Hermitian operators and isometries on symmetric operator spaces
aa r X i v : . [ m a t h . OA ] J a n HERMITIAN OPERATORS AND ISOMETRIES ON SYMMETRICOPERATOR SPACES
JINGHAO HUANG AND FEDOR SUKOCHEV
Abstract.
Let M be an atomless semifinite von Neumann algebra (or an atomic von Neumannalgebra with all atoms having the same trace) acting on a (not necessarily separable) Hilbertspace H equipped with a semifinite faithful normal trace τ . Let E ( M , τ ) be a symmetricoperator space affiliated with M , whose norm is order continuous and is not proportional tothe Hilbertian norm k·k on L ( M , τ ). We obtain general description of all bounded hermitianoperators on E ( M , τ ). This is the first time that the description of hermitian operators on asymmetric operator space (even for a noncommutative L p -space) is obtained in the setting ofgeneral (non-hyperfinite) von Neumann algebras. As an application, we resolve a long-standingopen problem concerning the description of isometries raised in the 1980s, which generalizes andunifies numerous earlier results. Introduction
The main purpose of this paper is to answer the following long-standing open question concern-ing isometries on a symmetric operator space (see e.g. [3, 23, 46, 92])If E (0 , ∞ ) is a separable symmetric function on (0 , ∞ ) and if ( M , τ ) is a semifinitevon Neumann algebra (on a separable Hilbert space) with a semifinite faithfulnormal trace τ , then how can one describe the family of surjective isometries onthe symmetric operator space E ( M , τ ) associated with E (0 , ∞ )?This is one of the most fundamental questions in the theory of symmetric operator/function/sequencespaces, and it has attracted a substantial amount of interest.The study of the above question has a very long history, initiated by Stefan Banach [10], whoobtained the general form of isometries between L p -spaces on a finite measure space in the 1930s.This result was extended by Lamperti [61] to certain Orlicz function spaces over σ -finite measurespaces. Representations of isometries between more general complex symmetric function spaceswere later obtained by Lumer and by Zaidenberg [37, 100, 101] (see Arazy’s paper [8] for thecase of complex sequence spaces). Precisely, Zaidenberg showed that under mild conditions onthe complex function spaces E (Ω , Σ , µ ) and E (Ω , Σ , µ ) over the atomless σ -finite measurespaces (Ω , Σ , µ ) and (Ω , Σ , µ ), any surjective isometry T between two complex symmetricfunction spaces E (Ω , Σ , µ ) and E (Ω , Σ , µ ) must be of the elementary form( T f )( t ) = h ( t )( T f )( t ) , f ∈ E , (1.1)where T is the operator induced by a regular set isomorphism from Ω onto Ω and h is ameasurable function on Ω [37, Theorem 5.3.5] (see also [100, 101]). Let (Γ , Σ , µ ) be a discretemeasure space on a set Γ with µ ( { γ } ) = 1 for every γ ∈ Γ. We denote by ℓ p (Γ), 1 ≤ p ≤ ∞ , the L p -space on (Γ , Σ , µ ) [63, p.xi]. Whereas ℓ p (Γ) is a well-studied object (see e.g. [43, 63, 85] andreferences therein) and the description of surjective isometries of ℓ p (Γ) follows yet from [86,99], thecase of arbitrary symmetric spaces E (Γ) for uncountable Γ remained untreated. The description ofisometries of these classical Banach spaces is a simple corollary of our general result, Theorem 1.1below. We also note that the study of isometries on real symmetric function spaces and those oncomplex symmetric function spaces have substantial differences (see e.g. the works of Braveman Mathematics Subject Classification.
Key words and phrases. surjective isometries; hermitian operators; semifinite von Neumann algebra; symmetricoperator spaces.Research supported by the Australian Research Council (FL170100052). and Semenov [18, 19], Jaminson, Kaminska and Lin [48], and Kalton and Randrianantoanina[55,56,75,76]). Throughout this paper, unless stated otherwise, we only consider complex Banachspaces and surjective linear isometries.A noncommutative version of Banach’s description on isometries between L p -spaces [10] wasobtained by Kadison [53] in the 1950s, who showed that a surjective isometry between two vonNeumann algebras can be written as a Jordan ∗ -isomorphism followed by a multiplication of aunitary operator. After the non-commutative L p -spaces were introduced by Dixmier [25] andSegal [83] in the 1950s, the study of L p -isometries was conducted by Broise [20], Russo [79],Arazy [5], Tam [96], etc. A complete description (for the semifinite case) was obtained in 1981 byYeadon [99], who proved that every isometry T : L p ( M , τ ) into −→ L p ( M , τ ), 1 ≤ p = 2 < ∞ , hasthe form T ( x ) = uBJ ( x ) , x ∈ M ∩ L p ( M , τ ) , (1.2)where u is a partial isometry in M , B is a positive self-adjoint operator affiliated with M and J is a Jordan ∗ -isomorphism from M onto a weakly closed ∗ -subalgebra of M (see [51, 52, 86, 97]for the case when M , M are of type III ).The isometries on general symmetric operator spaces on semifinite von Neumann algebras havebeen widely studied since the notion of symmetric operator spaces was introduced in the 1970s (seee.g. [28,29,57,72,73,91] and references therein). The question posed at the beginning of the paperindeed asks whether these isometries T have a natural description as in the cases of symmetricfunction spaces and noncommutative L p -spaces (see (1.1) and (1.2)). One of the most importantdevelopments in this area is due to the work of Sourour [90], who described isometries on separablesymmetric operator ideals, that is, when M is the ∗ -algebra B ( H ) of all bounded linear operatoron a separable Hilbert space H . Adopting Sourour’s techniques, the second author obtained thedescription of isometries on separable symmetric operator spaces affiliated with hyperfinite type II factors [92]. However, the approach used in [90] strongly relies on the matrix representation ofcompact operators on a separable Hilbert space H , which is not applicable for symmetric operatorspaces affiliated with general semifinite von Neumann algebras. In the latter case, only partialresults have been obtained. For example, the general form of isometries of Lorentz spaces on afinite von Neumann algebra was obtained in [23] (see also [68]). Under additional conditions on theisometries (e.g., disjointness-preserving, order-preserving, etc.), similar descriptions can be foundin [1, 23, 38, 46, 49, 50, 58, 67, 69, 77, 93].The following theorem answers the question stated at the outset of this paper. Theorem 1.1.
Let M and M be atomless von Neumann algebras (or atomic von Neumannalgebras whose atoms all have the same trace) equipped with semifinite faithful normal traces τ and τ , respectively. Let E ( M , τ ) and F ( M , τ ) be two symmetric operator spaces whose norms areorder continuous and are not proportional to k·k . If T : E ( M , τ ) → F ( M , τ ) is a surjectiveisometry, then there exist two nets of elements A i ∈ F ( M , τ ) , i ∈ I , disjointly supported fromthe right and B i ∈ F ( M , τ ) , i ∈ I , disjointly supported from the left, a surjective Jordan ∗ -isomorphism J : M → M and a central projection z ∈ M such that T ( x ) = k·k F − X i ∈ I J ( x ) A i z + B i J ( x )( − z ) , ∀ x ∈ E ( M , τ ) ∩ M , where the series is taken as the limit of all finite partial sums. In particular, if M is σ -finite,then the nets { A i } and { B i } are countable. If the trace τ is finite, then there exist elements A, B ∈ F ( M , τ ) such that T ( x ) = J ( x ) Az + BJ ( x )( − z ) , ∀ x ∈ E ( M , τ ) ∩ M . This extends numerous earlier results in this topic (see e.g. [8, 23, 49, 49, 50, 50, 65–68, 79, 86,90, 92, 97, 99]), and Theorem 1.1 yields the first description of surjective isometries on symmetricoperator spaces associated with non-hyperfinite algebras. On the other hand, we show that if M has atoms whose traces are different, then there exists a symmetric space E ( M , τ ) (whose normis not proportional to k·k ) and an isometry on E ( M , τ ) which is not in the form of (1.1) (see ERMITIAN OPERATORS AND ISOMETRIES ON SYMMETRIC OPERATOR SPACES 3
Example 6.2). This demonstrates that the assumption imposed on the von Neumann algebra issharp.Recall that the notion of hermitian operators on a Banach space was formulated by Lumer [65]in his seminal paper in the 1960s, for the purpose of extending Hilbert space type arguments toBanach spaces. This notion plays an important role in different fields such as operator theory onBanach spaces, matrix theory, optimal control theory and computer science (see e.g. [14,37,37,38,41, 65, 87, 98] and references therein).The main method used in this paper for the description of isometries is to establish and employthe general description of hermitian operators on the symmetric operator spaces E ( M , τ ). Thefollowing result is rather surprising as it shows that the stock of hermitian operators does notdepend on the symmetric space E ( M , τ ), and it is fully determined by the algebra M . Theorem 1.2.
Let E ( M , τ ) be a symmetric space on an atomless semifinite von Neumann algebra(or an atomic von Neumann algebra with all atoms having the same trace) M equipped with asemifinite faithful normal trace τ . Assume that k·k E is order continuous and is not proportionalto k·k . Then, a bounded operator T on E ( M , τ ) is a hermitian operator on E ( M , τ ) if and onlyif there exist self-adjoint operators a and b in M such that T x = ax + xb, x ∈ E ( M , τ ) . In particular, T can be extended to a bounded hermitian operator on the von Neumann algebra M . This idea to employ hermitian operators for description of isometries lurks in the backgroundof Lumer’s description of isometries on Orlicz spaces [37], however, the study of hermitian oper-ators on noncommutative spaces is substantially more difficult than that of function spaces, anddescriptions of hermitian operators are known only for very few operator spaces. For example,Sinclair obtained the general form of hermitian operators on a C ∗ -algebra [87] using earlier resultson derivations on operator algebras; Sourour obtained the general form of hermtitian operator onseparable operator ideal of B ( H ) when H is separable [90]; and the case for symmetric operatorspaces on hyperfinite type II factors was obtain by the second author [92] by adopting Sourour’sapproach. For more general von Neumann algebras, the form of a hermitian operator on a sym-metric space (even on noncommutative L p -spaces, see [88, Theorem 4] and [89, Theorem 4.2] forpartial results) was unknown. Theorem 1.2 yields the complete description of hermitian operatorson a symmetric operator space having order continuous norm by using a different approach tothose in [90, 92]. We note that our approach can be applied to the case of hermitian operators onnoncommutative L p -spaces, p ∈ [1 , ∞ ), p = 2, affiliated with a general semifinite von Neumannalgebra.The main ingredient of the proof of Theorem 1.2 is the following surprising observation: anybounded hermitian operator on E ( M , τ ) can be “reduced” to a bounded hermitian operator onthe so-called τ -compact ideal C ( M , τ ) (which is a C ∗ -algebra) and therefore, it can be writtenas the sum of a left-multiplication by a self-adjoint operator in M and a right-multiplication bya self-adjoint operator in M [87]. Having such a description at hand, we are able to describeisometries on symmetric operator spaces affiliated with M and infer Theorem 1.1. However, thestructure of a bounded hermitian operator on a von Neumann algebra is more complicated thanthat of a factor. This inference is, however, far from straightforward. As pointed out in [50, p.825],constructing a suitable Jordan ∗ -isomorphism from an isometry (even if this isometry is positiveand finiteness preserving [50]) is always ‘problematic’. A simple adaptation of proofs in [90, 92]does not yield Theorem 1.1. Many new techniques are required in the proof of Theorem 1.1,which are of interest on their own rights and has potential usage in the future study of hermitianoperators, Jordan ∗ -isomorphisms and isometries of vector-valued spaces.Finally, as an application of Theorem 1.1, we consider a variant of Pe lczy´nski’s problem onthe uniqueness of symmetric structure of operator ideals for symmetric structure of E ( M , τ )affiliated with a II -factor, which establishes a noncommutative version of a result by Abramovichand Zaidenberg for the uniqueness of symmetric structure of L p (0 ,
1) [2, Theorem 1] and itsgeneralizations due to Zaidenberg [100], and Kalton and Randrianantoanina [56, 75].
JINGHAO HUANG AND FEDOR SUKOCHEV
The authors would like to thank Professor Aleksey Ber for helpful discussions, and thankProfessors Evgueni Semenov and Mikhail Zaidenberg for their interest and points to the existingliterature and related problems in this field. We also thank Professor Dmitriy Zanin for pointingout a gap in our original proof of Corollary 4.1 in the earlier version of this paper and providingAppendix A. We thank Thomas Scheckter for his careful reading of this paper.2.
Preliminaries
In this section, we recall main notions of the theory of noncommutative integration, introducesome properties of generalised singular value functions and define noncommutative symmetricoperator spaces. For details on von Neumann algebra theory, the reader is referred to e.g. [15],[26], [54] or [94]. General facts concerning measurable operators may be found in [71], [83], [31](see also the forthcoming book [32]). For convenience of the reader, some of the basic definitionsare recalled.2.1. τ -measurable operators and generalised singular values. In what follows, H is a (notnecessarily separable) Hilbert space and ( B ( H ) , k·k ∞ ) is the ∗ -algebra of all bounded linear oper-ators on H , and is the identity operator on H . Let M be a von Neumann algebra on H . Let P ( M ) be the set of all projections of M . We denote by M p the reduced von Neumann algebra p M p generated by a projection p ∈ P ( M ).A linear operator x : D ( x ) → H , where the domain D ( x ) of x is a linear subspace of H , is saidto be affiliated with M if yx ⊆ xy for all y ∈ M ′ , where M ′ is the commutant of M . A linearoperator x : D ( x ) → H is termed measurable with respect to M if x is closed, densely defined,affiliated with M and there exists a sequence { p n } ∞ n =1 in the set P ( M ) of all projections of M such that p n ↑ , p n ( H ) ⊆ D ( x ) and − p n is a finite projection (with respect to M ) for all n .It should be noted that the condition p n ( H ) ⊆ D ( x ) implies that xp n ∈ M . The collection of allmeasurable operators with respect to M is denoted by S ( M ), which is a unital ∗ -algebra withrespect to strong sums and products (denoted simply by x + y and xy for all x, y ∈ S ( M )).Let x be a self-adjoint operator affiliated with M . We denote its spectral measure by { e x } . It iswell known that if x is a closed operator affiliated with M with the 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 x isclosed, densely defined, affiliated with M and e | x | ( λ, ∞ ) is a finite projection for some λ >
0. Itfollows immediately that in the case when M is a von Neumann algebra of type III or a type I factor, we have S ( M ) = M . For type II von Neumann algebras, this is no longer true. From nowon, let M be a semifinite von Neumann algebra equipped with a faithful normal semifinite trace τ . For any closed and densely defined linear operator x : D ( x ) → H , the null projection n ( x ) = n ( | x | ) is the projection onto its kernel Ker( x ). The left support l ( x ) is the projection onto theclosure of its range Ran( x ) and the right support r ( x ) of x is defined by r ( x ) = − n ( x ).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 . The collection of all τ -measurableoperators is a unital ∗ -subalgebra of S ( M ), denoted by S ( M , τ ). It is well known that a lin-ear operator x belongs to S ( M , τ ) if and only if x ∈ S ( M ) and there exists λ > τ ( e | x | ( λ, ∞ )) < ∞ . Alternatively, an unbounded operator x affiliated with M is τ -measurable(see [35]) if and only if τ (cid:16) e | x | ( n, ∞ ) (cid:17) → , n → ∞ . Definition 2.1.
Let a semifinite von Neumann algebra M be equipped with a faithful normal semi-finite trace τ and let x ∈ S ( M , τ ) . The generalised singular value function µ ( x ) : t → µ ( t ; x ) , t > , of the operator x is defined by setting µ ( t ; x ) = inf {k xp k ∞ : p ∈ P ( M ) , τ ( − p ) ≤ t } . 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 d x ( t ) = τ ( e x ( t, ∞ )) , t > , ERMITIAN OPERATORS AND ISOMETRIES ON SYMMETRIC OPERATOR SPACES 5 we have (see e.g. [35] and [64]) µ ( t ; x ) = inf { s ≥ d | x | ( s ) ≤ t } . Note that d x ( · ) is a right-continuous function (see e.g. [35] and [32]).Consider the algebra M = L ∞ (0 , ∞ ) of all Lebesgue measurable essentially bounded functionson (0 , ∞ ). Algebra M can be viewed as an abelian von Neumann algebra acting via multiplicationon the Hilbert space H = L (0 , ∞ ), with the trace given by integration with respect to Lebesguemeasure m. It is easy to see that the algebra of all τ -measurable operators affiliated with M canbe identified with the subalgebra S (0 , ∞ ) of the algebra of Lebesgue measurable functions whichconsists of all functions f such that m ( {| f | > s } ) is finite for some s >
0. It should also be pointedout that the generalised singular value function µ ( f ) is precisely the decreasing rearrangement µ ( f ) of the function | f | (see e.g. [11, 59]) defined by µ ( t ; f ) = inf { s ≥ m ( {| f | ≥ s } ) ≤ t } . For convenience of the reader, we also recall the definition of the measure topology t τ on thealgebra S ( M , τ ). For every ε, δ > , we define the set V ( ε, δ ) = { x ∈ S ( M , τ ) : ∃ p ∈ P ( M ) such that k x ( − p ) k ∞ ≤ ε, τ ( p ) ≤ δ } . The topology generated by the sets V ( ε, δ ), ε, δ > , is called the measure topology t τ on S ( M , τ )[32, 35, 71]. It is well known that the algebra S ( M , τ ) equipped with the measure topology isa complete metrizable topological algebra [71]. We note that a sequence { x n } ∞ n =1 ⊂ S ( M , τ )converges to zero with respect to measure topology t τ if and only if τ (cid:0) e | x n | ( ε, ∞ ) (cid:1) → n → ∞ for all ε > S ( M , τ ) of τ -compact operators is the space associated to the algebra of functionsfrom S (0 , ∞ ) vanishing at infinity, that is, S ( M , τ ) = { x ∈ S ( M , τ ) : µ ( ∞ ; x ) = 0 } . The two-sided ideal F ( τ ) in M consisting of all elements of τ -finite range is defined by F ( τ ) = { x ∈ M : τ ( r ( x )) < ∞} = { x ∈ M : τ ( s ( x )) < ∞} . Note that S ( M , τ ) is the closure of F ( τ ) with respect to the measure topology [31].A further important vector space topology on S ( M , τ ) is the local measure topology [31, 32]. Aneighbourhood base for this topology is given by the sets V ( ε, δ ; p ), ε, δ > p ∈ P ( M ) ∩ F ( τ ),where V ( ε, δ ; p ) = { x ∈ S ( M , τ ) : pxp ∈ V ( ε, δ ) } . It is clear that the localized measure topology is weaker than the measure topology [31, 32]. If { x α } ⊂ S ( M , τ ) is a net and if x α → α x ∈ S ( M , τ ) in local measure topology, then x α y → xy and yx α → yx in the local measure topology for all y ∈ S ( M , τ ) [31,32]. If 0 ≤ a i is an increasingnet in S ( M , τ ) and if a ∈ S ( τ ) is such that a = sup a i , then we write 0 ≤ a i ↑ a [31, p.212]. If { x i } is an increasing net in S ( M , τ ) + and x ∈ S ( M , τ ) + such that x i → x in the local measuretopology, then x i ↑ x (see e.g. [32, Chapter II, Proposition 7.6 (iii)]).2.2. Symmetric spaces of τ -measurable operators. Let E (0 , ∞ ) be a Banach space of real-valued Lebesgue measurable functions on (0 , ∞ ) (with identification m -a.e.), equipped with a norm k·k E . The space E (0 , ∞ ) is said to be absolutely solid if x ∈ E (0 , ∞ ) and | y | ≤ | x | , y ∈ S (0 , ∞ )implies that y ∈ E (0 , ∞ ) and k y k E ≤ k x k E . An absolutely solid space E (0 , ∞ ) ⊆ S (0 , ∞ ) is saidto be symmetric if for every x ∈ E (0 , ∞ ) and every y ∈ S (0 , ∞ ), the assumption µ ( y ) = µ ( x )implies that y ∈ E (0 , ∞ ) and k y k E = k x k E [59]. Without of loss generality, throughout this paper,we always assume that (cid:13)(cid:13) χ (0 , (cid:13)(cid:13) E (0 , ∞ ) = 1.We now come to the definition of the main object of this paper. Definition 2.2.
Let M be a semifinite von Neumann algebra equipped with a faithful normalsemi-finite trace τ . Let E be a linear subset in S ( M , τ ) equipped with a complete norm k·k E . Wesay that E is a symmetric space if for x ∈ E , y ∈ S ( M , τ ) and µ ( y ) ≤ µ ( x ) imply that y ∈ E and k y k E ≤ k x k E . JINGHAO HUANG AND FEDOR SUKOCHEV
Let E ( M , τ ) be a symmetric space. It is well-known that any symmetrically normed space E ( M , τ ) is a normed M -bimodule (see e.g. [31] and [32]). That is, for any symmetric operatorspace E ( M , τ ), we have k axb k E ≤ k a k ∞ k b k ∞ k x k E , a, b ∈ M , x ∈ E ( M , τ ). It is known thatwhenever E ( M , τ ) has order continuous norm k·k E , i.e., k x α k E ↓ ≤ x α ↓ ⊂ E ( M , τ ), we have E ( M , τ ) ⊂ S ( M , τ ) [31, 32, 46].The so-called K¨othe dual is identified with an important part of the dual space. If E ( M , τ ) ⊂ S ( M , τ ) is a symmetric space, then the K¨othe dual E ( M , τ ) × of E is defined by setting E ( M , τ ) × = ( x ∈ S ( M , τ ) : sup k y k E ≤ ,y ∈ E ( M ,τ ) τ ( | xy | ) < ∞ ) . The K¨othe dual E ( M , τ ) × can be identified as a subspace of the Banach dual E ( M , τ ) via the traceduality [31, p.228]. Recall that x ∈ L ( M , τ )+ M := { a ∈ S ( M , τ ) : µ ( a ) ∈ L (0 , ∞ )+ L ∞ (0 , ∞ ) } can be equipped with a norm k x k L + L ∞ = R µ ( s ; x ) ds and x ∈ L ( M , τ ) ∩ M := { a ∈ S ( M , τ ) : µ ( a ) ∈ L (0 , ∞ ) ∩ L ∞ (0 , ∞ ) } can be equipped with a norm k x k L ∩ L ∞ := max {k x k , k x k ∞ } .In particular, ( L ( M , τ ) ∩ M ) × = L ( M , τ ) + M and L ( M , τ ) ∩ M = ( L ( M , τ ) + M ) × [31,Example 4].The carrier projection c E ∈ M of E ( M , τ ) is defined by setting c E := ∨{ p : p ∈ P ( E ) } . It is clear that c E is in the center Z ( M ) of M [31]. It is often assumed that the carrier projection c E is equal to . Indeed, for any symmetric function space E (0 , ∞ ), the carrier projection of thecorresponding operator space E ( M , τ ) is always (see e.g. [31, 57]). On the other hand, if M isatomless or is atomic and all atoms have equal trace, then any non-zero symmetric space E ( M , τ ) isnecessarily [31, 32]. In this case, whenever E ( M , τ ) has order continuous norm, then E ( M , τ ) × is isometrically isomorphic to E ( M , τ ) ∗ (see e.g. [29], [30, Proposition 6.4] or [31, Proposition47(v)]).There exists a strong connection between symmetric function spaces and operator spaces ex-posed in [57] (see also [31, 64]). The operator space E ( M , τ ) defined by E ( M , τ ) := { x ∈ S ( M , τ ) : µ ( x ) ∈ E (0 , ∞ ) } , k x k E ( M ,τ ) := k µ ( x ) k E is a complete symmetric space whenever ( E (0 , ∞ ) , k·k E ) is a complete symmetric function space on(0 , ∞ ) [57]. In particular, for any symmetric function space E (0 , ∞ ), F ( τ ) ⊂ E ( M , τ ) [31, Lemma18]. In the special case when E (0 , ∞ ) = L p (0 , ∞ ), 1 ≤ p ≤ ∞ , E ( M , τ ) is the noncommutative L p -spaces affiliated with M and we denote the norm by k·k p . We note that if E (0 , ∞ ) is separable (i.e.has order continuous norm), then E × ( M , τ ) is isometrically isomorphic to E ( M , τ ) ∗ [31, p.246].Recall that every separable symmetric sequence/function space E is fully symmetric, that is, if x ∈ E and y ∈ ℓ ∞ (resp. y ∈ S (0 , ∞ )) with Z t µ ( t ; y ) dt ≤ Z t µ ( t ; x ) dt, t ≥ y ≺≺ x ), then y ∈ E with k y k E ≤ k x k E (see e.g. [59, Chapter II,Theorem 4.10]or [32, Chapter IV, Theorem 5.7]).3. Hermitian operators
Let X be a Banach space. Recall that a semi-inner product (abbreviated s.i.p. ) on X is amapping h· , ·i of X × X into the field of complex numbers such that:(1) h x + y, z i = h x, z i + h y, z i for x, y, z ∈ X ;(2) h λx, y i = λ h x, y i for x, y ∈ X and λ ∈ C ;(3) h x, x i > = x ∈ X ;(4) |h x, y i| ≤ h x, x ih y, y i for any x, y ∈ X .When a s.i.p is defined on X , we call X a semi-inner-product space (abbreviated s.i.p.s. ). If X is a s.i.p.s., then h x, x i is a norm on X . On the other hand, every Banach space can be madeinto a s.i.p.s. (in general, in infinitely many ways) so that the s.i.p. is consistent with the norm, ERMITIAN OPERATORS AND ISOMETRIES ON SYMMETRIC OPERATOR SPACES 7 i.e., h x, x i = k x k for any x ∈ X [37]. By virtue of the Hahn–Banach theorem, this can beaccomplished by choosing one bounded linear functional f x for each x ∈ X such that k f x k = k x k and f x ( x ) = k x k ( f x is called a support functional of x ), and then setting h x, y i = f y ( x ) forarbitrary x, y ∈ X [14, 37, 65]. Given a linear transformation T mapping a s.i.p.s. into itself, wedenote by W ( T ) the numerical range of T , that is, {h T x, x i|h x, x i = 1 , x ∈ X } . Let T be anoperator on a Banach space ( X, k·k ). Although in principle there may be many different s.i.p.consistent with k·k , nonetheless if the numerical range of T relative to one such s.i.p. is real,then the numerical range relative to any such s.i.p. is real (see e.g. [37, p.107], [65, Section 6]and [14, p.377]). If this is the case, T is said to be a hermitian operator on X .From now now, unless stated otherwise, we always assume that M is an atomless semifinite vonNeumann algebra or an atomic semifinite von Neumann algebra with all atoms having the sametraces (without loss of generality, we assume that τ ( e ) = 1 for any atom e ∈ M ), and we assumethat τ is a semifinite faithful normal trace on M .In particular, when M is atomless (resp. atomic), the set E (0 , τ ( )) := { f ∈ S (0 , τ ( )) : µ ( f ) = µ ( x ) for some x ∈ E ( M , τ ) } (resp. ℓ E := { f ∈ ℓ ∞ : µ ( f ) = µ ( x ) for some x ∈ E ( M , τ ) } )is a symmetric function (resp. sequence) space [64, Theorem 2.5.3]. There exists a bijective corre-spondence between symmetric operator spaces and symmetric function/sequence spaces. There-fore, if k·k E on E ( M , τ ) is not proportional to k·k on L ( M , τ ), then k·k E is not proportional to k·k on L ( A , τ ) for any maximal abelian von Neumann subalgebra A of M .Sourour [90, Lemma 1] obtained Lemma 3.3 below in the setting of B ( H ) by using a resultdue to Schneider and Turner (see e.g. [82, Lemma 3.1] and [37, Lemma 9.2.7]). Arazy gave aself-contained alternative proof in the setting of complex sequence spaces [8]. In the proof of thefollowing lemma, we adopt Arazy’s proof. Due to the technical differences between the atomlesscase and atomic case, we provide a full proof below.Before proceeding to the proof of Lemma 3.3, we need the following well-known proposition.For the sake of completeness, we provide a short proof below. Proposition 3.1.
Let p ∈ F ( τ ) be a projection and let E ( M , τ ) be an arbitrary symmetricoperator space having order continuous norm. Then, k p k E τ ( p ) p ∈ E ( M , τ ) ∗ is a support functionalof p ∈ E ( M , τ ) , i.e., τ (cid:16) p · k p k E τ ( p ) p (cid:17) = k p k E = k p k E (cid:13)(cid:13)(cid:13) k p k E τ ( p ) p (cid:13)(cid:13)(cid:13) E ∗ . In particular, for any boundedhermitian operator T on E ( M , τ ) , we have τ ( T ( p ) p ) ∈ R . (3.1) Proof.
We only consider the case when M is atomless. The atomic case follows from the sameargument (see also [8] or [37, Theorem 5.2.13]).Note that k p k E ∗ = k p k E × = sup (Z τ ( p )0 µ ( s ; z ) ds : z ∈ E ( M , τ ) , k z k E = 1 ) . Since R τ ( p )0 µ ( s ; z ) dsτ ( p ) µ ( p ) = R τ ( p )0 µ ( s ; z ) dsτ ( p ) χ (0 ,τ ( p )) ≺≺ µ ( z ), z ∈ E ( M , τ ) (see e.g. [64, Section 3.6]),we obtain that R τ ( p )0 µ ( s ; z ) dsτ ( p ) k p k E ≤ k z k E = 1, and therefore, k p k E ∗ ≤ τ ( p ) k p k E . On the other hand, we have [31, Remark 3] τ ( p ) ≤ k p k E ∗ k p k E . Hence, τ ( p ) = k p k E ∗ k p k E , i.e. τ (cid:16) p · k p k E τ ( p ) p (cid:17) = k p k E = k p k E (cid:13)(cid:13)(cid:13) k p k E τ ( p ) p (cid:13)(cid:13)(cid:13) E ∗ . (cid:3) JINGHAO HUANG AND FEDOR SUKOCHEV
Corollary 3.2.
Let u ∈ F ( τ ) be a partial isometry and let E ( M , τ ) be an arbitrary symmetricoperator space having order continuous norm. Then, k u ∗ u k E τ ( u ∗ u ) u ∗ ∈ E ( M , τ ) ∗ is a support functionalof u ∈ E ( M , τ ) . In particular, for any bounded hermitian operator T on E ( M , τ ) , we have τ ( T ( u ) u ∗ ) ∈ R . Proof.
Since u ∈ F ( τ ), it follows that r ( u ) and l ( u ) are τ -finite projections. Hence, r ( u ) ∨ l ( u ) arealso τ -finite. Therefore, there exists a unitary element v in M r ( u ) ∨ l ( u ) such that v ∗ l ( u ) v = r ( u ) [95,Chapter XIV, Lemma 2.1]. Define v ′ := u + v · ( r ( u ) ∨ l ( u ) − r ( u )). Note that( r ( u ) ∨ l ( u ) − r ( u )) v ∗ u = ( r ( u ) ∨ l ( u ) − r ( u )) v ∗ l ( u ) u = ( r ( u ) ∨ l ( u ) − r ( u )) r ( u ) v ∗ u = 0 . We have ( v ′ ) ∗ v ′ = ( u + v · ( r ( u ) ∨ l ( u ) − r ( u ))) ∗ ( u + v · ( r ( u ) ∨ l ( u ) − r ( u )))= r ( u ) + ( r ( u ) ∨ l ( u ) − r ( u )) = r ( u ) ∨ l ( u ) . Hence, v ′ is a unitary element in M r ( u ) ∨ l ( u ) and therefore, v ′′ := v ′ + ( − r ( u ) ∨ l ( u ))is a unitary element in M . It follows that ( v ′′ ) ∗ T ( v ′′ · ) on E ( M , τ ) is also a bounded hermitianoperator [37, p.22]. Noting that v ′′ r ( u ) = u , we obtain τ ( T ( u ) u ∗ ) = τ (( v ′′ ) ∗ T ( v ′′ r ( u )) r ( u )) (3.1) ∈ R . (cid:3) Lemma 3.3.
Let E ( M , τ ) be a symmetric space affiliated with M , whose norm is order continuousand is not proportional to k·k . Let x , x ∈ F ( τ ) be two partial isometries such that l ( x ) ⊥ l ( x ) and r ( x ) ⊥ r ( x ) . Then, for any bounded hermitian operator T : E ( M , τ ) → E ( M , τ ) , we have τ ( T ( x ) x ∗ ) = 0 . Moreover, if x ∈ E ( M , τ ) and x ∈ L ( M , τ ) ∩ M with l ( x ) ⊥ l ( x ) and r ( x ) ⊥ r ( x ) , then τ ( T ( x ) x ∗ ) = 0 . Proof.
We only prove the case for atomless von Neumann algebra. The atomic case follows by asimilar argument.We first consider the case when x and x are two projections such that x x = 0.Since k·k E is not proportional to k·k , it follows that there exists a set of pairwise orthogonalprojections { e i } ≤ i ≤ n having the same trace such that k·k E on E ( A ) is not proportional to k·k on L ( A ), where A is the abelian weakly closed ∗ -algebra generated by { e i } ≤ i ≤ n . Let t , := τ ( T ( e ) e ) and t , := τ ( T ( e ) e ) . By Proposition 3.1, we obtain that τ ( T ( e ) e ) , τ ( T ( e ) e ) ∈ R . We claim that t i,j = t j,i when i, j = 1 , , · · · , n, and i = j . Define x θ := e + e iθ e , 0 ≤ θ ≤ π . In particular, τ ( x θ x ∗ θ ) = τ ( e + e ) ∈ R . By Proposition 3.1, we obtain that τ ( T ( x θ ) x ∗ θ ) ∈ R , i.e., e iθ t i,j + e − iθ t j,i ∈ R for all θ . Hence, t i,j = t j,i . (3.2)By [8, Lemma 4] (see also [3, 37]), there exists 1 < n < ∞ , x = P nk =1 x ( k ) e k , y = P nk =1 y ( k ) e k so that(1) x ( k ) ≥ y ( k ) ≥ k ;(2) k x k E = k y k E ∗ = τ ( xy ∗ ) = 1;(3) x and y are linearly independent. ERMITIAN OPERATORS AND ISOMETRIES ON SYMMETRIC OPERATOR SPACES 9
Let θ = ( θ , · · · , θ n ), 0 ≤ θ k ≤ π , we let x θ = n X k =1 e iθ k x ( k ) e k and y θ = n X k =1 e iθ k y ( k ) e k . By the assumption on x and y , we obtain that k x θ k E = k y θ k E ∗ = τ ( y ∗ θ x θ ) (2) = 1 . Therefore, by the definition of a hermitian operator, for any θ , we have0 = Im τ ( y ∗ θ T ( x θ ))= Im n X k,l =1 e i ( θ k − θ l ) x ( k ) y ( l ) t l,k (3.2) = 12 i X k = l x ( k ) y ( l )( e i ( θ k − θ l ) t l,k − e i ( θ l − θ k ) t k,l )= 12 i X k = l x ( k ) y ( l ) e i ( θ k − θ l ) t l,k − i X k = l x ( k ) y ( l ) e i ( θ l − θ k ) t k,l = 12 i X k = l x ( k ) y ( l ) e i ( θ k − θ l ) t l,k − i X k = l x ( l ) y ( k ) e i ( θ k − θ l ) t l,k = 12 i X k = l e i ( θ k − θ l ) t l,k ( x ( k ) y ( l ) − x ( l ) y ( k ))This implies that t l,k ( x ( k ) y ( l ) − x ( l ) y ( k )) = 0 for all k, l .Since x, y are linearly independent, it follows that there exists k = l such that x ( k ) y ( l ) − x ( l ) y ( k ) = 0 and thus t l,k = 0. Replacing in this argument x and y by x π = P x k e π ( k ) and y π = P y k e π ( k ) , respectively, where π is an arbitrary permutation of { , · · · , n } , we deduce that t k,l = 0 for every k = l . Therefore, we obtain that τ ( T ( p ) q ) = 0for any projections p, q with pq = 0 and τ ( p ) = τ ( q ) = τ ( e ). Clearly, if the space ( E ( A ) , k·k E ( M ,τ ) )on the algebra generated by e k , 1 ≤ k ≤ n , is not proportional to k·k , then the norm k·k E on E ( A ′ ) on the algebra generated by e ′ k , 1 ≤ k ≤ n , is not proportional to k·k as well, where τ ( e ′ k ) = τ ( e k ). This implies that for any k ∈ N , if p, q are τ -finite projections such that pq = 0and τ ( p ) = τ ( q ) = k τ ( e ), then τ ( T ( p ) q ) = 0 . (3.3)The case when τ ( p ) = τ ( q ) follows by standard approximation argument. Indeed, let p, q be τ -finite projections with pq = 0. For any ε >
0, there exist two sets of τ -finite projections { p i } ≤ i ≤ n and { q i } ≤ i ≤ m such that τ p − X ≤ i ≤ n p i , τ q − X ≤ i ≤ m q i ≤ ε and τ ( p i ) = τ ( q j ) = 12 k τ ( e ) , ≤ i ≤ n, ≤ j ≤ m for some k ∈ N . Note that τ ( T ( p ) q )= τ T ( p ) q − X ≤ i ≤ m q i + τ T p − X ≤ i ≤ n p i X ≤ i ≤ m q i + τ T X ≤ i ≤ n p i X ≤ i ≤ m q i (3.3) = τ T ( p ) q − X ≤ i ≤ n p i + τ T p − X ≤ i ≤ n p i X ≤ i ≤ m q i . Since (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) τ T ( p ) q − X ≤ i ≤ m q i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ τ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T ( p ) q − X ≤ i ≤ m q i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) T ( p ) q − X ≤ i ≤ m q i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) → ε → (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) τ T p − X ≤ i ≤ n p i X ≤ i ≤ m q i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k T k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p − X ≤ i ≤ n p i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) E (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X ≤ i ≤ m q i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) E × ≤ k T k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p − X ≤ i ≤ n p i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) E (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X ≤ i ≤ m q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) E × → ε → k·k E is order continuous, it follows that | τ ( T ( p ) q ) | = 0 . The general case when x and x are partial isometries in F ( τ ) such that l ( x ) ⊥ l ( x ) and r ( x ) ⊥ r ( x ) is reduced to the just considered case via the same argument as in Corollary 3.2.Therefore, we obtain that τ ( T ( x ) x ∗ )) = 0 , which completes the proof of the first assertion.Now, we prove the second assertion. Since E ( M , τ ) ⊂ S ( M , τ ), there exist two sequences { y n } and { y ′ n } in F ( τ ) such that 0 ≤ y n ↑ | x | and 0 ≤ y ′ n ↑ | x | and y n (resp. y ′ n ) aregenerated by spectral projections of | x | (resp. | x | ). Let x = u | x | and x = u | x | be the polardecompositions. By the first assertion of the lemma, we obtain that τ ( T ( u y n )( u y ′ m ) ∗ ) = 0 forevery n and m . Since u y n → n x in k·k E , it follows that T ( u y n ) → T ( x ) in k·k E , and therefore T ( u y n ) → T ( x ) weakly. Hence, τ ( T ( x )( u y ′ m ) ∗ ) = 0 for each m .Consider the special case when x ∈ F ( τ ). In this case, we may assume, in addition, that k y ′ m − x k ∞ →
0. We obtain that | τ ( T ( x ) | x | u ∗ ) | = | τ ( T ( x ) r ( x )( y ′ m − | x | ) u ∗ ) | ≤ k T ( x ) r ( x ) k k y ′ m − | x |k ∞ → . (3.4)For the general case, let p := E | x | ( δ, ∞ ), δ >
0, be a τ -finite spectral projection of | x | suchthat k| x | ( − p ) k L ∩ L ∞ ≤ ε . Hence, we obtain that | τ ( T ( x ) | x | u ∗ ) | (3.4) = | τ ( T ( x )( − p ) | x | u ∗ ) |≤ k T ( x ) k L + L ∞ k ( − p ) | x | u ∗ k L ∩ L ∞ ≤ k T ( x ) k L + L ∞ · ε. Since ε is arbitrarily taken, it follows that | τ ( T ( x ) x ∗ ) | = 0. (cid:3) The following result is a semifinite version of [90, Corollary 1].
Corollary 3.4.
Let E ( M , τ ) be a symmetric space having order continuous norm and k·k E is notproportional to k·k . Let T be a bounded hermitian operator on E ( M , τ ) . For any x ∈ E ( M , τ ) ,there exist y, z ∈ E ( M , τ ) such that T ( x ) = y + z and r ( y ) ≤ r ( x ) and l ( z ) ≤ l ( x ) . ERMITIAN OPERATORS AND ISOMETRIES ON SYMMETRIC OPERATOR SPACES 11
Proof.
Denote A := T ( x ). Note that A = l ( x ) Ar ( x ) + l ( x ) Ar ( x ) ⊥ + l ( x ) ⊥ Ar ( x ) + ( l ( x ) ⊥ Ar ( x ) ⊥ ) . Assume that l ( x ) ⊥ Ar ( x ) ⊥ = 0. Let p ∈ F ( τ ) be a τ -finite projection such that z = l ( x ) ⊥ Ar ( x ) ⊥ p =0. Let zp = u | zp | be the polar decomposition. Then, u ∗ l ( x ) ⊥ Ar ( x ) ⊥ p = u ∗ zp ≥ , i.e., τ ( T ( x ) r ( x ) ⊥ pu ∗ l ( x ) ⊥ ) = τ ( u ∗ l ( x ) ⊥ Ar ( x ) ⊥ p ) > l (cid:16) l ( x ) ⊥ upr ( x ) ⊥ (cid:17) ⊥ l ( x ) and r (cid:0) l ( x ) ⊥ upr ( x ) ⊥ (cid:1) ⊥ r ( x ). By Lemma 3.3 above, weobtain that τ ( u ∗ l ( x ) ⊥ Ar ( x ) ⊥ p ) = τ ( T ( x ) r ( x ) ⊥ pu ∗ l ( x ) ⊥ ) = 0 , which is a contradiction. Taking y = l ( x ) Ar ( x ) + l ( x ) ⊥ Ar ( x ) and z = l ( x ) Ar ( x ) ⊥ , we completethe proof (note that the choices of y and z are not necessarily unique). (cid:3) The following lemma shows that any bounded hermitian operator T on E ( M , τ ) (whose normis not proportional to k·k ) maps the set of all τ -finite projections to a uniformly bounded set in M , which should be compared with the estimates in [92, Remark 2.5]. Lemma 3.5.
Let E ( M , τ ) is an arbitrary symmetric operator space having order continuousnorm. Assume that k·k E is not proportional to k·k . Let T be a bounded hermitian operator on E ( M , τ ) . Then, k T ( p ) k ∞ ≤ k T k for any τ -finite projection p ∈ P ( M ) .Proof. Let p ∈ P ( M ) be an arbitrary τ -finite projection. By Corollary 3.4 above, we have T ( p ) = A p p + pB p (3.5)for some A p , B p ∈ E ( M , τ ) with r ( A p ) ≤ p and l ( B p ) ≤ p . We note that the choices of A p and B p are not necessarily unique. For two τ -finite projections q , q ∈ M such that q q = 0 and q + q = p , we have T ( p ) = T ( q + q ) = A q q + q B q + A q q + q B q = ( A q q + A q q ) p + p ( q B q + q B q ) . Therefore, we have q T ( q ) q = q A q q + q B q q = q T ( p ) q (3.6)and p ⊥ A p q = p ⊥ T ( p ) q = p ⊥ ( A q q + A q q ) pq = p ⊥ A q q = p ⊥ ( A q q + q B q ) (3.5) = p ⊥ T ( q ) . (3.7)Consider the following decomposition (see Corollary 3.4 or (3.5)) T ( p ) = p ⊥ T ( p ) + T ( p ) p ⊥ + pT ( p ) p = p ⊥ A p p + pB p p ⊥ + pT ( p ) p. We claim that (cid:13)(cid:13) p ⊥ A p (cid:13)(cid:13) ∞ ≤ k T k E → E . Assume by contradiction that (cid:13)(cid:13) p ⊥ A p (cid:13)(cid:13) ∞ > k T k E → E . Then,taking q = E | p ⊥ A p | ( k T k , ∞ ) = 0, we obtain that k T k k q k E < (cid:13)(cid:13) p ⊥ A p q (cid:13)(cid:13) E (3.7) = (cid:13)(cid:13) p ⊥ T ( q ) (cid:13)(cid:13) E ≤ k T ( q ) k E ≤ k T k k q k E , which is a contradiction. Arguing similarly, (cid:13)(cid:13) B p p ⊥ (cid:13)(cid:13) ∞ ≤ k T k . Now, we aim to show that k pT ( p ) p k ∞ ≤ k T k . Note that, by (3.5), we have pT ( p ) p = p ( A p + B p ) p. Let C := p ( A p + B p ) p . We claim that C is self-adjoint. Indeed, assume that p ( A p + B p ) p = a + ib ,where a, b are non-zero self-adjoint operators in E ( M , τ ) and r ( a ) , r ( b ) ≤ p . Let q = E b (0 , ∞ ) ≤ p (or q = E b ( −∞ ,
0) if E b (0 , ∞ ) = 0) so that qbq = 0, which is a τ -finite projection. We obtain that τ ( T ( q ) q ) = τ ( qT ( q ) q ) (3.6) = τ ( qpT ( p ) pq ) (3.5) = τ ( qp ( A p + B p ) pq ) = τ ( qaq + iqbq ) / ∈ R , which contradicts Proposition 3.1. Hence, b = 0. This implies that pT ( p ) p is self-adjoint. Recallthat for any q ≤ p , we have qT ( p ) q (3.6) = qT ( q ) q. Let q := E | pT ( p ) p | [ k T k + ε, ∞ ), ε >
0. If q = 0, then we have( k T k + ε ) k q k E = k ( k T k + ε ) q k E ≤ k pT ( p ) pq k E = k qT ( p ) q k E = k qT ( q ) p k E ≤ k T ( q ) k E ≤ k T k k q k E , which is a contradiction. Hence, E | pT ( p ) p | [ k T k + ε, ∞ ) = 0 for any ε >
0. This implies that k pT ( p ) p k ∞ ≤ k T k .Combining the estimates (cid:13)(cid:13) p ⊥ A p (cid:13)(cid:13) ∞ ≤ k T k , (cid:13)(cid:13) B p p ⊥ (cid:13)(cid:13) ∞ ≤ k T k and k pT ( p ) p k ∞ ≤ k T k , we obtainthat k T ( p ) k ∞ ≤ k T k for any τ -finite projection p . In particular, A p and B p can be taken such that k A p k ∞ , k B p k ∞ ≤ k T k (see the proof for Corollary 3.4). (cid:3) The following lemma is the key auxiliary tool in the proof of Proposition 3.8, which showsthat any bounded hermitian operator T on E ( M , τ ) is a bounded operator from ( F ( τ ) , k·k ∞ )into ( C ( M , τ ) , k·k ∞ ). By applying the generalized Gleason theorem [42, Theorem 5.2.4] (seealso [16,39,40,70] for related results), we succeed to prove all but one case in the following lemma.However, one should note that the case when a von Neumann algebra has I direct summand isexceptional, which can not be covered by the generalized Gleason theorem. This special case israther complicated and requires careful study of the restriction of a hermitian operator on thetype I summand.We denote by C ( M , τ ) the closure in the norm k·k ∞ of the linear span of τ -finite projectionsin M . Equivalently, C ( M , τ ) = { a ∈ S ( M , τ ) : µ ( a ) ∈ L ∞ (0 , ∞ ) , µ ( ∞ , a ) = 0 } [64, Lemma2.6.9]. Lemma 3.6.
Let E ( M , τ ) is an arbitrary symmetric operator space having order continuousnorm. Assume that k·k E is not proportional to k·k . Let T be a bounded hermitian operator on E ( M , τ ) . Then, T is a bounded operator from ( F ( τ ) , k·k ∞ ) into ( C ( M , τ ) , k·k ∞ ) . In particular, T extends to a bounded operator from C ( M , τ ) into C ( M , τ ) .Proof. There exists a decomposition M = M ⊕ M , where M has no type I direct summandand M is either 0 or the type I direct summand of M (see e.g. [15, Chapter III.1.5.12]).By Lemma 3.5, k T ( p ) k ∞ ≤ k T k for any τ -finite projection p ∈ M . Moreover, T ( p ) = A p + B p (see Corollary 3.4) for some A p , B p ∈ E ( M , τ ) with r ( A p ) ≤ p and l ( B p ) ≤ p . Since p ≤ M , itfollows that A p = A p M ∈ M and B p = M B p ∈ M . On the other hand, τ ( p ) < ∞ impliesthat T ( p ) ∈ F ( M ) ⊂ C ( M , τ ) ⊂ C ( M , τ ). It follows from [42, Theorem 5.2.4] that for any τ -finite projection p , T | P f ( M ) extends uniquely to a bounded linear operator from the reducedalgebra p M p into C ( M , τ ). We denote this operator by R p . Moreover, k R p | p M p k k·k ∞ →k·k ∞ ≤ k T k (see the proof of [42, Theorem 5.2.4]). Moreover, by the uniqueness of the extension R p [42,Theorem 5.2.4], we obtain that if p ≥ q , then the extension R p of T coincides with R q on q M q .We claim that R p coincide with T on p M p . Indeed, let x be a positive operator in p M p .Then, there exists a sequence of positive operators x n whose singular values are step functionssuch that x n ↑ x and k x n − x k ∞ →
0. Since R p coincides with T on all projection q ≤ p , it followsthat R p ( x n ) = T ( x n ). Since E ( M , τ ) has order continuous norm, it follows that T ( x n ) → T ( x ) in k·k E , and therefore, in the measure topology [31, Proposition 20]. On the other hand, the k·k ∞ - k·k ∞ -boundedness of R p implies that R p ( x n ) → R p ( x ) in k·k ∞ , and therefore, in the measuretopology [31, Proposition 20]. Hence, T ( x ) = R p ( x ). Since p is an arbitrary τ -finite projection, itfollows that T is a bounded linear operator from ( F ( τ ) ∩ M , k·k ∞ ) into ( C ( M , τ ) , k·k ∞ ).Now, we consider the case when M is a non-vanishing type I von Neumann direct summand(if M = 0, then the lemma follows from the above result). It is known that M can be writtenas M ⊗A , where M is the algebra of all 2 ⊗ A is a σ -finite commutative von ERMITIAN OPERATORS AND ISOMETRIES ON SYMMETRIC OPERATOR SPACES 13
Neumann algebra (see e.g. [54, 94] or [15, Chapter III.1.5.12]). For every element in the form of (cid:18) p
00 0 (cid:19) , where p is a projection in A such that τ ( ⊗ p ) < ∞ , we have (see Corollary 3.4) T (cid:18) p
00 0 (cid:19) = A p (cid:18) p
00 0 (cid:19) + (cid:18) p
00 0 (cid:19) B p . By Lemma 3.5, A p and B p are uniformly bounded. Assume that A p = (cid:18) a a a a (cid:19) and B p = (cid:18) b b b b (cid:19) . Without loss of generality, we may assume, in addition, that a , a , b , b are 0.Hence, T (cid:18) p
00 0 (cid:19) = (cid:18) a a (cid:19) (cid:18) p
00 0 (cid:19) + (cid:18) p
00 0 (cid:19) (cid:18) b b (cid:19) = (cid:18) ( a + b ) p b pa p (cid:19) . Recall that k A p k ∞ , k B p k ∞ ≤ k T k (see the proof of Lemma 3.5). We obtain that a , a , b , b ≤ k T k . Hence, for any x ∈ A whose singular value function is a step function, we have T (cid:18) x
00 0 (cid:19) = (cid:18) h x h xh x (cid:19) , where k h k ∞ , k h k ∞ , k h k ∞ ≤ k T k . Therefore, (cid:13)(cid:13)(cid:13)(cid:13) T (cid:18) x
00 0 (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) ∞ ≤ k T k k x k ∞ . For anyself-adjoint x ∈ F ( τ ) there exists a sequence of self-adjoint elements x n ∈ A such that | x n | ↑ | x | , k x n − x k ∞ → µ ( x n ) are step functions. Since E ( M , τ ) has order continuous norm, it followsthat (cid:13)(cid:13)(cid:13)(cid:13) T (cid:18) x n
00 0 (cid:19) − T (cid:18) x
00 0 (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) E → T (cid:18) x n
00 0 (cid:19) → T (cid:18) x
00 0 (cid:19) in measure [31, Proposition 20]. On the other hand, (cid:13)(cid:13)(cid:13)(cid:13) T (cid:18) x n
00 0 (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) ∞ ≤ k T k k x n k ∞ ≤ k T k k x k ∞ . Since the unit ball of M is closed in themeasure topology [31, Theorem 32], we obtain that (cid:13)(cid:13)(cid:13)(cid:13) T (cid:18) x
00 0 (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) ∞ ≤ k T k k x k ∞ for any self-adjoint operator (cid:18) x
00 0 (cid:19) ∈ F ( τ ) ∩ M . Since every element in F ( τ ) ∩ M is thecombination of two self-adjoint elements, we obtain that (cid:13)(cid:13)(cid:13)(cid:13) T (cid:18) x
00 0 (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) ∞ . k T k k x k ∞ for any operator (cid:18) x
00 0 (cid:19) ∈ F ( τ ) ∩ M . The same argument show that (cid:13)(cid:13)(cid:13)(cid:13) T (cid:18) x (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) ∞ . k T k k x k ∞ . For estimates of (cid:13)(cid:13)(cid:13)(cid:13) T (cid:18) x (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) ∞ and (cid:13)(cid:13)(cid:13)(cid:13) T (cid:18) x (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) ∞ , one only need to note that T ′ ( · ) := (cid:18)(cid:18) A A (cid:19) + M (cid:19) T (cid:18)(cid:18)(cid:18) A A (cid:19) + M (cid:19) · (cid:19) is also a hermitian operator on E ( M , τ ) and T ′ (cid:18) x
00 0 (cid:19) = T (cid:18) x (cid:19) . By taking the linear combination, we obtain that for any x ∈ F ( τ ) ∩ M , k T ( x ) k ∞ . k T k k x k ∞ . (cid:3) We prove below an analogue of Proposition 3.1 for the symmetric space C ( M , τ ). Proposition 3.7.
Let E ( M , τ ) is an arbitrary symmetric operator space having order continuousnorm. Assume that k·k E is not proportional to k·k . Let T be a bounded hermitian operator on E ( M , τ ) . Then, for any operator x ∈ C ( M , τ ) and a τ -finite projection p ∈ P ( M ) commutingwith | x | , we have h T x, pu ∗ i ( C ( M ,τ ) ,C ( M ,τ ) ∗ ) := τ ( T ( x ) pu ∗ ) ∈ R , where x = u | x | is the polar decomposition.Proof. We only consider the case when x is self-adjoint. The proof of the general case follows fromthe same argument by replacing Proposition 3.1 used below with Corollary 3.2.Let x n := P ≤ k ≤ n α k p k ∈ F ( τ ) be such that x n → x in k·k ∞ , where p k are τ -finite spectralprojections of x n which commute with p , and α k are real numbers. For each p k , we have h T p k , p i ( C ( M ,τ ) ,C ( M ,τ ) ∗ ) = h T ( pp k ) , pp k i ( C ( M ,τ ) ,C ( M ,τ ) ∗ ) + h T ( pp k ) , p − pp k i ( C ( M ,τ ) ,C ( M ,τ ) ∗ ) + h T ( p k − pp k ) , p i ( C ( M ,τ ) ,C ( M ,τ ) ∗ )(3.3) = h T ( pp k ) , pp k i ( C ( M ,τ ) ,C ( M ,τ ) ∗ ) = τ ( T ( pp k ) pp k ) (3.1) ∈ R . Hence, h T x n , p i ( C ( M ,τ ) ,C ( M ,τ ) ∗ ) ∈ R for every n . Moreover, we have |h T x, p i ( C ( M ,τ ) ,C ( M ,τ ) ∗ ) −h T x n , p i ( C ( M ,τ ) ,C ( M ,τ ) ∗ ) | ≤ k T k C ( M ,τ ) → C ( M ,τ ) k x − x n k ∞ k p k → , which shows that h T x, p i ( C ( M ,τ ) ,C ( M ,τ ) ∗ ) ∈ R . (cid:3) Proposition 3.8.
Let E ( M , τ ) is an arbitrary symmetric operator space having order continuousnorm. Assume that k·k E is not proportional to k·k . Let T be a bounded hermitian operator on E ( M , τ ) . Then, T can be extended to a bounded operator on C ( M , τ ) (still denoted by T ) and,for any operator x ∈ C ( M , τ ) , there exists a support functional x ′ in C ( M , τ ) ∗ of x such that h T x, x ′ i ∈ R . In particular, T is a hermitian operator on C ( M , τ ) .Proof. By Lemma 3.6, T can be extended to a bounded operator on C ( M , τ ).Without loss of generality, we assume, in addition, that k x k ∞ = 1. Let x = u | x | be thepolar decomposition. Recall that x ∈ C ( M , τ ). Hence, τ ( E | x | (1 − n , < ∞ for any n > C ( M , τ )) × = L ( M , τ ) (see e.g. [84, Lemma 8] and [31, Theorem 53]). Define x n = E | x | (1 − n , τ ( E | x | (1 − n , u ∗ ∈ L ( M , τ ) ⊂ C ( M , τ ) ∗ , n ≥
1. We have k x n k C ( M ,τ ) ∗ = k x n k = 1 [31,p.228]. Note that 1 − n ≤ τ ( xx n ) = τ ( | x | E | x | (1 − n , τ ( E | x | (1 − n , ≤ . (3.8)By Alaoglu’s theorem [24, p.130, Theorem 3.1], there exists a subnet { x i } of { x n } n converging tosome element x ′ ∈ C ( M , τ ) ∗ in the weak ∗ topology of C ( M , τ ) ∗ and k x ′ k C ( M ,τ ) ∗ ≤
1. On theother hand, we have k x ′ k C ( M ,τ ) ∗ = k x k ∞ k x ′ k C ( M ,τ ) ∗ ≥ x ′ ( x ) = lim i τ ( xx i ) (3.8) = 1 . Hence, k x ′ k C ( M ,τ ) ∗ = 1. This implies that x ′ is a support functional of x . Therefore, by taking p = E | x | (1 − n ,
1] in Proposition 3.7, we obtain that h T x, x ′ i = ( w ∗ ) − lim i h T x, x i i ∈ R . (cid:3) ERMITIAN OPERATORS AND ISOMETRIES ON SYMMETRIC OPERATOR SPACES 15
Recall that a derivation δ on an algebra A is a linear operator satisfying the Leibniz rule.Although it is known that a derivation from C ( M , τ ) into C ( M , τ ) is not necessarily inner [12,44](see [80, Example 4.1.8] for examples of non-inner derivations on K ( H )), it is shown recently thatevery derivation δ from an arbitrary von Neumann subalgebra of M into C ( M , τ ) is inner, i.e.,there exists an element a ∈ C ( M , τ ) such that δ ( · ) = [ a, · ] [13, 44]. On the other hand, everyderivation from C ( M , τ ) into C ( M , τ ) is spatial, i.e., it can be implemented by an element from M (see e.g. [80, Theorem 2] and [9, Theorem 4.1]). Lemma 3.9.
Every derivation δ from C ( M , τ ) into C ( M , τ ) is spatial. In particular, if δ is a ∗ -derivation, then the element implementing δ can be chosen to be self-adjoint.Proof. The first statement follows from [80, Theorem 2] (or [9, Theorem 4.1]) and the fact that C ( M , τ ) is a C ∗ -algebra. For the second statement, see e.g. [44, Chapter 3.4, Remark 3.4.1]. (cid:3) We now come to the main result of this section, which gives the full description of hermitianoperators on a symmetric space E ( M , τ ). Theorem 3.10.
Let E ( M , τ ) be a symmetric space affiliated with an atomless semifinite vonNeumann algebra (or an atomic von Neumann algebra with all atoms having the same trace) M equipped with a semifinite faithful normal trace τ . Assume that k·k E is order continuous and isnot proportional to k·k . Then, a bounded linear operator T on E ( M , τ ) is a hermitian operatoron E ( M , τ ) if and only if there exist self-adjoint operators a and b in M such that T x = ax + xb, x ∈ E ( M , τ ) . (3.9) In particular, T can be extended to a bounded hermitian operator on the von Neumann algebra M .Proof. The ‘if’ part of the theorem is obvious (see e.g. the argument in [90, p.71] or [38, p. 167]).By Corollary 3.8, T is a bounded hermitian operator on C ( M , τ ). Recall that any hermitianoperator T on a C ∗ -algebra A is the sum of a left-multiplication by a self-adjoint operator in A anda ∗ -derivation in A (see e.g. [87, p.213]). It follows from Lemma 3.9 that there exist self-adjointelements a, b ∈ M such that T x = ax + xb, x ∈ C ( M , τ ) . Noting that F ( τ ) ⊂ C ( M , τ ), we obtain that T x = ax + xb, x ∈ F ( τ ) . Since E ( M , τ ) has order continuous norm, it follows that F ( τ ) is dense in ( E ( M , τ ) , k·k E ) (seee.g. [31, Proposition 46] or [46, Remark 2.9]). For any x ∈ E ( M , τ ), there exists a sequence { y n } ⊂ F ( τ ) such that k y n − x k E →
0. Hence, we obtain that
T x = k·k F − lim n T ( y n ) = k·k F − lim n ( ay n + y n b ) = ax + xb, x ∈ E ( M , τ ) . This completes the proof. (cid:3) Isometries
The goal of this section is to answer the question posed in [23, 92] and stated at the outsetof this paper. Throughout this section, unless stated otherwise, we always assume that M is anatomless semifinite von Neumann algebra or an atomic semifinite von Neumann algebra with allatoms having the same trace, and we assume that τ is a semifinite faithful normal trace on M .Before proceeding to the proof of Theorem 4.4, we need the following auxiliary tool, whichextends [90, Corollary 2] and [92, Corollary 3.2]. Corollary 4.1.
Let ( M , τ ) be an atomless semifinite von Neumann algebra or an atomic vonNeumann algebra whose atoms having the same trace. Let E ( M , τ ) be a symmetric operator spacewhose norm is order continuous and is not proportional to k·k . Let T be a bounded hermitianoperator on E ( M , τ ) . Then, T is also a hermitian operator on E ( M , τ ) if and only if T ( y ) = ay + yb, ∀ y ∈ L ( M , τ ) ∩ M , for some self-adjoint operators a ∈ M w and b ∈ M − w , where w ∈ P ( Z ( M )) . Proof. ( ⇐ ). Note that T ( y ) = a y + yb , ∀ y ∈ L ( M , τ ) ∩ M . It follows from Theorem 3.10 that T is a hermitian operator.( ⇒ ). Recall that, by Theorem 3.10, we have T ( y ) = ay + yb , y ∈ M , for some self-adjointelements a, b ∈ M . Due to the assumption that T is also hermitian, there exist self-adjointoperators c, d ∈ M such that T ( y ) = cy + yd = a y + 2 ayb + yb , ∀ y ∈ M . The claim follows from Theorem A.6. (cid:3)
Remark 4.2.
Let T , a , b , w be defined as in Corollary 4.1. In particular, l ( a ) ≤ w and l ( b ) ≤ − w . Define z a := sup { p ∈ P ( Z ( M )) : p ≤ w, pa ∈ Z ( M ) } and z b := sup { p ∈ P ( Z ( M )) : p ≤ − w, pb ∈ Z ( M ) } . For any elements z , z ∈ { p ∈ P ( Z ( M )) : p ≤ w, pa ∈ Z ( M ) } and d ∈ M , we have ( z ∨ z ) ad = ( z + z − z z ) ad = z ad + z ( − z ) ad = z ad + z a ( − z ) d = dz a + d ( − z ) z a = d ( z ∨ z ) a. That is, z ∨ z ∈ { p ∈ P ( Z ( M )) : p ≤ w, pa ∈ Z ( M ) } . Hence, { p ∈ P ( Z ( M )) : p ≤ w, pa ∈ Z ( M ) } is an increasing net with the partial order ≤ of projections. Therefore, by Vigier’s theorem [64,Theorem 2.1.1], we obtain that z a a ∈ Z ( M ) . That is, z a ∈ { p ∈ P ( Z ( M )) : p ≤ w, pa ∈ Z ( M ) } . Arguing similarly, z b ∈ { p ∈ P ( Z ( M )) : p ≤ − w, pb ∈ Z ( M ) } . We have T ( y ) = a ( w − z a ) y + yb (( − w ) − z b ) + ( az a + bz b ) y, y ∈ L ( M , τ ) ∩ M . (4.1) In particular, (1) w − z a , ( − w ) − z b , z a + z b are pairwise orthogonal projections in Z ( M ) ; (2) if p ∈ Z ( M ) such that p ≤ w − z a and ap ∈ Z ( M , τ ) (or p ≤ ( − w ) − z b and bp ∈ Z ( M ) ),then p = 0 ; (3) w − z a (resp., ( − w ) − z b ) is the central support of a ( w − z a ) (resp., b (( − w ) − z b ) ). Remark 4.3.
Let M be a semifinite factor. It is an immediate consequence of Corollary 4.1 thatif T a bounded hermitian operator on M , then T is hermitian if and only if T is a left(or right)-multiplication by a self-adjoint operator in M (see also the proof of [90, Corollary 2] and [92,Corollary 3.2]). Let M and M be two von Neumann algebras. A complex-linear map J : M injective −→ M is called Jordan ∗ -isomorphism if J ( x ∗ ) = J ( x ) ∗ and J ( x ) = J ( x ) , x ∈ M (equivalently, J ( xy + yx ) = J ( x ) J ( y ) + J ( y ) J ( x ), x, y ∈ M ) (see e.g. [17, 86, 99]). A Jordan ∗ -isomorphismis called normal if it is completely additive (equivalently, ultraweakly continuous). Alternatively,we adopt the following equivalent definition: J ( x α ) ↑ J ( x ) whenever x α ↑ x ∈ M +1 (see e.g [26,Chapter I.4.3]). If J : M → M is a surjective Jordan ∗ -isomorphism, then J is necessarilynormal [78, Appendix A].The following theorem is the main result of this section. Due to the complicated nature of her-mitian operators on a von Neumann algebra distinct from a factor, the proof below is substantiallymore involved than those in [90, 92]. Theorem 4.4.
Let M and M be atomless von Neumann algebras (or atomic von Neumannalgebras whose atoms have the same trace) equipped with semifinite faithful normal traces τ and τ , respectively. Let E ( M , τ ) and F ( M , τ ) be two symmetric operator spaces whose norms areorder continuous and are not proportional to k·k . If T : E ( M , τ ) → F ( M , τ ) is a surjectiveisometry, then there exist two nets of elements A i ∈ F ( M , τ ) , i ∈ I , disjointly supported from ERMITIAN OPERATORS AND ISOMETRIES ON SYMMETRIC OPERATOR SPACES 17 the right and B i ∈ F ( M , τ ) , i ∈ I , disjointly supported from the left, and a surjective Jordan ∗ -isomorphism J : M → M and a central projection z ∈ M such that T ( x ) = k·k F − X i ∈ I J ( x ) A i z + B i J ( x )( − z ) , x ∈ E ( M , τ ) ∩ M , where the series is taken as the limit of all finite partial sums.Proof. The indices of von Neumann algebras M and M play no role in the proof below. So, toreduce the notation, we assume that M = M = M . We denote by L a (resp. R a ) the left (resp.right) multiplication by a ∈ M , that is, L a ( x ) = ax (resp. R a ( x ) = xa ) for all x ∈ S ( M , τ ). For any self-adjoint operator a ∈ M , T L a T − and T L a T − are hermitian on F ( M , τ ) (see e.g. [36, Lemma 2.3] or [47]).We divide the proof into several steps. Step 1.
We aim to prove that there exists z ∈ P ( Z ( M )) (does not depend on a below) suchthat T L a T − = L J ( a ) z + R J ( a )( − z ) , a = a ∗ ∈ M , (4.2)where J ( a ) , J ( a ) are self-adjoint operators in M . Let z ( f ) be the central support of f = f ∗ ∈ M .For any fixed b = b ∗ ∈ M , T L b T − y (4.1) = L J ( b ) y + R J ( b ) y + J ( b ) y, ∀ y ∈ L ( M , τ ) ∩ M , (4.3)for some self-adjoint operators J ( b ) , J ( b ) ∈ M , J ( b ) = J ( b ) ∗ ∈ Z ( M ) such that(1) the z ( J ( b )), z ( J ( b )) and z ( J ( b )) are pairwise orthogonal projections in Z ( M ) (see (1)and (3) in Remark 4.2);(2) if p ∈ Z ( M ) such that p ≤ z ( J ( b )) and J ( b ) p ∈ Z ( M , τ ) (or p ≤ z ( J ( b )) and J ( b ) p ∈ Z ( M )), then p = 0 (see (2) in Remark 4.2).By (4.3), for any self-adjoint a ∈ M , we have T L a T − y = L J ( a ) y + R J ( a ) y + J ( a ) y, ∀ y ∈ L ( M , τ ) ∩ M (4.4)and T L a + b T − y = L J ( a + b ) y + R J ( a + b ) y + J ( a + b ) y, ∀ y ∈ L ( M , τ ) ∩ M . (4.5)Now, we consider the reduced algebra M z ( J ( b )) ∧ z ( J ( a )) . For all y ∈ ( L ∩ L ∞ )( M z ( J ( b )) ∧ z ( J ( a )) , τ ),we have L J ( b ) y + R J ( a ) y = L J ( b ) y + R J ( b ) y + J ( b ) y + L J ( a ) y + R J ( a ) y + J ( a ) y (4.3) and (4.4) = T L b T − y + T L a T − y = T L b + a T − y (4.5) = L J ( a + b ) y + R J ( a + b ) y + J ( a + b ) y. By Theorem A.6, there exists central projection p ≤ z ( J ( b )) ∧ z ( J ( a ))such that J ( b ) p and J ( a )( z ( J ( b )) ∧ z ( J ( a )) − p )are in the center Z ( M ) of M . However, by (2) of Remark 4.2 (used twice), p = 0 = z ( J ( b )) ∧ z ( J ( a )) − p. That is, z ( J ( b )) ∧ z ( J ( a )) = 0 . Note that a, b are arbitrarily taken. Defining z := _ b = b ∗ ∈M z ( J ( b )) , we obtain that T L b T − y (4.3) = L J ( b ) y + R J ( b ) y + J ( b ) y = L J ( b )+ J ( b ) z y + R J ( b )+ J ( b )( − z ) y, ∀ y ∈ L ( M , τ ) ∩M . Replacing J ( b ) + J ( b ) z (resp., J ( b ) + J ( b )( − z )) with J ( b ) (resp., J ( b )), we obtain (4.2). Step 2.
Note that L J ( a ) z + R J ( a ) ( − z ) (4.2) = ( T L a T − ) = T L a T − = L J ( a ) z + R J ( a )( − z ) for every a = a ∗ ∈ M . By standard argument (see e.g. [92, p.117]), we obtain J ( · ) := J ( · ) z + J ( · )( − z )(4.6)is an injective Jordan ∗ -isomorphism on M . Let 0 ≤ a i ↑ a ∈ M . Clearly, J ( a i ) z ↑≤ J ( a ) z (seee.g. [46, Eq.(12)] and [17, p.211]). Since a i ↑ a , it follows that for any x ∈ E ( M , τ ), x ∗ a i x ↑ x ∗ ax [31, Proposition 1 (vi)] and x ∗ a i x → x ∗ ax in measure topology [31, Proposition 2 (iv)]. By the factthat t t / is an operator monotone function [27, Proposition 1.2], we obtain that ( x ∗ ( a − a i ) x ) is a decreasing net. Note that ( x ∗ ( a − a i ) x ) → x ∗ ( a − a i ) x ) ↓ E ( M , τ ) ∋ | ( a − a i ) / x | = ( x ∗ ( a − a i ) x ) ↓ . It follows from the order continuity of k·k E that k J ( a − a i ) zT ( x ) k F (4.2) = k T (( a − a i ) x ) k F = k ( a − a i ) x k E ≤ (cid:13)(cid:13)(cid:13) ( a − a i ) / (cid:13)(cid:13)(cid:13) ∞ (cid:13)(cid:13)(cid:13) ( a − a i ) / x (cid:13)(cid:13)(cid:13) E → x ∈ E ( M , τ ) such that T ( x ) is a τ -finite projection in F ( τ ) less than z . Therefore, J ( a i ) z → J ( a ) z in localized measure topology [31, Proposition 20]. Hence, J ( a i ) z ↑ J ( a ) z (see Section2.1). The same argument shows that J ( a i )( − z ) ↑ J ( a ) z ( − z ). Therefore, J ( M ) is weaklyclosed [46, Remark 2.16] and J : M → J ( M ) is a surjective (normal) Jordan ∗ -isomorphism. Step 3.
We claim that J is surjective. Let c = c ∗ ∈ M . Note that T − ( L cz + R c ( − z ) ) T andits square T − ( L c z + R c ( − z ) ) T are hermitian operators. Hence, by Corollary 4.1, there exists acentral projection z ′ ∈ M and c ′ = ( c ′ ) ∗ ∈ M such that T − ( L cz + R c ( − z ) ) T = L c ′ z ′ + R c ′ ( − z ′ ) . (4.7)Employing the argument used in steps 1 and 2 to (4.7), we obtain a normal injective Jordan ∗ -isomorphism J ′′ : M → M such that J ′′ ( c ) = c ′ . Moreover, for each c ∈ J ( M ), we have L J − ( c ) (4.2) = T − ( L cz + R c ( − z ) ) T (4.7) = L c ′ z ′ + R c ′ ( − z ′ ) . Hence, L J − ( c ) − c ′ z ′ = R c ′ ( − z ′ ) . In particular, c ′ ( − z ′ ) ∈ Z ( M ). Hence, L J − ( c ) = L c ′ z ′ + c ′ ( − z ′ ) = L c ′ = L J ′′ ( c ) . That is, for each c ∈ J ( M ), we have J ′′ ( c ) = J − ( c ) (see also [92, Remark 3.3] forthe case when M is a factor). Hence, J ′′ ( J ( M )) = J − ( J ( M )) = M . By the injectivity of J ′′ ,we obtain that J ( M ) = M . This proves the claim. Step 4.
Applying (4.2) to T ( x ), we obtain that T ( ax ) = J ( a ) zT ( x ) + T ( x )( − z ) J ( a )= J ( a ) zT ( x ) + T ( x ) J ( a )( − z ) (4.6) = J ( a ) zT ( x ) + T ( x ) J ( a )( − z )for all a = a ∗ ∈ M , x ∈ E ( M , τ ) ∩ M . For any τ -finite projection e in M , we have T ( xe ) = J ( x ) T ( e ) z + T ( e ) J ( x )( − z ) , x ∈ M . Let { e i } i ∈ I be a net of pairwise orthogonal τ -finite projections in M such that sup i e i = [31,Corollary 8] and let { λ α } be collection of all finite subsets of I , partially ordered by inclusion. By ERMITIAN OPERATORS AND ISOMETRIES ON SYMMETRIC OPERATOR SPACES 19 the order continuity of k·k E , we have lim α P e ∈ λ α xe i = x [30, Theorem 6.13], and, since T is anisometry, it follows that T ( x ) = k·k F − lim α X e i ∈ λ α T ( xe i ) = k·k F − lim α X e i ∈ λ α J ( x ) T ( e i ) z + T ( e i ) J ( x )( − z )= k·k F − X i ∈ I J ( x ) T ( e i ) z + T ( e i ) J ( x )( − z ) , ∀ x ∈ E ( M , τ ) ∩ M . Note that T ( e i ) = J ( e i ) T ( e i ) z + T ( e i ) J ( e i )( − z ) . Recall that Jordan ∗ -isomorphisms preserve the disjointness for projections (see e.g. [46, Proposi-tion 2.14]). Letting B i := T ( e i ) z = J ( e i ) T ( e i ) z and A i := T ( e i )( − z ) = T ( e i ) J ( e i )( − z ), wecomplete the proof. (cid:3) Uniqueness of symmetric structure
Let C E be the symmetric operator ideal in B ( H ) generated by a symmetric sequence space E . We say that C E has a unique symmetric structure if C E isomorphic to some ideal C F corre-sponding to a symmetric sequence space F implies that E = F with equivalent norms. At theInternational Conference on Banach Space Theory and its Applications at Kent, Ohio (August1979), Pe lczy´nski posed the following question concerning the symmetric structure of ideals ofcompact operators on the Hilbert space ℓ (see also [6, Question (B)] and [7, Problem A]): Doesthe ideal C E of compact operators corresponding to an arbitrary separable symmetric sequencespace E have a unique symmetric structure? For readers who are interested in this topic, we referto [6, 7, 45].We assume, in addition, that k e k E = 1 for any atom e ∈ M if M = B ( H ) equipped withthe standard trace; k k E = 1 if M is a type II -factor equipped with the unique faithful normaltracial state. Here, we consider an analogue of Pe lczy´nski’s problem in the sense of isometricisomorphisms. This assumption implies that if k·k E is proportional to k·k , then k·k E = k·k . Let F ( M , τ ) be a symmetric operator space. If a symmetric operator symmetric E ( M , τ ) isometricto F ( M , τ ) implies that E ( M , τ ) coincides with F ( M , τ ), then we say that F ( M , τ ) has a uniquesymmetric structure up to an isometry. The following corollary extends results in [2,56,74,75,100]to the noncommutative setting. Corollary 5.1.
Let M = B ( H ) be equipped with the standard trace τ or M be a II -factorequipped with the unique faithful normal tracial state τ . Let E ( M , τ ) and F ( M , τ ) be symmet-ric operator spaces whose norms are order continuous and are not proportional to the norm of L ( M , τ ) . Then, T is a surjective isometry from E ( M , τ ) to F ( M , τ ) if and only if there exist aunitary element u ∈ M and a trace-preserving Jordan ∗ -isomorphism J such that T ( x ) = uJ ( x ) , x ∈ M . (5.1) In particular, any symmetric space E ( M , τ ) (including the case when E ( M , τ ) = L ( M , τ ) ) hasa unique symmetric structure up to an isometry.Proof. By Theorem 4.4, it suffices to show that the Jordan ∗ -isomorphism is trace-preserving.Indeed, when M = B ( H ), this follows from the fact that every ∗ -automorphism on B ( H ) isinner [34, Corollary 5.42] (see also argument in [90, p.75]). When M is a II -factor, then thecorollary follows from the same argument in [92, p. 118–119].For the uniqueness of symmetric structure, we only need to show that if L ( M , τ ) is isometricto F ( M , τ ) (when M = B ( H ) or a II -factor), then F ( M , τ ) = L ( M , τ ) (with the same norm).Indeed, all other cases follow from (5.1) immediately.If there exists a surjective isometry T : L ( M , τ ) → F ( M , τ ), then F ( M , τ ) × is isometric to L ( M , τ ). That is, F ( M , τ ) is isometric to F ( M , τ ) × . In particular, both F ( M , τ ) and F ( M , τ ) × have the Fatou property and order continuous norms [31, Theorem 45]. Hence, F ( M , τ ) coincideswith F ( M , τ ) ×× with the same norm [31, Theorem 32]. If k·k F × is proportional to k·k , then thenorm of its K¨othe dual F ( M , τ ) ×× is also proportional to k·k . By the assumption that k e k E = 1 for any atom e ∈ M if M = B ( H ) (or k k E = 1 if M is a type II -factor), we obtain that F ( M , τ ) coincide with L ( M , τ ) and k·k F = k·k . Hence, we only need to consider the case when k·k F × is not proportional to k·k . Recalling that F ( M , τ ) is isometric to F ( M , τ ) × , by (5.1), F ( M , τ ) coincides with F ( M , τ ) × with k·k F = k·k F × , and therefore, for any x ∈ F ( M , τ ), by thedefinition of K¨othe dual, we have τ ( xx ∗ ) < ∞ , x ∈ F ( M , τ ) , i.e., F ( M , τ ) = F ( M , τ ) × ⊂ L ( M , τ ). On the other hand, by the definition of K¨othe dual, F ( M , τ ) ⊂ L ( M , τ ) implies that F ( M , τ ) × ⊃ L ( M , τ ). Hence, F ( M , τ ) = F ( M , τ ) × = L ( M , τ ) (in the sense of sets).Since k x k = τ ( xx ∗ ) ≤ k x k F k x k F × = k x k F , it follows that k x k ≤ k x k F = k x k F × . Moreover, k x k F × = sup k y k F ≤ | τ ( xy ) | ≤ sup k y k ≤ | τ ( xy ) | = k x k . We obtain that k x k = k x k F = k x k F × . This completes the proof. (cid:3) Final remarks
Remark 6.1.
Theorem 4.4 above covers all existing results of surjective isometries on symmetricoperator/function/sequence spaces in the literature. Indeed, (1) when τ is finite, we have T ( x ) = T ( z ) J ( x ) + J ( x ) T ( − z ) . This recovers and extends [23,Theorem 3.1], [50, Theorem 4.11] and [93, Theorem 6]. (2) when M = L ∞ (Ω , Σ , µ ) , where (Ω , Σ , µ ) is some σ -finite atomless measure space, we have T ( x ) = BJ ( x ) , x ∈ M , for some measurable function B on (Ω , Σ , µ ) . In particular,Zaidenberg’s results are recovered (see (1.1) and [100, 101]). (3) when M = (Γ , Σ , µ ) is a discrete measure space on a set Γ with µ ( { γ } ) = 1 for every γ ∈ Γ , we obtain that if T is an isometry on a symmetric space E (Γ) whose norm is ordercontinuous norm and is not proportional to k·k , then ( T x )( γ ) = A ( γ ) · x ( σ ( γ )) , x ∈ E (Γ) , γ ∈ Γ , where A ( γ ) are unimodular scalars and σ is a permutation on Γ . Indeed, by Theo-rem 4.4, there exists a surjective Jordan ∗ -isomorphism J on ℓ ∞ (Γ) . By the bijectivity anddisjointness-preserving property of J , we infer that J maps atoms onto atoms in ℓ ∞ (Γ) .Hence, J is generated by a permutation. This extends Arazy’s description of isometrieson E ( N ) [8, Theorem 1]. (4) when H is a (not necessarily separable) Hilbert space and M = B ( H ) , Theorem 4.4 recov-ers and extends Sourour’s result [90, Theorem 2]. (5) when M is a hyperfinite type II -factor, Theorem 4.4 extends [92, Theorem 4.1], whichwas established under the assumption that H is separable. (6) when E = F = L p , Theorem 4.4 extends and complements results in e.g. [5, 58, 86, 96, 99]. (7) when E and F are Lorentz spaces, [23, Theorem 6.1] (see also [21, 60, 68]) is recovered. (8) when T is a positive isometry, several results are recovered and extended (see e.g. [1,Theorem 1], [23, Theorem 3.1], [93, Theorem 6], [46, Corollaries 5.4 and 5.5] and [50]). (9) when M is an atomic von Neumann algebra, Theorem 4.4 extends the main result in [66],which was established under the assumption that M is a σ -finite von Neumann algebra. It is shown by Zaidenberg [101] (see also [37]) that under certain conditions, every surjectiveisometry between two symmetric function spaces on a σ -finite atomless measure space can berepresented in the form of (1.1) (see [8] for the case of symmetric sequence spaces). In thisrespect, the condition imposed on the von Neumann algebras in Theorems 3.10 and 4.4 is verynatural. One may expect that Theorems 3.10 and 4.4 (and [101, Theorem 1]) can be proved in thesetting of more general von Neumann algebra, e.g., M = M ⊕ M , where M is an atomless vonNeumann algebra (or M is an atomic von Neumann algebra with all atoms having the same tracebut the trace of an atom in M is different from that in M ) and M is an atomic von Neumannalgebra with all atoms having the same trace. See e.g. [37, Definition 5.3.1] or [11] for definitionof symmetric function spaces on general measure spaces. However, the following simple example ERMITIAN OPERATORS AND ISOMETRIES ON SYMMETRIC OPERATOR SPACES 21 shows that for an atomic von Neumann algebra (or measure space) whose atoms have differentmeasures, isometries may have different forms from (1.1). This demonstrates that the conditionimposed on the von Neumann algebras in Theorems 3.10 and 4.4 is sharp. For an arbitrary (notnecessarily atomless or atomic with all atoms having the same trace) semifinite von Neumannalgebras M , it is interesting to characterize those symmetric operator spaces E ( M , τ ) such thatall isometries on E ( M , τ ) have elementary forms. Example 6.2.
Let (Ω , m ) be a measure space consisting of two atoms e and e . Assume that m ( e ) = 1 and m ( e ) = 2 . Then, there exists a symmetric space E (Ω) which is not propor-tional to L (Ω) but it is isometric to the -dimensional Hilbert space. In particular, there existnon-elementary isometries and hermitian operators on E (Ω) which can not be written as a multi-plication of an element in L ∞ (Ω) .Proof. A non-trivial projection in E (Ω) must be e , e or e + e , where e (resp. e and e + e )denotes the indicator function on e (resp. e and e + e ).For an element ae + be , we define a norm by k ae + be k E := p | a | + c | b | , (6.1)where c is an arbitrary positive integer greater than 3. Indeed, this can be considered as the L -norm on a measure space having an atom of measure 1 and the other of measure √ c . We claimthat k·k E is symmetric. Indeed, assume that x := a e + a e ≥ y := b e + b e ≥ µ ( x ) ≤ µ ( y ).If b ≥ b , then there are 2 possible cases:(1) If a ≥ a , then b ≥ a and b ≥ a . In this case, we have k x k E ≤ k y k E .(2) a ≤ a . Since m ( e ) ≤ m ( e ) and b ≥ a , it follows that b ≥ a ≥ a . Hence, k x k E ≤ k y k E .If b ≤ b , then there are 2 possible cases:(1) If a ≥ a , then b ≥ a ≥ a and b ≥ a . Note that k x k E = a + ca and k y k E = b + cb .We obtain that k y k E − k x k E = cb − a + b − ca ≥ ( c − b − ( c − a ≥ a ≤ a , then b ≥ a and b ≥ a . Hence, k x k E ≤ k y k E .Consider the matrix T := − i √ √ √ i √ ·√ √ ! . That is, T ( e ) = − i √ e + i √ √ e , T ( e ) = √ √ e + √ e , T (1) = ( √ − i √ , i + √ √ ·√ ). For any a, b ∈ C , we have k T ( ae + be ) k E = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − ia √ √ b √ ! e + (cid:18) ia √ √ b √ (cid:19) e (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) E = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − ia √ √ b √ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + 3 (cid:12)(cid:12)(cid:12)(cid:12) ia √ √ b √ (cid:12)(cid:12)(cid:12)(cid:12) . By the Parallelogram law, we obtain that k T ( ae + be ) k E = | a | + 3 | b | = k ae + be k E . This implies that T is an isometry on E (Ω).Assume that T can be written as an elementary form, that is, there exists a element in E (Ω)and a Jordan isomorphism on L ∞ (Ω) such that T = BJ . Since J ( e + e ) = e + e , it followsthat J ( e ) = e , J ( e ) = e or J ( e ) = e , J ( e ) = e . However, T ( e ) = − i √ e + i √ √ e = BJ ( e ) for any B ∈ E (Ω). Hence, T cannot be written inan elementary form. Consider a hermitian operator T on E (Ω) defined by T := i · √ − i √ ! , i.e., T ( e ) = e − i √ e and T ( e ) = i · √ e + e . Assume that T ( x ) = ax for some a ∈ E (Ω). Itfollows that e − i √ e = T ( e ) = ae = λe for some number λ ∈ C , which is a contradiction. (cid:3) Remark 6.3.
Recall that Zaidenberg’s description of isometries on symmetric function spacesonly requires that a symmetric function space has a Fatou norm, which is a slightly weaker as-sumption than the requirement that this space has an order continuous norm [37, Theorem 5.3.5].Throughout this paper, we always consider symmetric spaces having order continuous norms. Itwill be interesting to verify Theorem 4.4 under a slightly weaker assumption that the symmetricoperator spaces have Fatou norms only. This problem is yet unsolved. There are some partialanswers in this direction obtained in [3, 4, 22, 62] in the setting of B ( H ) . Remark 6.4.
The structure of (real or complex) symmetric sequence space (under the assump-tion that the spaces in question have the Fatou property, which is a stronger assumption thanthat of Fatou norm) has been discussed by Braverman and Semenov [18, 19], and by Arazy [8].Abramovich and Zaidenberg [2, Theorem 1] showed L p [0 , , ≤ p < ∞ , has a unique structure upto an isometry. The uniqueness of symmetric structure of separable complex symmetric functionspaces on [0 , was obtained by Zaidenberg [100]. By a generalized Zaidenberg’s theorem [75, The-orem 1 and Proposition 3] (see also [56, Theorem 7.2]), the uniqueness of the symmetric structureof separable real symmetric function spaces under some technical conditions is obtained by Ran-drianantoanina [75]. Remark 6.5.
Note that for the case when M is a II ∞ factor, Corollary 5.1 may fail becausethe Jordan ∗ -isomorphism J on M may not be trace-preserving. This is an oversight in the proofof [92, Theorem 4.1]. Indeed, letting R = ⊗ ≤ n< ∞ M be the hyperfinite II -factor equippedwith the faithful normal tracial state τ , we consider the hyperfinite II ∞ -factor M = B ( H ) ⊗ R equipped with the trace Tr ⊗ τ . Let φ be a ∗ -isomorphism from ( R , τ ) onto ( ⊗ ≤ n< ∞ M , τ ) ,and φ be a natural ∗ -isomorphism from B ( H ) ⊗ R onto B ( H ) ⊗ M ⊗ ⊗ ≤ n< ∞ M . Clearly, φ ⊗ φ is a ∗ -isomorphism on M which does not preserves traces Tr ⊗ τ . Indeed, φ maps atomsin B ( H ) ⊗ R to atoms in B ( H ) ⊗ M ⊗ ⊗ ≤ n< ∞ M . Let p ∈ B ( H ) be an atom. In particular, Tr( p ) = (Tr ⊗ τ )( p ⊗ R ) = 1 and (Tr ⊗ τ )( φ ( p ) ⊗ ⊗ ≤ n< ∞ M ) = . This oversight in [92, Theorem4.1] is rectified by Theorem 4.4 above. It is natural to compare this result with [37, Theorem 5.3.5](see also (1.1) and [21, Main Theorem]), where the set-isomorphism may not necessarily preservethe measure. Appendix
A.In this appendix, we extend [92, Theorem 3.1] and [90, Lemma 2] to the setting of arbitraryvon Neumann algebras. Our technique is different from that used in [90, 92]. We are grateful toDmitriy Zanin for providing us with a correction of our initial argument and allowing us to use itin this paper.Let M be a von Neumann algebra. Let p be a projection in M . We denote by z ( p ) the centralsupport of p . Lemma A.1. [81, Theorem 1.10.7] Let p, q ∈ M be projections such that pyq = 0 , ∀ y ∈ M . We have z ( p ) z ( q ) = 0 . Lemma A.2.
Let a, b, e, f ∈ M be self-adjoint and such that ey + yf = ayb, ∀ y ∈ M . We have (1) [ a, b ] = 0 . ERMITIAN OPERATORS AND ISOMETRIES ON SYMMETRIC OPERATOR SPACES 23 (2) [ b, [ a, y ]] = 0 for every y ∈ M . Proof.
Setting y = , we obtain e + f = ab. Taking adjoint, we obtain e + f = ba. Thus, ab = ba. This proves the first assertion.Substituting f = ab − e, we obtain[ e, y ] = [ a, y ] b, y ∈ M . Taking adjoints, we obtain [ e, y ] = b [ a, y ] , y ∈ M . Comparing the right hand sides, we establish the second assertion. (cid:3)
Lemma A.3.
Let a, b ∈ M be commuting self-adjoint elements such that [ b, [ a, y ]] = 0 , ∀ y ∈ M . We have [ p, [ q, y ]] = 0 , ∀ y ∈ M for all spectral projections p and q of a and b, respectively.Proof. Note that for all y ∈ M , we have[ b n , [ a, y ]] = [ b, [ b n − , [ a, y ]]] + [[ b, [ a, y ]] , b n − ] = [ b, [ b n − , [ a, y ]]] = · · · = [ b, [ b, · · · [ b, [ a, y ]] · · · ]] = 0 . By linearity, [ P ( b ) , [ a, y ]] = 0 , y ∈ M , for every polynomial P. Since polynomials are norm-dense in the algebra of continuous functions,it follows that [ x, [ a, y ]] = 0 , y ∈ M , for every x in the C ∗ -algebra generated by b. By weak continuity of our equation,[ x, [ a, y ]] = 0 , y ∈ M , for every x in the von Neumann algebra generated by b. In particular,[ q, [ a, y ]] = 0 , y ∈ M . Using the Leibniz rule [ q, [ a, y ]] + [ a, [ y, q ]] + [ y, [ q, a ]] = 0and taking into account that [ q, a ] = 0 , we have[ a, [ q, y ]] = 0 , y ∈ M . Repeating the argument in the first paragraph, we complete the proof. (cid:3)
Lemma A.4. If p, q ∈ M are commuting projections such that [ p, [ q, y ]] = 0 , ∀ y ∈ M , (A.1) then z ( p ) z ( q ) z ( − p ) z ( − q ) = 0 . Proof.
Denote z ′ := z ( p ) z ( q ) z ( − p ) z ( − q ) . Assume by contradiction that z ′ = 0 . By (A.1), we have[ pz ′ , [ qz ′ , y ]] = 0 , y ∈ M z ′ and z ( pz ′ ) z ( qz ′ ) z ( z ′ − pz ′ ) z ( z ′ − qz ′ ) [54, Proposition 5.5.3] = z ′ · z ( p ) z ( q ) z ( − p ) z ( − q ) = z ′ . Hence, by passing to the reduced von Neumann algebra M z ′ , we may assume without loss ofgenerality that z ′ = . In other words, z ( p ) = z ( q ) = z ( − p ) = z ( − q ) = . (A.2) Obviously, the assumption (A.1) is equivalent to(A.3) pqy + ypq = pyq + qyp. Replacing y in (A.3) with ( − q ) y ( − p ) , we obtain0 + 0 = p ( − q ) y ( − p ) q + 0 , y ∈ M . By Lemma A.1, we have z ( p ( − q )) · z (( − p ) q ) = 0 . (A.4)Let w := z ( p ( − q )) , w =: z (( − p ) q ) . We have w w = 0 . Set w := − w − w . By (A.1),we have [ pw , [ qw , y ]] = 0 , y ∈ w M , (A.5) [ pw , [ qw , y ]] = 0 , y ∈ w M , [ pw , [ qw , y ]] = 0 , y ∈ w M . Step 1:
We claim that qw ≤ pw , pw ≤ qw and pw = qw . (A.6)Note that z (( w − pw ) · qw ) [54, Proposition 5.5.3] = w · z (( − p ) q ) = w · w = 0 . Hence, ( w − pw ) · qw = 0 . (A.7)In other words, qw ≤ pw . Similarly, z ( pw · ( w − qw )) [54, Proposition 5.5.3] = z ( p ( − q )) · w = w · w = 0 . Hence, pw · ( w − qw ) = 0 . In other words, pw ≤ qw . Arguing similarly, we have pw ≤ qw and qw ≤ pw . This completes the proof of (A.6).
Step 2:
We claim that w = 0 , w = 0 , w = 0 . (A.8)We only prove the first equality. Proofs of the other 2 are similar.By (A.5), we have [ pw , [ qw , ( w − pw ) y ]] = 0 , y ∈ M w . Since ( w − pw ) · qw = 0 , (A.9)it follows that for all y ∈ M w , we have0 = [ pw , [ qw , ( w − pw ) y ]] = [ pw , qw ( w − pw ) y − ( w − pw ) yqw ] (A.9) = − [ pw , ( w − pw ) yqw ] . Since pw · ( w − pw ) = 0 and since qw ≤ pw , it follows that( w − pw ) yqw = 0 , y ∈ M w . By Lemma A.1, we have z ( w − pw ) · z ( qw ) = 0 . In other words, w = z ( − p ) z ( q ) w [54, Proposition 5.5.3] = z ( w − pw ) · z ( qw ) = 0 . This proves the first equality of (A.8).Finally, = w + w + w = 0, which is impossible. Hence, z = 0. This completes the proof. (cid:3) ERMITIAN OPERATORS AND ISOMETRIES ON SYMMETRIC OPERATOR SPACES 25
Lemma A.5. If a, b ∈ M are commuting elements such that [ b, [ a, y ]] = 0 , ∀ y ∈ M , then there exists a central projection z such that both a ( − z ) and bz are central.Proof. Since a commutes with b , it follows from Lemmas A.3 and A.4 that z ( p ) z ( q ) z ( − p ) z ( − q ) = 0for arbitrary spectral projection p (respectively, q ) of a (respectively, b ).Denote, for brevity, z q = z ( q ) z ( − q ) . We have z ( pz q ) · z ( z q − pz q ) [54, Proposition 5.5.3] = z ( p ) z ( − p ) z q = z ( p ) z ( − p ) z ( q ) z ( − q ) = 0 . Therefore,0 = z ( pz q ) z ( z q − pz q ) ≥ z ( pz q )( z q − pz q ) ≥ ( z q − pz q ) z ( pz q )( z q − pz q ) ≥ ( z q − pz q ) pz q ( z q − pz q ) = 0 . This implies that pz q ≥ z ( pz q ) ≥ pz q , that is, pz q = z ( pz q ) ∈ Z ( M ) . (A.10)Thus, pz ( q ) z ( − q ) = pz q is a central projection. By the Spectral Theorem, az ( q ) z ( − q ) ∈ Z ( M ).Define z ′ ∈ P ( Z ( M )) by − z ′ = _ q z ( q ) z ( − q ) , where the supremum is taken over all spectral projections q of b . We have a ( − z ′ ) = a · W q z ( q ) z ( − q ) ∈ Z ( M ). On the other hand, we have (see e.g. [94, Chapter V, Proposition 1.1]) z ′ = ^ ( − z ( q ) z ( − q )) . In particular, z ′ ≤ − z ( q ) z ( − q )(A.11)for every q. Thus, z ′ − z ( z ′ q ) z ( z ′ − z ′ q ) [54, Proposition 5.5.3] = z ′ · ( − z ( q ) z ( − q )) (A.11) = z ′ for every q . That is, z ( z ′ q ) z ( z ′ − z ′ q ) = 0for every q. Hence, z ′ q ∈ Z ( M ) for every spectral projection q of b (see e.g. the proof for (A.10)).By the Spectral Theorem, bz ′ ∈ Z ( M ). (cid:3) The following theorem is an immediate consequence of Lemmas A.2 and A.5. It should becompared with [92, Theorem 3.1] and [90, Lemma 2].
Theorem A.6.
Let a, b, e, f ∈ M be self-adjoint and such that ey + yf = ayb, ∀ y ∈ M . (A.12) Then there exists a central projection z such that a ( − z ) , e ( − z ) and bz, f z are central.Proof. By Lemmas A.2 and A.5, we obtain that there exists z ∈ P ( Z ( M )) such that a ( − z ) , bz ∈ Z ( M ). Replacing y with z in (A.12), we obtain that ez + f z = abz, ∀ y ∈ M z . We have f z = abz − ez . Hence, yf z (A.12) = ( abz − ez ) y = f zy, ∀ y ∈ M z . This implies that f z ∈ Z ( M ). The same argument shows that e ( − z ) ∈ Z ( M ). (cid:3) References [1] Yu. Abramovich,
Isometries of norm lattices , Optimizatsiya 43(60) (1988) 74–80 (in Russian).[2] Yu. Abramovich, M. Zaidenberg,
A rearrangement invariant space isometric to L p coincides with L p , Inter-action between functional analysis, harmonic analysis, and probability (Columbia, MO, 1994), 13–18, LectureNotes in Pure and Appl. Math., 175, Dekker, New York, 1996.[3] B. Aminov, V. Chilin,
Isometries and Hermitian operators on complex symmetric sequence spaces,
SiberianAdv. Math. 27 (2017), 239–252.[4] B. Aminov, V. Chilin,
Isometries of perfect norm ideals of compact operators,
Studia Math. 241 (2018), 87–99.[5] J. Arazy,
The isometries of C p , Israel J. Math. 22 (1975), 247–256.[6] J. Arazy,
Isomorphisms of unitary matrix spaces , Banach Space Theory and Its Applications (Bucharest,1981), pp. 1–6, Lecture Notes in Math., 991, Springer–Verlag, Berlin–New York 1983.[7] J. Arazy,
Basic sequences, embeddings, and the uniqueness of the symmetric structure in unitary matrixspaces,
J. Funct. Anal. 40 (1981), 302–340.[8] J. Arazy,
Isometries of complex symmetric sequence spaces,
Math. Z. 188 (1985), 427–431.[9] W. Arveson,
On groups of automorphisms of operator algebras,
J. Funct. Anal. 15 (1974), 217–243.[10] S. Banach,
Th´eorie des op´erations lin´eaires,
Warszawa (1932).[11] C. Bennett, R. Sharpley,
Interpolation of operators,
Academic Press, Boston, 1988.[12] A. Ber, J. Huang, G. Levitina, F. Sukochev,
Derivations with values in ideals of semifinite von Neumannalgebras,
J. Funct. Anal. 272 (12) (2017), 4984–4997.[13] A. Ber, J. Huang, G. Levitina, F. Sukochev,
Derivations with values in the ideal of τ -compact operatorsaffiliated with a semifinite von Neumann algebra, submitted for publication.[14] E. Berkson, Some types of Banach spaces, Hermitian operators, and Bade functions,
Trans. Amer. Math. Soc.116 (1965), 376–385.[15] B. Blackadar,
Operator algebras. Theory of C ∗ -algebras and von Neumann algebras , Encyclopaedia of Mathe-matical Sciences, 122. Operator Algebras and Non-commutative Geometry, III. Springer-Verlag, Berlin, 2006.[16] F. Botelho, J. Jamison, L. Moln´ar, Surjective isometries on Grassmann spaces,
J. Funct. Anal. 265 (2013),2226–2238.[17] O. Bratteli, D. Robinson,
Operator algebras and quantum statistical mechanics, I , New York, Springer-Verlag,1979.[18] M. Braverman, E. Semenov,
Isometries of symmetric spaces, (Russian) Dokl. Akad. Nauk SSSR 217 (1974),257–259.[19] M. Braverman, E. Semenov,
Isometries of symmetric spaces, (Russian) Vorone˘z. Gos. Univ. Trudy Nau˘cn.-Issled. Inst. Mat. VGU Vyp. 17 Teor. Operator. Uravneni˘ı (1975), 7–18.[20] M. Broise,
Sur les isomorphismes de certaines alg`ebres de von Neumann,
Ann. Sci. ´Ecole Norm. Sup. (3) 83(1966), 91—111.[21] N. Carothers, R. Haydon, P.-K. Lin,
On the isometries of the Lorentz function spaces,
Israel J. Math. 84(1993), 265–387.[22] J.-T. Chan, C.-K. Li, N.-S. Sze,
Isometries for unitarily invariant norms,
Linear Algebra Appl. 399 (2005),53–70.[23] V. Chilin, A. Medzhitov, F. Sukochev,
Isometries of non-commutative Lorentz spaces,
Math. Z. 200 (1989),527–545.[24] J. Conway,
A course in functional analysis,
Springer-Verlag, New York, 1997.[25] J. Dixmier,
Les fonctionelles lin´eaires sur l’ensemble des op´erateurs born´es d’un espace de Hilbert,
Ann. ofMath. 51 (1950), 387–408.[26] J. Dixmier,
Les algebres d’operateurs dans l’Espace Hilbertien, 2nd ed.,
Gauthier-Vallars, Paris, 1969.[27] P. Dodds, T. Dodds,
On a submajorization inequality of T. Ando,
Operator theory in function spaces andBanach lattices, 113–131, Oper. Theory Adv. Appl., 75, Birkh¨auser, Basel, 1995.[28] P. Dodds, T. Dodds, B. de Pagter,
Non-commutative Banach function spaces,
Math. Z. 201 (1989), 583–597.[29] P. Dodds, T. Dodds, B. de Pagter,
Noncommutative K¨othe duality , Trans. Amer. Math. Soc. 339 (1993),717–750.[30] P. Dodds, B. de Pagter,
The non-commutative Yosida–Hewitt decomposition theorem revisited,
Trans. Amer.Math. Soc. 364 (2012), 6425–6457.[31] P. Dodds, B. de Pagter,
Normed K¨othe spaces: A non-commutative viewpoint,
Indag. Math. 25 (2014), 206–249.[32] P. Dodds, B. de Pagter, F. Sukochev,
Theory of noncommutative integration, unpublished manuscript.[33] P. Dodds, B. de Pagter, F. Sukochev,
Sets of uniformly absolutely continuous norm in symmetric spaces ofmeasurable operators,
Trans. Amer. Math. Soc. 368 (2016), 4315–4355.[34] R. Douglas,
Banach algebra techniques in operator theory,
Academic Press, New York, 1972.[35] T. Fack, H. Kosaki,
Generalized s -numbers of τ -measurable operators, Pacific J. Math. 123 (2) (1986), 269–300.[36] R. Fleming, J. Jamison,
Hermitian operators and isometries on sums of Banach spaces,
Proc. EdinburghMath. Soc. 12 (1989), 169–191.[37] R. Fleming, J. Jamison,
Isometries on Banach spaces: function spaces , Monographs and Surveys in Pure andApplied Mathematics, 129. Chapman & Hall/CRC, Boca Raton, FL, 2003.
ERMITIAN OPERATORS AND ISOMETRIES ON SYMMETRIC OPERATOR SPACES 27 [38] R. Fleming, J. Jamison,
Isometries on Banach spaces: Vector-valued function spaces,
Vol. 2. Monographs andSurveys in Pure and Applied Mathematics, 138. Chapman & Hall/CRC, Boca Raton, FL, 2008.[39] G. Geh´er, P. ˇSemrl,
Isometries of Grassmann spaces,
J. Funct. Anal. 270 (2016), 1585–1601.[40] G. Geh´er, P. ˇSemrl,
Isometries of Grassmann spaces, II,
Adv. Math 332 (2018), 287–310.[41] J. Giles,
Classes of semi-inner-product spaces,
Trans. Amer. Math. Soc. 129 (1967), 436–446.[42] J. Hamhalter,
Quantum Measure Theory,
Springer Science, Business Media Dordrecht, 2003.[43] F. Hern´andez, B. Rodriguez-Salinas,
Lattice-embedding scales of L p spaces into Orlicz spaces, Israel J. Math.104 (1998), 191–220.[44] J. Huang,
Derivations with values into ideals of a semifinite von Neumann algebra , Ph.D. thesis, Universityof New South Wales, 2019.[45] J. Huang, O. Sadovskaya, F. Sukochev,
On Arazy’s problem concerning isomorphic embeddings of ideals ofcompact operators, submitted manuscript.[46] J. Huang, F. Sukochev, D. Zanin,
Logarithmic submajorization and order-preserving isometries,
J. Funct.Anal. 278 (2020), 108352.[47] J. Jamison, I. Loomis,
Isometries of Orlicz spaes of vector valued functions,
Math. Z. 193 (1986), 363–371.[48] J. Jamison, A. Kaminska, P.-K. Lin,
Isometries of Musielak–Orlicz spaces II , Studia Math. 104 (1993), 75–89.[49] P. de Jager, J. Conradie,
Isometries on non-commutative (quantum) Lorentz spaces associated with semi-finitevon Neumann algebras, arXiv:1907.07619v1, 2019.[50] P. de Jager, J. Conradie,
Isometries between non-commutative symmetric spaces associated with semi-finitevon Neumann algebras,
Positivity 24 (2020), 815–835.[51] M. Junge, Z. Ruan, D. Sherman,
A classification for -isometries of noncommutative L p -spaces , Israel J.Math. 150 (2005), 285–314.[52] M. Junge, D. Sherman, Noncommutative L p modules, J. Operator Theory 53 (2005), 3–34.[53] R. Kadison,
Isometries of operator algebras,
Ann. of Math. 54 (1951), 325–338.[54] R. Kadison, J. Ringrose,
Fundamentals of the theory of operator algebras. I , Pure and Applied Mathematics,100. Academic Press, Inc., New York, 1983.[55] N. Kalton, B. Randrianantoanina,
Isometries on rearrangement-invariant spaces,
C. R. Acad. Sci. Paris S´er.I Math. 316 (1993), 351–355.[56] N. Kalton, B. Randrianantoanina,
Surjective isometries on rearrangement-invariant spaces,
Quart. J. Math.Oxford (2) 45 (1994), 301–327.[57] N. Kalton, F. Sukochev,
Symmetric norms and spaces of operators , J. Reine Angew. Math. 621 (2008), 81–121.[58] A. Katavolos,
Are non-commutative L p spaces really non-commutative? Canad. J. Math. 6 (1976), 1319–1327.[59] S. Krein, Y. Petunin, E. Semenov,
Interpolation of linear operators.
Trans. Math. Mon., 54, AMS, Providence,1982.[60] A. Krygin, A. Medzhitov,
Isometries of Loretnz spaces,
Math. Anal. Algebra Prob. Theory, Collect. Sci. Works,Tashkent, 52–62 (1988).[61] J. Lamperti,
On the isometries of certain function spaces,
Pacific J. Math. 8 (1958), 459–466.[62] C.-K. Li, Y.-T. Poon, N.-S. Sze, Isometries for Ky Fan norms between matrix spaces, Proc. Amer. Math. Soc.133 (2004), 369–377.[63] J. Lindenstrauss, L. Tzafriri,
Classical Banach spaces. I. Sequence spaces.
Ergebnisse der Mathematik undihrer Grenzgebiete, Vol. 92. Springer-Verlag, Berlin-New York, 1977.[64] S. Lord, F. Sukochev, D. Zanin,
Singular traces: Theory and applications,
De Gruyter Studies in Mathematics,46. De Gruyter, Berlin, 2013.[65] G. Lumer,
Semi-inner-product spaces , Trans. Amer. Math. Soc. 100 (1961), 29–43.[66] A. Medzhitov,
Isometries of symmetric spaces on atomic von Neumann algebras,
Mathematical analysis,algebra and Geometry, Tashkent State Univ., 52–54 (1989).[67] A. Medzhitov, F. Sukochev,
Positive isometries of noncommutative symmetric spaces,
Izv. Akad. Nauk UzSSRSer. Fiz.-Mat. Nauk 3 (1987), 20–25 (in Russian).[68] A. Medzhitov, F. Sukochev,
Isometries of non-commutative Lorentz spaces,
Dokl. Acad. Nauk UzSSR 4 (1988),11–12 (in Russian).[69] C. le Merdy, S. Zadeh, ℓ -contractive maps on noncommutative L p -spaces, J. Operator Theory (2020), inpress.[70] M. Mori,
Isometries between projection lattices of von Neumann algebras,
J. Funct. Anal. 276 (2019), 3511–3528.[71] E. Nelson,
Notes on non-commutative integration,
J. Funct. Anal. 15 (1974), 103–116.[72] V. Ovˇcinnikov,
Symmetric spaces of measurable operators , Dok. Adad. Nauk SSSR 191 (1970), 448–451.(Russian)[73] V. Ovˇcinnikov,
Symmetric spaces of measurable operators , Trudy Inst. Math. WGU 3 (1971), 88–107. (Russian)[74] A. V. Potepun,
Hilbertian symmetric spaces,
Izv. Vyssh. Uchevn. Zaved. Math. 1 (1974), 90–95. Translatedin: Soviet Math. (Iz. VUZ) 18 (1974), 73–77.[75] B. Randrianantoanina,
Isometric classification of norms in rearrangement-invariant function spaces,
Com-ment. Math. Univ. Carolinae 38 (1997), 73–90.[76] B. Randrianantoanina,
Contractive projections and isometries in sequence spaces , Rocky Mountain J. Math.28 (1998), 323–340. [77] B. Randrianantoanina,
Injective isometries in Orlicz spaces,
Function spaces (Edwardsville, IL, 1998), 269–287,Contemp. Math., 232, Amer. Math. Soc., Providence, RI, 1999.[78] A. Rieckers, H. Roos,
Implementation of Jordan-isomorphisms for general von Neumann algebras,
Ann. Inst.H. Poincar´e Phys. Th´eor. 50 (1989), 95–113.[79] B. Russo,
Isometrics of L p -spaces associated with finite von Neumann algebras, Bull. Amer. Math. Soc. 74(1968), 228–232.[80] S. Sakai,
Derivations of W ∗ -algebras, Ann. of Math. 83 (1966), 273–279.[81] S. Sakai, C ∗ -algebras and W ∗ -algebras, Springer-Verlag, Berlin–Heidelberg–NewYork, 1971.[82] H. Schneider, R. Turner,
Matrices hermitian for an abosulte norm,
Linear Multilinear Algebra 1 (1973), 9–31.[83] I. Segal,
A non-commutative extension of abstract integration , Ann. of Math. 53 (1953), 401–457.[84] E. Semenov, F. Sukochev,
Sums and intersections of symmetric operator spaces , J. Math. Anal. Appl. 414(2014), 742–755.[85] E. Semenov, F. Sukochev, A. Usachev,
Geometry of Banach limits and their appications,
Russian Math.Surveys 75:4 (2020), 725–763.[86] D. Sherman,
Noncommutative L p structure encodes exactly Jordan structure, J. Funct. Anal. 221 (2005),150–166.[87] A. Sinclair,
Jordan homomorphisms and derivations on semisimple Banach algebras,
Proc. Amer. Math. Soc.24 (1970), 209–214.[88] A. Sourour,
On the isometries of L p (Ω , X ), Bull. Amer. Math. Soc. 83 (1977), 129–130.[89] A. Sourour, The isometries of L p (Ω , X ), J. Funct. Anal. 30 (1978), 276–285.[90] A. Sourour, Isometries of norm ideals of compact operators,
J. Funct. Anal. 43 (1981), 69–77.[91] F. Sukochev,
Symmetric spaces of measurable operators on finite von Neumann algebras,
Ph.D. Thesis (Rus-sian), Tashkent 1987.[92] F. Sukochev,
Isometries of symmetric operator spaces associated with AFD factors of type II and symmetricvector-valued spaces, Integral Equ. Oper. Theory 26 (1996), 102–124.[93] F. Sukochev, A. Veksler,
Positive linear isometries in symmetric operator spaces,
Integral Equ. Oper. Theory(2018), 90:58 1–15.[94] M. Takesaki,
Theory of operator algebras I , Springer-Verlag, New York, 1979.[95] M. Takesaki,
Theory of operator algebras III , Springer-Verlag, New York, 2003.[96] P. Tam,
Isometries of L p -spaces associated with semifinite von Neumann algebras, Trans. Amer. Math. Soc.254 (1979), 339–354.[97] K. Watanabe,
On isometries between noncommutative L p -spaces associated with arbitrary von Neumann al-gebra, J. Operator Theory 28 (1992), 267–279.[98] Y. Xu, Q. Ye,
Generalized Mercer kernels and reproducing kernel Banach spaces,
Mem. Amer. Math. Soc. 258(2019), no. 1243, vi+122 pp.[99] F. Yeadon,
Isometries of non-commutative L p spaces, Math. Proc. Cambridge Philos. Soc. 90 (1981), 41–50[100] M. Zaidenberg,
On isometric classification of symmetric spaces,
Soviet Math. Dokl. 18 (1977), 636–640.[101] M. Zaidenberg,
A representation of isometries on function spaces,
Mat. Fiz. Anal. Geom. 4 (1997), 339–347.
School of Mathematics and Statistics, University of New South Wales, Kensington, 2052, NSW,Australia
Email address : [email protected] School of Mathematics and Statistics, University of New South Wales, Kensington, 2052, NSW,Australia
Email address ::