The Microstates Free Entropy Dimension

of any DT–operator is 2

Ken Dykema^{1}, Kenley Jung^{2}, and Dimitri Shlyakhtenko^{3}

Received: January 5, 2005

Communicated by Joachim Cuntz

Abstract. Suppose that μ is an arbitrary Borel measure on ℂ with compact support and c > 0. If Z is a DT(μ, c)–operator as defined by Dykema and Haagerup in [6], then the microstates free entropy dimension of Z is 2.

2000 Mathematics Subject Classification: 46L54 (28A78)

1 Introduction.

DT–operators were introduced by Dykema and Haagerup in their work on invariant subspaces of certain operators in a II_{1} factor [5, 6]. A DT–operator Z is specified by two parameters, μ and c, where c > 0 and μ is a Borel probability measure on ℂ with compact support. Roughly, the operator Z is determined by stating that its ∗–distribution is the same as the limit ∗–distribution as N → ∞ of random matrices

Z_{N} = D_{N} + cT_{N},

where D_{N} are diagonal N × N matrices whose spectral measures converge to μ in distribution, while T_{N} is a strictly upper triangular random N × N matrix with i.i.d. Gaussian entries. Equivalently, (see [15], [12], [6] and the appendix of [7]), Z can be viewed as a sum Z = d + cT, where d is a normal operator with spectral measure μ contained in a diffuse von Neumann algebra A, and T is an A-valued circular operator with a certain covariance. Finally, a result of Śniady [14] shows that a DT(μ, c)– operator is one whose free entropy is maximized among all those operators having Brown measure equal to μ and with a fixed off–diagonality. If we write Z = d+cT as above, it is clear that W^{∗}(Z) ⊂ W^{∗}(d, T) ⊆ W^{∗}(A∪{T }), while a simple computation shows W^{∗}(A ∪ {T }) = L(𝔽_{2}). By Lemma 6.2 of [6], for any μ we may choose d having trace of spectral measure equal to μ and so that d, T ∈ W^{∗}(Z); by [7], A ⊆ W^{∗}(T), so we always have W^{∗}(Z) ≅ L(𝔽_{2}). Thus Z can be viewed as an interesting generator for this free group factor. In order to test the hypothesis that Voiculescu’s free entropy dimension δ_{0} [16, 17, 20] is the same for any sets of generators of a von Neumann algebra, it is important to decide whether the free entropy dimension of Z is 2 (L(𝔽_{2}) clearly has another set of generators of free entropy dimension 2). For another version of free entropy dimension, also defined by Voiculescu, called the non-microstates free entropy dimension [18], L. Aagaard has recently shown [1] that the dimension of Z is indeed 2. It is known by [4] that the non-microstates free entropy dimension dominates δ_{0} but at present it is open whether the reverse inequality holds. Thus, Aagaard’s result does not solve the question for the original microstates definition. In this paper, we show that, indeed, δ_{0}(Z) = 2. Our proof uses an equivalent packing number formulation of the microstates free entropy dimension, due to Jung [8]. In this approach, to get the nontrivial lower bound on δ_{0}(Z), one must have lower bounds on the ϵ–packing numbers of spaces of matricial microstates for Z, which are in turn obtained by lower bounds on the volume of ϵ–neighborhoods of these microstate spaces. The kth microstate space is the set Γ(Z; m, k, γ), for m, k ∈ ℕ and γ > 0, of all k × k complex matrices whose ∗–moments up to order m are γ–close to the values of the corresponding ∗–moments of Z, and the volumes are for Lebesgue measure λ_{k} on M_{k}(ℂ) viewed as a Euclidean space of real dimension 2k^{2} with coordinates corresponding to the real and imaginary parts of the entries of a matrix. In order to outline how we get these lower bounds on volumes, let us for convenience take Z equal to the DT(δ_{0}, 1)–operator T. A key result that we use is a recent one of Aagaard and Haagerup [2], showing that a certain ϵ–perturbation of T has Brown measure uniformly distributed on the disk of radius r_{ϵ} := 1/\sqrt{log(1 + ϵ^{−2})} centered at the origin; note how slowly this disk shrinks as ϵ approaches zero. Applying a result of Śniady [13] to this situation, we find matrices A_{k} ∈ M_{k}(ℂ) that lie in ϵ– neighborhoods of microstate spaces for T, whose eigenvalues are close to uniformly distributed (as k gets large) in the disk of radius r_{ϵ}. Thus, in order to get a lower bound on the volume of a 2ϵ–neighborhood of a microstate space for T, it will suffice to get a lower bound on the volume of a unitary orbit of an ϵ–neighborhood of A_{k}. Every element of M_{k}(ℂ) has an upper triangular matrix in its unitary orbit. Thus, letting T_{k}(ℂ) denote the set of upper triangular matrices in M_{k}(ℂ), there is a measure ν_{k} on T_{k}(ℂ) such that λ_{k}(O) = ν_{k}(O ∩ T_{k}) for every O ⊆ M_{k}(ℂ) invariant under unitary conjugation. Freeman Dyson identified such a measure ν_{k} (see Appendix 35 of [11]), and showed that if we view T_{k}(ℂ) as a Euclidean space of real dimension k(k − 1) with coordinates corresponding to the real and imaginary parts of the matrix entries lying on and above the diagonal, then ν_{k} is absolutely continuous with respect to Lebesgue measure on T_{k}(ℂ) and has density given at B = (b_{ij})_{1≤i,j≤k} ∈ T_{k}(ℂ) by

1Research supported by the Alexander von Humboldt Foundation and NSF grant DMS–0300336.

2Research supported by an NSF Postdoctoral Fellowship.

3Research supported by a Sloan Foundation fellowship and NSF grant DMS-0355226.

Documenta Mathematica 10 (2005) 247–261

C_{k} ∏_{1≤p<q≤k} |b_{pp} − b_{qq}|^{2},  (1)

where the constant is

C_{k} = \frac{π^{k(k−1)/2}}{∏_{j=1}^{k} j!} .  (2)

We will use this measure of Dyson to find lower bound on the volume of unitary orbits of an ϵ–neighborhood of A_{k}, and we may take A_{k} to be upper triangular. However, so far we only have information about the eigenvalues of A_{k}, namely the diagonal part of it. Loosely speaking, in order to get a handle on the part strictly above the diagonal, we use a result of Dykema and Haagerup [6] to realize T as an upper triangular matrix

T = \frac{1}{\sqrt{N}} [\begin{matrix}T_{11} & T_{12} & ⋯ & T_{1N} \\ 0 & T_{22} & ... & ⋮ \\ ⋮ & ... & ... & T_{N−1,N} \\ 0 & ⋯ & 0 & T_{NN} \end{matrix} ]

of operators where each T_{ii} is a copy of T, each T_{ij} for i < j is circular and the family (T_{ij})_{1≤i≤j≤N} is ∗–free. Thus, A_{k} can be taken to be of the form

[\begin{matrix}B_{11} & B_{12} & ⋯ & B_{1N} \\ 0 & B_{22} & ... & ⋮ \\ ⋮ & ... & ... & B_{N−1,N} \\ 0 & ⋯ & 0 & B_{NN} \end{matrix} ]

where each B_{ii} is upper triangular, where we have good knowledge of the eigenvalue distributions of each B_{ii} and where the B_{ij} for i < j approximate ∗–free circular elements. Using the strengthened asymptotic freeness results of Voiculescu [19], we find enough approximants for these B_{ij}. Although we still have no real knowledge about the entries of the B_{ii} lying above the diagonal, these parts are of negligibly small dimension as N gets large, and we are able to get good enough lower bounds. The techniques we use for estimating integrals of the quantity (1) over certain regions are taken from [9].

2 Microstates for Z with well–spaced spectral densities

The following lemma is an application of the result of Aagaard and Haagerup [2] mentioned in the introduction in order to make perturbations of general DT–operators having Brown measure that is relatively well spread out. For an element a of a noncommutative probability space (M, τ), we write ∥a∥_{2} for τ(a^{∗}a)^{1/2}.

Lemma 2.1. Let μ be a compactly supported Borel probability measure on ℂ and let c > 0. Let Z be a DT(μ, c)–operator in a W^{∗}–noncommutative probability space (M, τ). Let us write

μ = ν + ∑_{i=1}^{s} a_{i}δ_{z_{i}}

for some s ∈ {0} ∪ ℕ ∪ {∞}, z_{i} ∈ ℂ and a_{i} > 0, where ν is a diffuse measure and where z_{i} ≠ z_{j} if i ≠ j. Consider the W^{∗}–noncommutative probability space

(M̃, τ̃) = (M, τ) ∗ (L(𝔽_{2}), τ_{𝔽_{2}}).

Then for every ϵ > 0, there is Z̃_{ϵ} ∈ M̃ such that ∥Z̃_{ϵ} − Z∥_{2} ≤ ϵc and where the Brown measure of Z̃_{ϵ} is equal to

σ_{ϵ} := ν + ∑_{i=1}^{s} a_{i}ρ_{i,ϵ},

where ρ_{i,ϵ} is the probability measure that is uniform distribution on the disk centered at z_{i} and having radius

r_{i} := c\sqrt{\frac{a_{i}}{log(1 + a_{i}ϵ^{−2})}}.

Finally, if δ > 0 and if

X_{δ} = {(w_{1}, w_{2}) ∈ ℂ^{2} | |w_{1} − w_{2}| < δ},

then

(σ_{ϵ} × σ_{ϵ})(X_{δ}) ≤ (ν × ν)(X_{δ}) + 2 ∑_{i=1}^{s} min(a_{i}, δ^{2}c^{−2} log (1 + a_{i}ϵ^{−2})). (3)

Proof. By results from [6], taking projections onto local spectral subspaces of Z, we find projections p_{j} ∈ M (for 0 ≤ j < s + 1) such that

• ∑_{j=0}^{s} p_{j} = 1,

• p_{0} + p_{1} + ⋯ + p_{k} is Z–invariant for all integers k such that 0 ≤ k < s + 1,

• τ(p_{k}) = \begin{cases}|ν| & if k = 0 \\ a_{k} & if 1 ≤ k < s + 1, \end{cases}

• In (p_{k}Mp_{k}, τ(p_{k})^{−1}τ↾_{p_{k}Mp_{k}}), p_{k}Zp_{k} is DT(|ν|^{−1}ν, c\sqrt{|ν|}) if k = 0 and is DT(δ_{z_{k}}, c\sqrt{a_{k}}) if 1 ≤ k < s + 1.

Let Y ∈ M̃ be centered circular such that Y and Z are ∗–free and τ̃(Y^{∗}Y) = 1. Let

Z̃_{ϵ} = Z + ϵ ∑_{i=1}^{s} a_{i}^{−1/2}cp_{i}Y p_{i}.  (4)

Then ∥Z̃_{ϵ} − Z∥_{2}^{2} = ϵ^{2}c^{2} ∑_{i=1}^{s} a_{i} ≤ ϵ^{2}c^{2}. On the other hand, Z̃_{ϵ} is upper triangular with respect to the projections p_{0}, p_{1}, …; the Brown measure of Z̃_{ϵ} is, therefore, equal to the Brown measure of its diagonal part

p_{0}Zp_{0} + ∑_{i=1}^{s} (p_{i}Zp_{i} + ϵ a_{i}^{−1/2}cp_{i}Y p_{i}).  (5)

But in (p_{i}M̃p_{i}, a_{i}^{−1}τ̃↾_{p_{i}M̃p_{i}}), the operator ϵ a_{i}^{−1/2}cp_{i}Y p_{i} is a centered circular operator of second moment ϵ^{2}c^{2} that is ∗–free from the DT(δ_{z_{i}}, c\sqrt{a_{i}}) operator p_{i}Zp_{i}. Therefore, the random variable

p_{i}Zp_{i} + ϵ a_{i}^{−1/2}cp_{i}Y p_{i}  (6)

has the same ∗–distribution as z_{i}I + c\sqrt{a_{i}}(T + ϵ a_{i}^{−1/2}Y), where T is a DT(δ_{0}, 1)– operator that is ∗–free from Y. By [2], the Brown measure of the random variable (6) is equal to ρ_{i,ϵ}. This yields σ_{ϵ} for the Brown measure of the operator (5), hence of Z̃_{ϵ} itself. Finally, we have

(σ_{ϵ} × σ_{ϵ})(X_{δ}) ≤ (ν × ν)(X_{δ}) + 2 ∑_{i=1}^{s} a_{i}(σ_{ϵ} × ρ_{i,ϵ})(X_{δ})  (7)

and

(σ_{ϵ} × ρ_{i,ϵ})(X_{δ}) = ∫_{ℂ} ρ_{i,ϵ}(w + δ 𝔻)dσ_{ϵ}(w) ≤ min(1, δ^{2}r_{i}^{−2}),  (8)

where 𝔻 is the unit disk in ℂ. Taken together, (7) and (8) yield the inequality (3).

The next lemma uses a result of Śniady [13] to find matrix approximants of the operators appearing in Lemma 2.1. In the following lemma and throughout this paper, for a matrix A ∈ M_{k}(ℂ) we let |A|_{2} = tr_{k}(A^{∗}A)^{1/2}, where tr_{k} is the normalized trace on M_{k}(ℂ). Moreover, by the eigenvalue distribution of A ∈ M_{k}(ℂ) we mean its Brown measure, which is just the probability measure that is uniformly distributed on its list of eigenvalues λ_{1}, …, λ_{k}, where these are listed according to (general) multiplicity, i.e. a value z is listed dim ⋃_{n=1}^{∞} ker((A − zI)^{n}) times.

Lemma 2.2. Let μ be a compactly supported Borel probability measure on ℂ and let c > 0. Then there exists a sequence ⟨y_{k}⟩_{k=1}^{∞} such that for any ϵ > 0, there exists a sequence ⟨z_{k,ϵ}⟩_{k=1}^{∞} such that

• y_{k}, z_{k,ϵ} ∈ M_{k}(ℂ),

• ∥y_{k}∥ and ∥z_{k,ϵ}∥ remain bounded as k → ∞,

• lim sup_{k→∞} |y_{k} − z_{k,ϵ}|_{2} ≤ ϵc,

• y_{k} converges in ∗–moments as k → ∞ to a DT(μ, c)–operator,

• the eigenvalue distribution of z_{k,ϵ} converges weakly as k → ∞ to the measure σ_{ϵ} described in Lemma 2.1.

Proof. Let Z be a DT(μ, c)–operator, let Ỹ be the operator ∑_{i=1}^{s} a_{i}^{−1/2}cp_{i}Y p_{i} appearing in (4) in the proof of the preceding lemma, so that Z̃_{ϵ} = Z + ϵỸ. Since Z can be constructed in L(𝔽_{2}) and since free group factors can be embedded in the ultrapower R^{ω} of the hyperfinite II_{1} factor, there are bounded sequences ⟨y_{k}⟩_{k=1}^{∞} and ⟨d_{k}⟩_{k=1}^{∞} such that y_{k}, d_{k} ∈ M_{k}(ℂ) and such that the pair y_{k}, d_{k} converges in ∗– moments to the pair Z, Ỹ. Letting z̃_{k} = y_{k} + ϵd_{k}, we have that z̃_{k} converges in ∗–moments to Z̃_{ϵ} as k → ∞. By Theorem 7 of [13], there is a sequence ⟨z_{k,ϵ}⟩_{k=1}^{∞} with z_{k,ϵ} ∈ M_{k}(ℂ) such that ∥z_{k,ϵ}−z̃_{k,ϵ}∥ tends to zero and the eigenvalue distribution of z_{k,ϵ} converges weakly as k → ∞ to the Brown measure of Z̃_{ϵ}, namely, to σ_{ϵ}.

Suppose that λ = ⟨λ_{j}⟩_{j=1}^{k} is a finite sequence of complex numbers. For each j, write λ_{j} = a_{j} + ib_{j}, a_{j}, b_{j} ∈ ℝ. Define Q_{ϵ} = ∏_{j=1}^{k}[a_{j} − ϵ, a_{j} + ϵ] and R_{ϵ} = ∏_{j=1}^{k}[b_{j} − ϵ, b_{j} + ϵ]. Set

E_{ϵ}(λ) = ∫_{R_{ϵ}} ( ∫_{Q_{ϵ}} ∏_{\begin{matrix}1≤i,j≤k \\ i≠j \end{matrix}} (|s_{i} − s_{j}| + |t_{i} − t_{j}|^{2})^{1/2} ds)dt,

where ds = ds_{1} ⋯ ds_{k} and dt = dt_{1} ⋯ dt_{k}. The following lemma proves lower bounds for certain asymptotics of the quantities E_{ϵ}(λ). We will apply this lemma to the case when λ is the eigenvalue sequence of matrices like the z_{k,ϵ} found in Lemma 2.2.

Lemma 2.3. Let μ and c be as in Lemma 2.1. For each ϵ > 0 and k ∈ ℕ, let λ^{(k,ϵ)} = ⟨λ_{1}^{(k,ϵ)}, …, λ_{n(k)}^{(k,ϵ)}⟩ be a finite sequence of complex numbers and assume that for every ϵ > 0,

sup_{k∈ℕ, 1≤j≤n(k)} |λ_{j}^{(k,ϵ)}| < ∞

and the probability measures

\frac{1}{n(k)} ∑_{j=1}^{n(k)} δ_{λ_{j}^{(k,ϵ)}}  (9)

converge weakly to the measure σ_{ϵ} of Lemma 2.1 as k → ∞. Let

f(ϵ) = lim inf_{k→∞} n(k)^{−2} log(E_{ϵ}(λ^{(k,ϵ)})).

Then

lim inf_{ϵ→0} (\frac{f(ϵ)}{| log ϵ|}) ≥ 0.  (10)

Proof. Note that we must have n(k) → ∞ as k → ∞. Given ϵ > 0 small, take 1 ≥ δ > 3ϵ. Define

W_{k,ϵ} = {(i, j) ∈ {1, …, n(k)}^{2} | i ≠ j, |λ_{i}^{(k,ϵ)} − λ_{j}^{(k,ϵ)}| < δ}.

Writing for each 1 ≤ j ≤ k, λ_{j}^{(k,ϵ)} = a_{j} + ib_{j} where a_{j}, b_{j} ∈ ℝ define Q_{ϵ,k} = ∏_{j=1}^{n(k)}[a_{j} − ϵ, a_{j} + ϵ], R_{ϵ,k} = ∏_{j=1}^{n(k)}[b_{j} − ϵ, b_{j} + ϵ], and K_{ϵ,k} = Q_{ϵ,k} × R_{ϵ,k}. Now

E_{ϵ}(λ^{(k,ϵ)}) = ∫_{K_{ϵ,k}} ∏_{i≠j}(|s_{i} − s_{j}|^{2} + |t_{i} − t_{j}|^{2})^{1/2}dsdt

≥ (δ − \sqrt{8}ϵ)^{n(k)^{2}−#W_{k,ϵ}} ∫_{K_{ϵ,k}} ∏_{(i,j)∈W_{k,ϵ}}(|s_{i} − s_{j}|^{2} + |t_{i} − t_{j}|^{2})^{1/2}dsdt

≥ (δ − 3ϵ)^{n(k)^{2}−#W_{k,ϵ}} \begin{matrix}( ∫_{Q_{ϵ,k}} ∏_{(i,j)∈W_{k,ϵ}} |s_{i} − s_{j}|ds)· \\ ( ∫_{R_{ϵ,k}} ∏_{(i,j)∈W_{k,ϵ}} |t_{i} − t_{j}|dt), \end{matrix}

where ds = ds_{1} ⋯ ds_{n(k)} and dt = dt_{1} ⋯ dt_{n(k)}. We now wish to find a lower bounds for the two integrals in the above expression. By Fubini’s Theorem we can assume a_{1} ≤ a_{2} ≤ ⋯ ≤ a_{n(k)}. Let

[−ϵ, ϵ]_{<}^{n(k)} = {(x_{1}, …, x_{n(k)}) ∈ [−ϵ, ϵ]^{n(k)} | x_{1} < x_{2} < ⋯ < x_{n(k)}}.

Then by the change of variables [−ϵ, ϵ]_{<}^{n(k)} ∋ (x_{1}, …, x_{n(k)}) ↦ (a_{1} + x_{1}, …, a_{n(k)} + x_{n(k)}) ∈ Q_{ϵ,k} and Selberg’s Integral Formula it follows that

∫Q_{ϵ,k} ∏_{(i,j)∈W_{k,ϵ}} |s_{i} − s_{j}|ds ≥ ∫_{[−ϵ,ϵ]_{<}^{n(k)}} ∏_{(i,j)∈W_{k,ϵ}} |x_{i} − x_{j}|dx_{1} ⋯ dx_{n(k)}

≥ (2ϵ)^{−(n(k)^{2}−n(k)−#W_{k,ϵ})} · ∫_{[−ϵ,ϵ]_{<}^{n(k)}} ∏_{i≠j} |x_{i} − x_{j}|dx_{1} ⋯ dx_{n(k)}

= \frac{(2ϵ)^{−(n(k)^{2}−n(k)−#W_{k,ϵ})}}{n(k)!} · ∫_{[−ϵ,ϵ]^{n(k)}} ∏_{i≠j} |x_{i} − x_{j}|dx_{1} ⋯ dx_{n(k)}

= \frac{(2ϵ)^{n(k)+#W_{k,ϵ}}}{n(k)!} · ∏_{j=0}^{n(k)−1} \frac{Γ(j + 2)Γ(j + 1)^{2}}{Γ(n(k) + j + 1)} ,

The same lower bound applies to ∫_{R_{ϵ,k}} ∏_{(i,j)∈W_{k,ϵ}} |t_{i} − t_{j}|dt so that combining these two we get

E_{ϵ}(λ^{(k,ϵ)}) ≥ (δ − 3ϵ)^{n(k)^{2}−#W_{k,ϵ}}(\frac{(2ϵ)^{n(k)+#W_{k,ϵ}}}{n(k)!} · ∏_{j=0}^{n(k)−1} \frac{Γ(j + 2)Γ(j + 1)^{2}}{Γ(n(k) + j + 1)} )^{2}

≥ (δ − 3ϵ)^{n(k)^{2}}(\frac{(2ϵ)^{n(k)+#W_{k,ϵ}}}{n(k)!} · ∏_{j=0}^{n(k)−1} \frac{Γ(j + 2)Γ(j + 1)^{2}}{Γ(n(k) + j + 1)} )^{2}.

Documenta Mathematica 10 (2005) 247–261

Using

lim_{k→∞} n(k)^{−2} log( ∏_{j=0}^{n(k)−1} \frac{Γ(j + 2)Γ(j + 1)^{2}}{Γ(n(k) + j + 1)} ) = −2 log 2,

we find

f(ϵ) ≥ log(δ − 3ϵ) + 2 log(2ϵ) lim sup_{k→∞} \frac{#W_{k,ϵ}}{n(k)^{2}} − 4 log 2.

Since the measures (9) converge weakly to σ_{ϵ}, by standard approximation techniques one sees

lim_{k→∞} \frac{#W_{k,ϵ}}{n(k)^{2}} = (σ_{ϵ} × σ_{ϵ})(X_{δ}),

where X_{δ} is as in Lemma 2.1. As ϵ → 0 choose δ = \frac{1}{| log ϵ|}, so that δ^{2} log(1+aϵ^{−2}) → 0 for all a > 0 and \frac{ϵ}{δ} → 0 and \frac{log δ}{log ϵ} → 0. Using the upper bound (3) and the fact that ν is diffuse, we get

lim_{ϵ→0}(σ_{ϵ} × σ_{ϵ})(X_{δ}) = 0.

Now one easily verifies that (10) holds.

3 The Main Result

Before beginning the main result first a few comments on a packing formulation for microstates free entropy dimension are in order. If X = {x_{1}, …, x_{n}} is an n-tuple of selfadjoint elements in a tracial von Neumann algebra, then the free entropy dimension (as defined by Voiculescu [17]) is given by the formula

δ_{0}(X) = n + lim sup_{ϵ→0} \frac{χ(x_{1} + ϵs_{1}, …, x_{n} + ϵs_{n} : s_{1}, …, s_{n})}{| log ϵ|}

where {s_{1}, …, s_{n}} is a semicircular family free from X. The packing formulation found in [8] and modified slightly in [10] (to remove the norm restriction on microstates), is

δ_{0}(X) = lim sup_{ϵ→0} \frac{ℙ_{ϵ}(X)}{| log ϵ|},

where

ℙ_{ϵ}(X) = inf_{m∈ℕ, γ>0} lim sup_{k→∞} k^{−2} log P_{ϵ}(Γ(X; m, k, γ)).  (11)

Here, Γ(X; m, k, γ) ⊆ (M_{k}(ℂ)_{s.a.})^{n} is the microstate space of Voiculescu [16], but taken without norm restriction, as considered in [3], and P_{ϵ} is the packing number with respect to the metric arising from the normalized trace. Let Y = {y_{1}, …, y_{n}} be an arbitrary n-tuple of (possibly nonselfadjoint) elements in a tracial von Neumann algebra. Now the definition of ℙ_{ϵ} makes perfect sense for the set Y if we replace the microstate space in (11) with the non-selfadjoint ∗-microstate space Γ(Y; m, k, γ) ⊆ (M_{k}(ℂ))^{n}, which is the set of all n–tuples of k × k matrices whose ∗–moments up to order m approximate those of Y within tolerance of γ. Let us (temporarily) denote the quantity so obtained by \overline{ℙ_{ϵ}}(Y) and define

\overline{δ_{0}}(Y) = lim sup_{ϵ→0} \frac{\overline{ℙ_{ϵ}}(Y)}{| log ϵ|}.  (12)

It is easy to see that if X is a set of selfadjoints, then \overline{ℙ_{ϵ}}(X) ≥ ℙ_{ϵ}(X) ≥ \overline{ℙ_{2ϵ}}(X) and that in the nonselfadjoint setting the quantity (12) is a ∗-algebraic invariant, so that

δ_{0}(Re(y_{1}),Im(y_{1}), …, Re(y_{n}), Im(y_{n})) =

= lim sup_{ϵ→0} \frac{ℙ_{ϵ}(Re(y_{1}), Im(y_{1}), …, Re(y_{n}), Im(y_{n}))}{| log ϵ|}

= lim sup_{ϵ→0} \frac{\overline{ℙ_{ϵ}}(Re(y_{1}), Im(y_{1}), …, Re(y_{n}), Im(y_{n}))}{| log ϵ|}

= lim sup_{ϵ→0} \frac{\overline{ℙ_{ϵ}}(Y)}{| log ϵ|} = \overline{δ_{0}}(Y),

where Re(y_{i}) and Im(y_{i}) are the real and imaginary parts of y_{i}. Moreover, if X is set of selfadjoints, then

δ_{0}(X) = lim sup_{ϵ→0} \frac{ℙ_{ϵ}(X)}{| log ϵ|} = lim sup_{ϵ→0} \frac{\overline{ℙ_{ϵ}}(X)}{| log ϵ|} = \overline{δ_{0}}(X).

The following notational conventions, which will be used in the remainder of this paper, are, therefore, justified: for any finite set of operators Y (selfadjoint or otherwise) in a tracial von Neumann algebra we will write ℙ_{ϵ}(Y) for the packing quantity derived from the nonselfadjoint microstates (that was denoted \overline{ℙ_{ϵ}}(Y) above) and we will write δ_{0}(Y) for the free entropy dimension of Y that was denoted \overline{δ_{0}}(Y) above. In the proof of the main result, we will use E_{ϵ}(A) for A ∈ M_{k}(ℂ) to mean E_{ϵ}(λ), where λ = ⟨λ_{j}⟩_{j=1}^{k} are the eigenvalues of A listed according to general multiplicity (see the description immediately before Lemma 2.2). Notice that this is independent of the choice of λ since E_{ϵ}(λ ∘ σ) = E_{ϵ}(λ) for any permutation σ of {1, …, k}.

Theorem 3.1. Let Z be a DT(μ, c)–operator, for any compactly supported Borel probability measure μ on the complex plane and any c > 0. Then δ_{0}(Z) = 2.

Proof. Obviously δ_{0}(Z) ≤ 2 so it suffices to show the reverse inequality. We may without loss of generality assume c = 1 (see Proposition 2.12 of [6]). Fix N ∈ ℕ with N ≥ 2. By Theorem 4.12 of [6],

[\begin{matrix}B_{11} & B_{12} & ⋯ & B_{1N} \\ 0 & B_{22} & ... & ⋮ \\ ⋮ & ... & ... & B_{N−1,N} \\ 0 & ⋯ & 0 & B_{NN} \end{matrix} ] ∈ M ⊗ M_{N}(ℂ)  (13)

is a DT(μ, 1)–operator where {B_{11}, …, B_{NN}} ∪ ⟨B_{ij}⟩_{1≤i<j≤N} is a ∗-free family in M, the B_{ii} are DT(μ, \frac{1}{\sqrt{N}})–operators, and each B_{ij} is circular with φ(|B_{ij}^{2}|) = \frac{1}{N} . From this we see that finding microstates for Z is equivalent to finding microstates for the operator (13) in M ⊗ M_{N}(ℂ). Consider the sequence ⟨y_{k}⟩_{k=1}^{∞} constructed in Lemma 3.2 and for each ϵ > 0 small enough, the corresponding sequence ⟨z_{k,ϵ}⟩_{k=1}^{∞}. Let R > 1, m ∈ ℕ, γ > 0 and take γ′ = γ/16^{m}(R+1)^{m} > 0. By Corollary 2.11 of [19] there exist k×k complex unitary matrices u_{1k}, u_{2k}, …, u_{kk} such that {u_{1k}y_{k}u_{1k}^{∗}, …, u_{Nk}y_{k}u_{Nk}^{∗}} is an (m, γ′)–∗– free family in M_{k}(ℂ). Also,by an application of Corollary 2.14 of [19], there exists a set Ω_{k} ⊂ Γ_{R}(⟨B_{ij}⟩_{1≤i<j≤N}; m, k, γ′) such that for any ⟨η_{ij}⟩_{1≤i<j≤N} ∈ Ω_{k},

{u_{1k}y_{k}u_{1k}^{∗}, …, u_{Nk}y_{k}u_{Nk}^{∗}} ∪ ⟨η_{ij}⟩_{1≤i<j≤N}

is an (m, γ′)-∗ free family and such that

lim inf_{k→∞} (k^{−2} · log(vol(Ω_{k})) + \frac{N(N − 1)}{2} · log k) ≥

≥ χ(⟨ReB_{ij}⟩_{1≤i<j≤N}, ⟨ImB_{ij}⟩_{1≤i<j≤N}) > −∞,

where the volume is computed with respect to the product of the Euclidean norm k^{1/2}| · |_{2}. Since the operator (13) is a copy of Z, for any ⟨η_{ij}⟩_{1≤i<j≤N} ∈ Ω_{k} we have

[\begin{matrix}u_{1k}y_{k}u_{1k}^{∗} & η_{12} & ⋯ & η_{1N} \\ 0 & u_{2k}y_{2}u_{2k}^{∗} & ... & ⋮ \\ ⋮ & ... & ... & η_{N−1,N} \\ 0 & ⋯ & 0 & u_{Nk}y_{k}u_{Nk}^{∗} \end{matrix}] ∈ Γ(Z; m, Nk, γ).

Because every complex matrix can be put into an upper-triangular form with respect to an orthonormal basis, we can find for each 1 ≤ j ≤ N, a k×k unitary matrix v_{jk} such that v_{jk}u_{jk}z_{k,ϵ}u_{jk}^{∗}v_{jk}^{∗} is upper triangular. Observe now that for any ⟨η_{ij}⟩_{1≤i<j≤n} ∈ Ω_{k}, the product of matrices

[\begin{matrix}v_{1k} & 0 & ⋯ & 0 \\ 0 & v_{2k} & ... & ⋮ \\ ⋮ & ... & ... & 0 \\ 0 & ⋯ & 0 & v_{Nk} \end{matrix}] [\begin{matrix}u_{1k}y_{k}u_{1k}^{∗} & η_{12} & ⋯ & η_{1N} \\ 0 & u_{2k}y_{k}u_{2k}^{∗} & ... & ⋮ \\ ⋮ & ... & ... & η_{N−1,N} \\ 0 & ⋯ & 0 & u_{Nk}y_{k}u_{Nk}^{∗} \end{matrix}] ·

· [\begin{matrix}v_{1k}^{∗} & 0 & ⋯ & 0 \\ 0 & v_{2k}^{∗} & ... & ⋮ \\ ⋮ & ... & ... & 0 \\ 0 & ⋯ & 0 & v_{Nk}^{∗}. \end{matrix}]

is also an element of Γ(Z; m, Nk, γ) and is equal to

[\begin{matrix}v_{1k}u_{1k}y_{k}u_{1k}^{∗}v_{1k}^{∗} & v_{1k}η_{12}v_{2k}^{∗} & ⋯ & v_{1j}η_{1N}v_{2k}^{∗} \\ 0 & v_{2j}u_{2k}y_{k}u_{2k}^{∗}v_{2k}^{∗} & ... & ⋮ \\ ⋮ & ... & ... & v_{(N−1),k}η_{N−1,N}v_{Nk}^{∗} \\ 0 & ⋯ & 0 & v_{Nk}u_{Nk}y_{k}u_{Nk}^{∗}v_{Nk}^{∗} \end{matrix}] .

Moreover,

|v_{jk}u_{jk}z_{k,ϵ}u_{jk}^{∗}v_{jk}^{∗} − v_{jk}u_{jk}y_{k}u_{jk}^{∗}v_{jk}^{∗}|_{2} = |z_{k,ϵ} − y_{k}|_{2}

and lim sup_{k→∞} |z_{k,ϵ} − y_{k}|_{2} ≤ ϵ/\sqrt{N}. Therefore, for k sufficiently large and for each 1 ≤ j ≤ N we have |v_{jk}u_{jk}z_{k,ϵ}u_{jk}^{∗}v_{jk}^{∗} − v_{jk}u_{jk}y_{k}u_{jk}^{∗}v_{jk}^{∗}|_{2} ≤ ϵ. Set d_{jk} = v_{jk}u_{jk}z_{k,ϵ}u_{jk}^{∗}v_{jk}^{∗}, and denote by G_{k} the set of all Nk × Nk matrices of the form

[\begin{matrix}d_{1k} & v_{1k}η_{12}v_{2k}^{∗} & ⋯ & v_{1j}η_{1N}v_{Nk}^{∗} \\ 0 & d_{2k} & ... & ⋮ \\ ⋮ & ... & ... & v_{(N−1),k}η_{N−1,N}v_{Nk}^{∗} \\ 0 & ⋯ & 0 & d_{Nk} \end{matrix}]

where ⟨η_{ij}⟩_{1≤i<j≤N} ∈ Ω_{k}. Notice that each d_{jk} is upper triangular and its eigenvalue distribution is exactly the same as that of z_{k,ϵ}. For k sufficiently large, the set G_{k} lies in the ϵ-neighborhood of Γ(Z; m, Nk, γ). Let θ(G_{k}) denote the unitary orbit of G_{k} in M_{Nk}(ℂ). We will now find lower bounds for the ϵ-packing numbers of θ(G_{k}) and thus, ones for Γ(Z; m, Nk, γ). Denote by H_{k} ⊂ M_{Nk}(ℂ) all matrices of the form

[\begin{matrix}0 & v_{1k}η_{12}v_{2k}^{∗} & ⋯ & v_{1j}η_{1N}v_{Nk}^{∗} \\ 0 & 0 & ... & ⋮ \\ ⋮ & ... & ... & v_{(N−1),k}η_{N−1,N}v_{Nk}^{∗} \\ 0 & ⋯ & ⋯ & 0 \end{matrix}]

where ⟨η_{ij}⟩_{1≤i<j≤N} ∈ Ω_{k}. Notice that H_{k} is isometric to the space of all matrices of the form

[\begin{matrix}0 & η_{12} & ⋯ & η_{1N} \\ 0 & 0 & ... & ⋮ \\ ⋮ & ... & ... & η_{N−1,N} \\ 0 & · & ⋯ & 0 \end{matrix}]

where ⟨η_{ij}⟩_{1≤i<j≤N} ∈ Ω_{k}. It follows that H_{k} must also have the same volume as the above subspace, computed in the obvious ambient Hilbert space of block upper triangular matrices obeying the above decomposition. Recall that for n ∈ ℕ, T_{n}(ℂ) denotes the set of uppertriangular matrices in M_{n}(ℂ); let T_{n,<}(ℂ) denote the matrices in T_{n}(ℂ) that have zero diagonal, i.e. the strictly upper triangular matrices in M_{n}(ℂ). Denote by W_{k} the subset of T_{Nk,<}(ℂ) consisting of all matrices x such that |x|_{2} < ϵ and x_{ij} = 0 whenever 1 ≤ p < q ≤ N and (p−1)k < i ≤ pk and (q−1)k < j ≤ qk. Thus, W_{k} consists of N × N diagonal matrices whose diagonal entries are strictly upper triangular k × k matrices. Denote by D_{k} the subset of diagonal matrices x of M_{Nk}(ℂ) such that |x|_{2} < ϵ\sqrt{2}. It follows that if f_{k} is the matrix

[\begin{matrix}d_{1k} & 0 & ⋯ & 0 \\ 0 & d_{2k} & ... & ⋮ \\ ⋮ & ... & ... & 0 \\ 0 & ⋯ & ⋯ & d_{Nk} \end{matrix}]

then f_{k} + D_{k} + W_{k} + H_{k} ⊂ N_{3ϵ}(G_{k}), where the 3ϵ neighborhood is taken in the ambient space T_{Nk}(ℂ) with respect to the metric induced by | · |_{2}. Now observe that the space of diagonal Nk × Nk matrices and T_{Nk,<}(ℂ) are orthogonal subspaces of T_{Nk}(ℂ). Let θ_{3ϵ}(G_{k}) denote the 3ϵ–neighborhood of the unitary orbit θ(G_{k}) of G_{k}. Thus, denoting by dX Lebesgue measure on T_{Nk}(ℂ) where X = ⟨x_{ij}⟩_{1≤i≤j≤k}, using Dyson’s formula we have

vol(θ_{3ϵ}(G_{k})) ≥ C_{Nk} · ∫_{f_{k}+D_{k}+W_{k}+H_{k}} ∏_{1≤i<j≤Nk} |x_{ii} − x_{jj}|^{2}dX

= C_{Nk} · vol(W_{k} + H_{k}) · ∫_{f_{k}+D_{k}} ∏_{1≤i<j≤Nk} |x_{ii} − x_{jj}|^{2}dx_{11} ⋯ dx_{(Nk)(Nk)}

≥ C_{Nk} · vol(W_{k} + H_{k}) · E_{ϵ}(z_{k,ϵ} ⊗ I_{N}),  (14)

where the constant C_{Nk} is as in 2 and where vol(θ_{3ϵ}(G_{k})) is computed in M_{Nk}(ℂ) and vol(W_{k} + H_{k}) is computed in T_{Nk,<}(ℂ), both being Euclidean volumes corresponding to the norms (Nk)^{1/2}|·|_{2}. Clearly θ_{3ϵ}(G_{k}) ⊂ N_{4ϵ}(Γ(Z; m, Nk, γ)), so (14) gives a lower bound on vol(N_{4ϵ}(Γ(Z; m, Nk, γ)) as well. Using (14) and the standard volume comparison test, we have

P_{ϵ}(Γ(Z;m, Nk, γ)) ≥ \frac{vol(N_{4ϵ}(Γ(Z; m, Nk, γ)))}{vol(B_{6ϵ})}

≥ C_{Nk} · E_{ϵ}(z_{k,ϵ} ⊗ I_{N}) · vol(W_{k} + H_{k}) · \frac{Γ((Nk)^{2} + 1)}{π^{(Nk)^{2}}(6(Nk)^{1/2}ϵ)^{2(Nk)^{2}}} ,

where B_{6ϵ} is a ball in M_{Nk}(ℂ) of radius 6ϵ with respect to |·|_{2}, and we are computing volumes corresponding to the Euclidean norm (Nk)^{1/2}| · |_{2}. Since W_{k} and H_{k} are orthogonal, we have vol(W_{k} + H_{k}) = vol(W_{k})vol(H_{k}), where each volume is taken in the subspace of appropriate dimension. But W_{k} is a ball of radius (Nk)^{1/2}ϵ in a space of real dimension Nk(k − 1), so

vol(W_{k} + H_{k}) = \frac{π^{\frac{Nk(k−1)}{2}} ((Nk)^{1/2}ϵ)^{Nk(k−1)}}{Γ(\frac{Nk(k−1)}{2} + 1)} · (N^{1/2})^{k^{2}N(N−1)}vol(Ω_{k}).

Documenta Mathematica 10 (2005) 247–261

Applying Stirling’s formula, we find

ℙ_{ϵ}(Z; m, γ) ≥ lim inf_{k→∞}(Nk)^{−2} log P_{ϵ}(Γ(Z; m, Nk, γ))
≥ lim inf_{k→∞}(Nk)^{−2} log(E_{ϵ}(z_{k,ϵ} ⊗ I_{N}))

+ lim inf_{k→∞} ( \begin{matrix}(Nk)^{−2} log(C_{Nk}) + \frac{1}{2N} log k + \frac{1}{N} log ϵ \\ − \frac{1}{2N} log(\frac{Nk(k − 1)}{2}) + log((Nk)^{2}) − log k \\ − 2 log ϵ + (Nk)^{−2} log(vol(Ω_{k}))) + K_{1} \end{matrix}

= lim inf_{k→∞}(Nk)^{−2} log(E_{ϵ}(z_{k,ϵ} ⊗ I_{N}))

+ lim inf_{k→∞} ((Nk)^{−2} log(C_{Nk}) + \frac{1}{2} log k)

+ lim inf_{k→∞} ((Nk)^{−2} log(vol(Ω_{k})) + (\frac{1}{2} − \frac{1}{2N}) log k)

+(2 − N^{−1})| log ϵ| + K_{2}

= lim inf_{k→∞}(Nk)^{−2} log(E_{ϵ}(z_{k,ϵ} ⊗ I_{N}))

+N^{−2}χ(⟨ReB_{ij}⟩_{1≤i<j≤N}, ⟨ImB_{ij}⟩_{1≤i<j≤N})

+ (2 − N^{−1})| log ϵ| + K_{3},

where K_{1}, K_{2} and K_{3} are constants independent of ϵ, m and γ. Taking m → ∞ and γ → 0, we get

ℙ_{ϵ}(Z) ≥ lim inf_{k→∞}(Nk)^{−2} log(E_{ϵ}(z_{k,ϵ} ⊗ I_{N}))
+N^{−2}χ(⟨ReB_{ij}⟩_{1≤i<j≤N}, ⟨ImB_{ij}⟩_{1≤i<j≤N})
+ (2 − N^{−1})| log ϵ| + K_{3}.

Since the eigenvalue distribution of z_{k,ϵ} ⊗ I_{N} converges as k → ∞ to the measure σ_{ϵ} of Lemma 2.1, dividing by | log ϵ| and applying Lemma 2.3 now yields

δ_{0}(Z) = lim sup_{ϵ→0} \frac{ℙ_{ϵ}(Z)}{| log ϵ|} ≥ lim inf_{ϵ→0} \frac{f(ϵ)}{| log ϵ|} + 2 − N^{−1} ≥ 2 − N^{−1}.

Since N was arbitrary, it follows that δ_{0}(Z) ≥ 2, thereby completing the proof.

Acknowledgement. A significant part of this research was conducted during the 2004 Free Probabilty Workshop at the Banff International Research Station, and the authors would like to thank the organizers and sponsors for providing them with that opportunity to work together. K.D. would like to thank Joachim Cuntz, Siegfried Echterhoff and the Mathematics Institute at the Westfälische Wilhelms–Universität Münster for their generous hospitality during his year–long visit, which included much of the time he was working on this project.

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K. Dykema Mathematisches Institut Westfälische Wilhelms–Universität Münster Einsteinstr. 62, 48149 Münster, Germany dykema@math.uni-muenster.de permanent address: Department of Mathematics Texas A&M University College Station, TX 77843-3368, USA kdykema@math.tamu.edu K. Jung and D. Shlyakhtenko Department of Mathematics University of California Los Angeles, CA 90095-3840, USA kjung@math.ucla.edu shlyakht@math.ucla.edu 262

Documenta Mathematica 10 (2005)