Alberti--Uhlmann problem on Hardy--Littlewood--Pólya majorization
aa r X i v : . [ m a t h . F A ] A p r ALBERTI–UHLMANN PROBLEM ONHARDY–LITTLEWOOD–P ´OLYA MAJORIZATION
J. HUANG AND F. SUKOCHEV
Dedication to the 90th birthday of Professor Armin Uhlmann
Abstract.
We fully describe the doubly stochastic orbit of a self-adjoint el-ement in the noncommutative L -space affiliated with a semifinite von Neu-mann algebra, which answers a problem posed by Alberti and Uhlmann [2]in the 1980s, extending several results in the literature. It follows furtherfrom our methods that, for any σ -finite von Neumann algebra M equipped asemifinite infinite faithful normal trace τ , there exists a self-adjoint operator y ∈ L ( M , τ ) such that the doubly stochastic orbit of y does not coincidewith the orbit of y in the sense of Hardy–Littlewood–P´olya, which confirmsa conjecture by Hiai [33]. However, we show that Hiai’s conjecture fails fornon- σ -finite von Neumann algebras. The main result of the present paperalso answers the (noncommutative) infinite counterparts of problems due toLuxemburg [58] and Ryff [71] in the 1960s. Contents
1. Introduction 22. Preliminaries 62.1. τ -measurable operators and generalized singular value functions 62.2. Classic Hardy–Littlewood–P´olya majorization and submajorization 72.3. Hardy–Littlewood–P´olya majorization in the infinite setting 93. Alberti–Uhlmann problem in the setting of finite von Neumann algebras 104. On Hiai’s conjecture: Ryff’s problem in the infinite setting 135. A decomposition theorem 166. Alberti–Uhlmann problem in the setting of infinite von Neumannalgebras 187. Extreme points of Ω( y ): Luxemburg’s problem in the infinite setting 21Appendix A. Technical results 27References 35 Mathematics Subject Classification.
Key words and phrases. spectral scales; doubly stochastic operators; extreme points; majoriza-tion; noncommutative L -space; semifinite von Neumann algebras.Fedor Sukochev was supported by the Australian Research Council (FL170100052). Introduction
The partial order ≺ for real n -vectors (customarily termed Hardy–Littlewood–P´olya majorization) was introduced in the early 20th century, by Muirhead [60],Lorenz [57], Dalton [18] and Schur [78] (see also a fundamental monograph byHardy, Littlewood and P´olya [32]). Hardy–Littlewood–P´olya majorization plays avital role in the study of function spaces, Banach lattices and interpolation the-ory and has important applications in stochastic analysis, numerical analysis, geo-metric inequalities, matrix theory, statistical theory, optimization and economictheory [59].The starting point of this paper is a well-known result in classic analysis dueto Hardy, Littlewood and P´olya: x ≺ y ∈ R n if and only if x belongs to the con-vex hull Ω( y ) of the set of all permutations of y [32] (see also [59, p.10] and [67]),i.e., the convex hull of { P y : P is a permutation matrix } . Alternatively, one cansay that { P y : P is a permutation matrix } is the set of all extreme points of { x ∈ R n : x ≺ y } . This result has an important noncommutative counterpart,which states that, in the setting of n × n matrices, a Hermitian matrix A be-longs to the doubly stochastic orbit of another Hermitian matrix B ( A is saidto be more chaotic than B for physical interpretation [2]) if and only if the vec-tor λ ( A ) of eigenvalues of A is majorized by that of B in the sense of Hardy–Littlewood–P´olya (denoted by A ≺ B ) [67]. This, in turn, is closely related tothe Birkhoff–von Neumann theorem identifying extreme points of doubly stochas-tic matrices with permutation matrices [8]. The (Hardy–Littlewood–P´olya) orbitΩ( B ) := { A is a Hermitian n × n matrix : A ≺ B } can be described in terms ofunitary mixing and convex functions. We record these well-known results as follows. Theorem . [2, Theorem 2.2] [4] Let A, B be Hermitian n × n matrices. Thefollowing conditions are equivalent:(a) A is in the doubly stochastic orbit of B , that is, A = T B for some doublystochastic operator T (i.e., a positive linear map which preserves the traceand the identity);(b) A is majorized by B , that is A ≺ B ;(c) A is in the convex hull of elements C which are unitarily equivalent to B ,i.e., C and B have the same eigenvalues;(d) A is in the convex hull of elements which are unitarily equivalent to B andcommute with A ;(e) P nk =1 ( λ ( A ) − t ) + ≤ P nk =1 ( λ ( B ) − t ) + for any t ∈ R and Tr( A ) = Tr( B ) ;(f ) for any convex function f on the real axis, we have Tr( f ( A )) ≤ Tr( f ( B )) .Hence, Tr stands for the standard trace of a matrix. Alberti and Uhlmann developed a unitary mixing theory which is motivated byphysical problems related to irreversible processes, the classification of mixed states(in the sense of Gibbs and von Neumann), and general diffusions [2, p.8]. One ofthe central problems considered by Alberti and Uhlmann is as follows [2, Chapter3, p.58]:“We,. . . , may ask ourselves how to formulate a variant of Theo-rem 1.1 in the setting of von Neumann algebras?” In [2, Theorem 2.2], the authors use any concave function on the real axis, Tr( f ( A )) ≥ Tr( f ( B ))”. Actually, it is equivalent with condition ( f ) by taking − f . LBERTI–UHLMANN PROBLEM 3
This paper addresses a number of seemingly disparate open problems, and weshall see that they are in fact deeply intertwined. These questions are loosely con-centrated around Alberti–Uhlmann attempts [2] to extend majorization theory forgeneral von Neumann algebras and Hiai–Nakamura attempts [33, 34, 36] to extendthe Hardy–Littlewood–P´olya majorization to the setting of strictly infinite semifi-nite von Neumann algebras. We explain below the connection of these two themeswith classical themes in analysis and algebra.The notion of majorization in the setting of Lebesgue measurable functions on(0 ,
1) is due to Hardy, Littlewood and P´olya [31] (see also [58, 71]). As we mentionabove, x ≺ y ∈ R n if and only if x belongs to the convex hull Ω( y ) of the set of allpermutations of y [32, 59, 67]. In 1967, Luxemburg (see [58, Problem 1]) asked fora continuous counterpart of this result: how to describe extreme points of the set Ω( f ) of all elements majorized by an integrable function f on a finite measure space? The special case for finite Lebesgue measure spaces was resolved by Ryff [72–74],who showed that the set of all extreme point of the orbit Ω( f ) coincides with theset of all measurable functions having the same decreasing rearrangement with f . However, in full generality, Luxemburg’s problem was answered only recentlyin [19]. In particular, this provides a variant of the equivalence between (b) and(c) in Theorem 1.1 for arbitrary finite measure spaces. Ryff [71] also asked for theequivalence between (a) and (b) in Theorem 1.1 in the setting of finite Lebesguemeasure spaces, i.e., whether the doubly stochastic orbit and the orbit in the senseof Hardy–Littlewood–P´olya coincide, and answered affirmatively later in [72]. Thisresult was extended by Day [20] to arbitrary finite measure spaces.In the particular case of finite von Neumann algebras, Alberti–Uhlmann problemcan be viewed as the noncommutative counterpart of problems due to Luxemburgand Ryff, which has been widely studied during the past decades. Recall that theHardy–Littlewood–P´olya majorization in finite matrices is defined in terms of vec-tors of eigenvalues of finite matrices. The eigenvalue function of a self-adjoint oper-ator, the analogue of vector of eigenvalues of a Hermitian finite matrices, in the non-commutative L -space affiliated with a finite von Neumann algebra was introducedby Murray and von Neumann [61] (also by Grothendieck [30], by Ovˇcinnikov [64]and by Petz [66]). In terms of eigenvalues functions, Kamei [48, 49] and Hiai [33]defined Hardy–Littlewood–P´olya majorization in this setting. Since then, severalmathematicians have made contribution to Alberti–Uhlmann problem for the pos-itive core of a finite von Neumann algebra [13, 33, 35, 36, 82, 83]. In the noncommu-tative L -space affiliated with a finite von Neumann algebra, the extreme pointsof the orbit (in the sense of Hardy–Littlewood–P´olya) of a self-adjoint operatorwere fully characterized in [19]. In Section 3, we complement results in [19, 33, 35]by proving the equivalence between (a) and (b) in Theorem 1.1 in this setting fi-nite von Neumann algebra (see Theorem 3.5), which provides a resolution of thenoncommutative counterpart of Ryff’s question for doubly stochastic orbits in thissetting.In the full generality, Alberti–Uhlmann problem can be viewed as a noncommu-tative and infinite version of Luxemburg’s problem and Ryff’s problem at the sametime. Due to the complicated nature of an infinite von Neumann algebra, Alberti–Uhlmann problem (even in the setting of infinite measure spaces) is substantiallymore difficult than Luxemburg’s problem and Ryff’s problem. From now on, wefocus on Alberti–Uhlmann problem in the strictly infinite setting. This setting J. HUANG AND F. SUKOCHEV however is plagued by numerous technical difficulties and below, we explain theirorigin and earlier attempts to overcome those.In 1946, Birkhoff [8] showed that the extreme points of the set of all doublystochastic matrices are permutation matrices and asked for an extension of thisresult to the infinite-dimensional case [9, Problem 111]. Since then, doubly sto-chastic operators have been actively studied by many mathematicians such as Is-bell [39, 40], Hiai [33, 35], Kaftal and Weiss [44, 46, 46] (see also [92]), Kendall [50],Komiya [53], Rattray and Peck [68], R´ev´esz [70], Sakamaki and Takahashi [77],and Tregub [88, 89]. In particular, the extreme points of infinite stochastic matri-ces are permutation matrices [39, 40, 50, 70]. However, the description of extremepoints of doubly stochastic operator on l ∞ (or a von Neumann algebra) is stillunclear (see [33, 88, 89] and references therein). The notion of Hardy–Littlewood–P´olya majorization for infinite sequences has been discussed by various authors(see [59, p.25] and references therein). A natural question in this area is about theequivalence between conditions (a) and (b) in Theorem 1.1 in the infinite setting,that is, does there exist an element y ∈ l ∞ such that the doubly stochastic orbit of y does not coincide with its orbit in the sense of Hardy–Littlewood–P´olya? (see nextparagraph for a more general question by Hiai [33]). As far as we know, there areno prior known examples showing that these two conditions are not equivalent inthe general infinite setting. Nevertheless, there are results giving partial answersto this question under additional assumptions such as the vectors are positive anddecreasing [46, 53, 77]. In Section 4, we provide a negative answer to this questionin the full generality.In the setting of finite von Neumann algebras, Hiai [33, Theorem 4.7 (1)] showedthat the doubly stochastic orbit of a positive operator coincides with its Hardy–Littlewood–P´olay orbit. Due to the lack of natural meaning of Hardy–Littlewood–P´olya majorization in the infinite case, there are several different extensions of thisnotion in the literature (see e.g. [2, 33, 35, 52, 63, 80]). Hiai and Nakamura [35] (seealso [59, p. 25] and [63, 80]) provided a natural definition of Hardy–Littlewood–P´olya majorization in terms of eigenvalue functions in the setting of infinite vonNeumann algebras:Let L ( M , τ ) be the noncommutative L -space affiliated with asemifinite von Neumann algebra equipped with a faithful normalsemifinite trace τ . Let x, y ∈ L ( M , τ ) be self-adjoint. Then, x is said to be majorized by y (denoted by x ≺ y ) in the sense ofHardy–Littlewood–P´olya if x + ≺≺ y + , x − ≺≺ y − and τ ( x ) = τ ( y ) [35, p. 7]. Here, ≺≺ stands for the Hardy–Littlewood–P´olyasubmajorization.In the finite setting, the above definition coincides with the classic definition ofHardy–Littlewood–P´olya majorization [35]. It turns out that doubly stochasticoperators behave quite differently in the infinite setting from the finite setting andHiai couldn’t extend [33, Theorem 4.7 (1)] to inifinite von Neumann algebras. Heconjectured that [33, p. 40]):Because the set of all doubly stochastic operators on a von Neu-mann algbra is no longer BW-compact [5] when the trace is infinite,the assumption that the trace is finite in [33, Theorem 4.7 (1)] seemsessential. LBERTI–UHLMANN PROBLEM 5
Hiai didn’t provide an example showing that the “finite trace” condition is sharp.In Section 4, we provide several examples confirming Hiai’s conjecture. Moreover,we show that for a semifinite von Neumann algebra M , the doubly stochastic orbitof y coincides with its Hardy–Littlewood–P´olya orbit for any self-adjoint element y ∈ L ( M , τ ) if and only if M is not σ -finite equipped with a faithful normalsemifinite infinite trace τ . In particular, this result shows that Hiai’s conjecture istrue when M is σ -finite and τ is infinite but this conjecture fails for non- σ -finiteinfinite von Neumann algebras.The following theorem is the main result of the present paper, which answers(extensions of) problems due to Luxemburg [58], due to Ryff [71], due to Hiai [33],and due to Alberti and Uhlmann [2] in the setting of infinite von Neumann algebras(all notations are introduced in Section 2). It extends numerous existing results inthe literature such as [19, 33, 35, 44, 45, 53, 71–74, 77]. Theorem . Let M be a semifinite von Neumann algebra equipped with asemifinite infinite faithful normal trace τ . For any y ∈ L ( M , τ ) h and any x ∈ L ( M , τ ) h , the following conditions are equivalent:(a). there exists two semifinite von Neumann algebra ( A , τ ) and ( B , τ ) with M ⊂ A , B and the restriction of τ (and τ ) on M coincides with τ , andthere exists a (normal) doubly stochastic operator ϕ : A → B such that ϕ ( y ) = x .(b). x ∈ Ω( y ) , i.e., x ≺ y (see Proposition 6.1 and Theorems 6.2 and 6.4);(c). τ ( x ) = τ ( y ) , τ (( x − r ) + ) ≤ τ (( y − r ) + ) and τ (( − x − r ) + ) ≤ τ (( − y − r ) + ) for all r ∈ R (see Proposition 2.3);(d). τ ( f ( x )) ≤ τ ( f ( y )) for all convex function f on R with f (0) = 0 such that f ( x ) and f ( y ) are integrable (see Proposition 5.2);In addition, the extreme points of Ω( y ) are those elements x ∈ L ( M , τ ) h satisfyingthat (see Theorem 7.1) for x + and y + and for any t ∈ (0 , ∞ ) , one of the followingoptions holds:(i). λ ( t ; x + ) = λ ( t ; y + ) ;(ii). λ ( t ; x + ) = λ ( t ; y + ) with the spectral projection E x + { λ ( t ; x + ) } being an atomin M and Z { s ; λ ( s ; x + )= λ ( t ; x + ) } λ ( s ; y + ) ds = λ ( t ; x + ) τ ( E x + ( { λ ( t ; x + ) } )) , and, for x − and y − , for any t ∈ (0 , ∞ ) , one of the following options holds:(i). λ ( t ; x − ) = λ ( t ; y − ) ;(ii). λ ( t ; x − ) = λ ( t ; y − ) with the spectral projection E x − { λ ( t ; x − ) } being an atomin M and Z { s ; λ ( s ; x − )= λ ( t ; x − ) } λ ( s ; y − ) ds = λ ( t ; x − ) τ ( E x − ( { λ ( t ; x − ) } )) . Moreover, if M is a semifinite infinite factor, then x ∈ { uyu ∗ : u ∈ U ( M ) } k·k [35,Theorem 3.5]; A and B can be chosen to be M if and only if M is not σ -finite (seeTheorems 4.6 and 6.4). The authors would like to thank Jean-Christophe Bourin, Thomas Scheckter andDmitriy Zanin for their helpful discussion and useful comments.
J. HUANG AND F. SUKOCHEV Preliminaries
In this section, we recall some notions of the theory of noncommutative integra-tion. In what follows, H is a Hilbert space and B ( H ) is the ∗ -algebra of all boundedlinear operators on H equipped with the uniform norm k·k ∞ , and is the identityoperator on H . Let M be a von Neumann algebra on H . We denote by P ( M )the collection of all projections in M and by U ( M ) the collection of all unitaryelements. For details on von Neumann algebra theory, the reader is referred toe.g. [42, 43] or [86]. General facts concerning measurable operators may be foundin [62] and [79] (see also [87, Chapter IX] and the forthcoming book [28]). Forconvenience of the reader, some of the basic definitions are recalled.2.1. τ -measurable operators and generalized singular value functions. Aclosed, densely defined operator x : D ( X ) → H with the domain D ( x ) is said tobe affiliated with M if yx ⊆ xy for all Y ∈ M ′ , where M ′ is the commutant of M . A closed, densely defined operator x : D ( x ) → H affiliated with M is saidto be measurable if there exists a sequence { p n } ∞ n =1 ⊂ P ( M ), such that p n ↑ , p n ( H ) ⊆ D ( X ) and − p n is a finite projection (with respect to M ) for all n .The collection of all measurable operators with respect to M is denoted by S ( M ),which is a unital ∗ -algebra with respect to strong sums and products (denotedsimply by x + y and xy for all x, y ∈ S ( M )).Let X be a self-adjoint operator affiliated with M . We denote its spectralmeasure by { E X } . It is well known that if X is an operator affiliated with M withthe polar decomposition X = U | X | , then U ∈ M and E ∈ M for all projections E ∈ { E | X | } . Moreover, X ∈ S ( M ) if and only if E | X | ( λ, ∞ ) is a finite projectionfor some λ >
0. It follows immediately that in the case when M is a von Neumannalgebra of type III or a type I factor, we have S ( M ) = M . For type II vonNeumann algebras, this is no longer true. From now on, let M be a semifinite vonNeumann algebra equipped with a faithful normal semifinite trace τ .An operator x ∈ S ( M ) is called τ -measurable if there exists a sequence { p n } ∞ n =1 in P ( M ) such that p n ↑ , p n ( H ) ⊆ D ( x ) and τ ( − p n ) < ∞ for all n . Thecollection S ( M , τ ) of all τ -measurable operators is a unital ∗ -subalgebra of S ( M ).It is well known that a linear operator x belongs to S ( M , τ ) if and only if x ∈ S ( M )and there exists λ > τ ( E | x | ( λ, ∞ )) < ∞ . Alternatively, an unboundedoperator x affiliated with M is τ -measurable (see [29]) if and only if τ (cid:16) E | x | (cid:0) n, ∞ (cid:1)(cid:17) → , n → ∞ . The set of all self-adjoint elements in S ( M , τ ) is denoted by S ( M , τ ) h , which isa real linear subspace of S ( M , τ ). The set of all positive elements in S ( M , τ ) h isdenoted by S ( M , τ ) + . Definition . Let a semifinite von Neumann algebra M be equipped with a faithfulnormal semi-finite trace τ and let x ∈ S ( M , τ ) . The generalized singular valuefunction µ ( x ) : t → µ ( t ; x ) of the operator x is defined by setting µ ( s ; x ) = inf {k xp k ∞ : p ∈ P ( M ) with τ ( − p ) ≤ s } . An equivalent definition in terms of the distribution function of the operator x is the following. For every self-adjoint operator x ∈ S ( M , τ ) , setting the spectraldistribution function of x by d x ( t ) = τ ( E x ( t, ∞ )) , t > . LBERTI–UHLMANN PROBLEM 7
It is clear that the function d ( x ) : R → [0 , τ ( )] is decreasing and the normality ofthe trace implies that d ( x ) is right-continuous. We have (see e.g. [29]) µ ( t ; x ) = inf { s ≥ d | x | ( s ) ≤ t } . (1)It is obvious [29, Remark 3.3] that d ( | x | ) = d ( µ ( x )) . (2)An element x ∈ S ( M , τ ) is said to be τ -compact if µ ( t ; x ) → t → ∞ . Wedenote by S ( M , τ ) the subspace of S ( M , τ ) consisting of all τ -compact elementsin S ( M , τ ).Consider the algebra M = L ∞ (0 , ∞ ) of all Lebesgue measurable essentiallybounded functions on (0 , ∞ ). The algebra M can be seen as an abelian von Neu-mann algebra acting via multiplication on the Hilbert space H = L (0 , ∞ ), withthe trace given by integration with respect to Lebesgue measure m. It is easy tosee that the algebra of all τ -measurable operators affiliated with M can be identi-fied with the subalgebra S (0 , ∞ ) of the algebra of Lebesgue measurable functions L (0 , ∞ ) which consists of all functions x such that m ( {| x | > s } ) is finite for some s >
0. It should also be pointed out that the generalized singular value function µ ( x ) is precisely the decreasing rearrangement µ ( x ) of the function | x | (see e.g. [54])defined by µ ( t ; x ) = inf { s ≥ m ( {| x | ≥ s } ) ≤ t } . If M = B ( H ) (respectively, l ∞ ) and τ is the standard trace Tr (respectively, thecounting measure on N ), then it is not difficult to see that S ( M ) = S ( M , τ ) = M . In this case, for x ∈ S ( M , τ ) we have µ ( n ; x ) = µ ( t ; x ) , t ∈ [ n, n + 1) , n ≥ . The sequence { µ ( n ; x ) } n ≥ is just the sequence of singular values of the operator x .2.2. Classic Hardy–Littlewood–P´olya majorization and submajorization.
Let ( L (0 , ∞ ) , k·k L (0 , ∞ ) ) be the L -space of Lebesgue measurable functions on thesemi-axis (0 , ∞ ). The pair L ( M , τ ) = { x ∈ S ( M , τ ) : µ ( x ) ∈ L (0 , ∞ ) } , k x k L ( M ,τ ) := k µ ( x ) k L (0 , ∞ ) defines a Banach bimodule affiliated with M [47] (see also [26, 55]). For brevity,we denote k·k L ( M ,τ ) by k·k . Clearly, L ( M , τ ) ⊂ S ( M , τ ). Using the extendedtrace τ : S ( M , τ ) + → [0 , ∞ ] to a linear functional on S ( M , τ ) , denoted again by τ , the noncommutative L -space can be defined by L ( M , τ ) = { x ∈ S ( M , τ ) : τ ( | x | ) < ∞} . (see e.g. [24, 55]). Let us denote L ( M , τ ) h = { x ∈ L ( M , τ ) : x = x ∗ } and L ( M , τ ) + = L ( M , τ ) ∩ S ( M , τ ) + . If x, y ∈ S ( M , τ ), then x is said to be submajorized (in the sense of Hardy–Littlewood–P´olya) by y , denoted by x ≺≺ y , if Z t µ ( s ; x ) ds ≤ Z t µ ( s ; y ) ds for all t ≥ x, y ∈ S (0 , ∞ ), x ≺≺ y if and only if R t µ ( s ; x ) ds ≤ R t µ ( s ; y ) ds , t ≥
0. For any x, y ∈ S ( M , τ ), we have [23, 29] µ ( xy ) ≺≺ µ ( x ) µ ( y ) . (3) J. HUANG AND F. SUKOCHEV
Assume that M is a finite von Neumann algebra equipped with a faithful normalfinite tracial state τ . We note that the space S ( M , τ ) is the set of all densely definedclosed linear operators x affiliated with M . However, if the trace τ is infinite, thenthere are densely defined closed linear operators which are not τ -measurable.We introduce the notion of spectral scales (see [66], see also [3, 22, 28, 35, 36]).If x ∈ S ( M , τ ) h , then the spectral scales (also called eigenvalue functions ) λ ( x ) :[0 , → ( −∞ , ∞ ] λ ( t ; x ) = inf { s ∈ R : d ( s ; x ) t } , t ∈ [0 , . The spectral scale λ ( x ) is decreasing right-continuous functions. If x ∈ S ( M , τ ) + ,then it is evident that λ ( t ; x ) = µ ( t ; x ) for all t ∈ [0 , M = L ∞ (0 ,
1) and τ ( f ) = R f dm , where m the Lebesgue measureon (0 , S ( M , τ ) h consists of all real measurable functions f on (0 , f , λ ( f ) coincides with the right-continuous equimeasurable nonincreasingrearrangement δ f of f (see e.g. [35]): λ ( t ; f ) = δ f ( t ) = inf { s ∈ R : m ( { x ∈ X : f ( x ) > s } ) t } , t ∈ [0 , . We note that for every x ∈ L ( M , τ ) h , we have (see e.g. [22], [66, Proposition 1]and [28, Chapter III, Proposition 5.5])(4) τ ( x ) = Z λ ( t ; x ) dt. For every f, g ∈ L ( M , τ ) , g is said to be majorized by f in the sense of Hardy–Littlewood–P´olya (written by g ≺ f ) if Z s λ ( t ; g ) dt Z s λ ( t ; f ) dt for all s ∈ [0 ,
1) and Z λ ( t ; g ) dt = Z λ ( t ; f ) dt. For a self-adjoint element y ∈ L ( M , τ ) h , we denoteΩ( y ) := { x ∈ L ( M , τ ) h : x ≺ y } . We note [36, Theorem 3.2] that λ ( x ) − λ ( y ) ≺ λ ( x − y ) , ∀ x, y ∈ L ( M , τ ) h . (5)Let M be a semifinite von Neumann algebra. For every x ∈ S ( M , τ ) + and s ∈ R , if e ∈ P ( M ) is such that E x ( s, ∞ ) ≤ e ≤ E x [ s, ∞ ) , then (see [22] or [28, Chapter III, Proposition 2.10 and Lemma 7.10]): µ ( t ; xe ) = µ ( t ; x ) for all t ∈ [0 , τ ( e )) , (6)and µ ( t ; xe ⊥ ) = µ ( t + τ ( e ); x ) for all t ∈ [0 , τ ( e ⊥ )) . (7) LBERTI–UHLMANN PROBLEM 9
Hardy–Littlewood–P´olya majorization in the infinite setting.
Thedefinition of classic Hardy–Littlewood–P´olya majorization can not be extendedto the infinite setting directly. We adopt the definition suggested by Hiai andNakamura [35] (see also [63, 77]). For definition in the setting of positive infinitesequences, see [41, 44–46, 53, 56, 92]).
Definition . Let M be a semifinite von Neumann algebra equipped with a semifi-nite faithful normal trace τ . Let x, y ∈ L ( M , τ ) h . We say that x is majorizedby y (in the sense of Hardy–Littlewood–P´olya, denoted by x ≺ y ) if x + ≺≺ y + , x − ≺≺ y − and τ ( x ) = τ ( y ) . When τ is finite, Definition 2.2 coincides with the classic definition of Hardy–Littlewood–P´olya majorization [35, Proposition 1.3].We show the equivlance between (b) and (c) in Theorem 1.2 (similarly resultsfor positive operators can be found in [33, 63, 80]). Proposition . Let M be a semifinite von Neumann algebra equipped with asemifinite faithful normal trace τ . Let x, y ∈ L ( M , τ ) h . Then, x ≺ y if and onlyif τ (( x − t ) + ) ≤ τ (( y − t ) + ) and τ (( − x − t ) + ) ≤ τ (( − y − t ) + ) for all t ∈ R (or R + )and τ ( x ) = τ ( y ) .Proof. By [33, Proposition 2.3] (see also [85, Theorem 4]), x + ≺≺ y + implies that τ (( x − t ) + ) = τ (( x + − t ) + ) ≤ τ (( y + − t ) + ) = τ (( y + − t ) + ) for all t >
0. The sameargument shows that τ (( x − − t ) + ) ≤ τ (( y − − t ) + ) for all t >
0. When t ≤
0, bythe spectral theorem, we have τ (( x − t ) + ) = ∞ = τ (( y − t ) + )and τ (( − x − t ) − ) = ∞ = τ (( − y − t ) − ) , which prove the “ ⇒ ” implication.The “ ⇐ ” implication follows immediately from [33, Proposition 2.3]. (cid:3) Remark . If τ is finite, then the sufficient condition in Proposition 2.3 is equiva-lent with τ (( x − t ) + ) ≤ τ (( y − t ) + ) for all t ∈ R and τ ( x ) = τ ( y ) (see [35, Proposition1.2. (1) and Proposition 1.3]). Proposition . [35, Proposition 1.1] (see also [28, 55]) If x ∈ L ( M , τ ) h , thenfor every < s < τ ( ) , Z s µ ( t ; x + ) dt = sup { τ ( xa ) : a ∈ M , ≤ a ≤ , τ ( a ) = s } . If M is non-atomic, then the above equality holds under the condition that theelement a varies in the set of projections e ∈ P ( M ) with τ ( e ) = s . We draw reader’s attention that [29, Lemma 4.1] contains an inaccuracy. Namely,the second assertion in [29, Lemma 4.1] is false in general (see [28, Chapter III,Remark 9.7]).
Remark . For any x ∈ L ( M , τ ) h , λ ( x ) is defined by λ ( s ; x ) = inf { t ∈ R : τ ( e ( t, ∞ ) ( x )) ≤ s } . We note that whenever τ ( ) = ∞ , λ ( x ) ≥ for any x ∈ L ( M , τ ) h . Otherwise, > λ ( s ; x ) = inf { t ∈ R : τ ( e ( t, ∞ ) ( x )) ≤ s } for some s > .Hence, there exists s ′ < such that τ ( e ( s ′ , ∞ ) ( x )) ≤ s , i.e., τ ( e ( −∞ ,s ′ ] ( x )) = ∞ .This implies that x is not τ -compact, which contradicts with the assumption that x ∈ L ( M , τ ) . In particular, by the definition of eigenvalue functions, we obtainthat λ ( x ) = λ ( x + ) = µ ( x + ) for any x ∈ L ( M , τ ) h [35, p.5]. Proposition . If τ ( ) = ∞ . Let a, b ∈ L ( M , τ ) h . We have Z t λ ( s ; ( a + b ) + ) ds ≤ Z t λ ( s ; a + ) ds + Z t λ ( s ; b + ) ds and Z t λ ( s ; ( a + b ) − ) ds ≤ Z t λ ( s ; a − ) ds + Z t λ ( s ; b − ) ds. In particular, if a, b ≺ c ∈ L ( M , τ ) h , then a + b ≺ c .Proof. Without loss of generality, we may assume that M is non-atomic (seee.g. [55, Lemma 2.3.18]).Let t > e ∈ P ( M ) such that τ ( e ) = t , by Proposition 2.5,we have τ (( a + b ) e ) = τ ( ae ) + τ ( be ) ≤ Z t λ ( s ; a ) ds + Z t λ ( s ; b ) ds. By Proposition 2.5, we have Z t λ ( s ; a + b ) ds ≤ Z t λ ( s ; a ) ds + Z t λ ( s ; b ) ds. By Remark 2.6, we complete the proof for the first inequality. The second inequalityfollows by taking − a and − b . The last assertion is a straightforward consequenceof the above results. (cid:3) Alberti–Uhlmann problem in the setting of finite von Neumannalgebras
We recall that definition of doubly stochastic operators on von Neumann alge-bras, which were introduced by Tregub [88, 89], Kamei [49] and Hiai [33].
Definition . [33] Let A and B be semifinite von Neumann algebras equippedwith semifinite faithful normal traces τ and τ . A positive linear map ϕ : A → B is said to be doubly stochastic if ϕ ( A ) = B and τ ◦ ϕ = τ on A + . Throughout this section, we always assume that M is a finite von Neumannalgebra equipped with a faithful normal tracial state τ . We denote by DS ( M ) theset of all doubly stochastic operators on M . Using a recent advance in [19], weprovide a complete resolution of Alberti–Uhlmann problem in this setting.Note that ϕ is k·k -bounded on L ( M , τ ) ∩ M . Hence, it can be extended to abounded linear map on L ( M , τ ) (denoted by the same ϕ ) [33, Section 4]. Whenthe trace τ is finite, every doubly stochastic operator ϕ : M → M is normal [33,Remark 4.2 (2)], i.e., ϕ ( x i ) ↑ ϕ ( x ) if x i ↑ x ∈ M + . For any y ∈ L ( M , τ ) h , wedefine Ω( y ) := { x ∈ L ( M , τ ) h : x ≺ y } . The following result is folklore. Due to the lack of suitable references, we providea short proof below.
Proposition . Let y ∈ L ( M , τ ) h . Then, Ω( y ) is a convex set which is closedin L ( M , τ ) . LBERTI–UHLMANN PROBLEM 11
Proof.
By (2.7), Ω( y ) is convex. Let { x n } be a sequence in Ω( y ) converging in L .We define x := k·k − lim n →∞ x n . Hence, for any t ∈ (0 , Z t λ ( s ; x ) − λ ( s ; x n ) ds P rop. . ≤ Z t λ (cid:0) s ; λ ( x ) − λ ( x n ) (cid:1) ds (5) ≤ Z t λ ( s ; x − x n ) ds ≤ Z λ ( s ; ( x − x n ) + ) ds + Z λ ( s ; ( x − x n ) − ) ds = k x − x n k → . (8)Since x n ≺ y , it follows that for any t ∈ (0 , Z t λ ( s ; y ) ds ≥ Z t λ ( s ; x n ) ds ≥ Z t λ ( s ; x ) ds − k x − x n k → Z t λ ( s ; x ) ds as n → ∞ . Hence, R t λ ( s ; y ) ≥ R t λ ( s ; x ) ds , t ∈ (0 , Z λ ( s ; x n ) − λ ( s ; x ) ds ≤ k x − x n k → . Hence, τ ( y ) = τ ( x n ) ≤ τ ( x ), which together with R t λ ( s ; y ) ds ≥ R t λ ( s ; x ) ds impliesthat x ≺ y . (cid:3) Remark . Note that Ω( y ) is k·k -bounded. Moreover, for each ε > , thereexists a δ > (see e.g. [27, Theorem 3.1], [25, Lemma 6.1], [37]) such that if e ∈ P ( M ) with τ ( e ) < δ , then for all x ∈ Ω( y ) , we have | τ ( xe ) | = | τ ( x + e ) | + | τ ( x − e ) | P rop. . ≤ Z δ λ ( s ; x + ) ds + Z δ λ ( s ; x − ) ds ≤ Z δ λ ( s ; y + ) + λ ( s ; y − ) ds < ε, which shows that Ω( y ) is relatively weakly compact (see e.g. [27, Corollary 4.5], seealso [1, 69, 91]). Since Ω( y ) is a k·k -closed convex subset in L ( M , τ ) , it followsthat Ω( y ) is weakly closed [16, Chapter V., Theorem 1.4]. Then, by the Krein–Milman theorem, Ω( y ) is σ ( L , L ∞ ) -closure of the convex hull of all extreme pointson Ω( y ) . Again, by [16, Chapter V., Theorem 1.4], Ω( y ) is L -closure of the convexhull of all extreme points on Ω( y ) . The assumption that the operators are positive plays no role in the proof of [33,Theorem 4.7. (1)]. So, the “only if” part below follows from the same argument.Using a result by Day [20], we provide below a similar (but simpler) proof forcompleteness. We note that the special case for finite factors was described in [35](see also [80, Theorem 2.18]). The following extends [49] (see also [80, Theorem2.18] and [33, Theorem 4.7. (1)]).
Theorem . Let x, y ∈ L ( M , τ ) , then x ≺ y if and only if there exists a ϕ ∈ DS ( M ) such that x = ϕ ( y ) .Proof. ⇒ . Let A (resp. B ) be the commutative von Neumann subalgebra of M generated by all spectral projections of y (resp. x ). Since ( A , τ ) and ( B , τ ) are twonormalized measure spaces, by [20, Theorem 4.9], there exists a doubly stochasticoperator ψ : A → B such that x = ψ ( y ). Taking the conditional expectation E : M → A [21, Proposition 2.1], we define ϕ : M → M by ϕ ( z ) = ψ ◦ E , z ∈ M .Clearly, ϕ is a doubly stochastic operator and ϕ ( y ) = x . ⇐ . Assume that there exists a ϕ ∈ DS ( M ) such that x = ϕ ( y ). By the definitionof doubly stochastic operators, we have τ ( x ) = τ ( y ). Let y n := ye y ( − n, ∞ ) + 11 − τ ( e y ( − n, ∞ )) Z τ ( e y ( − n, ∞ )) λ ( s ; y ) ds. In particular, y n ≺ y (see e.g. [84, p.198]). By [33, Theorem 4.5 (2)], we have ϕ ( y n + n ) ≺ y n + n , i.e., ϕ ( y n ) ≺ y n ≺ y . By [33, Proposition 4.1], we have k x − ϕ ( y n ) k = k ϕ ( y − y n ) k ≤ k y − y n k → . It follows from Proposition 3.2 that x ≺ y . (cid:3) Combining Theorem 3.4, [36, Proposition 1.3] and [19, Theorem 1.1] (with Krein–Milman theorem), we present the finite von Neumann algebra version of [2, Theorem2.2], which provides a complete answer to Alberti and Uhlmann’s problem [2, Chap-ter 3] in this setting.
Theorem . For any x, y ∈ L ( M , τ ) h , the following conditions are equivalent:(a). x is in the doubly stochastic orbit of y ;(b). x ≺ y ;(c). x belongs to the L -closure of the convex hull of all elements x ∈ L ( M , τ ) satisfying that for any t ∈ (0 , , one of the following options holds:(i). λ ( t ; x ) = λ ( t ; y ) ;(ii). λ ( t ; x ) = λ ( t ; y ) with the spectral projection E x { λ ( t ; x ) } being an atomin M and Z { s ; λ ( s ; x )= λ ( t ; x ) } λ ( s ; y ) ds = λ ( t ; x ) τ ( E x ( { λ ( t ; x ) } )) . (d). x belongs to the L -closure of the convex hull of all elements x ∈ L ( A , τ ) satisfying that for any t ∈ (0 , , one of the following options holds:(i). λ ( t ; x ) = λ ( t ; y ) ;(ii). λ ( t ; x ) = λ ( t ; y ) with the spectral projection E x { λ ( t ; x ) } being an atomin A and Z { s ; λ ( s ; x )= λ ( t ; x ) } λ ( s ; y ) ds = λ ( t ; x ) τ ( E x ( { λ ( t ; x ) } )) . Here, A is an abelian von Neumann subalgebra of M such that x ∈ L ( A , τ ) .(e). τ ( x ) = τ ( y ) and τ (( x − r ) + ) ≤ τ (( y − r ) + ) for all r ∈ R ;(f ). τ ( f ( x )) ≤ τ ( f ( y )) for all convex function f on R ;(g). τ ( | x − r | ) ≤ τ ( | y − r | ) for all r ∈ R ;(h). R t λ ( s ; f ( x )) ds ≤ R t λ ( s ; f ( y )) ds for all convex function f on R .If, in addition, M is a factor, then co { uyu ∗ : u ∈ U ( M ) } k·k coincides with thedoubly stochastic orbit of y [35, Theorem 2.3]. Since x ≺ y is equivalent with x ≺ λ ( y ) and [19, Theorem 1.1] holds for x ∈ L ( A , τ ) and λ ( y ) ∈ L (0 ,
1) (see the the beginning of [19, Section 3]), it follows that the extreme points of { z ≺ λ ( y ) : z ∈ L ( A , τ ) } are those satisfying conditions (i) and (ii). LBERTI–UHLMANN PROBLEM 13 On Hiai’s conjecture: Ryff’s problem in the infinite setting
Throughout this section, we always assume that L ∞ (0 , ∞ ) is the space of all(classes of) bounded real Lebesgue measurable function on (0 , ∞ ). Let x, y ∈ L (0 , ∞ ). Recall that x is said to be majorized by y (denoted by x ≺ y ) in thesense of Hardy–Littlewood–P´olya if x + ≺≺ y + , x − ≺≺ y − and R R + xdt = R R + ydt .Following Birkhoff’s result [8] and his problem 111 [9], numerous results ondoubly stochastic operators in the infinite setting appeared (see e.g. [33, 39, 40,50, 53, 65, 68, 70, 75–77, 93] and references therein). Hiai’s proof of Theorem 4.7in [33] relies on the so-called BW-compactness [5] of DS ( M ). We present severalexamples below showing that Hiai’s conjecture (stated in the introduction) is truein the setting of σ -finite infinite von Neumann alagebras.Recall that any doubly stochastic operator on M can be canonically extendedto a linear map on L ( M , τ ) + M [33, Section 4]. Hence, by a doubly stochasticoperator ϕ on M , we always mean that ϕ acts on L ( M , τ ) + M .The first example concerning on normal doubly stochastic operators (for a doublystochastic operator ϕ on a von Neumann algebra M , ϕ is said to be normal if ϕ ( x i ) ↑ ϕ ( x ) if x i ↑ x ∈ M + ). Hiai [33] showed that a doubly stochastic operatoron a finite von Neuman algebra is necessarily normal and he gave an exampleof doubly stochastic operator on an infinite von Neumann algebra, which is notnormal. We show that one cannot expect that for any x ≺ y ∈ L (0 , ∞ ), thereexists a normal doubly stochastic operator such that ϕ ( y ) = x . Example 4.1.
There exists x ≺ y ∈ L (0 , ∞ ) such that there exists no normaldoubly stochastic operator ϕ such that ϕ ( y ) = x . Let y = µ ( y ) be an arbitrarystrictly positive integrable function on (0 , ∞ ) . We define x by x ( t ) = y ( t − , t ≥ , and x ( t ) = 0 , ≤ t < . Clearly, x ≺ y . Assume that there exists a normaldoubly stochastic operator ϕ : L ∞ (0 , ∞ ) → L ∞ (0 , ∞ ) such that (the extension of ϕ on L (0 , ∞ ) ) ϕ ( y ) = x . Then, for any indicator function χ (0 ,n ] , n ≥ , we have µ ( n ; y ) ϕ ( χ (0 ,n ] ) ≤ ϕ ( y ) . In particular, ϕ ( χ (0 ,n ] ) = ϕ ( χ (0 ,n ] ) χ [1 , ∞ ) . Note that ϕ ( χ (0 ,n ] ) ≤ ϕ ( χ R + ) ≤ χ R + .We obtain that ϕ ( χ (0 ,n ] ) ≤ χ [1 , ∞ ) . Since χ (0 ,n ] ↑ χ R + , it follows that χ R + = ϕ ( χ R + ) = sup ϕ ( χ (0 ,n ] ) ≤ χ [1 , ∞ ) , which isa contradiction. The second example shows that if one consider L ( M , τ ) + M (or simply M ),then there exists x ≺ y such that there exists no doubly stochastic operator ϕ suchthat ϕ ( y ) = x . Example 4.2.
Let y := χ R + and x := χ (1 , ∞ ) . Clearly, x ≺ y . Assume that x = ϕ ( y ) for some doubly stochastic operator ϕ . Then, χ R + = ϕ ( y ) = x = χ (1 , ∞ ) ,which is a contradiction. The following theorem shows that even if we restrict x, y in L (0 , ∞ ) and relaxthe restriction (i.e. normality) on ϕ , there are still element x with x ≺ y , whichare not in the doubly stochastic orbit of y , which again confirms Hiai’s conjecture.In Lemma 4.5, we will construct a similar but much more complicated example ina more general setting. Example 4.3.
Let M = l ∞ equipped with the standard trace. There exist x ≺ y ∈ ℓ such that there exists no doubly stochastic operator ϕ : l ∞ → l ∞ with ϕ ( y ) = x .Let y := (0 , , , · · · , n , · · · ) and x := (1 , , · · · , n , · · · ) . Assume that thereexists a doubly stochastic operator ϕ on l ∞ such that ϕ ( y ) = x . Since every doublystochastic operator is bounded on ℓ (with respect to the L -norm), it follows thatthe restriction of ϕ on ℓ can be represented as an infinite matrix ( a ij ) (see e.g. [81,Example 4.13]). Since ϕ preserves the trace, it follows that P ∞ i =1 a ij = 1 for every j . Moreover, since ϕ ( P ni =1 e n ) ≤ ϕ ( ) = and ϕ is positive, it follows that P nj =1 a ij ≤ for every n and a ij ≥ for any ≤ i, j < ∞ .By ϕ ( y ) = x , we obtain that P ∞ j =1 a ij y j = x i , i.e., P ∞ j =2 a ij j − = i − . When i = 1 , we have ∞ X j =2 a j j − = 1 . Since P nj =1 a j ≤ , it follows that a = 1 and a j = 0 whenever j = 2 . Recallthat P ∞ i =1 a i = 1 . We obtain that a i = 0 whenever i = 1 . Arguing inductively,we show that a n,n +1 = 1 , n ≥ and a i,j = 0 , i + 1 = j. In particular, a i = 0 for any i ≥ . Hence, ϕ ( e ) = 0 , which contradicts with theassumption that ϕ preserves the trace. Remark . It is shown in [45, Corollary 6.1, S(ii ′ )] that if x = µ ( x ) , y = µ ( y ) ∈ ℓ such that x ≺ y , then there exists a doubly stochastic operator ϕ suchthat x = ϕ ( y ) (see also [53, Corollary 2] and [77, Theorems 4 and 5] for similarresults). We show below that for any σ -finite infinite von Neumann algebra, there alwaysexists y such that the doubly stochastic orbit of y does not coincide with the orbitof y in the sense of Hardy–Littlewodd–P´olya. Before proceeding, we need thefollowing generalization of Example 4.3. The key instrument of the proof belowis the description of extreme points of Ω( y ) in the setting when atoms in l ∞ havedifferent traces (see [19], see also Theorem 7.1). Lemma . Let M = l ∞ equipped with a semifinite faithful normal trace τ suchthat τ ( e ) < τ ( e ) < · · · < τ ( e n ) < · · · , where e , e , · · · are the unit elements of l ∞ . There exist x ≺ y ∈ ℓ such that there exists no doubly stochastic operator ϕ : l ∞ → l ∞ with ϕ ( y ) = x .Proof. For the sake of convenience, we denote w n = τ ( e n ) for any n ≥ y := (cid:18) , w , w , · · · , n − w n , · · · , (cid:19) ∈ ℓ and x := w , w ( w − w ) + w w w , w ( w − w ) + w w w , · · · , n − w n − ( w n − − w ) + w n − w n w n − , · · · ! . LBERTI–UHLMANN PROBLEM 15
Note that µ ( y ) = P n ≥ n − w n χ [ P n − k =2 w k , P nk =2 w k ) . Define an averaging opera-tor [55, 84] E : L ∞ (0 , ∞ ) → L ∞ (0 , ∞ ) by E ( f ) = X n ≥ w n Z P nk =1 w k P n − k =1 w k f ( s ) ds · χ [ P n − k =1 w k , P nk =1 w k ) . In particular, E ( µ ( y )) = µ ( x ). Recall that averaging operators are doubly stochas-tic operators (see e.g. [55, Lemma 3.6.2], [84, p.198] and [93]). We obtain that x ≺ y (in particular, x is an extreme point of Ω( y ), see Theorem 7.1).Assume that there exists a doubly stochastic operator ϕ on l ∞ such that ϕ ( y ) = x . For every n , we denote ϕ ( e n ) = ( a ,n , a ,n , · · · , a k,n , · · · ) . Since ϕ is positive, it follows that a kn ≥ k, n ≥
1. Since ϕ preserves the traceand τ ( e n ) = w n , it follows that w n = ∞ X k =1 a k,n w k . Moreover, since ϕ ( ) = , it follows that ∞ X n =1 a k,n ≤ k ≥ ϕ on ℓ is continuous in k·k , it follows that ϕ is normalon ℓ (see e.g. [26, Proposition 2 (iii)]). We may view the restriction of ϕ on ℓ asan infinite matrix ( a k,n ).Since P ∞ n =1 a ,n ≤ ∞ X n =1 a ,n y n = ∞ X n =2 a ,n y n = x = y > y n , n > , it follows that a , = 1 and a ,n = 0 when n = 2. Arguing inductively, we obtainthat ( a k,n ) = · · · w − w w w w · · · w − w w w w · · · ... ... ... ... . . . Hence, ϕ ( e ) = 0, which contradicts with the assumption that ϕ preserves thetrace. (cid:3) Now, we are ready to show that Hiai’s conjecture is true for any σ -finite infinitevon Neumann algebra. Theorem . Let M be a σ -finite von Neumann algebra equipped with a semifiniteinfinite faithful normal trace. There are x, y ∈ L ( M , τ ) such that there exists nodoubly stochastic operator ϕ on M such that ϕ ( y ) = x .Proof. Since M is σ -finite, it follows that there exists a sequence { p n } n ≥ of disjoint τ -finite projections such that ∨ p n = . Moreover, we may assume that τ ( p n ) isstrictly increasing. The k·k -closure of the linear span of { p n } can be viewed as the ℓ -sequence space generated by atoms { p n } . By Lemma 4.5, there exists x, y ∈ ℓ such that there exists no doubly stochastic operator ϕ : ℓ → ℓ with ϕ ( y ) = x .Assume that there exists a doubly stochastic operator ϕ : M → M such that ϕ ( y ) = x . Let E be the conditional expectation from M to ℓ ∞ (the von Neumannsubalgebra of M generated by { p n } ). Clearly, E ◦ ϕ : ℓ ∞ → ℓ ∞ is a doublystochastic operator which satisfies that E ◦ ϕ ( y ) = E ( x ) = x . This contradicts withthe assumption imposed on x and y . (cid:3) A decomposition theorem
In view of Theorem 4.6, to study Alberti–Uhlmann problem in the infinite set-ting, we have to relax the restriction on the doubly stochastic operators. In thenext section, we will show that there exists a normal doubly stochastic operator ϕ on a larger algebra such that ϕ ( y ) = x provided x ≺ y . Before proceeding to thisresult, we prove a decomposition theorem for Hardy–Littlewood-P´olay majorizationin this section.Recall a result due to Ryff that [6, Chapter 2, Corollary 7.6] (see also [72]) forany 0 ≤ f ∈ L (0 , ∞ ), there exists a measure-preserving transformation σ (i.e., m ( σ − ( E )) = m ( E ) for any measurable set E ) from the support s ( f ) of f onto thesupport of µ ( f ) such that f = µ ( f ) ◦ σ. If f ∈ L ( X, ν ) h (( X, ν ) is a finite non-atomic measure space), then there exists ameasure-preserving transformation σ from X onto (0 , ν ( X )) such that [72] f = λ ( f ) ◦ σ. The following proposition is the key to extend Theorem 1.1 to the infinite setting(see also [12, Lemma 3.2], [84, Proposition 19] and [11, Lemma 3.2] for similarresults).
Proposition . If x ≺ y ∈ L (0 , ∞ ) , then there exist sequences of disjoint sets { A n ⊂ R + ⊕ R + } and { B n ⊂ R + ⊕ R + } such that m ( A n ) = m ( B n ) < ∞ , ∪ A n = ∪ B n = R + ⊕ R + and ( x ⊕ χ B n ≺ ( y ⊕ χ A n (on R + ⊕ R + ) for every n ≥ .Proof. Let σ x + : s ( x + ) → (0 , m ( s ( x + ))) , σ y + : s ( y + ) → (0 , m ( s ( y + ))) ,σ x − : s ( x − ) → (0 , m ( s ( x − ))) , σ y − : s ( y − ) → (0 , m ( s ( y − )))be measure-preserving transformations such that x + = λ ( x + ) ◦ σ x + , x − = λ ( x − ) ◦ σ x − , y + = λ ( y + ) ◦ σ y + , y − = λ ( y − ) ◦ σ y − . Define X := { s : µ ( s ; y + ) > µ ( s ; x + ) } ,X := { s : µ ( s ; y + ) < µ ( s ; x + ) } ,X := { s : µ ( s ; y + ) = µ ( s ; x + ) } . Let { Γ k : m (Γ k ) < ∞} k ≥ be a partition of X . Since τ ( y + ) ≥ τ ( x + ), it followsthat there exists a collection { Γ k : m (Γ k ) < ∞} k ≥ of disjoint subsets of X suchthat Z Γ k ∪ Γ k µ ( y + ) − µ ( x + ) = 0 . LBERTI–UHLMANN PROBLEM 17
We denote by Γ := X \ ( ∪ k ≥ Γ k ) (it may be the empty set). We also define apartition { Γ k : m (Γ k ) < ∞} k ≥ of X . For brevity, we numerate { Γ k ∪ Γ k } k ≥ ∪{ Γ k } k ≥ as { ∆ k } k ≥ and let A := Γ .Similarly, we construct a partition { Ω k : m (Ω k ) < ∞} k ≥ of { s : µ ( s ; y − ) >µ ( s ; x − ) } and a decomposition { Ω k : m (Ω k ) < ∞} k ≥ of { s : µ ( s ; y − ) > µ ( s ; x − ) } such that Z Ω k ∪ Ω k µ ( y − ) − µ ( x − ) = 0 . We denote by Ω := { s : µ ( s ; y − ) > µ ( s ; x − ) } \ ( ∪ k ≥ Ω k )(it may be the empty set). We also define a partition { Ω k : m (Ω k ) < ∞} k ≥ of { s : µ ( s ; y − ) = µ ( s ; x − ) } . For brevity, we numerate { Ω k ∪ Ω k } k ≥ ∪ { Ω k } k ≥ as { Λ k } k ≥ and let B := Ω .We consider couples (cid:16) ( σ x + ) − (∆ k ∩ (0 , m ( s ( x + )))) , ( σ y + ) − (∆ k ∩ (0 , m ( s ( y + )))) (cid:17) and (cid:16) ( σ x − ) − (Λ k ∩ (0 , m ( s ( x − )))) , ( σ y − ) − (Λ k ∩ (0 , m ( s ( y − )))) (cid:17) . Note that the measures of elements in any couple above may not be the same. Wecan find sequences { ∆ cxk } , { ∆ cyk } , { Λ cxk } and { Λ cyk } of disjoint sets of finite measuresin (0 , ∞ ) such that( m ⊕ m )(( σ x + ) − (∆ k ∩ (0 , m ( s ( x + )))) ⊕ ∆ cxk ) = ( m ⊕ m )(( σ y + ) − (∆ k ∩ (0 , m ( s ( y + )))) ⊕ ∆ cyk ) , and( m ⊕ m )(( σ x − ) − (Λ k ∩ (0 , m ( s ( x − )))) ⊕ Λ cxk ) = ( m ⊕ m )(( σ y − ) − (Λ k ∩ (0 , m ( s ( y − )))) ⊕ Λ cyk ) . Now, consider A := (0 , ∞ ) \ ( ∪ k ∆ k ) and B := (0 , ∞ ) \ ( ∪ k Λ k ). Since τ ( y ) = τ ( x ), it follows that there exist partitions {X k : m ( X k ) < ∞} and {Y k : m ( Y k ) < ∞} of A and B , respectively, such that Z X k µ ( y + ) − µ ( x + ) = Z Y k µ ( y − ) − µ ( x − ) . We consider couples (cid:0) ( σ x + ) − ( X k ∩ (0 , m ( s ( x + )))) , ( σ y + ) − ( X k ∩ (0 , m ( s ( y + )))) (cid:1) and (cid:0) ( σ x − ) − ( Y k ∩ (0 , m ( s ( x − )))) , ( σ y − ) − ( Y k ∩ (0 , m ( s ( y − )))) (cid:1) . Note that the measures of elements in any couple may not be the same. We canfind sequences {A cxk } , {A cyk } , {B cxk } and {B cyk } of disjoint sets of finite measures in(0 , ∞ ) such that( m ⊕ m ) (cid:0) ( σ x + ) − ( X k ∩ (0 , m ( s ( x + )))) ⊕ A cxk (cid:1) = ( m ⊕ m ) (cid:0) ( σ y + ) − ( X k ∩ (0 , m ( s ( y + )))) ⊕ A cyk (cid:1) and( m ⊕ m ) (cid:0) ( σ x − ) − ( Y k ∩ (0 , m ( s ( x − )))) ⊕ B cxk (cid:1) = ( m ⊕ m ) (cid:0) ( σ y − ) − ( Y k ∩ (0 , m ( s ( y − )))) ⊕ B cyk (cid:1) . Now, we numerate the following disjoint couples of subsets in ⊕ k =1 (0 , ∞ ) by( X k , Y k ), (cid:16) ( σ x + ) − (∆ k ∩ (0 , m ( s ( x + )))) ⊕ ∆ cxk ⊕ ∅ ⊕ ∅ , ( σ y + ) − (∆ k ∩ (0 , m ( s ( y + )))) ⊕ ∆ cyk ⊕ ∅ ⊕ ∅ (cid:17) , (cid:16) ( σ x − ) − (Λ k ∩ (0 , m ( s ( x − )))) ⊕ Λ cxk ⊕ ∅ ⊕ ∅ , ( σ y − ) − (Λ k ∩ (0 , m ( s ( y − )))) ⊕ Λ cyk ⊕ ∅ ⊕ ∅ (cid:17) , (cid:16) ( σ x + ) − ( X k ∩ (0 , m ( s ( x + )))) ∪ ( σ x − ) − ( Y k ∩ (0 , m ( s ( x − )))) ⊕ ∅ ⊕ ( A cxk ∪ B cxk ) ⊕ ∅ , ( σ y + ) − ( X k ∩ (0 , m ( s ( y + )))) ∪ ( σ y − ) − ( Y k ∩ (0 , m ( s ( y − )))) ⊕ ∅ ⊕ ( A cyk ∪ B cyk ) ⊕ ∅ (cid:17) . In particular, the measures of X k and Y k are the same.Since there exists a measure-preserving isomorphism between (0 , ∞ ) and(0 , ∞ ) ⊕ ⊕ ((0 , ∞ ) \ s ( x )) (or (0 , ∞ ) ⊕ ⊕ (0 , ∞ ) \ s ( y )) , we may view ( X k ) (and ( Y k )) as disjoint subsets of (0 , ∞ ) ⊕ (0 , ∞ ). Moreover, sincethe complements ( ∪ ( X k )) c (resp. ( ∪ ( X k )) c ) of ∪ ( X k ) (resp. ∪ ( Y k )) has infinitemeasure, it follows that there exists a partition { P n } (resp. { Q n } ) of ( ∪ ( X k )) c (resp. ( ∪ ( Y k )) c ) with ( m ⊕ m )( P n ) = 1 (resp. ( m ⊕ m )( Q n ) = 1).Numerating ( X k ) and ( P n ) (resp. ( Y k ) and ( Q n )), we complete the proof. (cid:3) Now, we prove the equivalence between (b) and (d) in Theorem 1.2.
Proposition . Let x, y ∈ L (0 , ∞ ) . Then, x ≺ y if and only if τ ( f ( x )) ≤ τ ( f ( y )) for any convex function f on the real axis with f (0) = 0 such that f ( x ) and f ( y ) are integrable. Here, τ ( f ) stands for R f dm ( m is the Lebesgue measure).Proof. ⇒ . By Proposition 5.1, there exist sequences of disjoint sets { A n ⊂ R + ⊕ R + } and { B n ⊂ R + ⊕ R + } such that m ( A n ) = m ( B n ) < ∞ , ∪ A n = ∪ B n = R + ⊕ R + and ( x ⊕ χ B n ≺ ( y ⊕ χ A n for every n . By the classic result (see e.g. [36, Proposition 1.3]), we have( τ ⊕ τ )( f (( x ⊕ χ B n )) ≤ ( τ ⊕ τ )( f (( y ⊕ χ A n ))for every n . Hence, we have( τ ⊕ τ )( f ( x ⊕ X n ( τ ⊕ τ )( f (( x ⊕ χ B n )) ≤ X n ( τ ⊕ τ )( f (( y ⊕ χ A n )) = ( τ ⊕ τ )( f ( y ⊕ . Since f (0) = 0, it follows that τ ( f ( x )) ≤ τ ( f ( y )). ⇐ . Let f ( t ) = t . Then, we have τ ( x ) ≤ τ ( y ). Let f ( t ) = − t . We have τ ( − x ) ≤ τ ( − y ). Hence, τ ( x ) = τ ( y ). On the other hand, we take any non-decreasing continuous convex function f on R + with f ( t ) = 0 when t <
0. We have τ ( f ( x + )) ≤ τ ( f ( y + )). Hence, x + ≺≺ y + (see [36, Proposition 1.2]). Now, we takeany non-decreasing continuous convex function g on R + with g ( t ) = 0 when t < f ( t ) = g ( − t ) on R . We have τ ( g ( x − )) = τ ( f ( − x − )) = τ ( f ( x )) ≤ τ ( f ( y )) = τ ( f ( − y − )) = τ ( g ( y − )) . Hence, τ ( g ( x − )) ≤ τ ( g ( y − )), i.e., x − ≺≺ y − (see [36, Proposition 1.2]). (cid:3) Alberti–Uhlmann problem in the setting of infinite von Neumannalgebras
Let L ( M , τ ) be the noncommutative L -space affiliated with a semifinite vonNeumann algebra M equipped with a faithful normal semifinite trace τ . Through-out this section, we always assume that τ ( ) = ∞ .The following proposition for positive operators can be found in [33, Theorem4.5]. LBERTI–UHLMANN PROBLEM 19
Proposition . Let y ∈ L ( M , τ ) h and let ϕ ∈ DS ( M ) . Then, ϕ ( y ) ≺ y .Proof. Since ϕ ( y ) − s = ϕ ( y − s ) ≤ ϕ (( y − s ) + ) for every s ∈ R , it follows that τ (( ϕ ( y ) − s ) + ) = τ ( p ( ϕ ( y ) − s ) p ) ≤ τ ( pϕ (( y − s ) + ) p ) ≤ τ ( ϕ (( y − s ) + )) = τ (( y − s ) + )for every s ≥ R + , where p := s (( ϕ ( y ) − s ) + ). The same argument shows that τ (( − ϕ ( y ) − s ) + ) ≤ τ ( ϕ (( − y − s ) + )) = τ (( − y − s ) + )for every s ≥ R + . Since ϕ is trace-preserving, it follows from Proposition 2.3 thatthe assertion holds. (cid:3) Now, we present the proof of (b) ⇒ (a) in Theorem 1.2. Theorem . If M is a σ -finite von Neumann algebra, then for any x ≺ y ∈ L ( M , τ ) h , there exists a normal doubly stochastic operator ϕ : M ⊕ L ∞ (0 , ∞ ) →M ⊕ L ∞ (0 , ∞ ) such that ϕ ( y ⊕
0) = x ⊕ .Proof. Without loss of generality, we may assume that M is atomless [55, Lemma2.3.18]. For every x ∈ Ω( y ), there exists an abelian non-atomic von Neumannsubalgebra N x of s ( x + ) M s ( x + ) containing all spectral projections of x + , and a ∗ -isomorphism V from S ( N x , τ ) onto S ([0 , τ ( s ( x + )) , m ) such that λ ( V ( f )) = λ ( f )for every f ∈ S ( N x , τ ) h (see [14, Lemma 1.3], see also [7, Proposition 3.1], [17]and [15, Lemma 4.1]). Similarly, we have a ∗ -isomorphism V from S ( N x , τ ) onto S ([0 , τ ( s ( x − )) , m ) such that λ ( V ( f )) = λ ( f ) for every f ∈ S ( N x , τ ) h , where N x is an abelian non-atomic von Neumann subalgebra of s ( x − ) M s ( x − ) containing allspectral projections of x − . We denote V x := V x ⊕ V x . There exists a commutative atomless von Neumann subalgebra N x of M n ( x ) , onwhich τ is again semifinite.The same argument show that there exists such a trace-preserving ∗ -isomorphism V y : S ( N y ⊕ N y ) → S (0 , τ ( s ( y ))), where N y (resp. N y ) is an atomless commu-tative reduced von Neumann subalgebra of M s ( y + ) (resp. M s ( y − ) ) containing allspectral projections of y + (resp. y − ). N y denotes a commutative atomless vonNeumann subalgebra of M n ( y ) . There exists a trace-preserving normal conditionalexpectation E : M → N y ⊕ N y ⊕ N y [21, Proposition 2.1].Since M is σ -finite, it follows that N is isomorphic to a subalgebra of L ∞ (0 , ∞ ).Hence, there are trace-preserving ∗ -isomorphisms U and U such that L ∞ (0 , τ ( s ( x ))) ⊕ L ∞ (0 , ∞ ) ⋍ U N x ⊕ N x ⊕ N x ⊕ L ∞ (0 , ∞ )and L ∞ (0 , τ ( s ( y ))) ⊕ L ∞ (0 , ∞ ) ⋍ U N y ⊕ N y ⊕ N y ⊕ L ∞ (0 , ∞ ) ⊂ E M ⊕ L ∞ (0 , ∞ ) . Hence, without loss of generality, we may assume that y ∈ L (0 , τ ( s ( y ))), x ∈ L (0 , τ ( s ( x ))) and it suffices to show that there exists a doubly stochastic operator ϕ from L ∞ (0 , τ ( s ( y ))) ⊕ L ∞ (0 , ∞ ) to L ∞ (0 , τ ( s ( x ))) ⊕ L ∞ (0 , ∞ ) whose L -extensionmaps y to x .By Proposition 5.1, there exist { A n ⊂ (0 , τ ( s ( y ))) ⊕ R + } and { B n ⊂ (0 , τ ( s ( x ))) ⊕ R + } such that m ( A n ) = m ( B n ) < ∞ , ∪ A n = (0 , τ ( s ( y ))) ⊕ R + and ∪ B n =(0 , τ ( s ( x ))) ⊕ R + and ( x ⊕ χ B n ≺ ( y ⊕ χ A n for every n ≥
0. By Theorem 3.5 (seealso [20]), there are normal doubly stochastic operators ϕ n : L ∞ ( B n ) → L ∞ ( A n ) with ϕ n ( y A n ) = x B n . By taking the direct sum Φ of all ϕ n , we obtain a doublystochastic operatorΦ(= ⊕ n ≥ ϕ n ) := L ∞ (0 , τ ( s ( x ))) ⊕ L ∞ (0 , ∞ ) → L ∞ (0 , τ ( s ( y ))) ⊕ L ∞ (0 , ∞ )such that Φ( y ) = x . Moreover, since every ϕ n is normal, it follows that the directsum Φ is also normal. Indeed, let z i ↑ z ∈ L ∞ (0 , τ ( s ( y ))) ⊕ L ∞ (0 , ∞ ). For any n ≥
1, we have ϕ n ( z i χ A n ) ↑ ϕ ( zχ A n ) . Since Φ( z i ) = ⊕ n ≥ ϕ n ( z i χ A n )and Φ( z ) = ⊕ n ≥ ϕ n ( zχ A n ) , it follows that that Φ( z i ) ↑ Φ( z ), which implies that Φ is normal. (cid:3) It is interesting to see that Hiai’s conjecture is not true in the setting of non- σ -finite semifinite von Neumann algebras. The proof is similar with that of Theo-rem 6.2. Before proceeding to the proof of this result, we need the following lemma.We would like to thank Dr. Dmitriy Zanin for his help with the proof of followinglemma. Lemma . Let A be an atomless abelian semifinite infinite von Neumann algebraequipped with a semifinite faithful normal trace τ . If A is not σ -finite, then A − s ( x ) ⋍ A (9) for any x ∈ S ( A , τ ) . Here, we denote by A p the reduced von Neumann subalgebra p A p of A , p ∈ P ( A ) .Proof. Since x is τ -compact, it follows that s ( x ) is a σ -finite projection, and theatomless commutative algebra A := A s ( x ) is isomorphically isomorphic (the ∗ -isomorphism is denoted by U ) to L ∞ (0 , s ( x )) [10, Theorem 9.3.4].Since A − s ( x ) is not σ -finite, we can construct a commutative subalgebra A of A − s ( x ) is isomorphically isomorphic to L ∞ (0 , s ( x )).Arguing inductively, we may construct a sequence {A n } of disjoint commutativesubalgebras of A , isomorphically isomorphic to L ∞ (0 , τ ( s ( x ))). We denote by U n the ∗ -isomorphism between A − s ( x ) and A n .Let A := A −⊕ n A n . Since is not σ -finite in A , it follows that − ⊕ n A n isnot σ -finite. Since A is a reduced algebra of A , it follows that the restriction of τ on A is semifinite, faithful and normal.Now, we can define a trace-preserving ∗ -isomorphism α : ⊕ n ≥ A n → ⊕ n =1 A n by setting α ( x ) = (cid:26) x, x ∈ A ,U n +1 ( U − n ( x )) , x ∈ A n , n ≥ . (cid:3) Theorem . If M is not a σ -finite von Neumann algebra, then for any x ≺ y ∈ L ( M , τ ) h , there exists a normal doubly stochastic operator ϕ : M → M such that ϕ ( y ) = x . LBERTI–UHLMANN PROBLEM 21
Proof.
Without loss of generality, we may assume that M is atomless [55, Lemma2.3.18]. Note that x and y are τ -compact. For every x ∈ Ω( y ), there ex-ists an abelian atomless von Neumann subalgebra N x of s ( x + ) M s ( x + ) contain-ing all spectral projections of x + , and a ∗ -isomorphism V from S ( N x , τ ) onto S ([0 , τ ( s ( x + )) , m ) such that λ ( V ( f )) = λ ( f ) for every f ∈ S ( N x , τ ) h (see [14,Lemma 1.3], see also [7, Proposition 3.1], [17] and [15, Lemma 4.1]). Similarly,we have a ∗ -isomorphism V from S ( N x , τ ) onto S ([0 , τ ( s ( x − )) , m ) such that λ ( V ( f )) = λ ( f ) for every f ∈ S ( N x , τ ) h , where N x is an abelian atomless vonNeumann subalgebra of s ( x − ) M s ( x − ) containing all spectral projections of x − .We denote V x := V x ⊕ V x . The same argument show that there exists such a trace-preserving ∗ -isomorphism V y : S ( N y ⊕N y ) → S (0 , τ ( s ( y ))), where N y (resp. N y ) is an atomless commutativereduced von Neumann subalgebra of M s ( y + ) (resp. M s ( y − ) ) containing all spectralprojections of y + (resp. y − ).Since M is not σ -finite, it follows that there exists an atomless non- σ -finite com-mutative von Neumann subalgebra A of M − s ( x ) ∨ s ( y ) . It follows from Lemma 6.3that A ≃ A ⊕ L ∞ (0 , a )for any 0 < a ≤ ∞ .Note that s ( x ) ∨ s ( y ) − s ( y ) is a σ -finite projection. Let B y (resp. B x )be a commutative atomless von Neumann subalgebra of M s ( x ) ∨ s ( y ) − s ( y ) (resp. M s ( x ) ∨ s ( y ) − s ( x ) ). We have A ⊕ B y ≃ A (resp. A ⊕ B x ≃ A ). There exists a trace-preserving conditional expectation E : M → N ⊕ N ⊕ A ⊕ B y [21, Proposition 2.1]. Hence, there are trace-preservingisomorphisms U and U such that L ∞ (0 , τ ( s ( x ))) ⊕ L ∞ (0 , ∞ ) ⊕ A ⋍ U N x ⊕ N x ⊕ A ⊕ B x ⊂ M and L ∞ (0 , τ ( s ( y ))) ⊕ L ∞ (0 , ∞ ) ⊕ A ⋍ U N y ⊕ N y ⊕ A ⊕ B y ⊂ E M where the restrictions of U and U coincide with V x and V y . Hence, withoutloss of generality, we may assume that y ∈ L (0 , τ ( s ( y ))), x ∈ L (0 , τ ( s ( x ))).The argument in Theorem 6.2 infers that there exists a normal doubly stochasticoperator ϕ from L ∞ (0 , τ ( s ( y ))) ⊕ L ∞ (0 , ∞ ) to L ∞ (0 , τ ( s ( x ))) ⊕ L ∞ (0 , ∞ ) whose L -extension maps y to x . (cid:3) Extreme points of Ω( y ) : Luxemburg’s problem in the infinitesetting In this section, we consider Luxemburg’s problem [58, Problem 1] in the (noncom-mutative) infinite setting, characterizing the extreme points of Ω( y ), y ∈ L ( M , τ ) h when M is a semifinite von Neumann equipped with a semifinite infinite faithfulnormal trace τ . The main idea steming from [19] still works for the infinite setting.However, the description of extreme points of Ω( y ) is much more complicated thanthat in the finite setting. We show that the case of an infinite von Neumann alge-bra can be reduced into the case of at most countably many finite von Neumannalgebras and we can apply the same idea used in [19]. The theorem below is the main result in this section.
Theorem . Let y ∈ L ( M , τ ) h . Then, x ∈ L ( M , τ ) is an extreme point of Ω( y ) if and only if for x + and y + , for any t ∈ (0 , ∞ ) , one of the following optionsholds:(i). λ ( t ; x ) = λ ( t ; y ) ;(ii). λ ( t ; x ) = λ ( t ; y ) with the spectral projection E x { λ ( t ; x ) } being an atom in M and Z { s ; λ ( s ; x )= λ ( t ; x ) } λ ( s ; y ) ds = λ ( t ; x ) τ ( E x ( { λ ( t ; x ) } )) . and, for x − and y − , for any t ∈ (0 , ∞ ) , one of the following options holds:(i). λ ( t ; − x ) = λ ( t ; − y ) ;(ii). λ ( t ; − x ) = λ ( t ; y ) with the spectral projection E − x { λ ( t ; − x ) } being an atomin M and Z { s ; λ ( s ; − x )= λ ( t ; − x ) } λ ( s ; − y ) ds = λ ( t ; − x ) τ ( E − x ( { λ ( t ; − x ) } )) . The following lemma allows us to reduce the problem to the setting of the positivecore of a finite von Neumann algebra.
Lemma . Let y ∈ L ( M , τ ) h . If x ∈ L ( M , τ ) h is an extreme point of Ω( y ) ,then τ ( x + ) = τ ( y + ) and τ ( x − ) = τ ( y − ) .Proof. Assume by contradiction that τ ( x + ) < τ ( y + ). Since τ ( x + ) − τ ( x − ) = τ ( x ) = τ ( y ) = τ ( y + ) − τ ( y − ), it follows that τ ( x − ) < τ ( y − ). Let A := (cid:26) t ∈ (0 , ∞ ] : Z t λ ( t ; y + ) dt = Z t λ ( t ; x + ) dt (cid:27) . If sup A = ∞ , then τ ( y + ) = τ ( x + ). Hence, sup A < ∞ . We define t ′ := sup A . Inparticular, t ′ ∈ A . Similarly, we define B := n t : R t λ ( t ; y − ) dt = R t λ ( t ; x − ) dt o and B ∋ s ′ := sup B < ∞ . For any t > t ′ , we have Z t λ ( t ; y + ) dt > Z t λ ( t ; x + ) dt and for any t > s ′ , we have Z t λ ( t ; y − ) dt > Z t λ ( t ; x − ) dt. By Lemma A.3, λ ( x ± ) is a step function on ( t ′ , ∞ ) (resp. ( s ′ , ∞ )). Assume that λ ( x + ) > R + . Since x is τ -compact, it follows λ ( x + ) decreases to 0 at infinity.By Lemma A.3, x is not an extreme point. Hence, λ ( x + ) has a finite support. Thesame argument shows that λ ( x − ) has a finite support.Recall that Z t ′ λ ( t ; y + ) dt = Z t ′ λ ( t ; x + ) dt, Z s ′ λ ( t ; y − ) dt = Z s ′ λ ( t ; x − ) dt. We claim that 0 < λ ( t ′ − ; y + ) ≤ λ ( t ′ − ; x + )(10) LBERTI–UHLMANN PROBLEM 23 and 0 < λ ( s ′ − ; y − ) ≤ λ ( s ′ − ; x − ) . Otherwise, λ ( t ′ − ; y + ) > λ ( t ′ − ; x + ), i.e., there exists δ > λ ( y + ) > λ ( x + )on ( t ′ − δ, t ′ ). Since x ≺ y , it follows that R t ′ λ ( t ; x + ) dt < R t ′ λ ( t ; y + ) dt , which is acontradiction with t ′ ∈ A . The case for x − and y − follows from the same argument.We claim that for any ε > λ ( t ′ + ε ; x + ) < λ ( t ′ − ; x + )(11)and λ ( s ′ + ε ; x − ) < λ ( s ′ − ; x − ) . Otherwise, λ ( t ′ + ε ; x + ) = λ ( t ′ − ; x + ) (10) ≥ λ ( t ′ − ; y + ) ≥ λ ( t ′ + ε ; y + ) for some ε > Z t ′ + εt ′ λ ( s ; x + ) ds ≥ Z t ′ + εt ′ λ ( s ; y + ) ds. This contradicts with t ′ = sup A (if we take “=” in the above inequality) or x ≺ y (if we take “ > ” in the above inequality). The case for x − follows from the sameargument.Let’s consider λ ( t ′ ; x + ). There are three possible situations:(1) λ ( t ′ ; x + ) = 0;(2) λ ( t ′ ; x + ) > λ ( t ′ + ε ; x + ) = λ ( t ′ ; x + ) for some ε > λ ( t ′ ; x + ) > λ ( t ′ + ε ; x + ) < λ ( t ′ ; x + ) for any ε > λ ( x + ), situation (3) is impossible. Con-sider situation (2). We claim that E x + (0 , δ ) = 0 for some sufficiently small δ > λ ( x + ) satisfies the conditions in Lemma A.3. That is, x is not an ex-treme point of Ω( y ). Hence, m := inf { λ ( x + ) > } >
0. Similarly, we obtain that m := inf { λ ( x − ) > } >
0. If situation (1) is true, then, by (11), we can define m > λ ( t ′ − ; x + ) (resp. m := λ ( s ′ − ; x − ) > s ( x ) is τ -finite and the trace τ is infinite. Take any non-zero τ -finiteprojections P , P ≤ − s ( x ) and P ⊥ P . Without loss of generality, we mayassume that τ ( P ) ≤ τ ( P ). Let C := 1 τ ( P ) Z τ ( s ( x + ))+ τ ( P )0 λ ( t ; y + ) dt − Z τ ( s ( x + ))0 λ ( t ; x + ) dt ! > ,C := 1 τ ( P ) Z τ ( s ( x − ))+ τ ( P )0 λ ( t ; y − ) dt − Z τ ( s ( x − ))0 λ ( t ; x − ) dt ! > ,C := 1 τ ( P ) Z τ ( s ( x + ))+ τ ( P )0 λ ( t ; y + ) dt − Z τ ( s ( x + ))0 λ ( t ; x + ) dt ! > , and C := 1 τ ( P ) Z τ ( s ( x − ))+ τ ( P )0 λ ( t ; y − ) dt − Z τ ( s ( x − ))0 λ ( t ; x − ) dt ! > . There exists a δ > δ < min { C , C , C , C , m , m } We define x := x + δ · (cid:18) P − τ ( P ) τ ( P ) P (cid:19) and x := x − δ · (cid:18) P − τ ( P ) τ ( P ) P (cid:19) . Clearly, τ ( x ) = τ ( x ) = τ ( x ) = τ ( y ) and and x + x = x . Note that t R t λ ( s ; y ) ds is a concave function and Z τ ( s ( x + ))0 λ ( s ; y ) ds ≥ Z τ ( s + )0 λ ( s ; ( x + )) ds = Z τ ( s + )0 λ ( s ; ( x )) ds, Z τ ( s ( x + ))+ τ ( P )0 λ ( s ; y ) ds > Z τ ( s ( x + ))+ τ ( P )0 λ ( s ; x ) ds = Z τ ( s + )0 λ ( s ; x ) ds + δτ ( P ) . Moreover, by spectral theorem, we obtain that λ (( x ) + ) is a positive constanton [ τ ( s ( x + )) , τ ( s ( x + )) + τ ( P )) and vanishes on [ τ ( s ( x + )) + τ ( P ) , ∞ ). Hence, λ (( x ) + ) ≺≺ λ ( y ). Arguing similarly to ( x ) − , ( x ) + and ( x ) − , we obtain that x , x ≺ y , which shows that x is not an extreme point of Ω( y ). Hence, τ ( x + ) = τ ( y + ) and τ ( x − ) = τ ( y − ). (cid:3) Now, we present the proof of Theorem 7.1.
Proof of Theorem 7.1. “ ⇒ ”. Assume that x ∈ extr(Ω( y )). By Lemma 7.2, weobtain that τ ( x + ) = τ ( y + ) and τ ( x − ) = τ ( y − ). It suffices to prove the case for x + and y + . Hence, we may always assume that x, y ≥ . We set A = (cid:26) s ∈ (0 , ∞ ) : Z s λ ( t ; y ) − λ ( t ; x ) dt > (cid:27) . Since λ ( y ) , λ ( x ) ∈ L (0 , ∞ ), it follows that the mapping f : s R s λ ( t ; y ) − λ ( t ; x ) dt is continuous. Moreover, noting that f (0) = f ( ∞ ) = 0, we infer that A is an openset, i.e., A = ∪ i ( a i , b i ), where a i , b i A . By Lemma A.2, λ ( x ) is a step functionon ( a i , b i ). Moreover, we claim that b i < ∞ for every i . Otherwise, ( a i , b i ) = ( a i , ∞ ). By R ∞ λ ( t ; y ) − λ ( t ; x ) dt = 0 and R s λ ( t ; y ) − λ ( t ; x ) dt > s ∈ ( a i , ∞ ), we obtain that λ ( x ) > , ∞ ). Since x ∈ L ( M , τ ) (therefore, τ -compact), it follows that λ ( x ) decreases to 0 on ( a, ∞ ).By Lemma A.3, we obtain that x is not an extreme point. Hence, ( a i , b i ) is finite forany i . Now, the statement of the theorem follows by the same argument in [19, p.20–24]. Indeed, the case when λ ( x ) takes only two values or more on ( a i , b i ) can beinferred by Lemma A.3 directly.“ ⇐ ” The proof of this part is also similar with that in [19]. However, there aresome technical details which are somewhat different from that in [19]. We providethe full proof below and technical lemmas in Appendix A.Let y ∈ L ( M , τ ). Let x, x , x ∈ Ω( y ) with x = x + x . Assume that x satisfiesthat for every t ∈ (0 , ∞ ), one of the followings holds:(i). λ ( t ; x ) = λ ( t ; y ); LBERTI–UHLMANN PROBLEM 25 (ii). λ ( t ; x ) = λ ( t ; y ) with the spectral projection E x { λ ( t ; x ) } being an atom in M and Z { s ; λ ( s ; x )= λ ( t ; x ) } λ ( s ; y ) ds = λ ( t ; x ) τ ( E x ( { λ ( t ; x ) } )) . and one of the following options holds:(i). λ ( t ; − x ) = λ ( t ; − y );(ii). λ ( t ; − x ) = λ ( t ; y ) with the spectral projection E − x { λ ( t ; − x ) } being anatom in M and Z { s ; λ ( s ; − x )= λ ( t ; − x ) } λ ( s ; − y ) ds = λ ( t ; − x ) τ ( E − x ( { λ ( t ; − x ) } )) . We claim that x = x = x , i.e., x is an extreme point of Ω( y ).For any t such that λ ( t ; y ) = λ ( t ; x ), we denote [ t , t ) = { s : λ ( s ; x ) = λ ( t ; x ) } , t < t . In particular, we have Z t λ ( s ; y ) ds = Z t λ ( s ; x ) ds and Z t λ ( s ; y ) ds = Z t λ ( s ; x ) ds. Since x , x ∈ Ω( y ) and Z t λ ( s ; 2 x + ) ds = Z t λ ( s ; ( x + x ) + ) ds P rop. . ≤ Z t λ ( s ; ( x ) + ) ds + Z t λ ( s ; ( x ) + ) ds ≤ Z t λ ( s ; y + ) ds, it follows that Z t λ ( s ; x + ) ds = Z t λ ( s ; ( x ) + ) ds = Z t λ ( s ; ( x ) + ) ds = Z t λ ( s ; y + ) ds. (12)The same argument with t replaced with t yields that Z t λ ( s ; x + ) ds = Z t λ ( s ; ( x ) + ) ds = Z t λ ( s ; ( x ) + ) ds = Z t λ ( s ; y + ) ds. (13)Let e := E x ( λ ( t ; x + ) , ∞ ) and e := E x [ λ ( t ; x + ) , ∞ ). In particular, e − e = E x { λ ( t ; x + ) } . Observe that τ ( e ) = t and τ ( e ) = t (due to the assumptionthat [ t , t ) = { s : λ ( s ; x + ) = λ ( t ; x + ) } ). By Proposition 2.5 and the definition ofspectral scales λ ( x ), we have2 Z t λ ( s ; x + ) ds = τ (2 x + e ) = τ (( x + x ) + e ) = τ ( e E x (0 , ∞ )( x + x ) E x (0 , ∞ ) e ) ≤ τ ( e E x (0 , ∞ )(( x ) + + ( x ) + ) E x (0 , ∞ ) e ) ≤ τ ( e ( x ) + e ) + τ ( e ( x ) + e ) P rop. . ≤ Z t λ ( s ; ( x ) + ) ds + Z t λ ( s ; ( x ) + ) ds (12) = 2 Z t λ ( s ; x + ) ds. We obtain that τ ( x e ) = R t λ ( s ; x ) ds = R t λ ( s ; x ) ds = τ ( x e ). By Corol-lary A.7, we have E x ( λ ( t ; x ) , ∞ ) ≤ e ≤ E x [ λ ( t ; x ) , ∞ )and E x ( λ ( t ; x ) , ∞ ) ≤ e ≤ E x [ λ ( t ; x ) , ∞ ) . Similar argument with t replaced with t yields that E x ( λ ( t ; x ) , ∞ ) ≤ e ≤ E x [ λ ( t ; x ) , ∞ )and E x ( λ ( t ; x ) , ∞ ) ≤ e ≤ E x [ λ ( t ; x ) , ∞ ) . In particular, e = E x ( λ ( t ; x ) , ∞ )+ q for some subprojection q of E x { λ ( t ; x ) } .Since q commutes with E x { λ ( t ; x ) } , it follows that q commutes with any spectralprojection of x . Hence, e commutes with x . The same argument implies thatboth e and e commute with x and with x . Moreover, the atom e := e − e ∈P ( M ) satisfies that e ≤ E x [ λ ( t ; x ) , λ ( t ; x )] and e ≤ E x [ λ ( t ; x ) , λ ( t ; x )] . By the spectral theorem, λ ( x e ) = λ ( x ) and λ ( x e ) = λ ( x ) on (0 , t ) (see (6)).On the other hand, λ ( t ; x e ) = λ ( t ; x e e ) (7) = λ ( t + t ; x e ) (6) = λ ( t + t ; x )and λ ( t ; x e ) = λ ( t ; x e e ) (7) = λ ( t + t ; x e ) (6) = λ ( t + t ; x )for all t ∈ [0 , t − t ). Since e is an atom, it follows that λ := λ ( t ; x e ) = λ ( t + t ; x )and λ := λ ( t ; x e ) = λ ( t + t ; x ) for every t ∈ [0 , t − t ). Combining (12) and(13), we have Z t t λ ( s ; x ) ds = Z t t λ ( s ; x ) ds = λ ( t − t )= Z t t λ ( s ; x ) ds = λ ( t − t ) = Z t t λ ( s ; y ) ds. (14)Hence, λ = λ = λ ( t ; x ).Let A = { s : λ ( s ; x ) = λ ( t ; x ) for some t such that λ ( t ; x ) = λ ( t ; y ) } . Notethat (14) holds any interval [ t , t ) := { s : λ ( s ; x ) = λ ( t ; x ) } for some t such that λ ( t ; x ) = λ ( t ; y ). For any t ∈ [0 , ∞ ) \ A , we have Z t λ ( s ; 2 x + ) ds P rop. . ≤ Z t λ ( s ; ( x ) + ) ds + Z t λ ( s ; ( x ) + ) ds ≤ Z t λ ( s ; 2 y + ) ds ( i ) = Z t λ ( s ; 2 x + ) ds. Hence, Z t λ ( s ; 2 x + ) χ [0 , ∞ ) \ A ds ≤ Z t λ ( s ; ( x ) + ) χ [0 , ∞ ) \ A ds + Z t λ ( s ; ( x ) + ) χ [0 , ∞ ) \ A ds ≤ Z t λ ( s ; 2 y + ) χ [0 , ∞ ) \ A ds = Z t λ ( s ; 2 x + ) χ [0 , ∞ ) \ A ds, LBERTI–UHLMANN PROBLEM 27 which implies that λ ( y ) χ [0 , ∞ ) \ A = λ ( x ) χ [0 , ∞ ) \ A = λ ( x ) χ [0 , ∞ ) \ A = λ ( x ) χ [0 , ∞ ) \ A a.e.. Hence, by right-continuity, λ ( x + ) = λ (( x ) + ) = λ (( x ) + ).The same argument shows that λ (( x ) − ) = λ (( x ) − ) = λ ( x − ). By Proposi-tion A.8, we obtain that x = x = x . That is, x is an extreme point of Ω( y ). (cid:3) Appendix A. Technical results
Throughout this appendix, we always assume that M is a semifinite vonNeumann algebra equipped with a semifinite infinite faithful normal trace τ .Some of the results in this section are well-known for positive operators (seee.g. [13, 15, 28, 38]). Main ideas used in this section come from [19]. However,dealing with technical obstacles in the infinite setting requires additional care.The following is a noncommutative analogue of Ryff’s Proposition stated in [73].The case for finite von Neumann algebras can be found in [19]. One should notethat λ ( t ; x ) = 0 for some t < ∞ does not implies that E x { } is not trivial. Lemma
A.1 . Let y ∈ L ( M , τ ) h . If x ∈ Ω( y ) = { x ∈ L ( M , τ ) h : x ≺ y } satisfiesthat (i) 0 ≤ λ ( s i +1 ; x + ) < λ ( s i ; x + ) (or ≤ λ ( s i +1 ; x − ) < λ ( s i ; x − ) ) for some
For the sake of convenience, we denote a i = λ ( s i ; x + ), i = 1 , , ,
4. Notethat a > a > a > a ≥ p := E x + [ a + a , a + a ) and p := E x + [ a + a , a + a ). We denote T = τ ( p ) and T = τ ( p ). Observe that T , T < ∞ and p p = 0. Set u := p − T T p . It is clear that τ ( u ) = 0. Assume that δ > δ < min { a − a , ( a − a ) T T , (2 a − a − a ) T T , a + a − a } . Let x := x + δu and x := x − δu. By the spectral theorem, λ ( s ; x ) = λ ( s ; x ) = λ ( s ; x ) for s [ s , s ] and λ ( s ; x i ) λ ( s ; x ) for s ∈ [ s , s ], i = 1 ,
2. We assert that x , x ≺ y .Note that for i = 1 ,
2, we have Z ∞ λ ( t ; ( x i ) + ) dt − Z ∞ λ ( t ; ( x i ) − ) dt = τ ( x i ) = τ ( x ± δu ) = τ ( x ) = τ ( y ) . Since λ ( s ; x i ) = λ ( s ; x ) for s [ s , s ] and i = 1 ,
2, it follows that Z s s λ ( t ; ( x i ) + ) dt − Z s s λ ( t ; x + ) dt = Z ∞ λ ( t ; ( x i ) + ) dt − Z ∞ λ ( t ; x + ) dt = 0 , i = 1 , . Hence, for i = 1 ,
2, we have Z s λ ( t ; ( x i ) + ) dt = Z s λ ( t ; x + ) dt Z s λ ( t ; y + ) dt, s [ s , s ] , where the last inequality follows from the assumption that x ∈ Ω( y ) . On the otherhand, since λ ( x ) is decreasing, it follows that Z s λ ( t ; ( x i ) + ) dt Z s λ ( t ; x ) dt + λ ( s ; x + )( s − s ) ( ii ) Z s λ ( t ; y + ) dt, s ∈ [ s , s ] . Hence, x , x ∈ Ω( y ) and x = ( x + x ) . That is, x extr (Ω( y )). (cid:3) The following lemma extends [19, Lemma 3.2].
Lemma
A.2 . Let y ∈ L ( M , τ ) h . If x ∈ extr(Ω( y )) and R s λ ( t ; y ± ) dt > R s λ ( t ; x ± ) dt for all s ∈ ( t , t ) , then for any s ∈ ( t , t ) , λ ( x ± ) is a constanton [ s, s + ε ) for some ε > . In particular, λ ( x ± ) is a step function on [ t , t ) .Here, ≤ t < t ≤ ∞ .Proof. We only consider the case for x + and y + . The case for x − and y − followsfrom the same argument. For the sake of simplicity, we may assume that x, y arepositive. Note that if λ ( t ; x + ) = 0, then λ ( x + ) is a constant function on ( t , t ).Hence, we may assume that λ ( t ; x + ) >
0. If λ ( t ; x + ) = 0, by the right-continuity,there exists a t ′ ∈ ( t , t ] such that λ ( t ′ ; x + ) = 0 and λ ( t ′ − ε ; x + ) > ε >
0. Hence, without loss of generality, we may assume that λ ( t − ε ; x + ) > ε > s ∈ ( t , t ) such that ( s , s + ε ) is nota constancy interval of λ ( x ) for any ε > . Without loss of generality, we assumethat λ ( s + ε ; x ) >
0. Since λ ( x ) is right-continuous, one can choose s ∈ ( s , s + ε )such that Z s λ ( t ; y ) − λ ( t ; x ) dt ≥ , s ∈ [ s , s ) , and R s λ ( t ; y ) − λ ( t ; x ) dtλ ( s ; x ) ≥ s − s . Hence, for any s ∈ [ s , s ], we have Z s λ ( t ; y ) dt ≥ Z s λ ( t ; y ) dt ≥ Z s λ ( t ; x ) dt + λ ( s ; x )( s − s ) . Since y and x satisfy the assumptions in Lemma A.1, it follows that x extr(Ω( Y )) . (cid:3) The following lemma covers cases 1 and 2 in the proof of [19, Theorem 1.1].
Lemma
A.3 . Let y ∈ L ( M , τ ) h . If x ∈ Ω( y ) satisfies that ≤ λ ( s ; x + ) < λ ( s ; x + ) < λ ( s ; x + ) < λ ( s − ; x + ) (or ≤ λ ( s ; x − ) < λ ( s ; x − ) < λ ( s ; x − ) < λ ( s − ; x − ) ) for some ≤ s i
Assume by contradiction that x ∈ extr (Ω( y )). For the sake of convenience,we denote a i = λ ( s i ; x + ), i = 1 ,
2. By Lemma A.2, for every t ∈ ( s , s ), λ ( x + )is a constant on [ t, t + ε ) for some sufficiently small ε >
0. Now, let’s consider s .Assume that λ ( x + ) is not constancy on [ s , s + ε ) for any ε >
0. Then, by right-continuity, there exists a sequence of positive numbers ε n decreasing to 0 such that { λ ( s + ε n ; x + ) } strictly increases to λ ( s ; x + ). Then, we may replace s , s and s with s + ε n , s + ε n − and s + ε n − , respectively. By Lemma A.2, we obtain that λ ( x + ) are constancy on [ s , s + δ ), [ s + δ , s + δ ) and [ s + δ , s + δ ) for some δ , δ , δ >
0. Hence, without loss of generality, we may assume that λ ( s ; x + ) = a i on [ s i , s i +1 ), i = 1 , p := E x + { a } and p := E x + { a } . We denote T = τ ( p ) and T = τ ( p ).Observe that T , T < ∞ and p p = 0. Set u := p − T T p . It is clear that τ ( u ) = 0. Let C := s − s (cid:0)R s λ ( y ) − R s λ ( x ) (cid:1) . Clearly, C > s − s R s s λ ( x ) = a .Assume that δ > δ < min ( C − a , ( a − ) − a , a − a T T , ( a − a ) T T ) . Let x := x + δu and x := x − δu. By the spectral theorem, λ ( s ; x ) = λ ( s ; x ) = λ ( s ; x ) for s [ s , s ] and λ ( s ; x i ) λ ( s ; x ) for s ∈ [ s , s ], i = 1 ,
2. We assert that x , x ≺ y + .Note that Z ∞ λ ( t ; ( x ) + ) dt − Z ∞ λ ( t ; ( x ) − ) dt = τ ( x ) = τ ( x + δu ) = τ ( x ) = τ ( y ) . Since λ ( s ; x i ) = λ ( s ; x ) for s [ s , s ] and i = 1 ,
2, it follows that Z s s λ ( t ; ( x i ) + ) dt − Z s s λ ( t ; x + ) dt = τ ( x i ) − τ ( x ) = 0 , i = 1 , . Hence, for i = 1 ,
2, we have Z s λ ( t ; ( x i ) + ) dt = Z s λ ( t ; x + ) dt Z s λ ( t ; y + ) dt, s [ s , s ] , where the last inequality follows from the assumption that x ∈ Ω( y ) . Since s R s λ ( t ; y ) dt is a concave function, Z s λ ( t ; y + ) dt ≥ Z s λ ( t ; ( x ) + ) dt, Z s λ ( t ; y + ) dt ≥ Z s λ ( t ; ( x ) + ) dt and λ (( x ) + ) is constancy on [ s , s ), it follows that R s λ ( t ; x ) dt ≤ R s λ ( t ; y ) dt forany s ∈ [ s , s ). The same argument show that R s λ ( t ; x ) dt ≤ R s λ ( t ; y ) dt on forany s ∈ [ s , s ). Hence, x ∈ Ω( y ) (similarly, x ∈ Ω( y )) and x = ( x + x ) . Thatis, x extr (Ω( y )). (cid:3) Let x ∈ L ( M , τ ) + . Denote by N x the abelian von Neumann (reduced) algebragenerated by all spectral projections E x ( t, ∞ ) of x , t >
0. We define N := N x ⊕M − s ( x ) . In particular, x ∈ L ( N , τ ) h . Since x is τ -compact, it follows that τ isagain semifinite on N . Recall that (see [35, Proposition 1.1]) Z s λ ( t ; x ) dt = sup { τ ( xa ); a ∈ M , ≤ a ≤ , τ ( a ) = s } . (15) Assume that τ ( xe ) = R s λ ( t ; x ) dt for a projection e ∈ M with τ ( e ) = s . Let E N be the conditional expectation from L ( M , τ ) onto L ( N , τ ) [90] (see also [21,Proposition 2.1]). In particular, E N ( e ) ≤ and τ ( E N ( e )) = τ ( e ) = s . Moreover, τ ( E N ( e ) x ) [21, Prop. 2.1]= τ ( ex ) = Z s λ ( t ; x ) dt. (16)We note that E N ( f ), f ∈ P ( M ), is not necessarily a projection [83]. The proof ofthe following proposition is similar with [19, Proposition 3.3]. We provide a prooffor completeness. Proposition
A.4 . Under the above assumptions on e , we have E x ( λ ( s ; x ) , ∞ ) ≤ E N ( e ) ≤ E x [ λ ( s ; x ) , ∞ ) . (17) Proof.
We present the proof for the first inequality and a similar argument yieldsthat E N ( e ) ≤ E x [ λ ( s ; x ) , ∞ ).Without loss of generality, we may assume that E x ( λ ( s ; x ) , ∞ ) = 0. Since N x isa commutative algebra, M − s ( x ) ⊥ N x and 0 ≤ E N ( e ) ≤ , it follows that E N ( e ) E x ( λ ( s ; x ) , ∞ ) = E x ( λ ( s ; x ) , ∞ ) / E N ( e ) E x ( λ ( s ; x ) , ∞ ) / ≤ E x ( λ ( s ; x ) , ∞ ) . If E N ( e ) E x ( λ ( s ; x ) , ∞ ) = E x ( λ ( s ; x ) , ∞ ), then E N ( e ) ≥ E N ( e ) / E x ( λ ( s ; x ) , ∞ ) E N ( e ) / = E N ( e ) E x ( λ ( s ; x ) , ∞ )= E x ( λ ( s ; x ) , ∞ ) , (18)which proves the first inequality of (17).Now, we assume that E N ( e ) E x ( λ ( s ; x ) , ∞ ) < E x ( λ ( s ; x ) , ∞ ) . This implies that τ ( E N ( e ) E x ( λ ( s ; x ) , ∞ )) < τ ( E x ( λ ( s ; x ) , ∞ )) . (19)Since τ ( E x ( λ ( s ; x ) , ∞ )) ≤ s , it follows that λ ( t ; E N ( e ) E x ( λ ( s ; x ) , ∞ )) = 0 when t > s . Hence, Z s λ ( t ; E N ( e ) E x ( λ ( s ; x ) , ∞ )) dt = τ ( E N ( e ) E x ( λ ( s ; x ) , ∞ )) (19) < τ ( E x ( λ ( s ; x ) , ∞ )) = Z s λ ( t ; E x ( λ ( s ; x ) , ∞ )) dt. In particular, Z s − λ ( t ; E N ( e ) E x ( λ ( s ; x ) , ∞ )) dt > . (20)Assume that τ ( E N ( e ) E x ( λ ( s ; x ) , ∞ )) = a ≥ τ ( E N ( e ) E x ( −∞ , λ ( s ; x )]) = a ≥
0. Observe that a + a = s = τ ( E N ( e )). Moreover, by (2), for every LBERTI–UHLMANN PROBLEM 31 < t < τ ( E x ( λ ( s ; x ) , ∞ )), we have λ ( t ; x ) > λ ( s ; x ). Hence, Z s λ ( t ; x ) dt = Z s λ ( t ; x ) λ ( t ; E N ( e ) E x ( λ ( s ; x ) , ∞ )) dt + Z s λ ( t ; x )(1 − λ ( t ; E N ( e ) E x ( λ ( s ; x ) , ∞ )) dt (20) > Z s λ ( t ; x ) λ ( t ; E N ( e ) E x ( λ ( s ; x ) , ∞ )) dt + λ ( s ; x ) Z s (1 − λ ( t ; E N ( e ) E x ( λ ( s ; x ) , ∞ )) dt. Recall that R s λ ( t ; E N ( e ) E x ( λ ( s ; x ) , ∞ )) dt = τ ( E N ( e ) E x ( λ ( s ; x ) , ∞ )). The aboveinequality implies that Z s λ ( t ; x ) dt> Z ∞ λ ( t ; x ) λ ( t ; E N ( e ) E x ( λ ( s ; x ) , ∞ )) dt + λ ( s ; x ) (cid:16) s − τ ( E N ( e ) E x ( λ ( s ; x ) , ∞ )) (cid:17) = Z ∞ λ (cid:16) t ; x (cid:17) λ (cid:16) t ; E N ( e ) E x ( λ ( s ; x ) , ∞ ) (cid:17) dt + λ ( s ; x ) a . Since λ ( s ; x ) a = τ ( λ ( s ; x ) E N ( e ) E x ( −∞ , λ ( s ; x )]) ≥ τ ( E N ( e ) E x ( −∞ , λ ( s ; x )] x ), itfollows that Z s λ ( t ; x ) dt (3) > τ (cid:16) xE x ( λ ( s ; x ) , ∞ ) E N ( e ) (cid:17) + τ (cid:16) xE x ( −∞ , λ ( s ; x )] E N ( e ) (cid:17) = τ ( xE N ( e )) , which is a contradiction with (16). Hence, the equality E N ( e ) E x ( λ ( s ; x ) , ∞ ) = E x ( λ ( s ; x ) , ∞ ) holds, and therefore, by (18), we have E N ( e ) ≥ E x ( λ ( s ; x ) , ∞ ). (cid:3) Lemma
A.5 . Let x ∈ L ( M , τ ) + . Let < s < τ ( ) = ∞ and let a be in the unitball of M + such that τ ( a ) = s and τ ( xa ) = R s λ ( t ; x ) dt. If λ ( x ) is not a constantin any left neighborhood of s, then a = E x ( λ ( s ; x ) , ∞ ) . Proof.
Let N = N x ⊕ M − s ( x ) , where N x is the commutative weakly closed ∗ -subalgebra of M generated by the spectral projections of x . Clearly, the restrictionof τ to N is semifinite. There exists a conditional expectation E N from L ( M , τ )to L ( N , τ ) [21, Proposition 2.1]. In particular, for any z ∈ L ( M , τ ) + M , wehave E N ( z ) ≺≺ z (see e.g. [21, Proposition 2.1 (g)]). Moreover, for every z ∈ M and y ∈ L ( N , τ ), we have τ ( yz ) = τ ( E N ( yz )) = τ ( yE N ( z )) . (21)Since a is positive, it follows that λ ( a ) = µ ( a ) (see [35]) and, therefore, Z r λ ( t ; E N ( a )) dt Z r λ ( t ; a ) dt for all r ∈ (0 , ∞ ) and, τ ( E N ( a )) [21, Prop. 2.1. (h)]= τ ( a ) . Moreover, since E N is a contraction on M and k a k ∞ ≤
1, it follows that λ ( E N ( a )) y := ( x − λ ( s ; x )) + . Note that x λ ( s ; x ) + y. Therefore,(22) τ ( xa ) τ ( λ ( s ; x ) a ) + τ ( ya ) = s · λ ( s ; x ) + τ ( ya ) . Since λ ( x ) is a decreasing function, we have λ ( t ; y ) = (cid:26) λ ( t ; x ) − λ ( s ; x ) , if 0 < t < s ;0 , if s t < ∞ . (23)It follows from τ ( a ) = s and τ ( xa ) = R s λ ( t ; x ) dt that Z s λ ( t ; y ) dt (23) = Z s λ ( t ; x ) dt − Z s λ ( s ; x ) dt = τ ( xa ) − λ ( s ; x ) τ ( a ) (22) τ ( ya ) (21) = τ ( yE N ( a )) (3) Z ∞ λ ( t ; y ) λ ( t ; E N ( a )) dt (23) = Z s λ ( t ; y ) λ ( t ; E N ( a )) dt. Thus, Z s λ ( t ; y )(1 − λ ( t ; E N ( a ))) dt . Since y is positive and λ ( E N ( a )) , we conclude that λ ( t ; y )(1 − λ ( t ; E N ( a ))) > . Hence, λ ( t ; y )(1 − λ ( t ; E N ( a ))) = 0 for all t ∈ (0 , s ) . Recall that λ ( x ) is not aconstant in any left neighborhood of s . We obtain that λ ( y ) > , s ). Recallthat E N ( a ) ≥ τ ( E N ( a )) = τ ( a ) = s . We obtain that λ ( E N ( a )) = 1 on (0 , s )and λ ( E N ( a )) = 0 on [ s, ∞ ). This implies that E N ( a ) is a projection in N . Hence, E N ( a ) = E N ( a ) E N ( a ) = E N ( a · E N ( a )) and E N ( a ( − E N ( a ))) = 0. It followsthat τ ( a / ( − E N ( a )) a / ) = τ ( a ( − E N ( a ))) = τ ( E N ( a ( − E N ( a )))) = 0 . Therefore, a / ( − E N ( a )) a / = 0 and a / = E N ( a ) a / . By the assumptionthat a ≤ , we have E N ( a ) = E N ( a ) ≥ E N ( a ) aE N ( a ) = a. Recall that τ ( E N ( a )) = τ ( a ). Hence, τ ( E N ( a ) − a ) = 0. Due to the faithfulnessof the trace τ , we obtain that a = E N ( a ) ∈ P ( N ). Since N commutes with x ,it follows that a = E x { B } ⊕ q for some Borel set B ⊂ [0 ,
1] and some projection p ∈ M − s ( x ) . By [29, Remark 3.3] and the assumption that τ ( ax ) = R s λ ( t ; x ) dt ,we have Z s λ ( t ; x ) dt = τ ( ax ) = τ ( E x { B } x ) = Z ∞ χ B ( λ ( t ; x )) dt. LBERTI–UHLMANN PROBLEM 33
Moreover, since λ ( · ; x ) is decreasing and is non-constant in any left neighborhoodof s , it follows that B = ( λ ( s ; x ) , ∞ ). Since τ ( E x ( λ ( s ; x ) , ∞ )) = s , it follows that a = E x ( λ ( s ; x ) , ∞ ). (cid:3) Remark
A.6 . Let x ∈ L ( M , τ ) + . By (3) , for any a in the unit ball of M + with τ ( a ) < s , we have τ ( xa ) ≤ Z ∞ λ ( t ; x ) λ ( t ; a ) dt. Since λ ( x ) is a non-negative decreasing function, ≤ λ ( a ) ≤ and τ ( a ) = R ∞ λ ( t ; a ) dt < s , it follows that τ ( xa ) ≤ Z ∞ λ ( t ; x ) λ ( t ; a ) dt < Z s λ ( t ; x ) dt. (24) Then, by (15) , we obtain that Z s λ ( t ; x ) dt = sup { τ ( xa ); a ∈ M , ≤ a ≤ , τ ( a ) ≤ s } . Lemma A.5 together with (24) implies that if a is in the unit ball of M + suchthat τ ( a ) ≤ s and τ ( xa ) = R s λ ( t ; x ) dt, and λ ( x ) is not a constant in any leftneighborhood of s, then a = E x ( λ ( s ; x ) , ∞ ) . Corollary
A.7 . Let x ∈ L ( M , τ ) h . Let < s < τ ( ) = ∞ and let e ∈ P ( M ) besuch that τ ( e ) = s and τ ( xe ) = R s λ ( t ; x ) dt. Then, E x ( λ ( s ; x ) , ∞ ) ≤ e ≤ E x [ λ ( s ; x ) , ∞ ) . Proof.
By Lemma A.5, it suffices to prove the case when λ ( x ) is a constant on aleft neighbourhood of s . Denote by N := N x ⊕ M − s ( x ) , where N x is the reducedvon Neumann algebra generated by all spectral projections of x . Let E N be theconditional expectation from L ( M , τ ) onto L ( N , τ ). Let λ := λ ( s ; x ) and x := ( x − λ ) E x ( λ, ∞ ) . Recall that E x [ λ, ∞ ) ≥ E N ( e ) ≥ E x ( λ, ∞ ) (see (17)). In particular, E N ( e ) E x ( λ, ∞ ) = E x ( λ, ∞ ). Observing that E x ( λ, ∞ ) is the support of x , wehave τ ( x E x ( λ, ∞ ) eE x ( λ, ∞ )) = τ ( x e ) = τ ( x E N ( e ))= τ ( x E x ( λ, ∞ ) E N ( e ))= τ ( x E x ( λ, ∞ ))= τ (( x − λ ) E x ( λ, ∞ ))= τ ( xE x ( λ, ∞ )) − τ ( λE x ( λ, ∞ ))= Z τ ( E x ( λ, ∞ ))0 ( λ ( t ; xE x ( λ, ∞ )) − λ ) dt (6) = Z τ ( E x ( λ, ∞ ))0 ( λ ( t ; x ) − λ ) dt = Z τ ( E x (0 , ∞ ))0 λ ( t ; x ) dt. Note that 0 ≤ E x ( λ, ∞ ) eE x ( λ, ∞ ) ≤ and τ ( E x ( λ, ∞ ) eE x ( λ, ∞ )) ≤ τ ( E x ( λ, ∞ )).Since x ≥
0, it follows from Remark A.6 that E x ( λ, ∞ ) eE x ( λ, ∞ ) = E x (0 , ∞ ) = E x ( λ, ∞ ) . That is, e ≥ E x ( λ, ∞ ). Let e := e − E x ( λ, ∞ ) ∈ P ( M ). We have τ ( xe ) = τ ( x ( E x ( λ, ∞ ) + E x ( −∞ , λ ]) e ) = τ ( xE x ( λ, ∞ ) + xE x ( −∞ , λ ] e ) . Hence, by the assumption that λ = λ ( s ; x ), we obtain that τ ( xE x ( −∞ , λ ] e ) = Z s λ ( t ; x ) dt − Z τ ( E x ( λ, ∞ ))0 λ ( t ; x ) dt = Z sτ ( E x ( λ, ∞ )) λ ( t ; x ) dt = τ ( λe ) . We have τ ( e ( λ − xE x ( −∞ , λ ]) e ) = 0 . Therefore, e ( λ − xE x ( −∞ , λ ]) e = 0. Since λ − xE x ( −∞ , λ ] ≥
0, it follows that( λ − xE x ( −∞ , λ ]) / e = 0. Hence, (cid:18)Z t<λ ( λ − t ) dE xt + Z t>λ λdE xt (cid:19) e = ( λ − xE x ( −∞ , λ ]) e = 0and E x (cid:16) ( −∞ , λ ) ∪ ( λ, ∞ ) (cid:17) · e = (cid:18)Z t<λ λ − t dE xt + Z t>λ λ dE xt (cid:19) (cid:18)Z t<λ ( λ − t ) dE xt + Z t>λ λdE xt (cid:19) e = 0 . This implies that e ≤ E x { λ ( s ; x ) } , which completes the proof. (cid:3) The following proposition is similar to a well-known property of rearrangementsof functions, see [54, property 9 , p. 65] and [33, Theorem 3.5] and [19]. Proposition
A.8 . Let x, x , x ∈ L ( M , τ ) h be such that x = ( x + x ) / and λ (( x ) + ) = λ (( x ) + ) = λ ( x + ) and λ (( x ) − ) = λ (( x ) − ) = λ ( x − ) . Then, x = x = x .Proof. Fix θ ∈ (0 , λ (0; x + )). Define s by setting s := min { ≤ v ≤ ∞ : λ ( v ; x ) ≤ θ } . If s >
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Clearly, τ ( e ) = s . By (15), we have τ ( e ( x + x )) = τ ( e ( x + x ) + ) = Z s λ ( t ; ( x + x ) + ) dt = Z s λ ( t ; x + ) dt = Z s λ ( t ; ( x ) + ) dt + Z s λ ( t ; ( x ) + ) dt (15) ≥ τ ( e ( x ) + ) + τ ( e ( x ) + ) ≥ τ ( ex ) + τ ( ex )= τ ( e ( x + x )) . Hence, R s λ ( t ; ( x i ) + ) dt = τ ( ex i ) , i = 1 ,
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