Notes on Microstate Free Entropy of Projections

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44(2008), 49–89

Notes on Microstate Free Entropy of Projections

By

FumioHiai^1,2,∗and YoshimichiUeda^1,3,∗∗

Abstract

We study the microstate free entropyχproj(p1, . . . , pn) of projections, and estab- lish its basic properties similar to the self-adjoint variable case. Our main contribution is to characterize the pair-block freeness of projections by the additivity ofχproj(The- orem 4.1), in the proof of which a transportation cost inequality plays an important role. We also briefly discuss the free pressure in relation toχproj.

Introduction

The theory of free entropy, initiated and mostly developed by D. Voiculescu in his series of papers [20]–[25], has become one of the most essential disciplines of free probability theory. The microstate free entropy χ(X₁, . . . , X_n) introduced in [21] for self-adjoint non-commutative random variablesX₁, . . . , X_n is deﬁned as a certain asymptotic growth rate (as the matrix sizeN goes to∞) of the Euclidean volume of the set ofN×Nself-adjoint matrices (A₁, . . . , A_n) ap- proximating (X₁, . . . , X_n) in moments. It is this microstate theory that settled some long-standing open questions in von Neumann algebras (see the survey [26]). On the other hand, the microstate-free free entropyχ^∗(X₁, . . . , X_n) was also introduced in [23] based on the non-commutative Hilbert transform and

Communicated by Y. Takahashi. Received June 16, 2006. Revised July 4, 2007.

2000 Mathematics Subject Classiﬁcation(s): Primary 46L54; Secondary 15A52, 60F10, 94A17.

1Supported in part by Japan Society for the Promotion of Science, Japan-Hungary Joint Project.

2Supported in part by Grant-in-Aid for Scientiﬁc Research (B)17340043.

3Supported in part by Grant-in-Aid for Young Scientists (B)17740096.

∗Graduate School of Information Sciences, Tohoku University, Aoba-ku, Sendai 980-8579, Japan.

∗∗Graduate School of Mathematics, Kyushu University, Fukuoka 810-8560, Japan.

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Fumio Hiai and Yoshimichi Ueda

the notion of conjugate variables, avoiding use of microstates or so-called matrix integrals which are rather hard to handle. Although it is believed that both approaches should be uniﬁed and give the same quantity, only the in- equalityχ ≤χ^∗ is known to hold true due to Biane, Capitaine and Guionnet [3] based on an idea of large deviation principle for several random matrices.

In his work [25] Voiculescu developed another kind of microstate-free approach to free entropy, the so-called free liberation theory, and introduced the mutual free informationi^∗ for subalgebras rather than random variables. He suggested there the need to apply the microstate approach to projection random variables because the usual microstate free entropy χ becomes always −∞ for projections whilei^∗does not in general. Following the suggestion, we here study the microstate free entropyχ_proj(p₁, . . . , p_n) of projections p₁, . . . , p_n in the same lines as in [21] and [22] to provide the basis for future research.

The large deviation principle for random matrices as mentioned above started with the paper of Ben Arous and Guionnet [2] and has been almost completed in the single random matrix case (corresponding to the study of χ(X) for single random variableX), see the survey [8]. We note that such large deviation principle played quite an important role not only for the foundation of free entropy theory but also for getting free analogs of several probability theoretic inequalities (see [14] and the references therein). Recently, one more large deviation was shown in [12] for an independent pair of random projection matrices, including the explicit formula of the free entropyχ_proj(p, q) of a projection pair (p, q). This is one of a few large deviation results (indeed the ﬁrst full large deviation result) in the setting of several random matrices, though the method of the proof is based on the single variable case. Moreover, in [15]

we applied it to get a kind of logarithmic Sobolev inequality between the free entropy χ_proj(p, q) and the mutual free Fisher informationϕ^∗(W^∗(p) :W^∗(q)) (see [25]) for a projection pair. The large deviation result in [12] also plays a crucial role in our study ofχ_projhere.

The paper is organized as follows. After giving the deﬁnition and basic properties ofχ_proj(p₁, . . . , p_n) in§1, we recall in§2 the formula in the case of two variables. In§3 we introduce a certain functional calculus for a projection pair (p, q) and provide a technical tool of separate change of variable formula.

This tool is essential in §4 to prove the additivity theorem characterizing the pair-block freeness of projections by the additivity of their free entropy. §5 treats a free analog of transportation cost inequalities for tracial distributions of projections. Such a free analog is of interest by itself while its simplest case is needed in the proof of the above-mentioned additivity theorem. Finally,

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along the same lines as in [10], we introduce in§6 the notion of free pressure and compare its Legendre transform withχ_proj(p₁, . . . , p_n), thus giving a variational expression of free entropy for projections.

§1. Definition

Let U(N) be the unitary group of orderN. LetG(N, k) denote the set of allN×N orthogonal projection matrices of rankk, that is,G(N, k) is identiﬁed with the Grassmannian manifold consisting ofk-dimensional subspaces inC^N. With the diagonal matrix P_N(k) of the ﬁrst k diagonals 1 and the others 0, eachP ∈G(N, k) is diagonalized as

(1.1) P =U P_N(k)U^∗,

where U ∈ U(N) is determined up to the right multiplication of elements in U(k)⊕U(N −k). Hence G(N, k) is identiﬁed with the homogeneous space U(N)/(U(k)⊕U(N−k)), and we have a unique probability measureγ_G(N,k) onG(N, k) invariant under the unitary conjugationP →U P U^∗forU ∈U(N).

In the homogeneous space description, this is the unique probability measure on U(N)/(U(k)⊕U(N−k)) invariant under the left multiplication of elements in U(N) or in other words the induced measure from the Haar probability measure γ_U(N₎on U(N). Letξ_N,k: U(N)→G(N, k) be the (surjective continuous) map deﬁned by (1.1), i.e.,ξ_N,k(U) :=U P_N(k)U^∗. Then the measureγ_G(N,k)is more explicitly written as

(1.2) γ_G(N,k)=γ_U(N₎◦ξ⁻¹_N,k.

Throughout the paper (M, τ) is a tracial W^∗-probability space. Let (p₁, . . . , p_n) be an n-tuple of projections in (M, τ). Following Voiculescu’s proposal in [25, 14.2] we deﬁne thefree entropyχ_proj(p₁, . . . , p_n) of (p₁, . . . , p_n) as follows. Choosek_i(N)∈ {0,1, . . . , N}for eachN ∈Nand 1≤i≤nin such a way that k_i(N)/N →τ(p_i) asN → ∞ for 1≤i≤n. For each m∈Nand ε >0 we set

Γ_proj(p₁, . . . , p_n;k₁(N), . . . , k_n(N);N, m, ε) (1.3)

:=

(P₁, . . . , P_n)∈ⁿ

i=1

G(N, k_i(N)) : 1

NTr_N(P_i₁· · ·P_i_r)−τ(p_i₁· · ·p_i_r) < ε for all 1≤i₁, . . . , i_r≤n, 1≤r≤m

,

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where Tr_N stands for the usual (non-normalized) trace on theN×N matrices.

We then deﬁne

χ_proj(p₁, . . . , p_n) := lim

m→∞ε0

lim sup

N→∞

(1.4)

1 N²log

_n

i=1

γ_G(N,k_i_(N₎₎

Γ_proj(p₁, . . . , p_n;k₁(N), . . . , k_n(N);N, m, ε) . To justify the deﬁnition ofχ_proj, here arises a natural question whether or not the quantityχ_proj(p₁, . . . , p_n) depends on the particular choice ofk_i(N). The answer is the following:

Proposition 1.1. The above deﬁnition of χ_proj(p₁, . . . , p_n)is indepen- dent of the choices ofk_i(N)withk_i(N)/N →α_i for1≤i≤n.

Proof. For 1 ≤ i≤ n letl_i(N), N ∈ N, be another sequence such that l_i(N)/N→α_i asN → ∞. In what follows we will denote, for brevity, the microstate set in (1.3) by Γ(k(N), m, ε) withk(N) := (k₁(N), . . . , k_n(N)). More- over, let ξ_k(N)(U) :=

ξ_N,k₁_(N₎(U₁), . . . , ξ_N,k_n_(N)(U_n) forU = (U₁, . . . , U_n)∈ U(N)ⁿ, and consider the subset Γ(l(N), m, ε) := ξ_l(N₎◦ξ⁻¹

k(N)

Γ(k(N), m, ε) of_n

i=1G(N, l_i(N)). Since

ξ_N,l_i_(N)(U)−ξ_N,k_i_(N₎(U) =U

P_N(k_i(N))−P_N(l_i(N)) U^∗, we get

ξ_N,l

i(N)(U)−ξ_N,k_i_(N)(U)

1= |l_i(N)−k_i(N)|

N ,

where · 1 denotes the trace-norm with respect toN⁻¹Tr_N. For everym∈N and ε > 0, there exists N₀ ∈ N such that N⁻¹|l_i(N)−k_i(N)| < ε/m for all N ≥N₀and 1≤i≤n. Let us prove thatΓ(l(N), m, ε)⊂Γ(l(N), m,2ε) whenever N ≥N₀. Assume that N ≥N₀ and Q = (Q₁, . . . , Q_n) ∈ Γ(l(N), m, ε);

then there is U = (U₁, . . . , U_n) ∈ U(N)ⁿ so that Q = ξ_l(N)(U) and P = (P₁, . . . , P_n) :=ξ_k(N)(U) ∈Γ(k(N), m, ε). Since

Q_i−P_i1=ξ_N,l_i_(N₎(U_i)−ξ_N,k_i_(N₎(U_i)

1< ε

m, 1≤i≤n, we get for 1≤i₁, . . . , i_r≤nand 1≤r≤m

1

NTr_N(Q_i₁· · ·Q_i_r)− 1

NTr_N(P_i₁· · ·P_i_r) ≤

r j=1

Q_i_j−P_i_j₁< ε,

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and thus

1

NTr_N(Q_i₁· · ·Q_i_r)−τ(p_i₁· · ·p_i_r) <2ε, implyingQ ∈Γ(l(N), m,2ε). Settingγ_k(N):=_n

i=1γ_G(N,k_i_(N₎₎, we now have, thanks to (1.2),

γ_l(N₎

Γ(l(N), m,2ε) ≥γ_k(N₎Γ(l(N), m, ε)

=

γ_U(N) ^⊗n◦ξ⁻¹

l(N)◦ξ_l_(N₎◦ξ⁻¹

k(N)

Γ(k(N), m, ε)

≥

γ_U(N) ^⊗n◦ξ⁻¹

k(N)

Γ(k(N), m, ε)

=γ_k(N₎

Γ(k(N), m, ε) wheneverN ≥N₀. This implies that

lim sup

N→∞

1

N²logγ_l(N)

Γ(l(N), m,2ε) ≥lim sup

N→∞

1

N²logγ_k(N)

Γ(k(N), m, ε) , which says that the free entropy (1.4) given fork(N) is not greater than that forl(N). By symmetry we observe that both free entropies must coincide.

The following are basic properties of χ_proj. We omit their proofs, all of which are essentially same as in the case of self-adjoint variables in [21] or else obvious.

Proposition 1.2. Let p₁, . . . , p_n be projections in(M, τ).

(i) Negativity: χ_proj(p₁, . . . , p_n)≤0.

(ii) Subadditivity: for every 1≤j < n,

χ_proj(p₁, . . . , p_n)≤χ_proj(p₁, . . . , p_j) +χ_proj(p_j+1, . . . , p_n).

(iii) Upper semi-continuity: if a sequence(p^(m)₁ , . . . , p^(m)_n )ofn-tuples of projec- tions converges to (p₁, . . . , p_n)in distribution, then

χ_proj(p₁, . . . , p_n)≥lim sup

m→∞ χ_proj(p^(m)₁ , . . . , p^(m)_n ).

(iv) χ_proj(p₁, . . . , p_n)does not change when p_i is replaced by p^⊥_i :=1−p_i for somei.

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Remark1.3. We may adopt diﬀerent ways to introduce the free entropy of an n-tuple (p₁, . . . , p_n) of projections in (M, τ). For instance, consider two unitarily invariant probability measures γ_G(N)⁽¹⁾ and γ_G(N⁽²⁾ ₎ on G(N) :=

_N

k=0G(N, k) determined by the weights onG(N, k), 0≤k≤N, given as γ_G(N)⁽¹⁾ (G(N, k)) = 1

N+ 1, γ⁽²⁾_G(N)(G(N, k)) = 1 2^N

N k

. We set

Γ_proj(p₁, . . . , p_n;N, m, ε) :=

(P₁, . . . , P_n)∈G(N)ⁿ: 1

NTr_N(P_i₁· · ·P_i_r)−τ(p_i₁· · ·p_i_r) < ε for all 1≤i₁, . . . , i_r≤n, 1≤r≤m

, and deﬁne forj= 1,2

χ^(j)_proj(p₁, . . . , p_n) := lim

m→∞ε0

lim sup

N→∞

1 N² log

γ_G(N^(j) ₎

_⊗n

Γ_proj(p₁, . . . , p_n;N, m, ε) . It is fairly easy to see (similarly to the proof of Proposition 1.1) that both χ^(j)_proj(p₁, . . . , p_n),j= 1,2, coincide withχ_proj(p₁, . . . , p_n) given in (1.4).

§2. Case of Two Projections

Let (p, q) be a pair of projections in a tracialW^∗-probability space (M, τ) withα:=τ(p) andβ :=τ(q). Set

E₁₁:=p∧q, E₁₀:=p∧q^⊥, E₀₁:=p^⊥∧q, E₀₀:=p^⊥∧q^⊥, E:=1−(E₀₀+E₀₁+E₁₀+E₁₁).

Then E and E_ij are in the center of {p, q} and (E{p, q}E, τ|_E{p,q}_E) is isomorphic toL^∞((0,1), ν;M₂(C)), whereν is the measure on (0,1) determined by

τ(A) = 1 2

(0,1)

Tr₂(A(x))dν(x), A∈L^∞((0,1), ν;M₂(C))∼=E{p, q}E (hence ν((0,1)) = τ(E)). Under this isomorphism, EpE andEqE are repre- sented as

(EpE)(x) =

1 0 0 0

and (EqE)(x) =

x

x(1−x) x(1−x) 1−x

forx∈(0,1).

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In this way, the mixed moments of (p, q) with respect to τ are determined by ν and {τ(E_ij)}¹_i,j=0. Although ν is not necessarily a probability measure, we deﬁne the free entropy Σ(ν) by

Σ(ν) :=

(0,1)²

log|x−y|dν(x)dν(y) in the same way as in [20]. Furthermore, we set

(2.1) ρ:= min{α, β,1−α,1−β},

(2.2) C:=ρ²B

|α−β|

ρ ,|α+β−1| ρ

(meant zero ifρ= 0), where B(s, t) :=(1 +s)²

2 log(1 +s)−s²

2 logs+(1 +t)²

2 log(1 +t)−t² 2 logt

−(2 +s+t)²

2 log(2 +s+t) +(1 +s+t)²

2 log(1 +s+t) for s, t ≥ 0. With these deﬁnitions, the following formula of χ_proj(p, q) was obtained in [12] as a consequence of the large deviation principle for an independent pair of random projection matrices.

Proposition 2.1 ([12, Theorem 3.2, Proposition 3.3]). If τ(E₀₀)τ(E₁₁)

=τ(E₀₁)τ(E₁₀) = 0, then χ_proj(p, q) =1

4Σ(ν) +|α−β| 2

(0,1)

logx dν(x) +|α+β−1|

2

(0,1)

log(1−x)dν(x)−C,

and otherwiseχ_proj(p, q) =−∞. Moreover, χ_proj(p, q) = 0if and only ifpand q are free.

Note that the conditionτ(E₀₀)τ(E₁₁) =τ(E₀₁)τ(E₁₀) = 0 is equivalent to

(2.3)











τ(E₁₁) = max{α+β−1,0}, τ(E₀₀) = max{1−α−β,0}, τ(E₁₀) = max{α−β,0}, τ(E₀₁) = max{β−α,0}.

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When this is the case, the following must hold:

τ(E₀₁) +τ(E₁₀) =|α−β|, τ(E₀₀) +τ(E₁₁) =|α+β−1|, τ(E) = 2ρ.

In the case where χ_proj(p, q) = 0 (equivalently, p and q are free), the measureν was computed in [27] as

(2.4)

(x−ξ)(η−x)

2πx(1−x) 1_(ξ,η)(x)dx withξ, η:=α+β−2αβ±

4αβ(1−α)(1−β). It is also worthwhile to note [12] that lim sup in deﬁnition (1.4) can be replaced by lim in the case of two projections due to the large deviation result mentioned above.

In §4 the equivalence between the additivity of χ_proj and the freeness of projection pairs will be generalized to the “pair-block freeness result” for more than two projections. To do this, we need a kind of separate change of variable formula forχ_proj, which will be established in the next section.

§3. Separate Change of Variable Formula

LetN ∈Nandk, l∈ {0,1, . . . , N}. Assume that 0< k≤l andk+l≤N. Consider a pair (P, Q) of N×N projection random matrices with rank(P) = k and rank(Q) = l, which is assumed to be distributed under the measure γ_G(N,k)⊗γ_G(N,l) onG(N, k)×G(N, l). Then, by means of the so-called sine- cosine decomposition of two projections, we can represent such (P, Q) as follows:

P =U

I 0 0 0

⊕0⊕0

U^∗, (3.1)

Q=U

X

X(I−X) X(I−X) I−X

⊕I⊕0

U^∗ (3.2)

inC^N = (C^k⊗C²)⊕C^l−k⊕C^N^−k−l, whereU is anN×Nunitary matrix andX is ak×kdiagonal matrix with the diagonal entries 0≤x₁≤x₂≤ · · · ≤x_k ≤1.

Whenx₁, . . . , x_k are in (0,1) and mutually distinct, it is easy to see thatU is uniquely determined up to the right multiplication of unitary matrices of the

form

T 0 0 T

⊕V₁⊕V₂, T ∈T^k, V₁∈U(l−k), V₂∈U(N−k−l).

We denote byV(N, k, l) the subgroup of U(N) consisting of all unitary matrices of the above form so that U(N)/V(N, k, l) becomes a homogeneous space. Also,

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let [0,1]^k_≤ and (0,1)^k_< denote the sets of (x₁, . . . , x_k) satisfying 0≤x₁ ≤ · · · ≤ x_k ≤ 1 and 0 < x₁ < · · · < x_k < 1, respectively. We then consider the continuous map Ξ_N,k,l : U(N)/V(N, k, )×[0,1]^k_≤→G(N, k)×G(N, l) deﬁned by (3.1) and (3.2), that is,

Ξ_N,k,l([U], X) :=

U

I 0 0 0

⊕0⊕0

U^∗, U

X

X(I−X) X(I−X) I−X

⊕I⊕0

U

,

whereX = (x₁, . . . , x_k) in the right-hand side is regarded as a diagonal matrix as above. The set

(G(N, k)×G(N, l))₀:= Ξ_N,k,l

U(N)/V(N, k, l)×(0,1)^k_<

is open and co-negligible with respect toγ_G(N,k)⊗γ_G(N,l)inG(N, k)×G(N, l) thanks to [5, Theorem 2.2] (or [12, Lemma 1.1]) and moreover Ξ_N,k,l gives a smooth diﬀeomorphism between U(N)/V(N, k, l) × (0,1)^k_< and (G(N, k)×G(N, l))₀. Then we show the next lemma for later use.

Lemma 3.1. The measure (γ_G(N,k)⊗γ_G(N,l))◦Ξ_N,k,l coincides with γ_N,k,l⊗

1 Z_N,k,l

k i=1

x^l−k_i (1−x_i)^N−k−l

1≤i<j≤k

(x_i−x_j)² k i=1

dx_i

,

whereγ_N,k,l is the(unique)probability measure onU(N)/V(N, k, l)induced by the Haar probability measure onU(N)andZ_N,k,l is a normalization constant.

Proof. Let λ be the measure on U(N)/V(N, k, l)×(0,1)^k_< transformed from the restriction ofγ_G(N,k)⊗γ_G(N,l) to (G(N, k)×G(N, l))₀ by the inverse of Ξ_N,k,l, and µ be its image measure by the projection map ([U], X) →X. The disintegration theorem (see e.g. [16, Chapter IV,§6.5]) ensures that there is a µ-a.e. unique Borel map λ_(·) from (0,1)^k_< to the probability measures on U(N)/V(N, k, l) such that λ =

(0,1)^k_<λ_Xdµ(X). Note that ([U], X) → X splits into Ξ_N,k,l, (P, Q)→P QP and the map sendingP QP to the eigenvalues in increasing order. Henceµcoincides with the eigenvalue distribution ofP QP arranged in increasing order, which is known to be equal to the second compo- nent given in the lemma by [5, Theorem 2.2]. Therefore, it suﬃces to show that λ_X coincides withγ_N,k,l for µ-a.e. X ∈(0,1)^k_<. For each V ∈U(N), the unitary conjugation AdV×AdV : (P, Q)→(V P V^∗, V QV^∗) onG(N, k)×G(N, l) and the left-translation L_V : [U]→ V[U] := [V U] on U(N)/V(N, k, l) satisfy

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the relation Ξ_N,k,l◦(L_V ×id) = (AdV ×AdV)◦Ξ_N,k,l; hence, in particular, (G(N, k)×G(N, l))₀ is invariant under the action AdV ×AdV for every V ∈U(N). Then one can easily verify that

(0,1)^k<

U(N)/V(N,k,l)

f([U], X)d(λ_X◦L_V)([U])

dµ(X)

=

U(N)/V(N,k,l)×(0,1)^k<

f([U], X)dλ([U], X)

for any bounded Borel function f on U(N)/V(N, k, l)×(0,1)^k. This means thatλenjoys a new disintegrationλ=

(0,1)^k_<λ_X◦L_V dµ(X). The uniqueness of the disintegration says that forµ-a.e.X∈(0,1)^k_<one hasλ_X =λ_X◦L_V for allV ∈U(N). Sinceγ_N,k,l is a unique probability measure on U(N)/V(N, k, l) invariant under all L_V, it follows that λ_X =γ_N,k,l for µ-a.e. X ∈ (0,1)^k_< so that

λ=

(0,1)^k_<

γ_N,k,ldµ(X) =γ_N,k,l⊗µ, as required.

For a pair (p, q) of projections in (M, τ) we introduce a sort of functional calculus via the representation explained in§2. Letψbe a continuous increasing function from (0,1) into itself. With the notations in §2 we deﬁne a new projectionq(ψ;p) in{p, q} by

q(ψ;p) :=Eq(ψ;p)E+E₀₀+E₀₁+E₁₀+E₁₁, (Eq(ψ;p)E)(x) :=

ψ(x)

ψ(x)(1−ψ(x)) ψ(x)(1−ψ(x)) 1−ψ(x)

forx∈(0,1).

It is obvious that τ(q(ψ;p)) = τ(q). (The deﬁnition itself is possible for gen- eral Borel function from (0,1) into [0,1] but the above case is enough for our purpose.) The aim of this section is to prove the following change of variable formula for free entropy of projections.

Theorem 3.2. Letp₁, q₁, . . . , p_n, q_n, r₁, . . . , r_n be projections in(M, τ) and assume that χ_proj(p_i, q_i)>−∞for1≤i≤n. Letψ₁, . . . , ψ_n be continu- ous increasing functions from(0,1) into itself, and q_i(ψ_i;p_i)be the projection deﬁned from p_i,q_i andψ_i in the above manner for 1≤i≤n. Then we have χ_proj(p₁, q₁(ψ₁;p₁), . . . , p_n, q_n(ψ_n;p_n), r₁, . . . , r_n)

≥χ_proj(p₁, q₁, . . . , p_n, q_n, r₁, . . . , r_n)+

n i=1

χ_proj(p_i, q_i(ψ_i;p_i))−χ_proj(p_i, q_i) .

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Moreover, if ψ₁, . . . , ψ_n are strictly increasing, then equality holds true in the above inequality.

The proof goes on the essentially same lines as in [22] and it is divided into two steps; one is to analyze the case whenψ₁, . . . , ψ_n are all extended to C¹- diﬀeomorphisms from [0,1] onto itself and the other is to approximate, in two stages, the given ψ₁, . . . , ψ_n by C^∞-diﬀeomorphisms from [0,1] onto itself in such a way that the corresponding free entropies converge to those in question.

As the ﬁrst step let us prove the following special case of the theorem.

Lemma 3.3. Let p₁, q₁, . . . , p_n, q_n, r₁, . . . , r_n be as in Theorem 3.2. If ψ₁, . . . , ψ_n are C¹-diﬀeomorphisms from [0,1] onto itself with ψ_i(0) = 0 and ψ_i(1) = 1and moreoverψ_i(x)>0 for allx∈[0,1], then the equality assertion of Theorem 3.2holds true.

Proof. In the same way as in the proof of [22, Proposition 3.1] it suﬃces to show when n= 1; hence we assume n = 1 and write p=p₁, q = q₁ and ψ=ψ₁for brevity. Letνand{E_ij}¹_i,j=0be as in§2 for (p, q). By Propositions 1.2 (iv) and 2.1 we may assume that τ(p)≤τ(q)≤1/2 so that E₁₁=E₁₀= 0 by (2.3). We may further assume thatpis non-zero; otherwise there is nothing to do. With the polar decomposition (1−p)qp=v_p,q

pqp(p−pqp), we thus represent p,qand q(ψ;p) as follows:

p=v_p,q^∗ v_p,q, q=pqp+v_p,q

pqp(p−pqp) +

pqp(p−pqp)v_p,q^∗ +v_p,q(p−pqp)v_p,q^∗ +

q−pqp−(1−p)qp−pq(1−p)−v_p,q(p−pqp)v^∗_p,q

, q(ψ;p) =ψ(pqp) +v_p,q

ψ(pqp)(p−ψ(pqp))

+

ψ(pqp)(p−ψ(pqp))v_p,q^∗ +v_p,q(p−ψ(pqp))v_p,q^∗ +

q−pqp−(1−p)qp−pq(1−p)−v_p,q(p−pqp)v^∗_p,q

,

where ψ(pqp) means the functional calculus of pqp. Choose two sequences k(N), l(N) for N ≥ 2 in such a way that 0 < k(N) ≤ l(N) ≤ N/2 and k(N)/N→τ(p),l(N)/N →τ(q) asN → ∞. As explained at the beginning of this section, for each (P, Q)∈(G(N, k(N))×G(N, l(N)))₀ there is a unitary U ∈U(N), unique up toV(N, k(N), l(N)), for which we have (3.1) and (3.2).

Then we can deﬁne the map Φ_N,ψ on (G(N, k(N))×G(N, l(N)))₀by sending (P, Q) to (P, Q(ψ;P)) with

Q(ψ;P) :=U

ψ(X)

ψ(X)(I−ψ(X)) ψ(X)(I−ψ(X)) I−ψ(X)

⊕I⊕0

U^∗.

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With the polar decomposition (I−P)QP =V_P,Q

P QP(I−P QP) we have the following expressions:

Q=P QP +V_P,Q

P QP(P−P QP)

+

P QP(P−P QP)V_P,Q^∗ +V_P,Q(P−P QP)V_P,Q^∗ +

Q−P QP−(I−P)QP−P Q(I−P)−V_P,Q(P−P QP)V_P,Q^∗

, Q(ψ;P) =ψ(P QP) +V_P,Q

ψ(P QP)(P−ψ(P QP))

+

ψ(P QP)(P−ψ(P QP))V_P,Q^∗ +V_P,Q(P−ψ(P QP))V_P,Q^∗ +

Q−P QP−(I−P)QP−P Q(I−P)−V_P,Q(P−P QP)V_P,Q^∗

. Upon these expressions, what we now need is to approximate v_p,q and V_P,Q by polynomials ofp, qandP, Q, respectively, as stated in the next lemma very similarly to [11, 6.6.4].

Lemma 3.4. For eacht≥1andε >0one can ﬁndN₀, m₀∈N,ε₀>0 and a real polynomial Gin such a way that the following assertions hold:

• v_p,q−(1−p)qp·G(pqp)t< ε.

• For each N≥N₀, if(P, Q)∈(G(N, k(N))×G(N, l(N)))₀ satisﬁes

(3.3)

1

NTr_N((P QP)^m)−τ((pqp)^m)

< ε₀ for1≤m≤m₀, thenV_P,Q−(1−P)QP·G(P QP)_t< ε.

Here, · t denotes the Schatten t-norm with respect toτ as well as N⁻¹Tr_N. The proof of this technical lemma is essentially similar to that of [11, 6.6.4]

so that its sketch will be given later.

Proof of Lemma 3.3 (continued). Choose k₁(N), . . . , k_n(N) so that k_i(N)/N →τ(r_i) as N→ ∞, and set

Φ_N := Φ_N,ψ×

n

i=1

id_G(N,k_i_(N₎₎ on (G(N, k(N))×G(N, l(N)))₀×

n

i=1

G(N, k_i(N))

andγ_N :=γ_G(N,k(N₎₎⊗γ_G(N,l(N₎₎⊗_n

i=1γ_G(N,k_i_(N)). Letm∈Nandε >0 be arbitrary. In what follows, for brevity we write Γ_proj(p, q, r₁, . . . , r_n;N, m₀, ε₀) etc. without k(N), l(N), k₁(N), . . . , k_n(N). Thanks to Lemma 3.4 together with the expressions ofq(ψ;p) andQ(ψ;P) above, we can choose N₀, m₀ ∈N

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and ε₀ > 0 with m₀ ≥ m and ε₀ ≤ ε such that, for every N ≥ N₀, if (P, Q, R₁, . . . , R_n)∈Γ_proj(p, q, r₁, . . . , r_n;N, m₀, ε₀) and (P, Q)∈(G(N, k(N))× G(N, l(N)))₀, then Φ_N(P, Q, R₁, . . . , R_n) falls into Γ_proj(p, q(ψ;p), r₁, . . . , r_n; N, m, ε). Via Ξ_N,k(N),l(N₎ in the ﬁrst two coordinates, Lemma 3.1 enables us to estimate the Radon-Nikodym derivative dγ_N ◦Φ_N/dγ_N on a co-negligible subset of Γ_proj(p, q, r₁, . . . , r_n;N, m, ε) from below by the inﬁmum value of

1≤i<j≤k(N)

ψ(λ_i(P QP))−ψ(λ_j(P QP)) λ_i(P QP)−λ_j(P QP)

₂ k(N) i=1

ψ(λ_i(P QP)) (3.4)

×

k(N) i=1

ψ(λ_i(P QP)) λ_i(P QP)

_l(N)−k(N)k(N)

i=1

1−ψ(λ_i(P QP)) 1−λ_i(P QP)

_N_−k(N_)−l(N₎

for all (P, Q) ∈ (G(N, k(N))×G(N, l(N)))₀∩Γ_proj(p, q;N, m₀, ε₀) with the eigenvalue listλ₁(P QP), . . . , λ_k(N₎(P QP) in increasing order.

Letψ^[1](x, y) be the so-called divided diﬀerence of ψ, i.e.,

ψ^[1](x, y) :=

_{ψ(x)−ψ(y)}

x−y (x=y), ψ(x) (x=y).

Then, quantity (3.4) is rewritten in the coordinate (P, Q) as det_k(N₎2×k(N)² P⊗P·ψ^[1](P QP⊗P, P ⊗P QP)·P⊗P

!

×

det_k(N_)×k(N₎[P(P QP)⁻¹ψ(P QP)P] ^l(N)−k(N)

×

det_k(N_)×k(N₎[P(P−P QP)⁻¹(P−ψ(P QP)P] ^N−k(N^)−l(N⁾

= exp

Tr^⊗2_k(N)

P⊗P·log(ψ^[1](P QP ⊗P, P⊗P QP))·P⊗P

× exp

Tr_k(N)

P·log

(P QP)⁻¹ψ(P QP) ·P _l(N_)−k(N₎

× exp

Tr_k(N)

P·log

(P−P QP)⁻¹(P−ψ(P QP)) ·P _N_{−k(N)−l(N)} ,

where ψ^[1](P QP ⊗ P, P ⊗ P QP) is deﬁned on PC^N ⊗ PC^N while (P QP)⁻¹ψ(P QP) and (P−P QP)⁻¹(P−ψ(P QP)) are onPC^N. Letδ >0 be arbitrary. SinceψisC¹, logψ^[1](x, y) is continuous on [0,1]² so that there is a real polynomialL(x, y) on [0,1]² such thatlogψ^[1]−L_∞ < δ. Ifm ∈Nis larger than the degree ofL, then we have, for each (P, Q)∈Γ_proj(p, q;N, m, ε)

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with an arbitraryε>0, 1

N²Tr^⊗2_N

P⊗P ·logψ^[1](P QP ⊗P, P⊗P QP)·P⊗P

−τ^⊗2(p⊗p·logψ^[1](pqp⊗p, p⊗pqp)·p⊗p)

≤2δ+ 1

N²Tr^⊗2_N (P⊗P·L(P QP ⊗P, P⊗P QP)·P⊗P)

−τ^⊗2(p⊗p·L(pqp⊗p, p⊗pqp)·p⊗p)

≤2δ+Cε

withC >0 depending only onL(hence onδ). Therefore, for eachη >0 there arem₁∈Nandε₁>0 such that

exp

Tr^⊗2_k(N)

P⊗P·log

ψ^[1](P QP⊗P, P ⊗P QP)

·P⊗P (3.5)

≥exp

N²

"

τ^⊗2(p⊗p·logψ^[1](pqp⊗p, p⊗pqp)·p⊗p)−η

#

for all (P, Q)∈Γ_proj(p, q;N, m, ε) as long asm≥m₁ and 0< ε ≤ε₁. Since x⁻¹ψ(x) and (1−x)⁻¹(1−ψ(x)) are both bounded away from zero on [0,1]

due to the assumption onψ, the same argument works for the other two terms exp

Tr_k(N₎

P·log

(P QP)⁻¹ψ(P QP) ·P , exp

Tr_k(N₎

P·log

(P−P QP)⁻¹(P−ψ(P QP)) ·P . Therefore, for each η >0 there arem₂∈Nandε₂>0 such that

exp

Tr_k(N₎

P·log

(P QP)⁻¹ψ(P QP) ·P (3.6)

≥exp N

τ(p·log((pqp)⁻¹ψ(pqp))·p)−η , exp

Tr_k(N₎

P·log

(P−P QP)⁻¹(P−ψ(P QP)) ·P (3.7)

≥exp N

τ(p·log((p−pqp)⁻¹(p−ψ(pqp)))·p)−η

for all (P, Q)∈Γ_proj(p, q;N, m, ε) as long asm≥m₂and 0< ε ≤ε₂. Hence, whenever N ≥ N₀, m ≥max{m₀, m₁, m₂} and 0< ε < min{ε₀, ε₁, ε₂}, we have

1

N²logγ_N

Γ_proj(p, q(ψ;p), r₁, . . . , r_n;N, m, ε)

≥ 1

N²logγ_N Φ_N

Γ_proj(p, q, r₁, . . . , r_n;N, m, ε)

≥ 1

N²logγ_N

Γ_proj(p, q, r₁, . . . , r_n;N, m, ε)

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+τ^⊗2(p⊗p·logψ^[1](pqp⊗p, p⊗pqp)·p⊗p) +

l(N)

N −k(N) N

τ(p·log((pqp)⁻¹ψ(pqp))·p) +

1−l(N)

N −k(N) N

τ(p·log((p−pqp)⁻¹(p−ψ(pqp)))·p)−3η

= 1

N²logγ_N

Γ_proj(p, q, r₁, . . . , r_n;N, m, ε) +1

4

(0,1)²

log

ψ(x)−ψ(y) x−y

dν(x)dν(y) +1

2 l(N)

N −k(N)

N _(0,1)logψ(x) x dν(x) +1

2

1−l(N)

N −k(N)

N _(0,1)log1−ψ(x)

1−x dν(x)−3η.

Take the lim sup as N → ∞ and the limit as m → ∞, ε 0 in the above inequality. Sinceη >0 is arbitrary, we get

χ_proj(p, q(ψ;p), r₁, . . . , r_n)

≥χ_proj(p, q, r₁, . . . , r_n) +1 4

(0,1)²

log

ψ(x)−ψ(y) x−y

dν(x)dν(y) +τ(q)−τ(p)

2

(0,1)

logψ(x)

x dν(x) +1−τ(q)−τ(p) 2

(0,1)

log1−ψ(x) 1−x dν(x)

=χ_proj(p, q, r₁, . . . , r_n) +χ_proj(p, q(ψ;p))−χ_proj(p, q)

thanks to Proposition 2.1. The reverse inequality can be shown as well if we replace the inequalities (3.5)–(3.7) by their reversed versions. Hence we complete the proof of Lemma 3.3.

Proof of Lemma 3.4 (sketch). Given small α, β >0 we can estimate v_p,q−((1−p)qp)(

pqp(p−pqp) +α1)⁻¹^t_t (3.8)

≤ 1 2



ν((0, β)) +ν((1−β,1)) +ν([β,1−β])

α

β(1−β) +α _t





and

V_P,Q−(I−P)QP(

P QP(P−P QP) +αI)⁻¹^t_t (3.9)

≤ 1

N#{i:λ_i(P QP)< β}+ 1

N#{i:λ_i(P QP)>1−β}

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+k(N) N

α

β(1−β) +α _t

,

where 0< λ₁(P QP)<· · ·< λ_k(N₎(P QP)<1 are the eigenvalues ofP QP|_PCN

for (P, Q)∈(G(N, k(N))×G(N, l(N)))₀. For any η >0 let us choose β >0 so thatν((0,2β)) +ν((1−2β,1))< η^t. By (3.8) we get

(3.10)

v_p,q−((1−p)qp)(

pqp(p−pqp) +α1)⁻¹^t_t≤ η^t

2 +τ(E) 2

α

β(1−β) _t

.

Note that ν is non-atomic on (0,1) due to the assumption χ_proj(p, q)>−∞. Setξ_N,i:= min{x∈[0,1] :ν((0, x)) =iτ(E)/k(N)} for 1≤i≤k(N); then we get

τ((pqp)^m) = lim

N→∞

1 N

k(N)

i=1

ξ_N,i ^m for allm∈N.

Also choose a constantC >sup_N≥2N/k(N). By [11, 4.3.4] there arem₀ ∈N andε₀>0 such that, for everyN ∈Nand for every (λ₁, . . . , λ_k(N))∈(0,1)^k(N_< ⁾,

1

k(N)

k(N) i=1

λ^m_i − 1 k(N)

k(N) i=1

ξ_N,i ^m

<2Cε₀ for 1≤m≤m₀ implies

(3.11) 1

k(N)

i=1

λ_i−ξ_N,i^m< βη^t.

Assume (3.11). Set i₀ := #{i : λ_i < β} and i₁ := #{i : ξ_N,i < 2β}. If i₁ < i≤i₀, then λ_i−ξ_N,i=ξ_N,i−λ_i ≥β so that we get i₀ < i₁+k(N)η^t by (3.11). Since i₁τ(E)/k(N) ≤ ν((0,2β)) < η^t, we get i₀ < τ(E)⁻¹(1 + τ(E))k(N)η^t. If there is no i₁ < i ≤ i₀, then i₀ ≤ i₁ < τ(E)⁻¹k(N)η^t. Therefore, #{i : λ_i < β} < τ(E)⁻¹(1 +τ(E))k(N)η^t. Similarly, we have

#{i:λ_i>1−β}< τ(E)⁻¹(1 +τ(E))k(N)η^t. Now, choose N₀∈Nso that

1 N

k(N)

i=1

ξ_N,i ^m−τ((pqp)^m) < ε₀

for all 1 ≤ m ≤ m₀ and N ≥ N₀. We then conclude that if N ≥ N₀ and (P, Q)∈(G(N, k(N))×G(N, l(N)))₀satisﬁes (3.3), then

#{i:λ_i(P QP)< β}< 1 +τ(E)

τ(E) k(N)η^t, (3.12)

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#{i:λ_i(P QP)>1−β}< 1 +τ(E)

τ(E) k(N)η^t. (3.13)

Inserting (3.12) and (3.13) into (3.9) we get V_P,Q−(I−P)QP(

P QP(P−P QP) +αI)⁻¹^t_t (3.14)

≤1 +τ(E) τ(E) η^t+1

2

α

β(1−β) _t

.

Finally, letα >0 be so small asα/

β(1−β)< η, and choose a real polynomial G(x) such that|G(x)−(

x(1−x)+α)⁻¹|< ηfor allx∈[0,1]. Then by (3.10) and (3.14) we obtain

v_p,q−(1−p)qp·G(pqp)_t<2η and

V_P,Q−(I−P)QP·G(P QP)t<

1 τ(E)+3

2 _1/t

+ 1

η.

The proof is completed if η > 0 was chosen so small as

(1/τ(E) + 3/2)^1/t+ 1 η < ε.

For the second step we present two more technical lemmas. The proof of the next lemma should be compared with that of [22, Lemma 4.1].

Lemma 3.5. Let µ be a measure on [0,1] with no atom at 0 and 1, and assume the conditions

(0,1)²

log|x−y|dµ(x)dµ(y)>−∞, (3.15)

(0,1)

logx dµ(x)>−∞, (3.16)

(0,1)

log(1−x)dµ(x)>−∞. (3.17)

Ifψis a continuous increasing function from[0,1]onto itself withψ(0) = 0and ψ(1) = 1, then there exists a sequence of C^∞-diﬀeomorphisms ψ_j from [0,1]

onto itself with ψ_j(0) = 0 andψ_j(1) = 1such that (i) ψ_j(x)≥1/jfor all j∈Nandx∈[0,1], (ii) ψ_j−→ψ uniformly on [0,1],