ {1, n} we have that either E_{D}(C_{m}) = 0 or that C_{m} is the unit of M_{d}(A); and

(ii) for m ∈ {2, 3, …, n − 1} we have that E_{D}(C_{m}) = 0. Consider also some indices i_{1}, j_{1}, …, i_{n−1}, j_{n−1} ∈ {1, …, d} such that i_{1} ≠ j_{1}, …, i_{n−1} ≠ j_{n−1}. Then:

E_{D}( C_{1}V_{i_{1},j_{1}} ⋯ C_{n−1}V_{i_{n−1},j_{n−1}}C_{n} ) = 0.  (8.1)

Proof. We will denote by c_{i,j}^{(m)} the (i, j)-entry of C_{m} (1 ≤ i, j ≤ d, 1 ≤ m ≤ n). The hypotheses (i) and (ii) given above show that:

\{ \begin{matrix}φ(c_{i,i}^{(m)}) = 0 or c_{i,i}^{(m)} = I & if & m ∈ \{1, n\}, 1 ≤ i ≤ d \\ φ(c_{i,i}^{(m)}) = 0 & if & m ∈ \{2, 3, …, n − 1\}, 1 ≤ i ≤ d. \end{matrix}  (8.2)

In connection to this, let us also record the fact that:

φ(c_{i,j}^{(m)}) = 0, ∀ 1 ≤ m ≤ n, ∀ 1 ≤ i, j ≤ d such that i ≠ j,  (8.3)

which follows from R-cyclicity (φ(c_{i,j}^{(m)}) = k_{1}(c_{i,j}^{(m)}) = 0 for i ≠ j).

We will present the proof under the assumption that n ≥ 3. The (similar, and simpler) case n = 2 is left as an exercise to the reader.

If we write explicitly the (i, i)-entry of the scalar diagonal matrix on the left-hand side of (8.1), it becomes clear that what we have to do in this proof is to fix an i ∈ {1, …, d}, and show that

φ( c_{i,i_{1}}^{(1)}c_{j_{1},i_{2}}^{(2)} ⋯ c_{j_{n−2},i_{n−1}}^{(n−1)}c_{j_{n−1},i}^{(n)} ) = 0.  (8.4)

By using the relation between moments and non-crossing cumulants, the quantity on the left-hand side of (8.4) can be written as:

∑_{π∈NC(n)} k_{π}( c_{i,i_{1}}^{(1)}, c_{j_{1},i_{2}}^{(2)}, …, c_{j_{n−2},i_{n−1}}^{(n−1)}, c_{j_{n−1},i}^{(n)} ).  (8.5)

We will actually prove that every term of the sum in (8.5) is equal to 0.

So, besides i ∈ {1, …, d}, let us also fix a partition π ∈ NC(n), and let us examine the non-crossing cumulant k_{π}( c_{i,i_{1}}^{(1)}, c_{j_{1},i_{2}}^{(2)}, …, c_{j_{n−2},i_{n−1}}^{(n−1)}, c_{j_{n−1},i}^{(n)} ). Recall from Section 1.3 that this cumulant is defined as a product having as many factors as there are blocks in π. For the sake of brevity, we will denote it in the rest of the proof by just “k_{π}”.

Denoting by B the block of π which contains the number 2, we distinguish four cases:

Case 1. B = {2}. In this case, k_{π} has a factor “k_{1}(c_{j_{1},i_{2}}^{(2)})”, which is equal to 0 by (8.2), (8.3). So k_{π} itself is equal to 0.

Case 2. B = {1, 2}. In this case, k_{π} has a factor “k_{2}(c_{i,i_{1}}^{(1)}, c_{j_{1},i_{2}}^{(2)})”, which is equal to 0 by R-cyclicity and the hypothesis that i_{1} ≠ j_{1}. So again k_{π} = 0.

Case 3. B ∋ 3. In this case, k_{π} has a factor “k_{|B|}(…, c_{j_{1},i_{2}}^{(2)}, c_{j_{2},i_{3}}^{(3)}, …)”, which is equal to 0 by R-cyclicity and the hypothesis that i_{2} ≠ j_{2}. So again k_{π} = 0.

Case 4. B does not fall in any of the Cases 1-3. In this case B intersects {4, …, n}; let us denote m := min(B ∩ {4, …, n}). The set {3, 4, …, m − 1} is a union of blocks of π; because π is non-crossing, there is one of these blocks, B_{1}, which has to be an interval-block – say that B_{1} = [p, q] ∩ Z, with 3 ≤ p ≤ q ≤ m − 1 (≤ n − 1). The cumulant k_{π} has a factor k_{|B_{1}|}(⋯) corresponding to the block B_{1}. If B_{1} has only one element (i.e. p = q), then the factor k_{|B_{1}|}(⋯) is equal to 0 by the same argument as in Case 1; while if |B_{1}| > 1 (i.e. p < q), then the factor k_{|B_{1}|}(⋯) is equal to 0 by the same argument as in Cases 2, 3. Either way, k_{π} is equal to 0. QED

8.2 Theorem. Let A_{1}, …, A_{s} be a family of matrices in M_{d}(A), and let C denote the subalgebra of M_{d}(A) generated by {A_{1}, …, A_{s}} ∪ D. The family A_{1}, …, A_{s} is R-cyclic if and only if C is free from B, with amalgamation over D.

Proof. “⇒”. We will verify that C is free from B, with amalgamation over D, by using the definition of freeness with amalgamation. That is: we consider an alternating sequence X_{1}, X_{2}, …, X_{k} of matrices from B and from C, such that E_{D}(X_{1}) = ⋯ = E_{D}(X_{k}) = 0, and we want to show that E_{D}(X_{1}X_{2} ⋯ X_{k}) = 0.

If the alternating sequence of matrices considered above does not begin with a matrix from C, let us add on the left end of the sequence one more matrix, equal to the identity of M_{d}(A), and viewed as belonging to C. Let us also use this procedure at the right end of the alternating sequence. With these adjustments we can assume that k (the number of matrices in the sequence) is odd, k = 2n − 1, and that X_{1}, X_{2n−1} ∈ C. On the other hand the hypothesis which we have on X_{1} and X_{2n−1} has to be weakened to the fact that they either have zero D-expectation, or they are equal to the identity of M_{d}(A).

We re-denote the matrices X_{1}, X_{3}, …, X_{2n−1} by C_{1}, …, C_{n}(∈ C). The family C_{1}, …, C_{n} is R-cyclic, by Theorem 4.3.

On the other hand, let us look at the matrices X_{2}, X_{4}, …, X_{2n−2}, which belong to B and have D-expectation equal to 0. It is clear that each of these matrices belongs to span{V_{i,j} | 1 ≤ i, j ≤ d, i ≠ j}. An immediate argument with linear combinations allows us to assume without loss of generality that in fact we have X_{2} = V_{i_{1},j_{1}}, …, X_{2n−2} = V_{i_{n−1},j_{n−1}} for some i_{1}, j_{1}, …, i_{n−1}, j_{n−1} ∈ {1, …, d} such that i_{1} ≠ j_{1}, …, i_{n−1} ≠ j_{n−1}.

With the above adjustments, the product X_{1}X_{2} ⋯ X_{k} now reads: C_{1}V_{i_{1},j_{1}} ⋯ C_{n−1} V_{i_{n−1},j_{n−1}}C_{n}. The fact that this product has zero D-expectation is exactly what was proved in Lemma 8.1.

“⇐” In a different non-commutative probability space (N , ψ) we construct a family of elements {x_{i,j}^{(r)} | 1 ≤ i, j ≤ d, 1 ≤ r ≤ s} such that:

k_{n}(x_{i_{1},j_{1}}^{(r_{1})}, …, x_{i_{n},j_{n}}^{(r_{n})}) = 0  (8.6)

for every n ≥ 1 and 1 ≤ r_{1}, …, r_{n} ≤ s, 1 ≤ i_{1}, j_{1}, …, i_{n}, j_{n} ≤ d for which it is not true that j_{1} = i_{2}, …, j_{n−1} = i_{n}, j_{n} = i_{1}; and such that

k_{n}(x_{i_{n},i_{1}}^{(r_{1})}, x_{i_{1},i_{2}}^{(r_{2})}, …, x_{i_{n−1},i_{n}}^{(r_{n})}) =  (8.7)
(i_{n}, i_{n}) − entry of k_{n}^{(D)}(A_{r_{1}}V_{i_{1},i_{1}}, …, A_{r_{n−1}}V_{i_{n−1},i_{n−1}}, A_{r_{n}}),

for every n ≥ 1 and 1 ≤ r_{1}, …, r_{n} ≤ s, 1 ≤ i_{1}, …, i_{n} ≤ d. Such a construction is possible because one can in general construct families of elements with any prescribed family of scalar cumulants, via an abstract free product construction (see e.g. [22], Chapter 1).

For 1 ≤ r ≤ s, we consider the matrix X_{r} = [x_{i,j}^{(r)}]_{i,j=1}^{d} ∈ M_{d}(N). The Equations (8.6) and (8.7) tell us that the family X_{1}, …, X_{s} is R-cyclic.

Observe that for every n ≥ 1 and every 1 ≤ r_{1}, …, r_{n} ≤ s, 1 ≤ i_{1}, …, i_{n} ≤ d we have that:

k_{n}^{(D)}(X_{r_{1}}V_{i_{1},i_{1}}, …, X_{r_{n−1}}V_{i_{n−1},i_{n−1}}, X_{r_{n}}) =  (8.8)
k_{n}^{(D)}(A_{r_{1}}V_{i_{1},i_{1}}, …, A_{r_{n−1}}V_{i_{n−1},i_{n−1}}, A_{r_{n}}).

Indeed, the scalar diagonal matrices on both sides of the Equation (8.8) have their (j, j)- entry equal to the cumulant k_{n}(x_{j,i_{1}}^{(r_{1})}, x_{i_{1},i_{2}}^{(r_{2})}, …, x_{i_{n−1},j}^{(r_{n})}) (for the right-hand side this is just (8.7), while for the left-hand side we invoke Equation (7.9) from Remark 7.3). By taking linear combinations with respect to V_{i_{1},i_{1}}, …, V_{i_{n−1},i_{n−1}} in (8.8) we find that the family X_{1}, …, X_{s} has identical D-cumulants with the family A_{1}, …, A_{s}. Or in other words, the families A_{1}, …, A_{s} and X_{1}, …, X_{s} have identical D-distributions.

Now, our current hypothesis is that the algebra C generated by {A_{1}, …, A_{s}} ∪ D is free from B, with amalgamation over D. On the other hand, the same is true about the algebra C̃ ⊂ N generated by {X_{1}, …, X_{s}}∪D; this follows from the fact that the family X_{1}, …, X_{s} is R-cyclic, and the implication “⇒” (proved above!) of the current theorem. But then we are in the position to apply the Proposition 5.9, which gives us that the families A_{1}, …, A_{s} and X_{1}, …, X_{s} actually have identical B-distributions. The latter fact implies in turn that the families A_{1}, …, A_{s} and X_{1}, …, X_{s} have identical B-cumulants.

Finally, let us fix n ≥ 1, r_{1}, …, r_{n} ∈ {1, …, s} and i_{1}, j_{1}, …, i_{n}, j_{n} ∈ {1, …, d}, and suppose it is not true that j_{1} = i_{2}, …, j_{n−1} = i_{n}, j_{n} = i_{1}. Then:

k_{n}(a_{i_{1},j_{1}}^{(r_{1})}, …, a_{i_{n},j_{n}}^{(r_{n})})
= (i_{1}, j_{n}) − entry of k_{n}^{(B)}(A_{r_{1}}V_{j_{1},i_{2}}, …, A_{r_{n−1}}V_{j_{n−1},i_{n}}, A_{r_{n}}) (by Theorem 6.2)
= (i_{1}, j_{n}) − entry of k_{n}^{(B)}(X_{r_{1}}V_{j_{1},i_{2}}, …, X_{r_{n−1}}V_{j_{n−1},i_{n}}, X_{r_{n}})

(since the families A_{1}, …, A_{s} and X_{1}, …, X_{s} have identical B-cumulants)

= k_{n}(x_{i_{1},j_{1}}^{(r_{1})}, …, x_{i_{n},j_{n}}^{(r_{n})}) (again by Theorem 6.2)

= 0 (by Equation (8.6)). QED

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