Thompson-type formulae

View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Servicio de Difusión de la Creación Intelectual Available online at www.sciencedirect.com Journal of Functional Analysis 262 (2012) 1515–1528 www.elsevier.com/locate/jfa Thompson-type formulae Jorge Antezana a,c,1 , Gabriel Larotonda a,b,∗,2 , Alejandro Varela a,b,2 a Instituto Argentino de Matemática “Alberto P. Calderón”, CONICET, Saavedra 15, 3er piso (C1083ACA), Buenos Aires, Argentina b Instituto de Ciencias, Universidad Nacional de General Sarmiento, J.M. Gutierrez 1150, (B1613GSX) Los Polvorines, Buenos Aires, Argentina c Departamento de Matemática, Universidad Nacional de La Plata, Esquina 50 y 115 s.n. (1900) La Plata, Argentina Received 14 July 2011; accepted 9 November 2011 Available online 23 November 2011 Communicated by D. Voiculescu Abstract Let X and Y be two n × n Hermitian matrices. In the article Proof of a conjectured exponential formula (Linear Multilinear Algebra 19 (1986) 187–197) R.C. Thompson proved that there exist two n × n unitary matrices U and V such that ∗ ∗ eiX eiY = eiU XU +V Y V . In this note we consider extensions of this result to compact operators as well as to operators in an embeddable II1 factor.  2011 Elsevier Inc. All rights reserved. Keywords: Operator identity; Unitary operators; Functional calculus * Corresponding author at: Instituto de Ciencias, Universidad Nacional de General Sarmiento, J.M. Gutierrez 1150, (B1613GSX) Los Polvorines, Buenos Aires, Argentina. E-mail addresses: [email protected] (J. Antezana), [email protected] (G. Larotonda), [email protected] (A. Varela). 1 Partially supported by MTM-2008-05561-C02-02, 2009 SGR 1303, UNLP (11X585), CONICET (PIP 2009-435) and ANPCYT (PICT07-00808). 2 Partially supported by ANPCYT (PICTO 2008-00076 UNGS) and CONICET (PIP 2010-0757). 0022-1236/$ – see front matter  2011 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2011.11.011 1516 J. Antezana et al. / Journal of Functional Analysis 262 (2012) 1515–1528 1. Introduction In his 1986 paper [13], studying the product eiX eiY (with X, Y Hermitian matrices) R.C. Thompson considered the analytic map ξ(w) = eiX eiwY defined for some w ∈ C in a neighborhood of the unit interval. Using perturbation theory techniques, he derived a series of inequalities concerning the eigenvalues of X, Y and those of Z = log(eiX eiY ). The family of inequalities found by Thompson happened to relate to those proposed by A. Horn [7] as the complete solution of the following (seemingly) elementary problem: find necessary and sufficient conditions on the eigenvalues of the Hermitian matrices A, B, C in order to have U AU ∗ + V BV ∗ = C for some unitary matrices U and V . By that time, V.B. Lidskii had recently published the paper [12], announcing the proof of Horn’s conjecture (see Appendix A below for a brief exposition on the subject). Thus, Thompson’s computations lead him to conclude that there existed unitary matrices U, V such that Z = U XU ∗ + V Y V ∗ . However, details on Lidskii’s proof never saw the light, and for a very long time, Horn’s conjecture remained open and consequently, Thompson’s result was gently archived. It was not until twelve years later that a proof of Horn’s conjecture was given in two exceptional papers, the first one due to A. Klyachko [9] and the second one due to A. Knutson and T. Tao [10]. Later, Horn’s result was extended to the infinite dimensional setting by Bercovici et al. in two papers [2,3] that deal with the case of operators in an embeddable II1 factor and with compact operators respectively. Then, it is only natural to ask for extensions of Thompson’s formula on adequate infinite dimensional settings. In this paper, we provide generalizations of Thompson’s formula to the setting of compact operators, and to the setting of finite von Neumann algebras. Our motivation stems for the applications of Thompson’s identity to the study of the geometry of the Grassmannian manifold when it is endowed with a left-invariant metric induced by a unitarily invariant norm (see [14] and the Appendix at [1]). 2. Preliminaries Let H be a complex separable and infinite dimensional Hilbert space. In this paper B(H), B0 (H) and Bf (H) stand for the sets of bounded linear operators, compact operators and finite rank operators in H respectively. The unitary group of B(H) is indicated by U(H). If x ∈ B(H), then kxk stands for the√usual uniform norm, and we will use | · | to indicate the modulus of an operator, i.e. |x| = x ∗ x. We indicate with B(H)h (resp. B0 (H)h ) the real linear space of Hermitian elements (resp. Hermitian compact elements) of B(H). Given η, ζ ∈ H, by means of η ⊗ ζ we denote the rank one operator defined by η ⊗ ζ (ξ ) = hξ, ζ iη. On the other hand, throughout this paper Mn (C) denotes the algebra of complex n × n matrices, Gl(n) the group of all invertible elements of Mn (C), U(n) the group of unitary n × n matrices, and H(n) the real subalgebra of Hermitian matrices. Given x ∈ B(H) (or T ∈ Mn (C)), R(x) is the range or image of x, N (x) the null space of x, and σ (x) denotes the spectrum of x. If x is normal (i.e. xx ∗ = x ∗ x), then Ex (Ω) denotes the spectral measure of x associated to the (measurable) subset Ω of the complex plane. Let us give a precise statement of Thompson’s formula: Theorem (Thompson). Given X, Y ∈ H(n), there exist unitary matrices U, V ∈ Mn (C) such that eiX eiY = ei(U XU ∗ +V Y V ∗ ) . J. Antezana et al. / Journal of Functional Analysis 262 (2012) 1515–1528 1517 2.1. Some preliminaries on II1 factors Throughout this section, Rω denotes the ultrapower of the hyperfinite II1 factor, and M denotes any II1 factor that can be embedded in Rω . We are going to use the Greek letter τ to denote the normalized tracial state of M. Given an Hermitian element a ∈ M, it can be written as a= Z1 λa (t) de(t), 0 where λa is a non-increasing right continuous function, and e(·) is a spectral measure on [0, 1) such that τ (e(t)) = t. One of the characterizations of embeddable factors is the existence of a “sequence of matricial approximations” for any finite family of Hermitian elements. This notion is described more precisely in the following theorem: Theorem 2.1. Let a1 , . . . , ak be Hermitian elements of Rω . Then, there are integer numbers 1 6 n1 < n2 < · · · and Hermitian matrices X1(m) , . . . , Xk(m) ∈ Mnm (C) such that for every noncommutative polynomial p it holds that ¡ ¡ (m) ¡ ¢ (m) ¢¢ τ p(a1 , . . . , ak ) = lim τnm p X1 , . . . , Xk m→∞ where τnm is the normalized trace of Mnm (C). Moreover, the matrices can be taken so that for (m) each j ∈ {1, . . . k} we have that kXj k 6 kaj k for every m ∈ N. Given a matrix M ∈ Mn (C) whose eigenvalues arranged in non-increasing order are denoted by λ1 , . . . , λn , let λM denote the real-valued function defined in [0, 1) in the following way: λM (t) = n X j =1 λj χ[ j −1 , j ) . n n With this notation, the following result is a direct consequence of Theorem 2.1, and the reader is referred to [2] for a detailed proof: Corollary 2.2. Let a ∈ M be an Hermitian element, and {X (m) }m∈N a sequence of matricial approximations of a. Then, λX(m) −−−−→ λa almost everywhere. m→∞ Finally, we mention the following result valid in every finite factor: Proposition 2.3. (See Kamei [8].) Let a and b be Hermitian elements of a finite factor M. Then, the following statements are equivalent: 1. λa = λb ; 2. a belongs to the norm closure of the unitary orbit of b. 1518 J. Antezana et al. / Journal of Functional Analysis 262 (2012) 1515–1528 3. Thompson-type formulae for compact operators Throughout this section, given a compact operator x, the eigenvalues of x are arranged in non-increasing order with respect to their moduli, i.e., if i 6 j then |λi (x)| > |λj (x)|. Theorem 3.1. Given x, y ∈ B0 (H)h , there exist unitary operators uk and vk ∈ B(H), for k ∈ N, such that ∗ ∗ eix eiy = lim eiuk xuk +ivk yvk . (1) k→∞ Proof. Let x = u|x| and y = v|y| be polar decompositions of x and y, and X ¡ ¢ X ¡ ¢ |x| = λj |x| βj ⊗ βj and |x| = λj |y| ζj ⊗ ζj , j ∈N j ∈N spectral decompositions of |x| and |y| respectively. Recall that the eigenvalues are arranged in non-increasing order. Define xk = k X j =1 ¡ ¢ λj |x| βj ⊗ (uβj ) and yk = k X j =1 ¡ ¢ λj |y| ζj ⊗ (vζj ), and Sk = R(xk ) + R(yk ). Then xk (Sk ) ⊂ Sk and yk (Sk ) ⊂ Sk . So, xk , yk ∈ B(Sk ) ' Mn (C) (where n = dim(Sk )). On the other hand, eix eiy = lim eixk eiyk . (2) k→∞ Due to Thompson’s formula for matrices, there exist uk , vk unitary linear transformations in Sk (which means that uk u∗k = pSk and vk vk∗ = pSk , where pSk denotes the orthogonal projection onto Sk ) such that ∗ ∗ eixk eiyk = eiuk xk uk +ivk yk vk . (3) We can extend uk , vk ∈ B(Sk ) to the unitaries ũk = uk + pS ⊥ and ṽk = vk + pS ⊥ ∈ B(H). Then k k from the equality (3) valid in Sk we get the following in B(H) ∗ ∗ eixk eiyk = ei ũk xk ũk +i ṽk yk ṽk . (4) Since (ũk x ũ∗k + ṽk y ṽk∗ ) − (ũk xk ũ∗k + ṽk yk ṽk∗ ) → 0, using (2) we get ∗ ∗ k·k ei(ũk x ũk +ṽk y ṽk ) − eix eiy −−−−→ 0. ✷ k→∞ Since the unitary orbit of a fixed operator in B(H) is not closed in general [15], to avoid the limit in (1) we have to pay some price. The following theorem follows this path. Theorem 3.2. Given x, y ∈ B0 (H)h , there is an isometry w ∈ B(H), and unitary operators u and v such that ∗ ∗ eiwxw eiwyw = eiu(wxw ∗ )u∗ +iv(wyw ∗ )v ∗ . J. Antezana et al. / Journal of Functional Analysis 262 (2012) 1515–1528 1519 Remark 3.3. Another way to state the theorem follows: there is a bigger Hilbert space K containing H such that the extensions x̂, ŷ ∈ B(K) defined by ¶ µ ¶ µ x 0 H y 0 H x̂ = , ŷ = 0 0 KªH 0 0 KªH satisfy the identity ei x̂ ei ŷ = ei(ux̂u ∗ +v ŷv ∗ ) , for some unitary operators u and v acting on K. ✷ Let us roughly sketch the idea behind the proof. We know that there are unitary operators un , vn ∈ U(H) such that ∗ ∗ eix eiy = lim ei(un xun +vn yvn ) . n→∞ Let zn = un xu∗n + vn yvn∗ . Extending to a bigger space K the operators zn , un , vn , x and y as in the previous remark, we can conjugate the sequence {ẑn }n∈N with unitary operators wn acting on ∗ K so that eẑn = ewn ẑn wn , and the modified sequence {wn ẑn wn∗ }n∈N has a convergent subsequence. If ŝ denotes the limit of that subsequence, provided dim K ª H = ∞, we can always find two unitary operators û0 and v̂0 such that ŝ = û0 x̂ û∗0 + v̂0 ŷ v̂0∗ . As this limit ŝ satisfies that ei x̂ ei ŷ = ei ŝ , this would complete the proof. Since the proof of Theorem 3.2 is rather long, some technical parts are included in the next three lemmas: Lemma 3.4. Let {an }n∈N be a bounded sequence of finite rank normal operators, and let pn denote the orthogonal projection onto R(an ). If there exists a finite rank projection p such that k·k pn −−−−→ p, then {an }n∈N has a convergent subsequence. n→∞ k·k Proof. Since pn −−−−→ p, the operators sn := pn p + (1 − pn )(1 − p) converge to 1 as n → ∞. n→∞ We can suppose that for every n ∈ N, sn is invertible. Note also that pn sn = sn p. For each n ∈ N, let sn = un |sn | be the polar decomposition of sn . Then, straightforward computations show that pn un = un p. So, as the sequence {an }n∈N is bounded, {u∗n an un }n∈N is a bounded sequence of normal operators whose range is the finite dimensional subspace R(p). Therefore, it has a normk·k convergent subsequence. Since un −−−−→ 1, the original sequence {an }n∈N also has a convergent n→∞ subsequence. ✷ Lemma 3.5. Let z ∈ B0 (H)h be such that kzk 6 π , and let {wn }n∈N be a bounded sequence of Hermitian compact operators which satisfies: k·k (a) eiwn −−−−→ eiz ; n→∞ (b) There exists n0 ∈ N and ε > 0 such that µ [ ¶ ¶ µ [ (2kπ − ε, 2kπ + ε) = ∅. σ (wn ) ∩ n>n0 k∈Z,k6=0 Then, {wn }n∈N has a convergent subsequence. 1520 J. Antezana et al. / Journal of Functional Analysis 262 (2012) 1515–1528 Proof. Since kzk 6 π and the operators wn satisfy condition (b), there exists ε > 0 such that ±ε is not contained neither in the spectrum of any wn nor in the spectrum of z, and it satisfies ¶¶ µ µ ¡ ¢ ε = Ewn (−ε, ε) , pn = Eewn B1 2 sin 2 µ µ ¶¶ ¡ ¢ ε z p = Ee B1 2 sin = Ez (−ε, ε) , 2 where Bα (ρ) denotes the ball in C of radius ρ centered at α. Standard arguments of functional k·k calculus imply that pn −−−−→ p. If log denotes the principal branch of the complex logarithm, n→∞ then ¢ ¡ log (1 − pn ) + pn ewn = pn wn and ¡ ¢ log (1 − p) + pez = pz. k·k So, pn wn −−−−→ pz because the sequence {(1 − pn ) + pn ewn }n∈N converges in the norm topoln→∞ ogy to (1 − p) + pez , and the holomorphic functional calculus is continuous with respect to this topology. On the other hand, if qn = 1 − pn , the sequence {wn qn }n∈N satisfies the conditions of Lemma 3.4. Hence, it has a convergent subsequence {wnk qnk }k∈N . Therefore, {wnk }k∈N converges, which concludes the proof. ✷ The next lemma is a variation of Lemma 4.3 in [3], and its proof follows essentially in the same lines. We include a sketch of its proof for the sake of completeness. Lemma 3.6. Let x, y ∈ B0 (H)h , and suppose there exist unitary operators uk and vk , for k ∈ N such that s = lim uk xu∗k + vk yvk∗ , k→∞ for some s ∈ B0 (H). Then, there exist compact operators s̄, x̄, ȳ satisfying s̄ = x̄ + ȳ, σ (s̄) = σ (s), σ (x̄) = σ (x), and σ (ȳ) = σ (y) with the same multiplicity for every non-zero eigenvalue. Sketch of proof. Let xk = uk xu∗k , yk = vk yvk∗ , and sk = xk + yk . For each k ∈ N consider an SOT increasing sequence of projections {pk,n }n∈N such that dim R(pk,n ) = n, pk,n −−−−→ 1, and n→∞ °¢ ° ° ° ° ¡° εn := sup °(1 − pk,n )sk ° + °(1 − pk,n )xk ° + °(1 − pk,n )yk ° −−−−→ 0. n→∞ k∈N This last requirement can be achieved by choosing the projections in such a way that they capture for each n as many eigenvectors of xk and yk as it is possible, among those corresponding to the biggest eigenvalues (in modulus) of xk and yk . Now, consider a fixed increasing sequence of projections {qn }n∈N such that dim R(qn ) = n, SOT and qn −−−−→ 1, and for each k ∈ N define a unitary operator wk such that n→∞ wk pk,n wk∗ = qn . 1521 J. Antezana et al. / Journal of Functional Analysis 262 (2012) 1515–1528 Let s̄k = wk sk wk∗ , x̄k = wk xk wk∗ , and ȳk = wk yk wk∗ . Straightforward computations show that these operators satisfy the following inequalities: ks̄k − qn s̄k qn k 6 2εn , kx̄k − qn x̄k qn k 6 2εn , and kȳk − qn ȳk qn k 6 2εn . (5) Note that, for each n ∈ N, set {qn s̄k qn : k ∈ N} is bounded, hence totally bounded. So, the first inequality of (5) implies that the set {s̄k : k ∈ N} is totally bounded as well. Therefore, passing to a subsequence if necessary, we may assume that the sequence {s̄k } converges to a compact Hermitian operator s̄. The same argument can be applied to the sequences {x̄k } and {ȳk }, and we get the operators x̄, and ȳ, respectively. Clearly these operators satisfy s̄ = x̄ + ȳ, and standard arguments of functional calculus show that σ (s̄) = σ (s), σ (x̄) = σ (x), and σ (ȳ) = σ (y) with the same multiplicity for every non-zero eigenvalue. ✷ Proof of Theorem 3.2. Let z be any bounded and Hermitian operator such that eiz = eix eiy . For simplicity, we are going to prove the alternative version of the statement described in Remark 3.3, and without lost of generality, we are going to assume that kzk 6 π . Then note that, since eiz − 1 = eix eiy − 1 and the right hand is compact, then an elementary argument using the functional calculus of the entire map F (λ) = (eiλ − 1)λ−1 shows that z is also a compact operator. By Theorem 3.1, there are unitary operators un and vn such that: ∗ ∗ eiz = lim ei(un xun +vn yvn ) . n→∞ Let zn := un xu∗n + vn yvn∗ . Since x and y are compact, there exists M > 0 big enough such that for every j > M and every n ∈ N it holds that λj (|zn |) < π . For technical reasons, passing to a subsequence if necessary, we can assume that {λj (zn )}n∈N converges for every j ∈ {1, . . . , M}. Define n o ¡ ¢ Ω = m ∈ N: lim sup λm |zn | = 2kπ for some k ∈ N n→∞ o n ¡ ¢ = m ∈ N: lim λm |zn | = 2kπ for some k ∈ N . n→∞ (n) The second equality holds because #Ω < M. Let {ζj }j ∈N be an orthonormal basis of H such (n) that ζj is an eigenvector of λj (zn ). Then ­ (n) (n) ® lim |zn |ζj , ζj = 2kπ n→∞ for j ∈ Ω, and some k ∈ Z. (6) Let K = H ⊕ H, and extend x, y, z to K as: x̂ = µ x 0 0 0 ¶ H , H ŷ = µ y 0 0 0 ¶ H , H and ẑ = µ z 0 0 0 ¶ H . H 1522 J. Antezana et al. / Journal of Functional Analysis 262 (2012) 1515–1528 The unitary operators un and vn are also extended, but in this case as the identity in the second copy of H. Denote with ûn and v̂n these extensions. With these definitions, we get ẑn = ûn x̂ û∗n + v̂n ŷ v̂n∗ = µ zn 0 0 0 ¶ H . H Fix an orthonormal basis {βj }j ∈N of H, and define for each n ∈ N the unitary operator wn as the unique unitary operator in B(K) that satisfies ¡ ¢ wn ζj(n) ⊕ 0 = 0 ⊕ βj if j ∈ Ω, wn (0 ⊕ βj ) = ζj(n) ⊕ 0 ¡ ¢ wn ζj(n) ⊕ 0 = ζj(n) ⊕ 0 (n) wn (0 ⊕ βj ) = 0 ⊕ βj (n) if j ∈ Ω, if j ∈ / Ω, if j ∈ / Ω. (n) Consider the new sequence sn = wn ẑn wn∗ . Let p2π , ps , pc and p0 be the orthogonal projections such that: © (n) ª ¡ (n) ¢ R p2π = span ζj ⊕ 0: j ∈ Ω , ª ¢ © (n) ¡ /Ω , R ps(n) = span ζj ⊕ 0: j ∈ R(pc ) = span{0 ⊕ βj : j ∈ Ω}, R(p0 ) = span{0 ⊕ βj : j ∈ / Ω}. Note that, for each n ∈ N, the operator sn commutes with the four projections. Claim. There exists n0 large enough so that (n) 1. sn (p2π + p0 ) = 0 for every n ∈ N; 2. {|sn pc |}n∈N converges to an operator whose spectrum is contained in {2kπ: k ∈ Z}; 3. There exists ε > 0 such that ¶ µ [ µ [ ¶ ¡ (n) ¢ (2kπ − ε, 2kπ + ε) = ∅. ∩ σ sn ps k∈Z,k6=0 n>n0 The first item is clear, and the second item is a direct consequence of (6). In order to prove the third one, recall that for every j > M and every n ∈ N the eigenvalues λj (|zn |) are contained in (−π, π). On the other hand, we can take n0 large enough so that the sequences {λj (zn )}n∈N for j ∈ {1, . . . , M} are close to their limits. Note that, for j ∈ / Ω the limits are far from the integer multiples of 2π . These facts, all together, imply (3), and conclude the proof of the claim. Straightforward computations show that ∗ ∗ ∗ ∗ ∗ ei ẑ = lim ei(ûn x̂ ûn +v̂n ŷ v̂n ) = lim eiwn (ûn x̂ ûn +v̂n ŷ v̂n )wn , n→∞ n→∞ 1523 J. Antezana et al. / Journal of Functional Analysis 262 (2012) 1515–1528 which implies (n) ei ẑ = lim ei(sn ps ) , (7) n→∞ because (n) lim ei(sn (pc +p2π +p0 )) = 1. n→∞ (n) The identity (7) and the claim allow us to apply Lemma 3.5 to the sequence {sn ps }n∈N , and to obtain a convergent subsequence. Therefore, the sequence {sn }n∈N has a convergent subsequence {snk }k∈N . Let s be its limit, that is ¡ ¢ s = lim snk = lim wnk ûnk x̂ û∗nk + v̂nk ŷ v̂n∗k wn∗k . k→∞ k→∞ (8) Clearly, this limit satisfies the identity ei ẑ = eis . On the other hand, if we consider the restriction of (8) to S = R(1 − p0 ), then by Lemma 3.6 there are operators s̄, x̄, ȳ ∈ B(S) which have the same non-zero eigenvalues (counted with multiplicity) as the operators s, x̂, and ŷ. Extended as zero in S ⊥ (and using this notation), s̄, x̄ and ȳ become unitary equivalent to s, x̂, and ŷ respectively. Therefore, as s̄ = x̄ + ȳ, there exist two unitary operators u0 and v0 acting on K such that s = u0 x̂u∗0 + v0 ŷv0∗ . This concludes the proof. ✷ 4. Thompson-type formulae for operators in an embeddable II1 factor Throughout this section, let M be a II1 factor that can be embedded in Rω . We start with two technical lemmas. Lemma 4.1. Let a, b ∈ M be Hermitian elements, and let {(A(m) , B (m) )}m∈N be a sequence of matricial approximations. Then, for every polynomial p ∈ C[z, z̄] ¡ ¡ (m) (m) ¢¢ ¡ ¡ ¢¢ . τ p eia eib = lim τnm p eiA eiB m→∞ Proof. It is a straightforward consequence of Theorem 2.1. ✷ Let us recall the definition of decreasing rearrangements of functions: given a measurable function f : [0, 1) → R, its decreasing rearrangement f ∗ : [0, 1) → R is defined by ª ª¯ © ¯© f ∗ (t) = inf s: ¯ x: f (x) > s ¯ 6 t . Remark 4.2. Note that, given two functions f, g : [0, 1) → R, if they satisfy |{x: f (x) > s}| = |{x: g(x) > s}| for every s ∈ R, then f ∗ = g ∗ . The reader is referred to [4] for more details on decreasing rearrangements. ✷ 1524 J. Antezana et al. / Journal of Functional Analysis 262 (2012) 1515–1528 Lemma 4.3. Let f, g : [0, 1) → R be bounded non-increasing functions such that kgk∞ 6 π , and for any interval I of the unit circle S 1 it holds that Z1 0 ¡ ¢ χI eif (t) dt = Z1 0 ¡ ¢ χI eig(t) dt. (9) Then, there is a function ḡ : [0, 1) → R such that eif (t) = eig(t) , and ḡ ∗ = g. Proof. Let Ω = {t ∈ [0, 1): eif (t) = −1}, and divide it in two measurable sets Ω+ and Ω− such that ¯© ª¯ |Ω+ | = ¯ t ∈ [0, 1): g(t) = π ¯ ¯© £ ª¯ and |Ω− | = ¯ t ∈ 0, 1): g(t) = −π ¯. This is possible because |Ω| = |{t ∈ [0, 1): eig(t) = −1}| by (9). Define ḡ : [0, 1) → R as follows: ⎧ ⎨ f (t) − 2kπ ḡ(t) := π ⎩ −π if f (t) ∈ ((2k − 1)π, (2k + 1)π); if t ∈ Ω+ ; if t ∈ Ω− . The function ḡ clearly satisfies the identity eif (t) = eig(t) . So, for every arc I of the unit circle Z1 0 ¢ ¡ χI eig(t) dt = Z1 0 ¡ ¢ χI eif (t) dt, and therefore Z1 0 ¢ ¡ χI eig(t) dt = Z1 0 ¡ ¢ χI eig(t) dt. (10) The next (and last) step, is to prove that ḡ ∗ = g ∗ = g (almost everywhere). The last identity holds because g is decreasing and the decreasing rearrangements considered here are with respect to the Lebesgue measure. To prove that ḡ ∗ = g ∗ , it is enough to verify that for every s ∈ R ¯© ª¯ ª¯ ¯© ¯ x: ḡ(x) > s ¯ = ¯ x: g(x) > s ¯. (11) Note that, by construction, kḡk∞ 6 π . Hence, kḡk∞ = kgk∞ by (10). Moreover, also by construction, it holds that ¯© ª¯ ª¯ ¯© ¯ x: ḡ(x) = −π ¯ = ¯ x: g(x) = −π ¯. Therefore, the equality in (11) is apparent if s > kgk∞ or s 6 −π . On the other hand, if −π < s 6 kgk∞ , let I = {eit : s < t 6 π}. Then J. Antezana et al. / Journal of Functional Analysis 262 (2012) 1515–1528 ¯© ª¯ ¯ x: ḡ(x) > s ¯ = = Z1 0 Z1 0 This concludes the proof. 1525 ¯© ª¯ ¢ ¡ χI eig(t) dt − ¯ x: ḡ(x) = −π ¯ ¯© ª¯ ª¯ ¯© ¡ ¢ χI eig(t) dt − ¯ x: g(x) = −π ¯ = ¯ x: g(x) > s ¯. ✷ Theorem 4.4. Given a, b ∈ M Hermitian, there are two sequences of unitaries {un }n∈N and {vn }n∈N such that ∗ ∗ eia eib = lim ei(un aun +vn bvn ) , n→∞ where the convergence is with respect to the operator norm topology. Proof. Let {A(m) }m∈N and {B (m) }m∈N be sequences of matricial approximations of a and b respectively. By Thompson’s theorem, there are unitary matrices Um and Vm such that for each m∈N (m) eiA eiB (m) U ∗ +V B (m) V ∗ ) m m m (m) = ei(Um A . ∗ . By Theorem A.1, the functions λ Define Dm = Um A(m) Um∗ + Vm B (m) Bm A(m) , λB (m) , and λD (m) satisfy Eq. (A.2). Since the sequence of non-increasing functions {λD (m) }m∈N is uniformly bounded, by Helly’s selection theorem, there is a subsequence of this sequence that converges for all but almost countable many points t ∈ [0, 1). To simplify the notation, let us assume that the original sequence converges in this way, and let f be its limit. This limit is also non-increasing and bounded. Moreover, as λA(m) , λB (m) , and λD (m) satisfy (A.2) for every m ∈ N, by the dominated convergence theorem, λa , λb and f also satisfy those inequalities. Then, there are operators a 0 , b0 such that λa 0 = λa , λb 0 = λb , and λa 0 +b0 = f . (12) Let c ∈ M such that eia eib = eic and kck 6 π . Given a polynomial p, on one hand by Lemma 4.1: ¢¢ ¡ ¡ ¢¢ ¡ ¡ ¡ ¡ (m) (m) ¢¢ = τ p eia eib = τ p eic = lim τnm p eiA eiB m→∞ Z1 0 ¢ ¡ p eiλc (t) dt. (13) On the other hand, by the dominated convergence theorem, we obtain ¡ ¡ (m) ¢¢ ¡ ¡ (m) (m) ¢¢ lim τnm p eiA eiB = lim = lim τnm p eiD m→∞ m→∞ = Z1 0 ¡ ¢ p eif dt. m→∞ Z1 0 ¡ ¢ p eiλD(m) (t) dt (14) 1526 J. Antezana et al. / Journal of Functional Analysis 262 (2012) 1515–1528 Therefore, (13) and (14) imply that for every polynomial p Z1 0 ¡ ¢ p eiλc (t) dt = Z1 0 ¡ ¢ p eif (t) dt. Using standard arguments we obtain the same result replacing the polynomials by characteristic functions of arcs. Then, by Lemma 4.3, there is a function λ¯c such that eif = eiλc , and λ̄∗c = λc . Suppose that c= Z1 λc (t) de(t), 0 and define 0 c = Z1 λ̄c (t) de(t) and d = Z1 f (t) de(t). 0 0 0 Then, eic = eid , λd = f , and λc0 = λc . Combining these facts with Eq. (12), and using Proposi(b) (c) (d) tion 2.3, we get that there are sequences {u(a) n }n∈N , {un }n∈N , {un }n∈N , and {un }n∈N of unitary elements of M so that ¢∗ ¢¡ 0 ¢¡ ¡ d = lim u(d) , a + b0 u(d) n n n→∞ ¡ ¢ ¡ (a) ¢∗ a 0 = lim u(a) a un , n→∞ n ¡ ¢ ¡ ¢∗ b0 = lim u(b) b u(b) , and n n n→∞ ¡ ¢ 0 ¡ (c) ¢∗ c = lim u(c) . n c un n→∞ (c) (d) (b) (c) (d) (a) Finally, if we define un = un un un y vn = un un un we get ∗ ∗ eic = lim ei(un aun +vn bvn ) , n→∞ which concludes the proof. ✷ Appendix A. Brief review on Horn’s conjecture One of the most challenging problems in linear algebra has been to characterize the real ntuples α, β, and γ that are the eigenvalues of n × n Hermitian matrices A, B, and C such that C = A + B. In his remarkable 1962 paper [7], Alfred Horn found necessary condition on the n-tuples α, β, and γ and conjectured that this conditions were also sufficient. This conjecture remained open for several years, and it was solved at the end of the 20th century. Later on, these results were extended to operators in embeddable II1 factors. In this appendix, we briefly recall J. Antezana et al. / Journal of Functional Analysis 262 (2012) 1515–1528 1527 these results; for some really deep material on the subject, we point the reader to the nice surveys by R. Bhatia [5] and W. Fulton [6]. To begin with, we are going to fix some notation and conventions in order to state correctly the results in the finite dimensional setting. Firstly, the n-tuples will be considered arranged in non-increasing order, and by means of λ(A) we denote the vector of eigenvalues of a self-adjoint matrix, also arranged in non-increasing order. Clearly, one necessary condition that three n-tuples α, β, and γ have to satisfy in order to be the eigenvalues of Hermitian matrices A, B, and C such that C = A + B, is the next identity n X j =1 γj = n X j =1 αj + n X (A.1) βj . j =1 This equality is far from being sufficient. In [7], Horn prescribed sets of triples (I, J, K) of subsets of {1, . . . , n}, that we will always write in increasing order, and he proved that the system of inequalities n X γk 6 k∈K X i∈I αi + X βj , j ∈J are necessary. The triples (I, J, K) are defined by the following inductive procedure. Set ¾ ½ X X r(r + 1) X n + k . Ur := (I, J, K): i+ j= 2 i∈I j ∈J k∈K For r = 1 set T1n = U1n . If n > 2, set ½ Trn := (I, J, K) ∈ Urn : for all p < r and all (F, G, H ) ∈ Tpr , X f ∈F if + X g∈G ¾ p(p + 1) X + kh . jg 6 2 h∈H Then, system of inequalities considered by Horn runs over all the triples in the set Tn := Sn the n . He also conjectured that this system of inequalities, together with the identity (A.1), T k=1 k were sufficient. The proof of this conjecture is a consequence of several deep works of Klyachko, Knutson, and Tao (see [6,9–11]). Theorem A.1. Given α, β, γ ∈ Rn , the following statements are equivalent: 1. There are n × n Hermitian matrices A, B and C such that C = A + B and λ(A) = α, λ(B) P P = β, andPλ(C) = γ ; Pn n 2. Pnk=1 γk =P ni=1 αi + j =1 βj , and for every (I, J, K) in Tr , k∈K γk 6 β α + j i j ∈J i∈I Later on, this result was extended by Bercovici and Li in [2] to operators in an embeddable II1 factor M, i.e. a factor that can be embedded in the ultrapower of the hyperfinite factor. To state 1528 J. Antezana et al. / Journal of Functional Analysis 262 (2012) 1515–1528 correctly this generalization, we need to introduce some notations. Given n ∈ N, if I ⊆ {1, . . . , n}, then σI denotes the set [· (i − 1) i ¶ , . n n i∈I With this notation, the set T is defined by T := ∞ n−1 [ [© ª (σI , σJ , σK ): (I, J, K) ∈ Trn . n=1 r=1 Theorem A.2. Consider bounded non-increasing and right-continuous functions u, v, and w defined in the [0, 1). The following are equivalent: 1. There exist a, b ∈ M such that u = λa , v = λb and w = λa+b ; 2. The functions u, v, and w satisfy: Z1 u(t) dt + ω1 u(t) dt + v(t) dt = Z ω2 v(t) dt > Z1 w(t) dt, and 0 0 0 Z Z1 Z ω3 w(t) dt, ∀(ω1 , ω2 , ω3 ) ∈ T . (A.2) References [1] J. Antezana, G. Larotonda, A. Varela, Optimal paths for symmetric actions in the unitary group, preprint, 2011, arXiv:1107.2439v1. [2] H. Bercovici, W.S. Li, Eigenvalue inequalities in an embeddable factor, Proc. Amer. Math. Soc. 134 (2006) 75–80. [3] H. Bercovici, W.S. Li, D. Timotin, The Horn conjecture for sums of compact selfadjoint operators, Amer. J. Math. 131 (2009) 1543–1567. [4] C. Bennett, R. Sharpley, Interpolation of Operators, Pure Appl. Math., vol. 129, Academic Press Inc., Boston, MA, 1988. [5] R. Bhatia, Linear algebra to quantum cohomology: The story of Alfred Horn’s inequalities, Amer. Math. Monthly 108 (4) (2001) 289–318. [6] W. Fulton, Eigenvalues, invariant factors, highest weights, and Schubert calculus, Bull. Amer. Math. Soc. 37 (2000) 209–249. [7] A. Horn, Eigenvalues of sums of Hermitian matrices, Pacific J. Math. 12 (1962) 225–241. [8] E. Kamei, Majorization in finite factors, Math. Jpn. 28 (1983) 495–499. [9] A.A. Klyachko, Stable bundles, representation theory and Hermitian operators, Selecta Math. 4 (1998) 419–445. [10] A. Knutson, T. Tao, The honeycomb model of GLn (C) tensor products I: Proof of the saturation conjecture, J. Amer. Math. Soc. 12 (1999) 1055–1090. [11] A. Knutson, T. Tao, C. Woodward, The honeycomb model of GLn (C) tensor products II: Puzzles determine facets of the Littlewood–Richardson cone, J. Amer. Math. Soc. 17 (2004) 19–48. [12] B.V. Lidskii, Spectral polyhedron of a sum of two Hermitian matrices, Funktsional. Anal. i Prilozhen. 16 (2) (1982) 76–77 (in Russian), translation in Funct. Anal. Appl. 16 (1982) 139–140. [13] R.C. Thompson, Proof of a conjectured exponential formula, Linear Multilinear Algebra 19 (2) (1986) 187–197. [14] L. Qiu, Y. Zhang, C.-K. Li, Unitarily invariant metrics on the Grassmann space, SIAM J. Matrix Anal. Appl. 27 (2) (2005) 507–531. [15] D.V. Voiculescu, A non-commutative Weyl–von Neumann theorem, Rev. Roum. Math. Pures Appl. 21 (1976) 97– 113.