A CHARACTERIZATION OF FREENESS BY A FACTORIZATION PROPERTY OF R-TRANSFORM.

ALEXANDRU NICA*, DIMITRI SHLYAKHTENKO†, AND ROLAND SPEICHER

Abstract. Let M be a B-probability space. Assume that B itself is a D-probability space; then M can be viewed as a D-probability space as well. Let X ∈ M. We characterize freeness of X from B with amalgamation over D in terms of a certain factorization condition linking the B-valued and D-valued R-transforms of X. We give an application to random matrices.

1. Operator-valued R-transform.

1.1. Multiplicative functions. Let B be a unital algebra. Recall that a B-probability space (M, E : M → B) (see e.g. [8], [6]) is a pair consisting of an algebra M containing B as a unital subalgebra, and a conditional expectation E : M → B. In other words, E : M → B is unital and B-bilinear:

E(1) = 1, E(bmb′) = bE(m)b′, ∀b, b′ ∈ B, ∀m ∈ M.

Elements of M are called B-valued random variables.

Recall that a multiplicative map ⟨⋯ ⟩ : ⋃ M^{n} → B is a ℂ-multilinear map, (i.e., a sequence of maps M^{n} ∋ m_{1}, …, m_{n} ↦ ⟨m_{1}, …, m_{n}⟩ ∈ B) satisfying the B-linearity conditions

⟨bm_{1}, …, m_{n}⟩ = b⟨m_{1}, …, m_{n}⟩
⟨m_{1}, …, m_{n}b⟩ = ⟨m_{1}, …, m_{n}⟩b
⟨m_{1}, …, m_{k}b, m_{k+1}, …, m_{n}⟩ = ⟨m_{1}, …, m_{k}, bm_{k+1}, …, m_{n}⟩.

(Here m_{i} ∈ M, b ∈ B).

Given a non-crossing partition π ∈ NC(n) and an arbitrary multiplicative map ⟨⋯ ⟩, we can construct a bracketing π⟨⋯ ⟩ : M^{n} → B, defined recursively by

1_{k}⟨m_{1}, …, m_{k}⟩ = ⟨m_{1}, …, m_{k}⟩
(π ⊔ ρ)⟨m_{1}, …, m_{k}⟩ = π⟨m_{1}, …, m_{p}⟩ · ρ⟨m_{p+1}, …, m_{k}⟩
ins(p, ρ → π)(m_{1}, …, m_{k}) = π⟨m_{1}, …, m_{p}ρ⟨m_{p+1}, …, m_{p+q}⟩,
m_{p+q+1}, …, m_{k}⟩.

Here 1_{k} denotes the partition with the sole class {1, …, k}, π ⊔ ρ denotes disjoint union (with the equivalence classes of ρ placed after those of π), and ins(p, ρ → π) denotes the partition obtained from π by inserting the partition ρ after the p-th element of the set on which π determines a partition.

* Research supported by a grant of NSERC, Canada.

†Research supported by an NSF postdoctoral fellowship.

In other words, each partition π is interpreted as a recipe for placing brackets ⟨⋯ ⟩, and π⟨⋯ ⟩ is the value of the resulting expression.

1.2. Moments and R-transform. The B-probability space structure of the algebra M gives rise to one example of such a multiplicative map, namely, the moments map

[⋯]_{E} : ⋃ M^{n} → B,

given by

[m_{1}, …, m_{n}]_{E} = E(m_{1} ⋯ m_{n}).

The reason for the name is that, having fixed B-random variables X_{1}, …, X_{p} ∈ M, the following values of [⋯]_{E},

[b_{0}X_{i_{1}}b_{1}, ⋯, X_{i_{n−1}}b_{n−1}, X_{i_{n}}b_{n}]_{E}

are called B-valued moments of the family X_{1}, …, X_{n}.

In [6] and [7] the notion of B-valued R-transform was introduced (we follow the combinatorial approach of [6], see also [5]). Like the map [⋯], the R-transform map is a multiplicative map

{⋯ }_{E} : ⋃ M^{n} → B.

The following combinatorial formula actually determines {⋯ }_{E} uniquely:

[m_{1}, …, m_{n}]_{E} = ∑ all possible bracketings involving {⋯ }_{E}

= ∑_{π∈NC(n)} π{m_{1}, …, m_{n}}_{E}.

The uniqueness of the definition can be easily seen by observing that the right-hand side of the equation above involves {m_{1}, …, m_{n}}_{E} and that the rest of the terms are products of factors of smaller order (i.e., restrictions of {⋯ }_{E} to M^{k}, k < n).

It is important to note that [⋯] determines {⋯ } and vice-versa. Moreover, the value of [⋯]|_{M^{n}} depends only on {⋯ }|_{M∪···∪M^{n}}, and vice-versa.

1.3. Moment and cumulant series. Given a multiplicative function ⟨⋯ ⟩ : ⋃ M^{k} → B, and X_{1}, …, X_{n} ∈ B, consider the family of multilinear maps

M_{i_{1},...,i_{k}}^{⟨··· ⟩} : B^{k−1} → B, i_{1}, …, i_{k} ∈ {1, …, n}

given by

M_{i_{1},...,i_{k}}^{⟨··· ⟩}(b_{1}, …, b_{k−1}) = ⟨X_{i_{1}}b_{1}, ⋯, b_{k−1}X_{i_{k}}⟩.

In the particular examples above, we get the moment series of X_{1}, …, X_{n},

μ_{i_{1},...,i_{k}}^{X_{1},...,X_{n}} = M_{i_{1},...,i_{k}}^{[···]_{E}}

and the cumulant series,

k_{i_{1},...,i_{k}}^{X_{1},...,X_{n}} = M_{i_{1},...,i_{k}}^{{··· }_{E}}.

We will sometimes write

k_{B;i_{1},...,i_{k}}^{X_{1},...,X_{n}}

to emphasize that the series is valued in B.

1.4. Freeness with amalgamation. Let M_{1}, M_{2} ⊂ M be two subalgebras, each containing B. The importance of R-transform is apparent from the following theorem ([6], see also [1]):

Theorem. Let S_{1}, S_{2} be two subsets of M. Let M_{i} be the algebra generated by S_{i} and B, i = 1, 2. Then M_{1} and M_{2} are free with amalgamation over B iff whenever X_{1}, …, X_{n} ∈ S_{1} ∪ S_{2},

{X_{1}, …, X_{n}}_{E} = 0

unless either all X_{1}, …, X_{n} ∈ S_{1}, or all X_{1}, …, X_{n} ∈ S_{2}.

1.5. Canonical random variables. Let X_{1}, …, X_{n} ∈ M be fixed.

Then by [7] there exists a B-probability space (F, E_{B} : F → B), elements λ_{1}^{∗}, …, λ_{n}^{∗} ∈ F, and elements λ_{p}^{k} satisfying the following properties:

(i) λ_{j_{1}}^{∗}b_{1}λ_{j_{2}}^{∗}b_{2} … λ_{j_{k}}^{∗}b_{k}λ_{j}^{k} = k_{j_{1},...,j_{k},j}^{X_{1},...,X_{n}}(b_{1}, …, b_{k}), b_{1}, …, b_{k} ∈ B; (ii) Let w = b_{0}a_{1}b_{1}a_{2}b_{2} … a_{n}b_{n}, where b_{i} ∈ B, a_{i} = λ_{p}^{k} or a_{i} = λ_{p}^{∗}. Then E_{B}(w) = 0

unless w can be reduced to an element of B using relation 1.5(i). (By convention, we take λ_{p}^{0} = E_{B}(X_{p}) ∈ B). We should clarify that for each k, λ_{p}^{k} is just a formal variable, and we do not assume any relations between {λ_{k}^{p}}_{k,p}: for example, λ_{p}^{k} is not the k-th power of λ_{p}^{1}.

It is not hard to show that the properties listed above determine the restriction of E_{B} to the algebra generated by λ^{∗} and {λ_{p}^{k}}_{p,k}.

Let

Y_{j} = λ_{j}^{∗} + ∑_{k≥0} λ_{j}^{k}.

(This series is formal; however, Y_{1}, …, Y_{n} have moments, since each such moment involves only a finite number of terms from the series defining Y_{j}).

It turns out that

k_{i_{1},...,i_{k}}^{X_{1},...,X_{n}} = k_{i_{1},...,i_{k}}^{Y_{1},...,Y_{n}}, and μ_{i_{1},...,i_{k}}^{X_{1},...,X_{n}} = μ_{i_{1},...,i_{k}}^{Y_{1},...,Y_{n}}.

In other words, given a cumulant series, Y_{1}, …, Y_{n} is an explicit family of B-valued random variables, whose cumulant series is equal to the one given.

2. Freeness from a subalgebra.

2.1. D-cumulants vs. B-cumulants. Let now D ⊂ B be a unital subalgebra, and let F : B → D be a conditional expectation. If (M, E : M → B) is a B-probability space, then (M, F ∘ E : M → D) is a D-probability space.

Theorem 2.1. Let X_{1}, …, X_{n} ∈ M. Assume that the B-valued cumulants of X_{1}, …, X_{n} satisfy

k_{B;i_{1},...,i_{k}}^{X_{1},...,X_{n}}(d_{1}, …, d_{k−1}) ∈ D, ∀d_{1}, …, d_{k−1} ∈ D.

Then the D-valued cumulants of X_{1}, …, X_{n} are given by the restrictions of the B-valued cumulants:

k_{D;i_{1},...,i_{k}}^{X_{1},...,X_{n}}(d_{1}, …, d_{k−1}) = k_{B;i_{1},...,i_{k}}^{X_{1},...,X_{n}}(d_{1}, …, d_{k−1}), ∀d_{1}, …, d_{k−1} ∈ D.

Proof. Let N be the algebra generated by D and X_{1}, …, X_{n}. The condition on cumulants implies that

{⋯ }_{E}|_{⋃ N^{p}}

is valued in D. It follows from the moment-cumulant formula that [⋯]_{E}|_{⋃ N^{p}} is valued in D, and hence that

[⋯]_{F ∘E}|_{⋃ N^{p}} = [⋯]_{E}|_{⋃ N^{p}} = ∑_{π∈NC(p)} π{⋯ }_{E}|_{⋃ N^{p}}.

Since the moment-cumulant formula determines {⋯ }_{F ∘E}|_{⋃ N^{p}}, it follows that

{⋯ }_{F ∘E}|_{⋃ N^{p}} = {⋯ }_{E}|_{⋃ N^{p}}.

We record an equivalent formulation of the theorem above (which was implicit in the proof):

Theorem 2.2. Let N ⊂ M be a subalgebra, containing D. Assume that {⋯ }_{E}|_{⋃ N^{p}} is valued in D. Then

{⋯ }_{F ∘E}|_{⋃ N^{p}} = {⋯ }_{E}|_{⋃ N^{p}}.

In general, in the absence of the condition that k_{B}^{X_{1},...,X_{n}} restricted to ∪D^{p} is valued in D, the expression of D-valued cumulants of X_{1}, …, X_{n} in terms of the B-valued cumulants is quite complicated. Note, for example, that if X is a B-valued random variable, and b ∈ B, then the B-valued cumulant series of bX are very easy to describe. On the other hand, the D-valued cumulant series of bX can have a very complicated expression in terms of the D-valued cumulant series of X and b.

The sufficient condition in the theorem above is actually quite close to being necessary in the case that the conditional expectations are positive maps of ∗-algebras. As an illustration, consider the case that D ⊂ B consists of scalar multiples of 1, and F : B → D is such that τ = F ∘ E is a trace on M, satisfying τ(xy) = τ(yx) for all x, y ∈ M.

Recall that X is called a B-semicircular variable if its cumulant series is given by

k\underbrace{1, … 1}_{p}^{X}(b_{1}, …, b_{p−1}) = δ_{p,2}η(b_{1})

for some map η : B → B. It is easily seen that if X is B-semicircular, then η(b) = E(XbX).

Theorem 2.3. Let (M, E : M → B) be a B-probability space, such that M and B are C^{∗}-algebras. Let F : B → ℂ = D ⊂ B be a faithful state. Assume that τ = F ∘ E is a faithful trace on M. Let X be a B-semicircular variable in M. Then the distribution of X with respect to τ is the semicircle law iff E(X^{2}) ∈ ℂ.

Proof. If E(X^{2}) = k_{11}^{X}(1) ∈ ℂ, it follows that the B-valued cumulants of X, restricted to D = ℂ are valued in D. Hence by Theorem 2.2, the D-valued cumulant series of X are the same as the restriction of the B-valued cumulant series; hence the only scalar-valued cumulant of X which is nonzero is the second cumulant k_{11}^{X}, so that the distribution of X is the semicircle law.

Conversely, assume that the distribution of X is the semicircle law. Let η(b) = k_{11}^{X}(b), b ∈ B. Then we have

2τ(η(1))^{2} = 2τ(X^{2})^{2}
= τ(X^{4}) = F ∘ E(X^{4})
= F (E(η(η(1)))) + F (E(η(1)η(1)))
= τ(Xη(1)X) + τ(η(1)^{2})
= τ(η(1)XX) + τ(η(1)^{2})
= 2τ(η(1)^{2}),

so that

τ(η(1)) = τ(η(1)^{2})^{1/2}.

By the Cauchy-Schwartz inequality, we have that if η(1) ∉ ℂ,

τ(η(1)) = ⟨η(1), 1⟩ < ∥η(1)∥_{2} · ∥1∥_{2} = τ(η(1)^{2})^{1/2},

which is a contradiction. Hence η(1) ∈ ℂ.

We mention a corollary, which is of interest to random matrix theory. Let σ(x, y) = σ(y, x) be a non-negative function on [0, 1]^{2}, having at most a finite number of discontinuities in each vertical line. Let G(n) be an n×n random matrix with entries g_{ij}, so that {g_{ij} : i ≤ j} are independent complex Gaussian random variables, g_{ij} = \overline{g_{ji}}, the expectation E(g_{ij}) = 0 and the variance E(|g_{ij}|^{2}) = \frac{1}{n}σ(\frac{i}{n}, \frac{j}{n}). The matrices G(n) are called Gaussian Random Band Matrices. Let μ_{n} be the expected eigenvalue distribution of G(n), i.e.,

μ_{n}([a, b]) = \frac{1}{n} × expected number of eigenvalues of G(n) in [a, b].

Corollary 2.4. The eigenvalue distribution measures μ_{n} of the Gaussian Random Band Matrices G(n) converge weakly to the semicircle law iff ∫_{0}^{1} σ(x, y)dy is a.e. a constant, independent of x.

The proof of this relies on a result from [2], showing that G(n) has limit eigenvalue distribution μ, given as follows. Let X be the L^{∞}[0, 1]-semicircular variable in an L^{∞}[0, 1]- probability space (M, E : M → L^{∞}[0, 1]), so that E(XfX)(x) = ∫_{0}^{1} f(y)σ(x, y)dy. Let F : L^{∞}[0, 1] → ℂ denote the linear functional F (f) = ∫_{0}^{1} f(x)dx, and denote by τ the trace F ∘E on W^{∗}(X, L^{∞}[0, 1]). Then μ is the scalar-valued distribution of X with respect to τ, i.e.,

∫ t^{k}dμ(t) = τ(X^{k}).

It remains to apply Theorem 2.3, to conclude that μ is a semicircle law iff E(X^{2}) ∈ ℂ, i.e., ∫_{0}^{1} σ(x, y)dy is a constant function of x.

2.2. A characterization of freeness. We are now ready to state the main theorem of this note. The following theorem was earlier proved for B-valued semicircular variables in [3], and found many uses in operator algebra theory.

Theorem 2.5. Let X_{1}, …, X_{n} ∈ M. Assume that F : B → D satisfies the faithfullness condition that if b_{1} ∈ B and if F (b_{1}b_{2}) = 0 for all b_{2} ∈ B, then b_{1} = 0. Then X_{1}, …, X_{n} are free from B with amalgamation over D iff their B-valued cumulant series satisfies

k_{B;i_{1},...,i_{k}}^{X_{1},...,X_{n}}(b_{1}, …, b_{k−1}) = F (k_{B;i_{1},...,i_{k}}^{X_{1},...,X_{n}}(F (b_{1}), …, F (b_{k−1})). (2.1)

for all b_{1}, …, b_{k−1} ∈ B. In short, k = F ∘ k ∘ F. (Here the cumulant series are computed in the B-probability space (M, E : M → B)). Equivalently,

k_{B;i_{1},...,i_{k}}^{X_{1},...,X_{n}}(b_{1}, …, b_{k−1}) = k_{D;i_{1},...,i_{k}}^{X_{1},...,X_{n}}(F (b_{1}), …, F (b_{k−1})).

We note that in the case that M is a C^{∗}-probability space, the faithfulness assumption above is exactly the condition that the GNS representation of B with respect to the conditional expectation F is faithful.

Since X_{1}, …, X_{n} are free from B with amalgamation over D iff the algebra N generated by X_{1}, …, X_{n} and D is free from B over D, the theorem above can be equivalently stated as

Theorem 2.6. Let N ⊂ M be a subalgebra of M, containing D. Assume that F : B → D satisfies the faithfullness condition that if b_{1} ∈ B and if F (b_{1}b_{2}) = 0 for all b_{2} ∈ B, then b_{1} = 0. Then N is free from B over D iff

{n_{1}b_{1}, n_{2}b_{2}, …, n_{k}}_{E} = F {n_{1}F (b_{1}), n_{2}F (b_{2}), …, n_{k}}_{E}

for all k and n_{1}, …, n_{k} ∈ N, b_{1}, …, b_{k−1} ∈ B.

Proof. We prove the theorem in the first formulation.

Assume that the condition (2.1) is satisfied by the cumulant series of X_{1}, …, X_{n}. Let Y_{1}, …, Y_{n} be as in Section 1.5. Since the freeness of X_{1}, …, X_{n} from B with amalgamation over D is a condition on the B-moment series of X_{1}, …, X_{n}, and Y_{1}, …, Y_{n} have the same B-moment series as X_{1}, …, X_{n}, it is sufficient to prove that Y_{1}, …, Y_{n} ∈ F are free with amalgamation over D from B.

Since (2.1) is satisfied, λ_{j}^{0} = E_{B}(Y_{j}) ∈ D and hence Y_{1}, …, Y_{n} belong to the algebra L generated in F by λ_{j}^{∗} and λ_{p}^{q}, 1 ≤ j, p ≤ n, q ≥ 1 and D. Therefore, it is sufficient to prove that L is free from B with amalgamation over D.

Let w_{1}, …, w_{s} ∈ L, so that F ∘ E_{B}(w_{j}) = 0, and let b_{0}, …, b_{s} ∈ B, so that F (b_{j}) = 0 (allowing also b_{0} and/or b_{s} to be equal to 1). We must prove that

F ∘ E_{B}(b_{0}w_{1}b_{1} ⋯ w_{s}b_{s}) = 0.

Note that the factorization condition (2.1) as well as the definition of the generators of L (see 1.5(i) and 1.5(ii)) imply that E_{B}|_{L} has values in D. It follows that we may assume that E_{B}(w_{j}) = 0 (since F ∘ E_{B}(w_{j}) = E_{B}(w_{j}) ∈ D). By the definition of E_{B}, its kernel is spanned by irreducible non-trivial words in the generators λ_{j}^{∗} and λ_{p}^{q}. Then

W = b_{0}w_{1} ⋯ w_{s}b_{s}

is again a linear combination of words in the generators λ_{j}^{∗} and λ_{p}^{q}. By linearity, we may reduce to the case that W is a single word. If W is irreducible, it must be non-trivial (since each w_{i} is non-trivial), hence E_{B}(W) = 0, so that F ∘ E_{B}(W) = 0. So assume that W is not irreducible. Since each w_{i} is irreducible, this means that W contains a sub-word of the form

W = W_{1} · d_{0}λ_{i_{1}}^{∗}d_{1}λ_{i_{2}}^{∗}d_{2} ⋯
d_{s}b_{1}d_{s+1}λ_{i_{s+2}}^{∗} ⋯ b_{j} ⋯ λ_{i_{k}}^{∗}d_{k}b_{r}d_{k+1}λ_{i_{k+1}}^{∗}
⋯ d_{p}λ_{i_{p}}^{∗}d_{p+1}λ_{j}^{p} · W_{2}.

Using the relation (1.5(i)) and the factorization condition (2.1), we get that

W = W_{1}d_{0}k_{i_{1},...,i_{p},j}(d_{1}, …, F (d_{k}b_{r}d_{k+1}), …, d_{p+1})W_{2} = 0,

since F (d_{k}b_{r}d_{k+1}) = d_{k}F (b_{r})d_{k+1} = 0. Thus in any case, F ∘ E_{B}(W) = 0.

We have therefore seen that the factorization condition implies freeness with amalgamation.

To prove the other implication, assume that X_{1}, …, X_{n} are free with amalgamation over D from B. Let Z_{1}, …, Z_{n} be B-valued random variables, so that

k_{B;i_{1},...,i_{k}}^{Z_{1},...,Z_{n}}(b_{1}, …, b_{k−1}) = F (k_{D,i_{1},...,i_{k}}^{X_{1},...,X_{n}}(F (b_{1}), …, F (b_{k−1}))).

where k_{D} denote D-valued cumulants. (Note that the first occurrence of F is actually redundant, as k_{D,i_{1},...,i_{k}}^{X_{1},...,X_{n}}(F (b_{1}), …, F (b_{k−1}) ∈ D). Then by the first part of the proof, Z_{1}, …, Z_{n} are free from B with amalgamation over D. Moreover, by Theorem 2.2, the D-valued distributions of Z_{1}, …, Z_{n} and X_{1}, …, X_{n} are the same. By assumption, X_{1}, …, X_{n} are free with amalgamation over D from B. This freeness, together with the D-valued distribution of X_{1}, …, X_{n}, determines their B-valued distribution. Indeed, the freeness assumptions determine

F ∘ E(b′b_{0}X_{i_{1}}b_{1}X_{i_{2}} ⋯ b_{n−1}X_{i_{n}}b_{n}), b′, b_{i} ∈ B

which in view of the assumptions on F determines

E(b_{0}X_{i_{1}}b_{1}X_{i_{2}} ⋯ b_{n−1}X_{i_{n}}b_{n}), b_{i} ∈ B.

It follows that the B-valued distributions of X_{1}, …, X_{n} and Z_{1}, …, Z_{n} coincide. Hence the B-valued cumulants of X_{1}, …, X_{n} satisfy (2.1).

As an application, we have the following proposition (see e.g. [4, Lemma 2.7]):

Proposition 2.7. Let N ⊂ M be a subalgebra. Let B ⊂ C ⊂ D ⊂ M be subalgebras, and E_{C} : M → C, E_{B} : M → B, E_{D} : M → D be conditional expectations, so that E_{B} = E_{B} ∘ E_{C}, E_{C} = E_{C} ∘ E_{D}. Assume that E_{C} and E_{B} and E_{D} satisfy the faithfullness assumptions of Theorem 2.5. Assume that N is free from C with amalgamation over B, and also free from D with amalgamation over C. Then N is free from D with amalgamation over B.

Proof. Since N is free from D with amalgamation over C, we have that for all n_{j} ∈ N and d_{j} ∈ D,

{n_{1}d_{1}, n_{2}d_{2}, …, n_{k}}_{E_{D}} = E_{C}{n_{1}E_{C}(d_{1}), …, n_{k}}_{E_{C}}.

Since N is free from C with amalgamation over D, we get similarly that for all c_{j} ∈ C,

{n_{1}c_{1}, n_{2}c_{2}, …, n_{k}}_{E_{C}} = E_{B}{n_{1}E_{B}(c_{1}), …, n_{k}}_{E_{B}}.

Applying this with c_{j} = E_{C}(d_{j}) and combining with the previous equation gives

{n_{1}d_{1}, n_{2}d_{2}, …, n_{k}}_{E_{D}} = E_{B}{n_{1}E_{B}(d_{1}), …, n_{k}}_{E_{B}},

since E_{B} = E_{B} ∘ E_{C}. Hence N is free from D with amalgamation over B.

We mention as a corollary the following identity. Let B ⊂ C ⊂ D be C^{∗}-algebras, E_{B}^{N} : N → B, E_{B}^{C} : C → B and E_{C}^{D} : D → C be conditional expectations, having faithful GNS representations. Consider the reduced free product

(((N, E_{B}^{N}) ∗_{B} (C, E_{B}^{C})), E_{B}^{N} ∗ id) ∗_{C} (D, E_{C}^{D}),

where E_{B}^{N} ∗ id denotes the canonical conditional expectation from the free product (N, E_{B}^{N}) ∗_{B} (C, E_{B}^{C}) onto C. Then

(((N, E_{B}^{N}) ∗_{B} (C, E_{B}^{C})), E_{B}^{N} ∗ id) ∗_{C} (D, E_{C}^{D})
≅ (N, E_{B}^{N}) ∗_{B} (D, E_{B}^{C} ∘ E_{C}^{D}).

To see this, it is sufficient to prove that N ⊂ (((N, E_{B}^{N}) ∗_{B} (C, E_{B}^{C})), E_{B}^{N} ∗ id) ∗_{C} (D, E_{C}^{D}) is free from D with amalgamation over B, since both (((N, E_{B}^{N}) ∗_{B} (C, E_{B}^{C})), E_{B}^{N} ∗ id) ∗_{C} (D, E_{C}^{D}) and (N, E_{B}^{N}) ∗_{B} (D, E_{B}^{C} ∘ E_{C}^{D}) are generated by N and D as C^{∗}-algebras. But N is free from D over C, and from C over D, by construction. Hence by the proposition above, N is free from D over B.

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Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada

E-mail address: anica@math.uwaterloo.ca

Department of Mathematics, UCLA, Los Angeles, CA 90095, USA

E-mail address: shlyakht@math.ucla.edu

Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario K7L 3N6, Canada

E-mail address: speicher@mast.queensu.ca