Differentiation of Operator Functions in Non-Commutative Lp-Spaces B

ARTICLE IN PRESS

Journal of Functional Analysis 212 (2004) 28–75

Differentiation of operator functions in non-commutative Lp-spaces B. de Pagtera and F.A. Sukochevb, a Department of Mathematics, Faculty ITS, Delft University of Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands b School of Informatics and Engineering, Flinders University of South Australia, Bedford Park, 5042 SA, Australia

Received 14 May 2003; revised 18 September 2003; accepted 15 October 2003

Communicated by G. Pisier

Abstract

The principal results in this paper are concerned with the description of differentiable operator functions in the non-commutative Lp-spaces, 1ppoN; associated with semiﬁnite von Neumann algebras. For example, it is established that if f : R-R is a Lipschitz function, then the operator function f is Gaˆ teaux differentiable in L2ðM; tÞ for any semiﬁnite von Neumann algebra M if and only if it has a continuous derivative. Furthermore, if f : R-R has a continuous derivative which is of bounded variation, then the operator function f is Gaˆ teaux differentiable in any LpðM; tÞ; 1opoN: r 2003 Elsevier Inc. All rights reserved.

1. Introduction

Given an arbitrary semifinite von Neumann algebra M on the Hilbert space H; we associate with any Borel function f : R-R; via the usual functional calculus, the corresponding operator function a/f ðaÞ having as domain the set of all (possibly unbounded) self-adjoint operators a on H which are affiliated with M: This paper is concerned with the study of the differentiability properties of such operator functions f : Let LpðM; tÞ; with 1pppN; be the non-commutative Lp-space associated with ðM; tÞ; where t is a faithful normal semifinite trace on the von f Neumann algebra M and let M be the space of all t-measurable operators affiliated

Corresponding author. Fax: +61-88-201-29-04. E-mail address: sukochev@infoeng.ﬂinders.edu.au (F.A. Sukochev).

B. de Pagter, F.A. Sukochev / Journal of Functional Analysis 212 (2004) 28–75 29

f with M (see e.g. [20] or [26]). Fixing a self-adjoint operator aAM we are interested whether for any self-adjoint operator bALpðM; tÞ the difference f ða þ tbÞÀf ðaÞ; with tAR; belongs to LpðM; tÞ and further, whether the limit

f ða þ tbÞÀf ðaÞ lim t-0 t exists in the Lp-norm. If this limit exists and depends linearly and continuously on b; then we say that f is Gaˆ teaux differentiable at a along (the self-adjoint part of) LpðM; tÞ: In the special case that M is a type I factor and p ¼ 1; N (i.e., M ¼ LðHÞ with standard trace) this problem has been studied extensively. This study was started by Daletskii and Krein [11,12] and was substantially expanded in scope and methods in a series of papers by Birman and Solomyak [4–6]. It is instructive to recall the Daletskii–Krein formula for derivatives of matrix valued functions, which corresponds to the case that M ¼ Mn; the (ﬁnite dimensional) algebra of all n Â n complex matrices equipped with the standard trace Tr (in this situation LpðMn; TrÞ¼Mn with equivalent norms for all 1pppN). For self-adjoint 1 operators a; bAMn and C -function f : R-R; this formula at the point t ¼ 0is

Xn d f ðliÞÀf ðljÞ f ða þ tbÞ¼ eibej; ð1Þ dt l l t¼0 i; j¼1 i À j where ðl1; y; lnÞ is the sequence of eigenvalues of a; repeated according to multiplicity and ðe1; y; enÞ is the corresponding sequence of one dimensional eigenprojections. It should be noted that in this situation the Gaˆ teaux derivative is actually a Fre´ chet derivative. The starting point of the Birman–Solomyak theory is to view the summation in (1) as an integral with respect to the Mn-valued spectral product measure P#Q deﬁned on the ﬁnite product space f1; 2; yngÂf1; 2; yng; where PðfigÞx ¼ eix and QðfigÞx ¼ xei for all i ¼ 1; y; n and all xAMn: The integrand is the function

f ðlÞÀf ðmÞ c ðl; mÞ¼ f l À m and the integration process corresponds to the Schur–Hadamard multiplication of n the operator matrix ½eibeji; j¼1 by the scalar matrix n f ðliÞÀf ðljÞ

li À lj i; j¼1

(which is sometimes called a Loewner matrix of the function f ). Observe that, since we are dealing here with the ﬁnite dimensional situation, it is obvious that P#Q is actually a s-additive spectral measure in any of the spaces LpðMn; TrÞ: The latter fact fails in the inﬁnite dimensional setting, which is one of the main obstructions to be dealt with in the development of a general theory. ARTICLE IN PRESS

30 B. de Pagter, F.A. Sukochev / Journal of Functional Analysis 212 (2004) 28–75

In the case when H is a (infinite dimensional) separable Hilbert space and the von Neumann algebra ðM; tÞ coincides with the algebra ðLðHÞ; TrÞ of all bounded linear operators on H with standard trace Tr (in this case LpðM; TrÞ coincide with the Schatten ideals Cp for 1ppoN), Birman and Solomyak developed the theory of double operator integrals and successfully applied this to the perturbation and differentiation theory of operator functions in the Schatten–von Neumann ideals, putting special emphasis on the extreme cases p ¼ 1; N: In recent papers of the authors together with Witvliet [27,28] the double operator integration theory of Birman and Solomyak has been recast in the form of integration with respect to a finitely additive product spectral measure P#Q on an arbitrary Banach space X: This allowed the extension of the theory of double operator integrals beyond the scope of the type I factor von Neumann algebras. In particular, applying recent (vector-valued) extensions of the classical Marcinkiewicz multiplier theorem from [10] to (generalized) Schur–Hadamard multipliers in the spaces LpðM; tÞ; 1opoN; it was established in [28] that f ðaÞÀf ðbÞALpðM; tÞ whenever a; bAM are self-adjoint such that a À bALpðM; tÞ and f : R-R has a (weak) derivative of bounded variation. The present paper develops this theory further and is mostly concerned with the description of differentiable operator functions along the pre-dual of M (identified with the space L1ðM; tÞ) and along the spaces LpðM; tÞ; 1opoN: The explicit selection of the latter class and the techniques developed to treat this case are among the main novel features of the present paper. It is also worth to note that our technique yields the complete description of the class of operator functions f which differentiable along the space L2ðM; tÞ for all semifinite algebras M (earlier, in [28], we established that the operator function f remains Lipschitz in any L2ðM; tÞ if its real-valued prototype is Lipschitz). However, even in the case p ¼ 1; a significant part of our differentiation technique is entirely different from that of [6] (although the double operator integration theory and the appropriate integral decompositions of the divided difference cf underpin both approaches). For a better appreciation of the nature of the (new) obstacles appearing when dealing with non-atomic and non- hyperfinite algebras, we mention the following phenomenon, which has been noted already in a number of papers see, e.g., the Editor’s Note to [42] and also [1,30,31]). Although the results in [6] assert only Gaˆ teaux differentiability of operator functions along LðHÞ and C1; yet they automatically yield Fre´ chet differentiability. However, let us now consider the following simple example. Let M ¼ LNð0; 1Þ acting on H ¼ L2ð0; 1Þ via multiplication, the trace t on M being given by integration with respect to Lebesgue measure on ð0; 1Þ: Clearly, L1ðM; tÞ may be identified with L1ð0; 1Þ and the self-adjoint elements in L1ðM; tÞ correspond to the space R - L1 ð0; 1Þ of all real-valued functions in L1ð0; 1Þ: Let the function f : R R be given by f ðtÞ¼sin t: It is straightforward to verify that

f ða þ tbÞÀf ðaÞ 0 jj Á jj1 À lim ¼ f ðaÞb ð2Þ t-0 t A R / for all functions a; b L1 ð0; 1Þ: In other words, the function x f ðxÞ is Gaˆ teaux R / differentiable on L1 ð0; 1Þ: But it is not difﬁcult to see that the function x f ðxÞ is ARTICLE IN PRESS

B. de Pagter, F.A. Sukochev / Journal of Functional Analysis 212 (2004) 28–75 31

R not Fre´ chet differentiable on L1 ð0; 1Þ (the limit in (2) is not uniform on the R unit ball of L1 ð0; 1Þ). The same phenomenon occurs if we replace in this example p ¼ 1byanypA½1; NÞ: This example indicates the (unavoidable and significant) intrinsic difference between the techniques in the present paper and in [6]. Our approach here is based on the use of the generalized singular value function and its properties, and is designed to suit arbitrary semifinite von Neumann algebras ðM; tÞ: Now we survey briefly the contents of the paper. In the next section we briefly introduce some notation and terminology concerning the theory of non-commutative integration on semifinite von Neumann algebras and of the theory of symmetric operator spaces associated with such algebras, with special emphasis on spaces with order continuous norm. Concerning the inclusion of the latter theory, we note that the results and techniques needed to treat the L1-case happen to be applicable to any symmetric operator space associated with a separable rearrangement invariant Banach function space. A similar remark applies to the case of the Lp-spaces with 1opoN: all differentiation and perturbation results discussed in the present paper remain valid for any symmetric operator space associated with a separable rearrangement invariant Banach function space with non-trivial Boyd indices (see [18,25]). We collect a number of results concerning various notions of convergence of measurable operators in Section 3. In particular, we establish a direct connection between convergence in measure of (unbounded) self-adjoint t-measurable operators and the resolvent convergence of their left standard representations. In Section 4 we present a necessary extension of some of the results concerning double operator integration theory from [28] in the framework of order continuous symmetric operator spaces. Sections 5 and 6 contain a number of sufficient conditions for a function f : R-R to be operator Gaˆ teaux differentiable along L1ðM; tÞ and along any LpðM; tÞ; 1opoN respectively, for any semifinite von Neumann algebra ðM; tÞ: In particular, Corollary 6.10 and Proposition 6.11 contain the results announced in the abstract. Section 7 contains some additional discussion concerning the classes of differentiable operator functions from Sections 5 and 6. In particular, we prove ’1 that any function belonging to the homogeneous Besov class BN;1ðRÞ is operator differentiable along L1ðM; tÞ: This result complements the results given in [30,31] for the situation when M ¼ LðHÞ: Finally we show that the classes of functions which are operator differentiable along L1ðM; tÞ and along LpðM; tÞð1opoNÞ respectively, are different. In fact we exhibit a function f : R-R which is operator differentiable along LpðM; tÞ (1opoN), but which is not Lipschitz continuous in L1ðM; tÞ: We present below a short list of symbols used in this paper with the indication of the place where these symbols are introduced. E; EÂ: non-commutative symmetric space and its Ko¨ the dual (Section 2); Pa ; Qb ; Pa; Qb : spectral measures (Corollary 3.5, Section 4); L2 L2 E a # b PE QE: finitely additive product spectral measure (Section 4); ARTICLE IN PRESS

32 B. de Pagter, F.A. Sukochev / Journal of Functional Analysis 212 (2004) 28–75

a# b a # b JE ðP Q Þ: collection of all PE QE -integrable functions (Definition 4.1); 2 C0; A0; C1; A1: function spaces on R (Definitions 4.5 and 6.1); DEf ðaÞ:Gaˆ teaux derivative of f at a along Eh (Definition 5.15); À1 À1 À1 À1 C ðA0Þ; C ðA1Þ C ðC0Þ; C ðC1Þ: function spaces on R (Definition 5.1).

2. Preliminaries on non-commutative integration

In this section we will introduce some notation and collect some of the results concerning the theory of non-commutative integration which will be used throughout the paper. Let ðH; /Á; ÁSÞ be a complex Hilbert space and let M be a von Neumann algebra on H: The unit element in M will be denoted by 1: We assume that M is equipped with a semifinite faithful normal trace t : Mþ-½0; N: In this situation we will say that ðM; tÞ is a semifinite von Neumann algebra. For the general theory of von Neumann algebras we refer the reader to the books [15,22,23,35,37]. The set of all orthogonal projections in M will be denoted by Mp and the real subspace of all self-adjoint elements in M is denoted by Mh: For basics on the integration theory in semifinite von Neumann algebras we refer to [20,26,38]. If a densely defined operator x in the Hilbert space H is affiliated with the von Neumann algebra M; this will be denoted by xZM: The collection of all f t-measurable operators affiliated with M will be denoted by M: With respect to the f f measure topology, M is a complete topological -algebra and M is dense in M with respect to this topology. Suppose that a : DomðaÞ-H is a self-adjoint operator in the Hilbert space H: We will denote the spectral measure of a by ea; so ea : BðRÞ-LðHÞ; where BðRÞ is A a a the s-algebra of all Borel subsets of R: For all l R we will write el ¼ e ðÀN; l: Observe that if xZM; then ejxjðBÞAM for all BABðRÞ: The generalized singular value f function mðxÞ : ½0; N-½0; N of an operator xAM is defined by

X jxj mtðxÞ¼inffl 0:tð1 À el Þptg for all 0ptAR: Note that mtðxÞoN for all t40 and that m0ðxÞoN if and only if xAM; in which case m0ðxÞ¼jjxjj: A detailed study of the properties of generalized singular value functions can be found in [20]. We mention in particular that a N f - -N sequence fxngn¼1 in M converges to 0 in measure if and only if mtðxnÞ 0asn for all t40: For the general theory of rearrangement invariant Banach function spaces we refer the reader to the books [2,24]. We consider these spaces on the interval ð0; NÞ with Lebesgue measure dt: The space of all (equivalence classes of) real valued measurable functions on ð0; NÞ is denoted by L0ð0; NÞ: For f AL0ð0; NÞ we denote by f the decreasing rearrangement of the function j f j: Recall that a Banach space ðE; jj Á jjEÞ is called a rearrangement invariant Banach function space if f0gaEDL0ð0; NÞ and f AE; gAL0ð0; NÞ; g pf imply that gAE and ARTICLE IN PRESS

B. de Pagter, F.A. Sukochev / Journal of Functional Analysis 212 (2004) 28–75 33 jjgjjEpjj f jjE: Special examples of such Banach function spaces are the spaces Lpð0; NÞ; 1pppN equipped with their usual norm jj Á jjp: In particular, we recall that any such space satisfies L1-LNð0; NÞDEDðL1 þ LNÞð0; NÞ; with continuous embeddings. Furthermore we recall that the norm in E is said to be order continuous k k if fn 0inE implies that jj fnjjE 0; and that order continuity of the norm is equivalent to separability of the space E: A rearrangement invariant Banach function space E will be called symmetric,iff ; gAE and g!!f imply jjgjjE pjj f jjE: Here g!!f means that the function g is submajorized by f ; i.e., Z Z t t gðsÞ dsp f ðsÞ ds 0 0 for all t40: A separable rearrangement invariant Banach function space E is necessarily symmetric [24]. Let ðM; tÞ be a semifinite von Neumann algebra. For any symmetric Banach function space E on ð0; NÞ the corresponding non-commutative space E ¼ EðM; tÞ associated with ðM; tÞ is defined by f E ¼fxAM : mðxÞAEg;

A equipped with the norm given by jjxjjE ¼jjmðxÞjjE for all x E: It can be shown that ðE; jj Á jjEÞ is indeed a Banach space (see [16,17,36] for details), which is called a non- commutative symmetric space. In particular, if E ¼ Lpð0; NÞ; 1pppN; then EðM; tÞ¼LpðM; tÞ is the corresponding non-commutative Lp-space. The norm in LpðM; tÞ is usually denoted by jj Á jjp: We note that LNðM; tÞ¼M and for this þ reason the norm in M is also denoted by jj Á jjN: The trace t on M extends uniquely to a bounded linear functional on the space L1ðM; tÞ; which will be denoted by t as A well. The norm on L1ðM; tÞ is then also given by jjxjj1 ¼ tðjxjÞ for all x L1ðM; tÞ: / S Furthermore, if x; yAL2ðM; tÞ; then xyAL1ðM; tÞ and deﬁning x; y 2 ¼ tðy xÞ; / S the space ðL2ðM; tÞ; Á; Á 2Þ is a Hilbert space. It should be observed that every non- commutative symmetric space E ¼ EðM; tÞ satisﬁes

L1-LNðM; tÞDEDðL1 þ LNÞðM; tÞ; with continuous embeddings. For any non-commutative symmetric space E; the Ko¨ the dual is deﬁned by

Â f E ¼fyAM : xyAL1ðM; tÞ8xAEg and equipped with the norm given by

A jjyjjEÂ ¼ supfjtðxyÞj : x E; jjxjjEp1g A Â Â for all y E ; the space ðE ; jj Á jjEÂ Þ is a Banach space. As in the commutative situation, EÂ can be identiﬁed with a subspace of the Banach dual E via the duality ARTICLE IN PRESS

34 B. de Pagter, F.A. Sukochev / Journal of Functional Analysis 212 (2004) 28–75 pairing given by

/x; yS ¼ tðxyÞ¼tðyxÞð3Þ for all xAE and yAEÂ: Moreover, it turns out that EÂ ¼ EÂðM; tÞ; where EÂ is the (classical) Ko¨ the dual for E; so EÂ is itself a non-commutative (fully) symmetric space. For the details concerning this non-commutative duality theory we refer the reader to the paper [17]. In particular we recall that if E has order continuous norm, then E ¼ EðM; tÞ has order continuous norm as well (i.e., xak0inE implies that k Â jjxajjE 0), which implies that E ¼ E (see Proposition 3.6 and Theorem 5.11 in [17]; see also [18]). If E has order continuous norm, then L1-LNð0; NÞ is a dense subspace of E and this implies that L1-LNðM; tÞ is dense in E (see [17, Proposition 2.8]). We will need a strengthening of this approximation result. We denote by Mfin the set of all xAM for which there exists pAMp such that x ¼ pxp and tðpÞoN: It is easy to see that Mfin is a -subalgebra of M and it is clear that MfinDL1-LNðM; tÞ:

Lemma 2.1. Suppose that F is a symmetric Banach function space on ð0; NÞ and let F ¼ FðM; tÞ: If yAL1-LNðM; tÞ and E40; then there exists y1AMfin such that jjy À y jj E and jjy jj jjyjj : 1 L1-LN p 1 Fp F

Proof. Let yAL1-LNðM; tÞ and E40 be given. First we show that there exists p pAM such that pjyj¼jyjp; tðpÞoN and jjjyjÀpjyjjj pE: Indeed, let pn ¼ ÀÁ L1-LN ejyj 1 N n y yAL p N n; for ¼ 1; 2; : Since ÀÁ1ÂÃðM; tÞ; it follows that tð nÞo for n y p y ejyj 1 y 1 y all : Furthermore, 0pj jÀ ÀÁnÂÃj j¼ 0; n j jpn 1; which shows that jjj jÀ - -N jyj 1 k A pnjyjjjN 0asn : Since e 0; n jyj 0 and y L1ðM; tÞ; we also have 1 jjjyjÀp jyjjj ¼ t ejyj 0; jyj k0 n 1 n as n-N: This shows that jjjyjÀp jyjjj -0asn-N; so we can take p ¼ p n L1-LN n for a sufficiently large value of n: Similarly we can find qAMp such that qjyj¼jyjq; tðqÞ N and jjjyjÀqjyjjj E: Now define rA p by r ¼ p q and let o L1-LN p M 3 A N y1 ¼ ryr: It is clear that jjy1jjFpjjyjjF and that y1 Mfin; as tðrÞo : Observe that

jjjyjÀjyjrjj ¼ jjjyjð1 À rÞjj ¼ jjjyjð1 À pÞð1 À rÞjj L1-LN L1-LN L1-LN jjjyjð1 À pÞjj ¼ jjjyjÀjyjpjj E p L1-LN L1-LN p and similarly jjjyjÀrjyjjj E: Let y ¼ ujyj be the polar L1-LN p decomposition of y: Then y ¼ ujyj is the polar decomposition of y; so ARTICLE IN PRESS

B. de Pagter, F.A. Sukochev / Journal of Functional Analysis 212 (2004) 28–75 35 y ¼jyju: Hence, jjy À y jj jjy À yrjj þjjyr À ryrjj 1 L1-LN p L1-LN L1-LN jjy À yrjj þjjy À ryjj p L1-LN L1-LN ¼jjujyjÀujyjrjj þ jjjyju À rjyjujj L1-LN L1-LN jjjyjÀjyjrjj þ jjjyjÀrjyjjj 2E: p L1-LN L1-LN p

This completes the proof of the lemma. &

Corollary 2.2. If E is a separable symmetric Banach function space on ð0; NÞ and E ¼ EðM; tÞ; then Mfin is dense in E:

Proof. It follows from Lemma 2.1 that Mfin is dense in L1-LNðM; tÞ with respect to jj Á jj : Since there exists a constant K40 such that jjzjj Kjjzjj for all L1-LN Ep L1-LN zAL1-LNðM; tÞ; this implies that Mfin is dense in L1-LNðM; tÞ with respect to jj Á jjE: Moreover, L1-LNðM; tÞ is dense in E; as the norm in E is order continuous. Hence Mfin is dense in E: &

Lemma 2.3. If E is a separable symmetric Banach function space on ð0; NÞ and E ¼ EðM; tÞ; then A jjxjjE ¼ supfjtðxyÞj : y Mfin; jjyjjEÂ p1g: for all xAE:

Proof. As mentioned above, the order continuity of the norm in E implies that EÂ ¼ E; hence A Â jjxjjE ¼ supfjtðxyÞj : y E ; jjyjjEÂ p1g A A for all x E: This can also be formulated as jjxjjE ¼jjxjjEÂÂ for all x E: Now it follows from [17] Proposition 5.3(ii), that

A jjxjjE ¼ supfjtðxyÞj : y L1-LNðM; tÞ; jjyjjEÂ p1g for all xAE: Hence, it is sufﬁcient to show that for every yAL1-LNðM; tÞ and every A E40 there exists y1 Mfin such that jjy À y1jjEÂ pE and jjy1jjEÂ pjjyjjEÂ : Since there exists a constant K40 such that jjzjj Â Kjjzjj for all zAL LNð ; tÞ; the E p L1-LN 1- M existence of such y1AMfin is an immediate consequence of Lemma 2.1 applied to F ¼ EÂ: &

f Next we will discuss the representation of the operators in M as left and right multiplication operators on the Hilbert space L2ðM; tÞ: For aAM we deﬁne the ARTICLE IN PRESS

36 B. de Pagter, F.A. Sukochev / Journal of Functional Analysis 212 (2004) 28–75 linear operators La; Ra : L2ðM; tÞ-L2ðM; tÞ by LaðxÞ¼ax and RaðxÞ¼xa respectively, for all xAL2ðM; tÞ: This deﬁnes the mappings

- fL; fR : M LðL2ðM; tÞÞ given by fLðaÞ¼La and fRðaÞ¼Ra; respectively for all aAM: The mapping fL (respectively, fR) is a normal -isomorphism (respectively, -anti-isomorphism) from M onto

ML :¼ fLðMÞ; ðrespectively; MR :¼ fRðMÞÞ:

0 Furthermore, it is known that the commutant ML of ML is equal to MR (and 0 similarly MR ¼ ML) (see e.g. [15,23,35]). À1 À1 - For 0pLaAML we define tLðLaÞ¼tðfL ðLaÞÞ ¼ tðaÞ: Since fL : ML M is a normal -isomorphism, it is clear that tL : ML-½0; N is a semifinite faithful normal g trace. We denote by ML the -algebra of all tL -measurable operators on L2ðM; tÞ: f g Then fL extends (uniquely) to a -isomorphism from M onto ML: This fact is an immediate corollary of the following simple observation, whose proof follows immediately from the definition of the measure topology and from the well known fact that a -homomorphism is a norm isometry.

Lemma 2.4. Let ðM1; t1Þ and ðM2; t2Þ be two semiﬁnite von Neumann algebras on the Hilbert spaces H1 and H2 respectively. Suppose that f is a -isomorphism from M1 onto M2 which is trace preserving (i.e., t2ðfðaÞÞ ¼ t1ðaÞ for all 0paAM1). Then f is a homeomorphism with respect to the measure topologies in M1 and M2:

f Since M is the completion of M with respect to the measure topology, and g similarly for ML it is now clear from Lemma 2.4 applied to fL; that fL extends f f-g uniquely to a -isomorphic homeomorphism fL : M ML: g It is of course expected that the elements in ML may be identiﬁed as left f multiplication on L2ðM; tÞ by elements of M: To see the validity of this assertion, we f deﬁne for aAM

DomðLaÞ¼fxAL2ðM; tÞ : axAL2ðM; tÞg;

f where the products ax are taken in the algebra M: It is clear that DomðLaÞ is a linear subspace of L2ðM; tÞ: Furthermore, we deﬁne the linear operator

La : DomðLaÞ-L2ðM; tÞ by LaðxÞ¼ax: Note that if aAM; then this deﬁnition agrees with the previous of La; so there is no danger of confusion. The properties of the operators La are given in the following lemma whose proof based on standard arguments is omitted. ARTICLE IN PRESS

B. de Pagter, F.A. Sukochev / Journal of Functional Analysis 212 (2004) 28–75 37

f g Lemma 2.5. For any aAM the operator La is tL-measurable, i.e., LaAML: Moreover, f g the mapping a/La is a -isomorphism from M into ML; which is continuous with respect to the measure topologies.

f f It is now clear that fLðaÞ¼La for all aAM: Moreover, appropriately modiﬁed all of the above results remain valid if we replace La by Ra: We formulate these observations in the following proposition.

f g Proposition 2.6. The mapping a/La is a -isomorphism from M onto ML which is a homeomorphism for the measure topologies. Similarly, the mapping a/Ra is f g a -anti-isomorphism from M onto MR which is a homeomorphism for the measure topologies.

f Given a; bAMh; we denote the spectral measures of the self-adjoint operators L Ag and R Ag by Pa and Qb respectively. The following a ML b MR L2 L2 lemma describes the relation between these spectral measures and the spectral measures ea and eb: The proof is again omitted, since it follows via standard arguments.

f a b Lemma 2.7. Suppose that a; bAMh with spectral measures e ; e : BðRÞ-M: Then the spectral measures Pa and Qb of L and R are given by L2 L2 a b

a b P ðBÞ¼L a and Q ðBÞ¼R b L2 e ðBÞ L2 e ðBÞ respectively for all BABðRÞ:

3. Convergence of measurable operators

In this section we will collect some results concerning the convergence of self- adjoint and measurable operators which play an important role in later sections. Given a self-adjoint operator a on the Hilbert space H we will denote, as before, the a - spectralR measure of a by e : For any Borel function f : R C we will write f ðaÞ¼ a R fde: The space of all bounded continuous complex valued functions on R will be denoted by CbðRÞ and C0ðRÞ will denote the subspace of CbðRÞ consisting of all functions vanishing at inﬁnity. The space of all bounded complex valued Borel functions on R will be denoted by BðRÞ: Similar notation will be used for functions 2 N on R : Recall that a sequence fangn¼1 of self-adjoint operators on H is called À1 À1 resolvent strongly convergent to a self-adjoint operator a if ðl À anÞ x-ðl À aÞ x as n-N for all xAH and all lAC\R: We start by recalling the following well known result (see e.g. [32, Theorem VIII.20]). ARTICLE IN PRESS

38 B. de Pagter, F.A. Sukochev / Journal of Functional Analysis 212 (2004) 28–75

N Proposition 3.1. Let fangn¼1 and a be self-adjoint operators on the Hilbert space H: The following two statements are equivalent. N 1. The sequence fangn¼1 is resolvent strongly convergent to a: A N 2. For every f CbðRÞ the sequence f f ðanÞgn¼1 is strongly convergent to f ðaÞ: Next we recall some facts concerning products of spectral measures. Suppose that e1 : BðRÞ-LðHÞ and e2 : BðRÞ-LðHÞ are two spectral measures in the Hilbert space H which commute, i.e., e1ðB1Þe2ðB2Þ¼e2ðB2Þe1ðB1Þ for all B1; B2ABðRÞ: We 2 2 denote by BðR Þ the s-algebra of all Borel sets in R and by A0 we denote the collection of all Borel rectangles, i.e.,

A0 ¼fB1 Â B2 : B1; B2ABðRÞg:

2 Let A be the subalgebra of BðR Þ generated by A0: For B1 Â B2AA0 we deﬁne

ðe1#e2ÞðB1 Â B2Þ¼e1ðB1Þe2ðB2Þ:

Since e1 and e2 commute it follows that ðe1#e2ÞðB1 Â B2Þ is an orthogonal projection in H: It is easy to see that e1#e2 extends uniquely to a ﬁnitely additive spectral measure

e1#e2 : A-LðHÞ:

It is well known that (see e.g. [7, Theorem V.2.6]) in this situation e1#e2 extends uniquely to a spectral measure deﬁned on BðR2Þ; this extension will be denoted by e1#e2 as well, so

2 e1#e2 : BðR Þ-LðHÞ:

If gABðRÞ and f ðl; mÞ¼gðlÞ for all ðl; mÞAR2; then it is easy to verify that Z Z

fdðe1#e2Þ¼ gde1: ð4Þ R2 R Now we discuss the convergence of spectral integrals for sequences of product measures.

N N Proposition 3.2. Assume that fangn¼1 and fbngn¼1 are two sequences of self-adjoint operators on the Hilbert space H; resolvent strongly convergent to the self-adjoint operators a and b respectively. Furthermore we will assume that the spectral measures ean and ebn commute for all n. Then Z Z fdðean #ebn Þ- fdðea#ebÞð5Þ R2 R2

2 as n-N with respect to the strong operator topology in LðHÞ for all f ACbðR Þ: ARTICLE IN PRESS

B. de Pagter, F.A. Sukochev / Journal of Functional Analysis 212 (2004) 28–75 39

Proof. First we observe that the conditions of the proposition imply that the spectral measures ea and eb commute and so ea#eb is a well deﬁned spectral measure on 2 an bn À1 À1 BðR Þ: Indeed, since e and e commute, it follows that ðl1 À anÞ and ðl1 À bnÞ commute for all lAC\R: The resolvent strong convergence now implies that ðl1 À aÞÀ1 and ðl1 À bÞÀ1 commute for all lAC\R and via [19, Theorem XII.2.10] we may conclude that ea and eb commute. 2 We denote by R the collection of all f ACbðR Þ for which (5) holds. It is easy to 2 see that R is a uniformly closed subalgebra of CbðR Þ: Now take g; hACbðRÞ and let f ðl; mÞ¼gðlÞhðmÞ for all ðl; mÞAR2: It follows from Proposition 3.1 that gðanÞx-gðaÞx and hðbnÞx-hðbÞx as n-N for all xAH and hence gðanÞhðbnÞx-gðaÞhðbÞx for all xAH: Using (4) we ﬁnd that Z Z Z fdðean #ebn Þ¼ gdðean #ebn Þ hdðean #ebn Þ R2 R2 R2

¼ gðanÞhðbnÞ and similarly Z fdðea#ebÞ¼gðaÞhðbÞ: R2

2 This shows that f AR: We denote by R0 theP subalgebra of C0ðR Þ con- n A sisting of all functions f of the form f ðl; mÞ¼ i¼1 giðlÞhiðmÞ with gi; hi C0ðRÞ for i ¼ 1; y; n and nAN: From the above it follows that R0DR: Furthermore, 2 it is clear that R0 is conjugate closed, separates the points of R and that 2 for every ðl; mÞAR there exists a function f AR0 such that f ðl; mÞa0: Hence, by the 2 Stone–Weierstrass theorem, R0 is uniformly dense in C0ðR Þ: This implies that 2 C0ðR ÞDR: R f fdean #ebn f R For sake of convenience we will write Fnð Þ¼ R2 ð Þ and Fð Þ¼ a# b A 2 y R2 fdðe e Þ for all f CbðR Þ: For k ¼ 1; 2; we deﬁne the functions 2 gkAC0ðR Þ by l2 þ m2 g ðl; mÞ¼exp À : k k

2 Since 0pgkðl; mÞp1 and gkðl; mÞ-1ask-N for all ðl; mÞAR ; it follows that 2 FðgkÞx-x as k-N for all xAH: Moreover, since gkAC0ðR Þ the above implies that 2 FnðgkÞx-FðgkÞx as n-N for all kAN and all xAH: Now take f ACbðR Þ: Then 2 fgkAC0ðR Þ and so it follows from the above that Fnð fgkÞx-Fð fgkÞx as n-N for ARTICLE IN PRESS

40 B. de Pagter, F.A. Sukochev / Journal of Functional Analysis 212 (2004) 28–75 all kAN and all xAH: For all n; kAN and all xAH we have

jjFnð f Þx À Fð f Þxjj

pjjFnð f Þx À Fnð f ÞFnðgkÞxjj þ jjFnð f ÞFnðgkÞx À Fð f ÞFðgkÞxjj

þjjFð f ÞFðgkÞx À Fð f Þxjj

pjj f jjNjjx À FnðgkÞxjj þ jjFnð fgkÞx À Fð fgkÞxjj

þjjf jjNjjFðgkÞx À xjj:

Hence

lim sup jjFnð f Þx À Fð f Þxjjp2jj f jjNjjFðgkÞx À xjj: n-N

This holds for all k and since FðgkÞx-x as k-N we may conclude that Fnð f Þx-Fð f Þx as n-N for all xAH; by which the proof is ﬁnished. &

As before we assume that ðM; tÞ is a semiﬁnite von Neumann algebra on the f Hilbert space H and M is the algebra of all t-measurable operators equipped with the measure topology. In the proof of the next proposition we will make use of the following observation.

N - -N Lemma 3.3. Suppose that fangn¼1 is a sequence in M such that an 0 as n with respect to the measure topology and jjanjjNpK for all n and some constant 0pKAR: - - A Then jjanxjj2 0 as n N for all x L2ðM; tÞ:

Proof. Let xAL2ðM; tÞ be given. As mentioned in the beginning of Section 2, an-0 - in measure implies that mtðanÞ 0 for all t40: It follows from

mtðanxÞpjjanjjNmtðxÞpKmtðxÞ

- - that mtðanxÞ 0 for all t40asn N: Since

Z N 2 2 mtðxÞ dt ¼jjxjj2oN; 0 the dominated convergence theorem implies that

Z N 2 2 - jjanxjj2 ¼ mtðanxÞ dt 0 0 as n-N: & ARTICLE IN PRESS

B. de Pagter, F.A. Sukochev / Journal of Functional Analysis 212 (2004) 28–75 41

f As in Section 2, for any aAM we denote by La and Ra the operators of left and f right multiplication by a on L2ðM; tÞ respectively. If aAMh; then it follows from Proposition 2.6 that the operators La and Ra are self-adjoint on L2ðM; tÞ: We f observe furthermore that, if xAM has an inverse xÀ1ALðHÞ; then xÀ1AM: Indeed, since x is afﬁliated with M we have ux ¼ xu for all unitary uAM0: This implies that xÀ1u ¼ uxÀ1 for all unitary uAM0 and hence xÀ1AM:

N f Af Proposition 3.4. Suppose that fangn¼1 is a sequence in Mh and that a Mh such that - - - - an aasn N with respect to the measure topology. Then Lan La and Ran Ra as n-N resolvent strongly on L2ðM; tÞ:

Proof. We only prove the statement concerning the Lan ; the proof for Ran being similar. Let lAC\R be ﬁxed. From the observation made above it follows À1 À1 that ðl1 À anÞ AM for all n and ðl1 À aÞ AM: Furthermore, recall that À1 jjðl1 À anÞ jjNpjIm lj for all n: Since

À1 À1 À1 À1 ðl1 À anÞ Àðl1 À aÞ ¼ðl1 À anÞ ðan À aÞðl1 À aÞ ; it follows that

À1 À1 mtððl1 À anÞ Àðl1 À aÞ Þ

À1 À1 pjjðl1 À anÞ jjNjjðl1 À aÞ jjNmtðan À aÞ 1 p mtðan À aÞð6Þ jIm lj2 - - for all t40: Since an a in measure implies that mtðan À aÞ 0 for all t40; it follows from (6) that À1 À1 - mtððl1 À anÞ Àðl1 À aÞ Þ 0

À1 À1 for all t40asn-N: Hence ðl1 À anÞ -ðl1 À aÞ with respect to the measure topology. An application of Lemma 3.3 now yields that À1 À1 - jjðl1 À anÞ x Àðl1 À aÞ xjj2 0 ð7Þ as n-N for all xAL2ðM; tÞ: Finally, it follows from Proposition 2.6 that

À1 À1 ðl1 À a Þ x ¼ L À1 ðxÞ¼ðlI À L Þ ðxÞ n ðl1ÀanÞ an

À1 À1 and similarly ðl1 À aÞ x ¼ðlI À LaÞ x for all xAL2ðM; tÞ and all n: Conse- quently, (7) is the desired result. &

The following corollary plays an important role in the sequel. ARTICLE IN PRESS

42 B. de Pagter, F.A. Sukochev / Journal of Functional Analysis 212 (2004) 28–75

N N f Corollary 3.5. Suppose that fangn¼1 and fbngn¼1 are two sequences in Mh and that f a; bAMh such that an-a and bn-basn-N in measure. Deﬁne the spectral measures Pa ; Qb ; Pan ; Qbn : ðRÞ- ðL ð ; tÞÞ by L2 L2 L2 L2 B L 2 M

Pa ðdÞðxÞ¼eaðdÞx; Qb ðdÞðxÞ¼xebðdÞ; L2 L2

Pan ðdÞðxÞ¼ean ðdÞx; Qbn ðdÞðxÞ¼xebn ðdÞ L2 L2 for all xAL2ðM; tÞ and all dABðRÞ: Then Z Z fdðPan #Qbn Þ- fdðPa #Qb Þð8Þ L2 L2 L2 L2 R2 R2

2 strongly on L2ðM; tÞ as n-N for all f ACbðR Þ:

Proof. It follows from Lemma 2.7 that Pa ; Qb ; Pan and Qbn are the spectral L2 L2 L2 L2 measures of the operators La; Rb; Lan and Rbn respectively. It follows from - - - Proposition 3.4 that Lan La and Rbn Rb resolvent strongly as n N: Since the spectral measures Pan and Qbn commute for all n; we are in the situation to apply L2 L2 Proposition 3.2, which yields (8). &

We end this section by recalling another result concerning measure convergence. Let CðRÞ denote the space of all continuous complex valued functions on R: Since f every f ACðRÞ is bounded on bounded intervals, it is easy to see that f ðaÞAM for all f aAMh: The following proposition is due to Tikhonov [39].

Proposition 3.6. Suppose that ðM; tÞ is a semiﬁnite von Neumann algebra, that N f Af - -N fangn¼1 is a sequence in Mh and that a Mh such that an a in measure as n : Then f ðanÞ-f ðaÞ in measure as n-N for all f ACðRÞ:

4. Double operator integrals

In this section we prove the existence of double operator integrals in non- commutative symmetric spaces for a sufficiently large class of functions. For a detailed account of the theory of double operator integrals in Banach spaces and in non-commutative symmetric spaces we refer the reader to [28]. Suppose that ðM; tÞ is a semifinite von Neumann algebra on the Hilbert space H: Let E be a separable symmetric Banach function space on ð0; NÞ and denote by E ¼ EðM; tÞ the corresponding non-commutative space associated with ðM; tÞ: Suppose that a and b are two self-adjoint operators affiliated with M with spectral measures ea; eb : BðRÞ-LðHÞ: Note that ea and eb take their values in M: For every ARTICLE IN PRESS

B. de Pagter, F.A. Sukochev / Journal of Functional Analysis 212 (2004) 28–75 43

a b - BABðRÞ deﬁne the operators PEðBÞ; QEðBÞ : E E by

a a b b PEðBÞðxÞ¼e ðBÞx; QEðBÞðxÞ¼xe ðBÞ8xAE: ð9Þ

a b - a Then PE; QE : BðRÞ LðEÞ are two commuting spectral measures and jjPEðBÞjj; b jjQEðBÞjjp1 for all BABðRÞ: Denoting by A the algebra generated by all Borel rectangles in R2; let

a # b - PE QE : A LðEÞ be the ﬁnitely additive product spectral measure, i.e.,

a # b a b PE QEðA Â BÞ¼PEðAÞQEðBÞ for all A; BABðRÞ: First we consider the special case that E ¼ L2ð0; NÞ; i.e., a b E ¼ L2ðM; tÞ: In this case PE and QE take their values in the orthogonal projections of the Hilbert space L ð ; tÞ; and so the product measure Pa #Qb extends 2 M L2 L2 uniquely to a s-additive spectral measure on BðR2Þ; taking its values in the orthogonal projections of L2ðM; tÞ (see the remarks made in Section 3). We denote this extension by Pa #Qb as well, so L2 L2

Pa #Qb : ðR2Þ- ðL ð ; tÞÞ: L2 L2 B L 2 M

Consequently, for every jABðR2Þ the spectral integral Z T a;b ¼ j dðPa #Qb Þ j;2 L2 L2 R2

/ a;b is well deﬁned and the integration mapping j Tj;2 deﬁnes an algebra 2 homomorphism from BðR Þ into LðL2ðM; tÞÞ: It is easy to see that Z aðlÞbðmÞ dðPa #Qb Þ ðxÞ¼aðaÞxbðbÞð10Þ L2 L2 R2 for all xAL2ðM; tÞ whenever a; bABðRÞ: Now we return to the situation that E is an arbitrary symmetric Banach function space on ð0; NÞ with order continuous norm.

2 a # b Deﬁnition 4.1. A function jABðR Þ will be called PE QE -integrable if

a;b D Tj;2ðE-L2ðM; tÞÞ E-L2ðM; tÞ

a;b and if the restriction of Tj;2 to E-L2ðM; tÞ is continuous with respect to jj Á jjE: If this is the case, then the restriction ðT a;bÞ has a unique extension to a j;2 jE-L2ðM;tÞ ARTICLE IN PRESS

44 B. de Pagter, F.A. Sukochev / Journal of Functional Analysis 212 (2004) 28–75 bounded linear operator on E; which will be denoted by Z a;b a # b Tj;E ¼ j dðPE QEÞ: R2

a # b The collection of all PE QE-integrable functions in this sense will be denoted by a# b JEðP Q Þ:

For later reference we include the following lemma, the proof of which follows / a;b immediately from the fact that the integration mapping j Tj;2 is an algebra 2 homomorphism from BðR Þ into LðL2ðM; tÞÞ:

a# b 2 Lemma 4.2. The collection JEðP Q Þ is a subalgebra of BðR Þ and that the / a;b a# b integration mapping j Tj;E is an algebra homomorphism from JE ðP Q Þ into LðEÞ:

Remark 4.3. Using the notation and terminology of [28], it is easy to see that a 2 a# b a# b function jABðR Þ belongs to JEðP Q Þ if and only if j is P Q -integrable with respect to SE; where E Â S ¼ðE-L2ðM; tÞÞ#ðE -L2ðM; tÞÞ; i.e., J ðPa#QbÞ¼JðPa #Qb ; SEÞ: Moreover, E E E Z a;b E a # b Tj;E ¼ S À j dðPE QEÞ R2

a# b for all jAJE ðP Q Þ: Indeed, this follows immediately from Proposition 2.17(iii) in E > E [28]. Moreover, it follows from the same proposition that S DS a # b ; that S ¼ PE QE a # b a# b a # b f0g and that JðPE QE ÞDJEðP Q Þ; where JðPE QEÞ denotes the algebra of a # b all PE QE-integrable functions as deﬁned in [28], Deﬁnition 2.5.

Remark 4.4. If F is any other separable Banach function space on ð0; NÞ with corres- a# b a# b ponding non-commutative space F; and if jAJEðP Q Þ and jAJF ðP Q Þ; then a;b a;b Tj;E ¼ Tj;F : jE-F jE-F

a;b a;b Indeed, it is clear from the deﬁnition that the operators Tj;E and Tj;F coincide on E-F-L2ðM; tÞ and since the norms in E and F are order continuous, this intersection is dense in both E and F: a;b a;b Consequently, there is no danger of confusion if the operators Tj;E and Tj;F are a;b simply denoted by Tj ; as we will frequently do in the sequel. ARTICLE IN PRESS

B. de Pagter, F.A. Sukochev / Journal of Functional Analysis 212 (2004) 28–75 45

2 Next we will introduce an algebra A0 of Borel functions on R for which we show a# b that it is contained in JEðP Q Þ: We will use the following notation. If ðO; BÞ is a measurable space then MðO; BÞ will denote the algebra of all complex valued B- measurable functions on O and BðO; BÞ will be the subalgebra of MðO; BÞ consisting of all bounded functions. The sup-norm in BðO; BÞ is denoted by jj Á jjN: If ðO1; B1Þ and ðO2; B2Þ are two measurable spaces, then the s-algebra generated by all the measurable rectangles B1 Â B2 in O1 Â O2 with B1AB1 and B2AB2 is denoted by 2 B1#B2: As before, BðRÞ and BðR Þ will denote the algebras of complex valued bounded Borel functions on R and R2; respectively.

2 Deﬁnition 4.5. We denote by A0 the collection of all functions jABðR Þ for which there exists a s-ﬁnite measure space ðS; S; nÞ and functions a; bAMðR Â S; BðRÞ#SÞ satisfying aðÁ; sÞ; bðÁ; sÞABðRÞ for all sAS and Z

Ca;b ¼ jjaðÁ; sÞjjNjjbðÁ; sÞjjN dnðsÞoN; ð11Þ S such that Z jðl; mÞ¼ aðl; sÞbðm; sÞ dnðsÞð12Þ S for all ðl; mÞAR2: For every function jAA we deﬁne jjjjj ¼ inf C ; where the 0 A0 a;b inﬁmum is taken over all possible representations (11) of j: Furthermore, we denote by C0 the subset of A0 consisting of all functions j with the property that the functions a and b in the representation (12) can be chosen such that aðÁ; sÞ; bðÁ; sÞACbðRÞ for all sAS:

Note that if a; bAMðR Â S; BðRÞ#SÞ are such that aðÁ; sÞ; bðÁ; sÞABðRÞ for all sAS; satisfying (11), then (12) deﬁnes a function jAA0: The proof of the following lemma is straightforward and therefore omitted.

Lemma 4.6. A is a subalgebra of BðR2Þ; jjjjj jjjjj for all jAA and 0 Np A0 0 ðA ; jj Á jj Þ is a Banach algebra. Moreover, C is a closed subalgebra of A and all 0 A0 0 0 2 jAC0 are continuous on R :

Next we will show that the algebra A0; introduced in Deﬁnition 4.5 is a# b always contained in JEðP Q Þ: This proposition will play an important role in Section 5.

Proposition 4.7. Let ðM; tÞ be a semiﬁnite von Neumann algebra and let E be a separable Banach function space on ð0; NÞ with corresponding non-commutative space a# b E: Then A0DJEðP Q Þ for all a; bZM with a ¼ a ; b ¼ b ; and the integration ARTICLE IN PRESS

46 B. de Pagter, F.A. Sukochev / Journal of Functional Analysis 212 (2004) 28–75

/ a;b mapping j Tj is an algebra homomorphism from A0 into LðEÞ satisfying

jjT a;bjj jjjjj : j p A0

Proof. Take jAA0 and let j be given by (12) for some s-ﬁnite measure space ðS; S; nÞ and functions a; bABðR Â S; BðRÞ#SÞ: Since jABðR2Þ; the bounded a;b - linear operator Tj : L2ðM; tÞ L2ðM; tÞ is deﬁned by the usual spectral integral Z T a;b ¼ j dðPa #Qb Þ: j L2 L2 R2 In particular, Z /T a;bx; yS ¼ j d/ðPa #Qb Þx; yS j L2 L2 R2 for all x; yAL2ðM; tÞ (here /Á; ÁS denotes the duality pairing given by (3)). Since Z Z jaðl; sÞbðm; sÞj dnðsÞ dj/ðPa #Qb Þx; ySj C jjxjj jjyjj N; L2 L2 p a;b 2 2o R2 S where the constant Ca;b is given by (11), it follows from Fubini’s theorem that for all x; yAL2ðM; tÞ we have Z Z / a;b S / a # b S Tj x; y ¼ aðl; sÞbðm; sÞ dnðsÞ d ðPL QL Þx; y 2 2 2 ZRZ S ¼ aðl; sÞbðm; sÞ d/ðPa #Qb Þx; yS dnðsÞ: ð13Þ L2 L2 S R2

For sAS we deﬁne the bounded linear operator Ts : L2ðM; tÞ-L2ðM; tÞ by Z T ¼ aðl; sÞbðm; sÞ dðPa #Qb Þ: s L2 L2 R2

As observed in (10), this operator Ts is given by Tsx ¼ aða; sÞxbðb; sÞ for all xAL2ðM; tÞ: Consequently, (13) can now be written as Z / a;b S / S Tj x; y ¼ Tsx; y dnðsÞ: ð14Þ S

Note that the operators Ts are actually bounded on any symmetric space F; satisfying

jjTsjjLðFÞpjjaðÁ; sÞjjNjjbðÁ; sÞjjN: ð15Þ

a;b Next we will show that the operator Tj maps ðL1-LNÞðM; tÞ into itself. To this end, take xAðL1-LNÞðM; tÞ: It follows from (15) applied to F ¼ðL1-LNÞðM; tÞ ARTICLE IN PRESS

B. de Pagter, F.A. Sukochev / Journal of Functional Analysis 212 (2004) 28–75 47 that

j/T x; ySj jjT xjj jjyjj s p s L1-LN L1þLN jjaðÁ; sÞjj jjbðÁ; sÞjj jjxjj jjyjj p N N L1-LN L1þLN for all sAS and for all yAðL1-LNÞðM; tÞ: Consequently, for all yAðL1-LNÞðM; tÞ with jjyjj 1 we have L1þLN p Z a;b a;b / S tðjTj ðxÞyjÞ ¼ tðTj ðxÞyu Þ¼ Tsx; yu dnðsÞ Z S jjaðÁ; sÞjj jjbðÁ; sÞjj dnðsÞ jjxjj p N N L1-LN S ¼ C jjxjj ; a;b L1-LN

a;b a;b a;b where Tj ðxÞy ¼ ujTj ðxÞyj is the polar decomposition of Tj ðxÞy and we have used that jjyujj jjyjj 1: Since the Ko¨ the dual space ðL þ L1þLN p L1þLN p 1 Â LNÞðM; tÞ ¼ðL1-LNÞðM; tÞ; it now follows from Proposition 5.3 in [17] that a;b a;b T ðxÞAðL LNÞð ; tÞ and jjT ðxÞjj C jjxjj : j 1- M j L1-LN p a;b L1-LN A Now take x; y ðL1-LNÞðM; tÞ with jjyjjEÂ p1: Applying (15) to the space E; we ﬁnd that

/ S j Tsx; y jp jjTsxjjEjjyjjEÂ

p jjaðÁ; sÞjjNjjbðÁ; sÞjjNjjxjjE for all sAS: Hence Z / a;b S / S j Tj x; y jp j Tsx; y j dnðsÞpCa;bjjxjjE: S

a;b Using Proposition 5.3 in [17] once again, this implies that jjTj xjjEÂÂ pCa;bjjxjjE: A The order continuity of the norm in E implies that jjzjjEÂÂ ¼jjzjjE for all z E; and a;b A so we may conclude that jjTj xjjEpCa;bjjxjjE for all x ðL1-LNÞðM; tÞ: Since a;b ðL LNÞð ; tÞ is dense in ; the operator ðT Þ has a unique 1- M E j jðL1-LNÞðM;tÞ extension to a bounded linear operator TE : E-E satisfying jjTEjjpCa;b: Using that ðL1-LNÞðM; tÞ is dense in the space E-L2ðM; tÞ as well, it follows that TE a;b A a# b a;b coincides with Tj on E-L2ðM; tÞ: Consequently, j JEðP Q Þ and TE ¼ Tj;E : a# b Since the integration mapping is an algebra homomorphism on JEðP Q Þ; the proof is complete. & ARTICLE IN PRESS

48 B. de Pagter, F.A. Sukochev / Journal of Functional Analysis 212 (2004) 28–75

5. Differentiation of operator functions in order continuous symmetric operator spaces

The main objective of this section is to exhibit a sufficiently rich class of operator f f functions f : Mh-Mh which are Gaˆ teaux-differentiable along the space L1ðM; tÞ; where M is an arbitrary semifinite von Neumann algebra on a Hilbert space H; equipped with a faithful normal semifinite trace t: However, all of the results in the present section remain valid for the more general class of symmetric operator spaces with order continuous norm. Therefore, we will treat this more general situation. Throughout this section E ¼ Eð0; NÞ will be a symmetric Banach function space on ð0; NÞ which has order continuous norm (equivalently, E is separable). As before we denote by E ¼ EðM; tÞ the corresponding non-commutative space associated with M: Our first objective is to obtain a version of the perturbation formula from [28, Theorem 7.4], where such a formula was established for ðLp; LqÞ-interpolation spaces with 1oppqoN: It will be convenient to introduce the following notation. - 2- 2 For a Borel function f : R R we denote by cf any Borel function cf : R R which is given by f ðlÞÀf ðmÞ c ðl; mÞ¼ ð16Þ f l À m 2 a 2 for all ðl; mÞAR with l m: Note that cf ABðR Þ if and only if f is Lipschitz continuous on R: Furthermore, it is clear that there exists at most one continuous 1 function cf which satisfies (16), which is the case if and only if f AC ðRÞ: If B is a 2 subset of BðR Þ; then the statement ‘‘cf AB’’ will be interpreted as ‘‘there exists a a function cf AB which is given by (16) for all l m’’. With these conventions we introduce the following classes of functions.

Deﬁnition 5.1. If A is a subalgebra of BðR2Þ then we denote by CÀ1ðAÞ the linear - space of all Borel functions f : R R such that cf AA:

À1 Note in particular that all functions in C ðC0Þ are continuously differentiable on R with a bounded derivative. Recall that the algebra A0 has been introduced in Deﬁnition 4.5.

À1 2 f Proposition 5.2. Suppose that f AC ðA0Þ and that cf ACbðR Þ: If a; bAMh such that a À bAE; then f ðaÞÀf ðbÞAE and Ta;bða À bÞ¼f ðaÞÀf ðbÞ: cf

Proof. It follows from Proposition 4.7 that for any two self-adjoint operators c d c d c;d c; dZM we have c AJ ðP #Q Þ J ðP #Q Þ and that T ALðEÞ LðL2Þ f E - L2 cf - c;d c;d with jjT jj - ; jjT jj - Kf ; where Kf 40 is a constant which only depends cf E E cf L2 L2 p f on f (here L2 ¼ L2ðM; tÞ). Now assume that a; bAMh such that a À bAE: Since E ARTICLE IN PRESS

B. de Pagter, F.A. Sukochev / Journal of Functional Analysis 212 (2004) 28–75 49 has order continuous norm, E-L2ðM; tÞ is dense in E and so there exists a sequence N - -N fxngn¼1 in E-L2ðM; tÞh such that jjða À bÞÀxnjjE 0asn : Deﬁne an ¼ b þ xn f for all n: Since an; bAM and an À b ¼ xnAL2ðM; tÞ; it follows from [28, Theorem 7.4], applied to the Banach function space E ¼ L2ð0; NÞ; that

an;b T ðan À bÞ¼f ðanÞÀf ðbÞ: ð17Þ cf

Note that we are in a position to apply [28, Theorem 7.4], thanks to the fact that 2;2 an # b 2 f AF (in the notation of [31]), i.e., cf belongs to the space JðP Q ; S Þ of integrable functions [28, Section 4]. The latter conclusion indeed follows from Proposition 4.7 and Remark 4.3. Since f is continuous, we know from Tikhonov’s theorem (see Proposition 3.6) that f ðanÞ-f ðaÞ as n-N with respect to the measure f a;b topology in M: Putting z ¼ f ðaÞÀf ðbÞÀT ða À bÞ; it follows that cf

an;b a;b T ðan À bÞÀT ða À bÞ-z as n-N ð18Þ cf cf with respect to the measure topology. We will now show that z ¼ 0: To this end we write no an;b a;b an;b an;b T ðan À bÞÀT ða À bÞ¼ T ðan À bÞÀT ða À bÞ cf cf cf cf no þ Tan;bða À bÞÀTa;bða À bÞ ð19Þ cf cf and observe that

an;b an;b an;b Tc ðan À bÞÀTc ða À bÞ ¼ Tc ðxn Àða À bÞÞ f f E f E

p Kf jjðxn Àða À bÞÞjjE

an;b an;b for all n: This implies that jjT ðan À bÞÀT ða À bÞjj -0 and hence cf cf E

an;b an;b T ðan À bÞÀT ða À bÞ-0asn-N cf cf with respect to the measure topology. Consequently, it follows from (18) and (19) that

T an;bða À bÞÀT a;bða À bÞ-z as n-N ð20Þ cf cf with respect to the measure topology. For all n; k ¼ 1; 2; y we deﬁne nono an;b an;b a;b a;b yn;k ¼ T ða À bÞÀT ðxkÞ þ T ðxkÞÀT ða À bÞ cf cf cf cf ARTICLE IN PRESS

50 B. de Pagter, F.A. Sukochev / Journal of Functional Analysis 212 (2004) 28–75 and write no an;b a;b an;b a;b T ða À bÞÀT ða À bÞ¼yn;k þ T ðxkÞÀT ðxkÞ : cf cf cf cf

2 - - Since xkAL2ðM; tÞ; cf ACbðR Þ and an a as n N with respect to the measure topology, it follows from Corollary 3.5 that

an;b a;b - -N Tc ðxkÞÀTc ðxkÞ 0asn f f 2

an;b a;b and hence T ðxkÞÀT ðxkÞ-0 in measure as n-N for all k ¼ 1; 2; y : Now cf cf (20) implies that yn;k-z in measure as n-N for all k: Furthermore,

an;b an;b Tc ða À bÞÀTc ðxkÞ pKf jjða À bÞÀxkjjE f f E and

a;b a;b Tc ðxkÞÀTc ða À bÞ pKf jjxk Àða À bÞjjE; f f E hence

jjyn;kjjEp2Kf jjða À bÞÀxkjjE ð21Þ for all n and k: Since the measure topology is metrizable, we can ﬁnd a sequence ? - -N n1on2o such that ynk;k z in measure as k : It follows from (21) that - - -N jjynk;kjjE 0 and hence ynk;k 0 in measure as k : Consequently z ¼ 0: Now we may conclude from (17) and (18) that Ta;bða À bÞ¼f ðaÞÀf ðbÞ and the proof is cf complete. &

The following corollary follows immediately from Proposition 5.2, Deﬁnition 4.5 and the second assertion of Lemma 4.6.

Corollary 5.3. Let E be a separable symmetric Banach function space on ð0; NÞ and À1 2 f E ¼ EðM; tÞ: Suppose that f AC ðA0Þ and that cf ACbðR Þ: If a; bAMh such that a À bAE; then f ðaÞÀf ðbÞAE and there exists a constant Kf 40 such that A À1 jj f ðaÞÀf ðbÞjjEpKf jja À bjjE: In particular, this holds when f C ðC0Þ:

The proof of the main result of the present section (Theorem 5.16) is based on Lemma 5.14. In its turn, the proof of this lemma is based on Corollary 5.12. To establish the latter result we need some technical preparations which we will divide into a number of lemmas. As before, we assume that ðM; tÞ is a semiﬁnite von Neumann algebra on the Hilbert space H: By Mh we denote the collection of self-adjoint elements in M: Furthermore, let ðS; S; nÞ be a s-ﬁnite measure space, which, for simplicity, is ARTICLE IN PRESS

B. de Pagter, F.A. Sukochev / Journal of Functional Analysis 212 (2004) 28–75 51

s assumed to be complete. We will denote by LNðS; S; n; MÞ the space of all bounded functions f : S-M with the property that the functions s/f ðsÞx and s/f ðsÞx are Bochner measurable from S into H for all xAH: This class of measurable functions is also considered in a slightly different setting in [37, Section IV.7]. Clearly, s LNðS; S; n; MÞ contains the constant functions and has a complex vector space s structure. Furthermore, if f ALNðS; S; n; MÞ and we deﬁne f ðsÞ¼f ðsÞ for all s s sAS; then it is obvious that f ALNðS; S; n; MÞ: Moreover, LNðS; S; n; MhÞ s denotes the real subspace of LNðS; S; n; MÞ consisting of all functions f for which f ðsÞAMh for all sAS:

s Remark 5.4. In the deﬁnition of LNðS; S; n; MÞ the condition that the functions s/f ðsÞx are Bochner measurable for all xAH cannot be omitted (see e.g. [37, Example IV.7.6]).

Using the Pettis measurability criterion (see e.g. [14]) the proof of the next lemma follows via a standard argument.

s Lemma 5.5. Given f1; f2ALNðS; S; n; MÞ we deﬁne the function f1 f2 : S-M by s s ð f1 f2ÞðsÞ¼f1ðsÞf2ðsÞ for all sAS: Then f1 f2ALNðS; S; n; MÞ and LNðS; S; n; MÞ is a unital algebra with respect to this multiplication.

If ðO; SÞ is a measurable space, then we will denote by BRðO; SÞ the space of all bounded real valued S-measurable functions on O: If O ¼ JDR and S ¼ BðJÞ; the Borel subsets of J; then this space will be denoted simply by BRðJÞ: Similarly for subsets of R2:

s Lemma 5.6. Suppose that f ALNðS; S; n; MhÞ and jABRðRÞ: If we deﬁne the function s jð f Þ : S-Mh by jð f ÞðsÞ¼jð f ðsÞÞ for all sAS; then jð f ÞALNðS; S; n; MhÞ:

Proof. The function f is bounded, so there exists a closed bounded interval JDR such that sð f ðsÞÞDJ for all sAS: Hence, for every jABRðRÞ we have jð f ðsÞÞ ¼

ðjjJ Þð f ðsÞÞ for sAS: Therefore we may replace in our considerations the real line by the interval J: If jABRðJÞ; then jjjð f ðsÞÞjjpjjjjjN for all sAS; so it is clear that jð f Þ s is bounded. It remains to show that jð f ÞALNðS; S; n; MhÞ: To this end we deﬁne

s U ¼fjABRðJÞ : jð f ÞALNðS; S; n; MhÞg:

It is routine to show that U is a unital subalgebra of BRðJÞ which contains all polynomials and is closed under pointwise convergence of sequences. As well known (see e.g. [21, Section 11]), this sufﬁces to conclude that U ¼ BRðJÞ; which completes the proof of the proposition. &

s Corollary 5.7. Suppose that f ALNðS; S; n; MÞ and deﬁne j f jðsÞ¼jf ðsÞj for all sAS: s Then j f jALNðS; S; n; MhÞ: ARTICLE IN PRESS

52 B. de Pagter, F.A. Sukochev / Journal of Functional Analysis 212 (2004) 28–75

A s Proof. It follows from Lemma 5.5 that f f LNðS; S; n; MhÞ: Take K4ffiffiffi0 such that 2 p jj f ðsÞjj pK for all sAS: Now define the function jABRðRÞ by jðlÞ¼ l if lA½0; K and jðlÞ¼0 otherwise. Then j f j¼jðð f f ÞÞ and now it follows from the above s lemma that j f jALNðS; S; n; MhÞ: &

For any self-adjoint operator a : DomðaÞ-H with spectral measure ea : BðRÞ-LðHÞ we will write

a a el ¼ e ðÀN; l¼wðÀN;lðaÞ for all lAR: The following corollary is immediate from the above lemma and corollary.

A s A / j f ðsÞj Corollary 5.8. If f LNðS; S; n; MÞ; then for every l R the function s el s belongs to LNðS; S; n; MhÞ:

Recall that a von Neumann algebra M is called s-finite if every family of non-zero pairwise orthogonal projections in M is at most countable. If ðM; tÞ is a semifinite von Neumann algebra, then it is not difficult to see that M is s-finite if and only if N N there exists a sequence fpngn¼1 of orthogonal projections in M such that tðpnÞo for all n and pnm1: In this case we will simply say that ðM; tÞ is a s-finite von Neumann algebra.

Lemma 5.9. Suppose that ðM; tÞ is a s-ﬁnite von Neumann algebra and that s f ALNðS; S; n; MhÞ is such that f ðsÞX0 for all sAS: Then the function s/tð f ðsÞÞ from S into ½0; N is S-measurable.

Proof. First assume that p is an orthogonal projection in M with tðpÞoN: Then the mapping a/tðpapÞ from Mþ into ½0; NÞ is the restriction to Mþ of a positive normal functional on M; which we will denote by f : Hence there exists a sequence P p P N N 2 N N / S A fxkgk¼1 in H such that k¼1jjxkjj o and fpðxÞ¼ k¼1 xxk; xk for all x M (see e.g. [15, Theorem I.4.1]). Thus XN tðpf ðsÞpÞ¼ / f ðsÞxk; xkS k¼1 for all sAS: Since for all k the functions s// f ðsÞxk; xkS are by assumption S-measurable, this shows that the function s/tðpf ðsÞpÞ is S-measurable. N N Now let fpngn¼1 be a sequence of orthogonal projections in M such that tðpnÞo for all n and pnm1: Then 1 1 2 2 m tðpn f ðsÞpnÞ¼tð f ðsÞpnÞ¼t f ðsÞ pn f ðsÞ ntð f ðsÞÞ ARTICLE IN PRESS

B. de Pagter, F.A. Sukochev / Journal of Functional Analysis 212 (2004) 28–75 53 for all sAS: From the ﬁrst part of the proof we know that the functions s/tðpn f ðsÞpnÞ are S-measurable for all n ¼ 1; 2; y; from which it follows that the function s/tð f ðsÞÞ is S-measurable. &

Remark 5.10. The condition that ðM; tÞ is s-finite cannot be omitted in the above lemma as can be seen in the following example. We consider the unit interval ½0; 1 equipped with the counting measure, so the corresponding L2-space is H ¼ l2ð½0; 1Þ: þ Let M ¼ lNð½0; 1Þ; acting on l2ð½0; 1Þ via multiplication. The trace t : M -½0; N is defined by X tðxÞ¼ xð jÞ jA½0;1 for all 0pxAlNð½0; 1Þ: Let S ¼½0; 1 and S ¼ Bð½0; 1Þ equipped with Lebesgue measure n: Fix any set FD½0; 1 such that FeBð½0; 1Þ and define f : ½0; 1-lNð½0; 1Þ by f ðsÞ¼wF wfsg for all sA½0; 1: Clearly, f is a bounded function and since every xAl2ð½0; 1Þ is countably supported it is easy to see that the mapping s/f ðsÞx is Bochner measurable for all xAl2ð½0; 1Þ: However, a simple computation shows that tð f ðsÞÞ ¼ / wF ðsÞ for all sAS: Hence the function s tð f ðsÞÞ is not S-measurable in this case.

Lemma 5.11. Suppose that ðM; tÞ is a s-ﬁnite von Neumann algebra and that s / f ALNðS; S; n; MÞ: Then the function ðt; sÞ mtð f ðsÞÞ from ½0; NÞÂS into ½0; N is B½0; NÞ#S-measurable.

Proof. For lA½0; NÞ and sAS we deﬁne j f ðsÞj dðl; sÞ¼t 1 À el :

It follows from Corollary 5.8 and Lemma 5.9 that for each lA½0; NÞ the function s/dðl; sÞ is S-measurable. Recall that by deﬁnition

mtð f ðsÞÞ ¼ infflX0:dðl; sÞptg for all tX0: For a40 we put Aa ¼fðt; sÞA½0; NÞÂS : mtð f ðsÞÞoag: It is easy to verify that [ Aa ¼ fðt; sÞA½0; NÞÂS : dðl; sÞptg: 0ploa lAQ

For each lX0 the set fðt; sÞA½0; NÞÂS : dðl; sÞptg is the epigraph of the measurable function dðl; ÁÞ; so this set belongs to B½0; NÞ#S: Consequently, AaAB½0; NÞ#S for all a40; and this sufﬁces to prove the lemma. &

For the sake of reference we will now formulate the above result in the precise form in which it will be used. The orthogonal projection q in the semiﬁnite von ARTICLE IN PRESS

54 B. de Pagter, F.A. Sukochev / Journal of Functional Analysis 212 (2004) 28–75

N Neumann algebra ðM; tÞ will be called s-ﬁnite if there exists a sequence fpngn¼1 of orthogonal projections in M such that pnmq and tðpnÞoN for all n:

Corollary 5.12. Suppose that ðM; tÞ is a semiﬁnite von Neumann algebra and that s / f ALNðS; S; n; MÞ: Then the function ðt; sÞ mtðqf ðsÞqÞ from ½0; NÞÂS into ½0; N is B½0; NÞ#S-measurable for every s-ﬁnite orthogonal projection qAM:

Proof. As usual we consider qMq as a von Neumann algebra on the Hilbert space qðHÞ; equipped with the trace tq given by the restriction of t to qMq: Then ðqMq; tqÞ is a semifinite von Neumann algebra. If xAqMq then we can consider its ðqÞ generalized singular value function m ðxÞ with respect to ðqMq; tqÞ; and we can consider the generalized singular value function mðxÞ of x with respect to ðM; tÞ: ðqÞ A moment’s reflection shows that mt ðxÞ¼mtðxÞ for all tX0 and all xAqMq: Since q is a s-finite projection, it is clear that the von Neumann algebra ðqMq; tqÞ ðqÞ is s-finite. Therefore, by the above lemma, the function ðt; sÞ/mt ðqf ðsÞqÞ is B½0; NÞ#S-measurable from ½0; NÞÂS into ½0; N: Now the result of the corollary follows immediately from the above observation. &

Lemma 5.13. Let ðM; tÞ be a semifinite von Neumann algebra on the Hilbert space H and suppose that a is a self-adjoint operator on H which is affiliated with M: Let aABRðBðRÞ#SÞ and define f : S-Mh by f ðsÞ¼aða; sÞ for all sAS: Then s f ALNðS; S; n; MhÞ:

Proof. As before, we denote the spectral measure of a by ea : BðRÞ-LðHÞ: Since a aR is afﬁliated with M; we have e ðBÞAM for all BABðRÞ: Hence aða; sÞ¼ a A A - R aðl; sÞ de ðlÞ Mh and jjaða; sÞjjpjjajjN for all s S; so f : S Mh is a well-deﬁned bounded function. Fix xAH: For all ZAH and sAS we have Z / S / S a f ðsÞx; Z ¼ aða; sÞx; Z ¼ aðl; sÞ dex;ZðlÞ; R

a where ex;Z denotes the ﬁnite Borel measure on R given by

a / a S ex;ZðBÞ¼ e ðBÞx; Z for all BABðRÞ: This shows that the function s/f ðsÞx is weakly S-measurable on S: Next we will show that this function has separable range. To this end we deﬁne

W ¼ spanfea½l; mÞx : l; mAQg; which is a closed separable subspace of H: Let

F ¼fBABðRÞ : eaðBÞxAWg: ARTICLE IN PRESS

B. de Pagter, F.A. Sukochev / Journal of Functional Analysis 212 (2004) 28–75 55

It is easy to see that F contains the algebra R generated by all cells ½l; mÞ with l; mAQ: Moreover, if BnAF (n ¼ 1; 2; y) and BABðRÞ such that BnmB; then a a a e ðBnÞx-e ðBÞx as n-N; so e ðBÞxAW; i.e., BAF: Hence F is a monotone class and so F contains the s-algebra generated by R; which shows that F ¼ BðRÞ: Therefore, eaðBÞxAW for all BABðRÞ: From the deﬁnition of the spectral integral it now follows that f ðsÞx ¼ aða; sÞxAW for all sAS: Hence the function s/f ðsÞx is separable valued. Now the Pettis measurability criterion implies that this function is Bochner measurable. Since this holds for all xAH; we may conclude that f belongs to s LNðS; S; n; MhÞ: &

The following lemma is crucial for the proof of Theorem 5.16. Although the result of this lemma is similar to Lemma 6.8, the proof follows a completely different argument, using Corollary 5.12 and inequalities for generalized singular value functions.

Lemma 5.14. Let E be a separable symmetric Banach function space on ð0; NÞ and f f E ¼ EðM; tÞ: Suppose that a; bAMh and an; bnAMh (n ¼ 1; 2; y) such that an-a and - - A an;bn bn basn N with respect to the measure topology. If j C0; then jjTj ðxÞÀ a;b - -N A Tj ðxÞjjE 0 as n for all x E:

Proof. For the sake of simplicity we will present the proof for the case that bn ¼ b for all n: The adjustments for the general case are straightforward. Let jAC0 be given by Z jðl; mÞ¼ aðl; sÞbðm; sÞ dnðsÞ S

2 for all ðl; mÞAR ; where a; bABðBðRÞ#SÞ are such that aðÁ; sÞ; bðÁ; sÞACbðRÞ for all sAS and Z

Ca;b ¼ jjaðÁ; sÞjjNjjbðÁ; sÞjjN dnðsÞoN: S

a# b Note that it follows from Proposition 4.7 and Lemma 4.6 that jAJEðP Q Þ an # b and jAJE ðP Q Þ for all n; and that the corresponding operators on E satisfy a;b an;b jjTj jj; jjTj jjpCa;b: As in the proof of Proposition 4.7, we deﬁne for each an;b a;b an;b a;b sAS the operators Ts ; Ts ALðEÞ by Ts ðxÞ¼aðan; sÞxbðb; sÞ and Ts ðxÞ¼ a a; s xb b; s for all xA : It follows from (14) that ð Þ ð Þ E Z / an;b a;b S / an;b a;b S Tj ðxÞÀTj ðxÞ; y ¼ Ts ðxÞÀTs ðxÞ; y dnðsÞ ZS

¼ tð½aðan; sÞÀaða; sÞxbðb; sÞyÞ dnðsÞð22Þ S for all x; yAL1-LNðM; tÞ: ARTICLE IN PRESS

56 B. de Pagter, F.A. Sukochev / Journal of Functional Analysis 212 (2004) 28–75

Fix xAMfin and let p0AM be an orthogonal projection such that tðp0ÞoN and x ¼ p0xp0: It follows from Lemma 2.3 that for each n ¼ 1; 2; y there exists ynAMfin with jjynjjEÂ p1; such that

1 j/T an;bðxÞÀTa;bðxÞ; y SjXjjT an;bðxÞÀTa;bðxÞjj À : ð23Þ j j n j j E n

Our objective is to show that

an;b a;b lim /T ðxÞÀT ðxÞ; ynS ¼ 0: ð24Þ n-N j j

For each n ¼ 1; 2; y let pnAM be an orthogonal projection suchW that tðpnÞoN and A N yn ¼ pnynpn: Deﬁne the orthogonal projection q M by q ¼ n¼0 pn: Observe that x ¼ qxq and yn ¼ qynq for all n ¼ 1; 2; y and that q is s-ﬁnite. Next we observe that

jtð½aðan; sÞÀaða; sÞxbðb; sÞynÞj

¼jtð½aðan; sÞÀaða; sÞqxqbðb; sÞqynqÞj

¼jtðq½aðan; sÞÀaða; sÞqxqbðb; sÞqynqÞj

Z N p mtðq½aðan; sÞÀaða; sÞqxqÞmtðbðb; sÞqynqÞ dt 0 Z N p mtðq½aðan; sÞÀaða; sÞqxqÞjjbðÁ; sÞjjNmtðynÞ dt; 0 where we have used Proposition 3.10 in [17] (see also [9]). Deﬁning fnðsÞ¼ ½aðan; sÞÀaða; sÞx for all sAS; it follows from Corollary 5.12 and Lemma 5.13 that the function

/ ðt; sÞ mtðq½aðan; sÞÀaða; sÞqxqÞ is measurable on ½0; NÞÂS: Now it follows from (22) and the Fubini theorem that

/ an;b a;b S j Tj ðxÞÀTj ðxÞ; yn j Z

p jtð½aðan; sÞÀaða; sÞxbðb; sÞynÞj dnðsÞ S Z Z N p mtðq½aðan; sÞÀaða; sÞqxqÞjjbðÁ; sÞjjNmtðynÞ dt dnðsÞ S 0 ARTICLE IN PRESS

B. de Pagter, F.A. Sukochev / Journal of Functional Analysis 212 (2004) 28–75 57 Z N Z ¼ mtðq½aðan; sÞÀaða; sÞqxqÞjjbðÁ; sÞjjN dnðsÞ mtðynÞ dt 0 S Z N

¼ hnðtÞmtðynÞ dtpjjhnjjE jjmtðynÞjjEÂ 0

¼jjhnjjE jjynjjEÂ pjjhnjjE; where we have denoted Z

hnðtÞ¼ mtðq½aðan; sÞÀaða; sÞqxqÞjjbðÁ; sÞjjN dnðsÞ S - - for all tX0: Hence, to prove (24) it is sufﬁcient to show that jjhnjjE 0asn N: To this end we ﬁrst note that

mtðq½aðan; sÞÀaða; sÞqxqÞjjbðÁ; sÞjjN

pjjaðan; sÞÀaða; sÞjjjjbðÁ; sÞjjNmtðxÞ

p2jjaðÁ; sÞjjNjjbðÁ; sÞjjNmtðxÞð25Þ for all ðt; sÞA½0; NÞÂS and so Z

hnðtÞp 2mtðxÞ jjaðÁ; sÞjjNjjbðÁ; sÞjjN dnðsÞ S

p 2Ca;bmtðxÞð26Þ for all tX0: Since aðÁ; sÞACbðRÞ and an-a as n-N with respect to the measure f topology in M; it follows from Proposition 3.6 that aðan; sÞ-aða; sÞ as n-N in measure for all sAS: Hence q½aðan; sÞÀaða; sÞxq-0 in measure and so - mtðq½aðan; sÞÀaða; sÞxqÞ 0 as n-N for all t40 and all sAS: Since the right-hand side of (25) is n-integrable over S; we can apply the dominated convergence theorem to conclude that Z - mtðq½aðan; sÞÀaða; sÞxqÞjjbðÁ; sÞjjN dnðsÞ 0; S i.e., hnðtÞ-0asn-N for all t40: Since the Banach function space E has order - - continuous norm and mðxÞAE; it now follows from (26) that jjhnjjE 0asn N: This sufﬁces to prove (24) and via (23) this shows that lim jjTan;bðxÞÀT a;bðxÞjj ¼ 0 ð27Þ n-N j j E for all xAMfin: It follows from Lemma 2.2 that Mfin is dense in E: Since an;b A jjTj jjpCa;b for n ¼ 1; 2; y we may now conclude that (27) holds for all x E; which completes the proof of the lemma. & ARTICLE IN PRESS

58 B. de Pagter, F.A. Sukochev / Journal of Functional Analysis 212 (2004) 28–75

The following theorem is the main result of the present section. Suppose that f f : R-R is a continuous function and aAM is self-adjoint. It will be convenient to adopt the following terminology.

Deﬁnition 5.15. We say that f is Gaˆ teaux operator-differentiable along Eh at the - point a if there exists a bounded linear operator DEf ðaÞ : Eh Eh such that for every bAEh;

(i) f ða þ tbÞÀf ðaÞAEh for all tAR; (ii) the following limit exists in the norm topology on E f ða þ tbÞÀf ðaÞ lim ¼ DEf ðaÞb: t-0 t

In this case, the operator DEf ðaÞ is called the Gaˆ teaux derivative of f at a along Eh:

Theorem 5.16. Let ðM; tÞ be a semiﬁnite von Neumann algebra, let E be a separable À1 symmetric Banach function space on ð0; NÞ and E ¼ EðM; tÞ: If f AC ðC0Þ; then f is f a;a Gaˆteaux operator differentiable along Eh at every point aAMh and D f ðaÞ¼T : E cf

Proof. Take bAEh: It follows from Proposition 5.2 that

f ða þ tbÞÀf ðaÞ¼Taþtb;aðtbÞ cf for all tAR and so f ða þ tbÞÀf ðaÞ ¼ T aþtb;aðbÞ t cf for all ta0: By Lemma 5.14 we know that jjT aþtb;aðxÞÀTa;aðxÞjj -0ast-0 for all cf cf E xAE: Applying this to x ¼ b we ﬁnd that f ða þ tbÞÀf ðaÞ lim ¼ lim Taþtb;aðbÞ¼Ta;aðbÞ; t-0 t t-0 cf cf and we are done. &

6. Differentiability of operator functions in ðLp; LqÞ-interpolation spaces

It will be convenient to discuss ﬁrst some general results concerning double operator integrals in Banach spaces. For any function f : R-C we denote by V1 f the total variation of f over R: If V1 f oN then, by deﬁnition, f is of bounded variation. By V1ðRÞ we denote the space of all right-continuous functions f : R-C ARTICLE IN PRESS

B. de Pagter, F.A. Sukochev / Journal of Functional Analysis 212 (2004) 28–75 59 which are of bounded variation. Now we deﬁne m A 2 A A LNðV1 Þ¼ j BðR Þ : jðl; ÁÞ V1ðRÞ8l R; sup V1jðl; ÁÞoN lAR

A m and for j LNðV1 Þ let N jjjjjðmÞ ¼jjjjjN þ sup V1jðl; ÁÞo : lAR

m 2 m It is easy to see that LNðV1 Þ is a subalgebra of BðR Þ and that ðLNðV1 Þ; jj Á jjðmÞÞ is l actually a Banach algebra. The space LNðV1 Þ and the norm jj Á jjðlÞ are deﬁned similarly by interchanging the roles of l and m in the above.

2 Deﬁnition 6.1. We denote by A1 the collection of all functions jABðR Þ for which there exists a s-ﬁnite measure space ðS; S; nÞ and functions a; bAMðR2 Â 2 # A m A l A S; BðR Þ SÞ with aðÁ; Á; sÞ LNðV1 Þ and bðÁ; Á; sÞ LNðV1 Þ for all s S satisfying Z

Ka;b ¼ jjaðÁ; Á; sÞjjðmÞjjbðÁ; Á; sÞjjðlÞ dnðsÞoN ð28Þ S such that Z jðl; mÞ¼ aðl; m; sÞbðl; m; sÞ dnðsÞð29Þ S for all ðl; mÞAR2: For jAA we deﬁne jjjjj ¼ inf K ; where the inﬁmum is taken 1 A1 a;b 2 over all possible representations (29) of j: Furthermore, let C1 ¼ A1-CbðR Þ:

Lemma 6.2. With the notation introduced above we have:

(i) A is a subalgebra of BðR2Þ; jjjjj jjjjj for all jAA and ðA ; jj Á jj Þ is a 1 Np A1 1 1 A1 Banach algebra; (ii) A ; as introduced in Deﬁnition 4.5, is a subalgebra of A and jjjjj jjjjj for 0 1 A1 p A0 all jAA0:

Proof. The proof of (i) is analogous to the proof of Lemma 4.6 and therefore omitted. For the proof of (ii), suppose that jAA0 is given as in (12). Now e deﬁne eaðl; m; sÞ¼aðl; sÞ and bðl; m; sÞ¼bðm; sÞ for all ðl; m; sÞAR2 Â S: e A m e e A l Then aðÁ; Á; sÞ LNðV1 Þ with jjaðÁ; Á; sÞjjðmÞ ¼jjaðÁ; sÞjjN and bðÁ; Á; sÞ LNðV1 Þ with e A jjbðÁ; Á; sÞjjðlÞ ¼jjbðÁ; sÞjjN for all s S: Hence, Z Z e e jjaðÁ; Á; sÞjjðmÞjjbðÁ; Á; sÞjjðlÞ dnðsÞ¼ jjaðÁ; sÞjjNjjbðÁ; sÞjjN dnðsÞoN; S S ARTICLE IN PRESS

60 B. de Pagter, F.A. Sukochev / Journal of Functional Analysis 212 (2004) 28–75 which shows that jAA1: From this argument it is also clear that jjjjj jjjjj : & A1 p A0

Observe that it is an immediate consequence of the definition and of (i) in the above lemma that C1 is a closed subalgebra of A1: Let X be a Banach space and assume that P; Q : BðRÞ-LðXÞ are two bounded commuting spectral measures, with bounds CP and CQ respectively. By P#Q we denote the (finitely additive) product measure of P and Q defined on the algebra A generated by all Borel rectangles in R2: In Sections 2 and 3 of [28] the space JðP#QÞ of all P#Q-integrable functions has been introduced and studied, under the > additional technical assumption that SP#Q ¼f0g: For further details we refer to this paper. Furthermore, for the definition of a UMD-space and the properties of such Banach spaces we refer the reader to e.g. [33]. Now we recall the following result.

Proposition 6.3 (de Pagter et al. [28], Proposition 3.13). Assume that X is a UMD- > m D # A space and that SP#Q ¼f0g: Then LNðV1 Þ JðP QÞ and for all j A1 we have jjTjjjpCjjjjjL Vm ; where C40 is a constant only depending on CP; CQ and the Nð 1 Þ UMD-constant of the Banach space X:

l Clearly, there is also a corresponding result for the algebra LNðV1 Þ: Actually, in the situation of the above proposition, there is a much larger subalgebra of BðR2Þ which is contained in JðP#QÞ:

> Proposition 6.4. Assume that X is a UMD-space and that SP#Q ¼f0g: Then A DJðP#QÞ and for all jAA we have jjT jj Cjjjjj ; where C40 is a constant 1 1 j p A1 only depending on CP; CQ and the UMD-constant of the Banach space X:

Proof. Suppose that jAA1 is given by (29) as Deﬁnition 6.1. Now deﬁne the linear functional F : SP#Q-C by Z FðgÞ¼ jðl; mÞ d/P#Q; gS R2 for all gASP#Q: Observe that Z Z jaðl; m; sÞbðl; m; sÞj dnðsÞ dj/P#Q; gSj 2 R ZS 2 p jjaðÁ; Á; sÞjjNjjbðÁ; Á; sÞjjN dnðsÞ j/P#Q; gSjðR Þ ZS / # S 2 p jjaðÁ; Á; sÞjjðmÞjjbðÁ; Á; sÞjjðlÞ dnðsÞ j P Q; g jðR ÞoN: S ARTICLE IN PRESS

B. de Pagter, F.A. Sukochev / Journal of Functional Analysis 212 (2004) 28–75 61

Hence it follows via the Fubini–Tonelli theorem that Z Z FðgÞ¼ aðl; m; sÞbðl; m; sÞ dnðsÞ d/P#Q; gS 2 ZRZS ¼ aðl; m; sÞbðl; m; sÞ d/P#Q; gS dnðsÞ: ð30Þ S R2

It follows from Proposition 6.3 that aðÁ; Á; sÞ and bðÁ; Á; sÞ belong to JðP#QÞ and so aðÁ; Á; sÞbðÁ; Á; sÞAJðP#QÞ for all sAS: So for all sAS we can deﬁne TsALðXÞ by Z

Ts ¼ aðl; m; sÞbðl; m; sÞ dðP#QÞ: R2

Note that, by the multiplicativity of the integration mapping (see Proposition 2.8 in [28]), we have Ts ¼ TaðÁ;Á;sÞTbðÁ;Á;sÞ; so

2 jjTsjjpC jjaðÁ; Á; sÞjjðmÞjjbðÁ; Á; sÞjjðlÞ ð31Þ for all sAS: Now (30) can be written as Z

FðgÞ¼ /Ts; gS dnðsÞ S

0 for all gAS # : Denoting the nuclear norm on ðXÞ ¼ X#X by jj Á jj ; it P Q L s n1 follows from (31) that

j/T ; gSj C2jjaðÁ; Á; sÞjj jjbðÁ; Á; sÞjj jjgjj s p ðmÞ ðlÞ n1 for all sAS and so Z jFðgÞj C2 jjaðÁ; Á; sÞjj jjbðÁ; Á; sÞjj dnðsÞ jjgjj p ðmÞ ðlÞ n1 S ¼ C2K jjgjj ; a;b n1 where the constant Ka;b is defined by (28). This shows that the linear functional F is continuous with respect to the nuclear norm on SP#Q: As a UMD-space, X is 0 reflexive, so SP#Q is dense in LðXÞs with respect to this nuclear norm. Hence, F extends uniquely to a jj Á jj -bounded linear functional F : ðXÞ0 -C with n1 L s 2 jjFjjpC Ka;b: Using once again that X is reflexive, it follows that there exists A / S A 0 T LðXÞ such that T; g ¼ FðgÞ for all g LðXÞs and jjTjj ¼ jjFjj (for more details see [28, Remark 2.1]). From the definition ofRF and Definition 2.5 in [28] it is A # # now clear that j JðP QÞ and that T ¼ Tj ¼ R2 j dðP QÞ: The estimate of jjT jj follows immediately from the definition of jjjjj ; so the proof is complete. & j A1 ARTICLE IN PRESS

62 B. de Pagter, F.A. Sukochev / Journal of Functional Analysis 212 (2004) 28–75

Now we assume that ðM; tÞ is a semiﬁnite von Neumann algebra on the Hilbert space H and that E be a separable symmetric Banach function space on ð0; NÞ: Concerning E we shall make in the present section the following additional assumption:

* E is an ðLp; LqÞ-interpolation space for some 1oppqoN:

This implies that the corresponding non-commutative space E ¼ EðM; tÞ is an interpolation space for the couple ðLpðM; tÞ; LqðM; tÞÞ and the space ðL1-LNÞðM; tÞ is norm dense in E: We denote by Eh the self-adjoint part of E; i.e., Eh ¼fxAE : x ¼ xg: f a b Given a; bAMh we define the commuting spectral measures PE and QE in E by (9). > We note that under these assumptions the condition S a # b ¼ fg0 is satisfied (see PE QE Section 4 of [28]). Now we will show that the present assumptions on the space E a# b imply that A1DJEðP Q Þ; where A1 is the algebra defined in Definition 6.1 and a# b JEðP Q Þ is introduced in Definition 4.1.

Proposition 6.5. We assume that the Banach function space E on ð0; NÞ is an ðLp; LqÞ- interpolation space for some 1op; qoN and that E has order continuous norm. Then D a# b / a;b A1 JEðP Q Þ and the integration mapping j Tj is an algebra homomorphism from A1 into LðEÞ satisfying

jjT a;bjj C jjjjj ð32Þ j LðEÞp E A1

f for all self-adjoint a; bAMh; where the constant CE40 only depends on the space E:

Proof. Since LpðM; tÞ is a UMD-space for 1opoN (see [3]), it follows from D Pa #Qb Proposition 6.4 that A1 Jð Lp Lp Þ and hence by Remark 4.3 we have D a# b D a# b A1 JLp ðP Q Þ: Similarly A1 JLq ðP Q Þ and therefore

D a# b a# b A1 JLp ðP Q Þ-JLq ðP Q Þ: ð33Þ

Take jAA1 and denote the corresponding operators on LpðM; tÞ and LqðM; tÞ by a;b a;b Tj;p and Tj;q respectively. By Remark 4.4 we know that

a;b a;b A Tj;pðxÞ¼Tj;qðxÞ8x ðLp-LqÞðM; tÞ;

a;b a;b hence the pair ðTj;p; Tj;qÞ is admissible for ðLpðM; tÞ; LqðM; tÞÞ: Since E is an ðLpðM; tÞ; LqðM; tÞÞ-interpolation space, there exists an operator SALðEÞ such that

a;b a;b SðxÞ¼Tj;pðxÞ¼Tj;qðxÞ ARTICLE IN PRESS

B. de Pagter, F.A. Sukochev / Journal of Functional Analysis 212 (2004) 28–75 63 for all xAðLp-LqÞðM; tÞ: Using Remark 4.4 once again, it follows that SðxÞ¼ a;b A Tj;2ðxÞ for all x ðL1-LNÞðM; tÞ: Since ðL1-LNÞðM; tÞ is dense in E-L2ðM; tÞ a;b A we may conclude from this that SðxÞ¼Tj;2ðxÞ for all x E-L2ðM; tÞ: Conse- A a# b a;b quently, j JEðP Q Þ and S ¼ Tj : Moreover, there exists a constant K40 such that

a;b a;b a;b Tj pK max Tj;p ; jjTj;qjjLqðM;tÞ LðEÞ LpðM;tÞ and from Proposition 6.4 it follows that

T a;b C jjjjj and Ta;b C jjjjj ; j;p p p A1 j;q p q A1 LpðM;tÞ LqðM;tÞ where Cp and Cq are constants depending only on p and q respectively. A combination of these inequalities yields (32) and the proof is complete. &

Corollary 6.6. We assume that the separable Banach function space E on ð0; NÞ is an f ðLp; LqÞ-interpolation space for some 1op; qoN: Suppose that a; bAM are two À1 self-adjoint operators such that a À bAE: If f AC ðA1Þ; then

T a;bða À bÞ¼f ðaÞÀf ðbÞ: ð34Þ cf

In particular, f ðaÞÀf ðbÞAE and

jj f ðaÞÀf ðbÞjjEpCEKf jja À bjjE; where the constant CE 40 depends only on the space E and the constant Kf depends only on the function f :

Proof. In Theorem 7.4 of [28] it is shown that (34) holds for functions f for which there exists a function cf satisfying (16) such that

A a# b a# b cf JLp ðP Q Þ-JLq ðP Q Þ: ð35Þ

However, by (33) in the proof of Proposition 6.5, cf AA1 implies (35), which yields the desired result. &

À1 f Recall from Corollary 6.6 that for every f AC ðA1Þ and all self-adjoint a; bAM; the double operator integral Z Ta;b ¼ c dðPa #Qb Þ cf f E E R2 is a well-deﬁned bounded linear operator in E satisfying (34). ARTICLE IN PRESS

64 B. de Pagter, F.A. Sukochev / Journal of Functional Analysis 212 (2004) 28–75

In the proof of the next theorem we will use the following observation.

Lemma 6.7. Let ðM; tÞ be a semiﬁnite von Neumann algebra. N N (i) Suppose that 1pp0ppop1p and that fxngn¼1 is a sequence in ðL L Þð ; tÞ such that jjx jj -0 as n-N and sup jjx jj N: Then p0 - p1 M n p0 n n p1 o - - jjxnjjp 0 as n N: N (ii) Suppose that 1pp0oppp1pN and that fxngn¼1 is a sequence in ðL L Þð ; tÞ such that jjx jj -0 as n-N and sup jjx jj N: Then p0 - p1 M n p1 n n p0 o - - jjxnjjp 0 as n N:

Proof. We only indicate the proof of (i). Write

1 1 À y y ¼ þ p p0 p1 with 0pyo1: As is well known (see e.g. Exercise 7, Chapter 3 in the book [2]), we then have

jj f jj jj f jj1Àyjj f jjy pp p0 p1

A N N for all functions f Lp0 ð0; Þ-Lp1 ð0; Þ: Applying this inequality to the functions mðxnÞ this yields

jjx jj jjx jj1Àyjjx jjy ; n pp n p0 n p1 from which (i) follows immediately. &

Lemma 6.8. Let E be a separable ðLp; LqÞ-interpolation space on ð0; NÞ for some f f 1oppqoN and suppose that cAC1: If a; bAMh and an; bnAMh ðn ¼ 1; 2; yÞ such that an-a and bn-basn-N with respect to the measure topology, then an;bn a;b - -N A jjTc ðxÞÀTc ðxÞjjE 0 as n for all x E ¼ EðM; tÞ:

Proof. First we take xAðL1-LNÞðM; tÞ and deﬁne

an;bn a;b yn ¼ Tc ðxÞÀTc ðxÞ

y A D an # bn N for all n ¼ 1; 2; : Since c A1 JLr ðP Q Þ for all 1oro and similarly A a# b N c JLr ðP Q Þ for all 1oro ; it follows that

an;bn a;b yn ¼ Tc;r ðxÞÀTc;r ðxÞ ARTICLE IN PRESS

B. de Pagter, F.A. Sukochev / Journal of Functional Analysis 212 (2004) 28–75 65 for all 1oroN and all n ¼ 1; 2; y . This implies that ynALrðM; tÞ and, by Proposition 6.5, that

sup jjynjjr ¼ MroN n for all 1oroN: Furthermore, since c is a bounded continuous function on R2 and since an-a; bn-b as n-N with respect to the measure topology, it follows from Corollary 3.5 that Z Z

a b a b lim c dðP n #Q n ÞðxÞÀ c dðP #Q ÞðxÞ ¼ 0; n-N R2 R2 2

i.e., limn-N jjynjj2 ¼ 0: Consequently, Lemma 6.7 implies in particular that - - - jjynjjp 0 and jjynjjq 0asn N: Since E is an ðLpðM; tÞ; LqðM; tÞÞ-interpolation - -N space, we may conclude that jjynjjE 0asn : By this we have shown that

an;bn a;b lim T ðxÞ¼T ðxÞ8xAðL1-LNÞðM; tÞ: n-N c c

an;bn N By (32), the operators fTc gn¼1 are uniformly bounded on E: Since the space ðL1-LNÞðM; tÞ is dense in E we may therefore conclude that

lim Tan;bn ðxÞ¼T a;bðxÞ8xAE n-N c c and the proof is complete. &

The following theorem is the main result in the present section.

Theorem 6.9. Let E be a separable ðLp; LqÞ-interpolation space on ð0; NÞ for some À1 1oppqoN and suppose that f AC ðC1Þ: Then f is Gaˆteaux operator-differentiable f a;a along Eh at every point a ¼ a AM and D f ðaÞ¼T : E cf

Proof. For every bAEh and all tAR it follows from Corollary 6.6 that

f ða þ tbÞÀf ðaÞ¼T aþtb;aðtbÞ; cf and so for all ta0 we have

f ða þ tbÞÀf ðaÞ ¼ T aþtb;aðbÞ: t cf ARTICLE IN PRESS

66 B. de Pagter, F.A. Sukochev / Journal of Functional Analysis 212 (2004) 28–75

Since cf is continuous and cf AA1; it follows from Lemma 6.8 that

aþtb;a a;a - Tc ðxÞÀTc ðxÞ 0 f f E as t-0 for all xAE: Applying this to x ¼ b we ﬁnd that

f ða þ tbÞÀf ðaÞ lim ¼ lim Taþtb;aðbÞ¼Ta;aðbÞ; t-0 t t-0 cf cf which proves the theorem. &

The following corollary is the result announced in the abstract.

Corollary 6.10. If f : R-R is continuously differentiable and f 0 is of bounded variation, then f is Gaˆteaux operator-differentiable along LpðM; tÞh; 1opoN; at f a;a every point a ¼ a AM and DL f ðaÞ¼T : p cf

0 2 Proof. First note that the continuity and boundedness of f imply that cf ACbðR Þ: A m Furthermore, it follows from Proposition 8.5 in [28] that cf LNðV1 Þ: Hence À1 cf AC1; i.e., f AC ðC1Þ: Now the result follows immediately from the above theorem. &

We ﬁnish this section with the complete description of the class of operator- function which are differentiable along the space L2ðM; tÞ for all semiﬁnite algebras M:

Proposition 6.11. A Lipschitz function f : R-R is Gaˆteaux differentiable along Af L2ðM; tÞh at every point a ¼ a M for any semiﬁnite von Neumann algebra ðM; tÞ if a;a and only if f is continuously differentiable. In this case DL f ðaÞ¼T : 2 cf

Proof. Suppose f satisfies the above assumptions. Then it is established in [28, a;a Corollary 7.7] that T belongs LðL2ðM; tÞÞ for any semifinite von Neumann cf algebra ðM; tÞ: By repeating the argument identical to that of Theorem 6.9 (using Proposition 3.5 instead of Lemma 6.8) we complete the proof of sufficiency. The necessity of the assumption on f can be proved by a similar argumentation to that of in the first part of [42] (there the continuity of f 0 was established for Gaˆ teaux differentiability along LNð0; 1Þ) if to note that in such a setting the norm jj Á jj2 majorizes the norm jj Á jjN: This completes the proof of the proposition. & ARTICLE IN PRESS

B. de Pagter, F.A. Sukochev / Journal of Functional Analysis 212 (2004) 28–75 67

7. Examples

In this section we will present some classes of functions to which the results of Sections 6 and 5 apply. Furthermore we will give an example of a function which À1 À1 belongs to C ðC1Þ but not to C ðC0Þ: First we recall the definition of the ’1 homogeneous Besov space BN;1ðRÞ and introduce some relevant notation. For the details we refer the reader to the books [29,40,41]. N N We denote by Cc ðRÞ the space of all complex-valued C -functions on R with compact support. As usual, we will denote by SðRÞ the space of all rapidly decreasing CN-functions on R and its dual space of tempered distributions is denoted by S0ðRÞ: The Fourier transform of a function jASðRÞ is given by Z 1 ðFjÞðxÞ¼pffiffiffiffiffiffi jðxÞeÀixx dx: 2p R

The Fourier transformation on S0ðRÞ is denoted by F as well and is deﬁned by /Ff ; jS ¼ / f ; FjS for all jASðRÞ and all f AS0ðRÞ: N \ Let fcngn¼ÀN be a (from now on ﬁxed) resolution of the identity on R f0g , N i.e., cnACc ðRÞ such that: nÀ1 nþ1 (i) 0pcnðxÞp1 for all xAR; suppðcnÞ¼fxAR :2 pjxjp2 g and cnðxÞ40 nÀ1 nþ1 Pwhenever 2 ojxjo2 ; for all nAZ; N A \ (ii) n¼ÀN cnðxÞ¼1 for all x R f0g; (iii) ðkÞ Àkn there exist constants Ck40 such that jcn ðxÞjpCk2 for all xAR; nAZ and k ¼ 0; 1; y:

’1 The homogeneous Besov space BN;1ðRÞ is now deﬁned by () XN ’1 A 0 n À1 BN;1ðRÞ¼ f S ðRÞ : jj f jjB’1 ¼ 2 jjF ðcnFf ÞjjNoN : N;1 n¼ÀN

’1 0 Clearly, BN;1ðRÞ is a linear subspace of S ðRÞ and jj Á jjB’1 is a seminorm on N;1 ’1 BN;1ðRÞ: Note that jj f jjB’1 ¼ 0 if and only if f is a polynomial. We recall in N;1 particular the following result. For the convenience of the reader we also indicate the main steps of the proof.

A ’1 A Lemma 7.1. For every f BN;1ðRÞ there exist constants cn C and a polynomial P such that

XN À1 f ¼ fF ðcnFf ÞÀcn1gþP; ð36Þ n¼ÀN where the series is absolutely and uniformly convergent on compact subsets of R: A ’1 Moreover, every f BN;1ðRÞ is continuously differentiable. ARTICLE IN PRESS

68 B. de Pagter, F.A. Sukochev / Journal of Functional Analysis 212 (2004) 28–75

À1 Proof. For all nAZ put fn ¼ F ðcnFf Þ: By hypothesis we have fnALNðRÞ and nþ1 nþ1 Ffn ¼ cnFf : This implies in particular that suppðFfnÞD½À2 ; 2 ; so it follows from the Paley–Wiener–Schwartz theorem (see [34]) that fn is the restriction to R of an entire function of exponential type 2nþ1: Now it follows via a Bernstein-type theorem (see e.g. [8], Chapter 11) that

0 nþ1 jj fnjjNp2 jj fnjjN for all nAZ and so

XN XN 0 nþ1 À1 jj fnjjNp 2 jjF ðcnFf ÞjjNoN: n¼ÀN n¼ÀN P N 0 Consequently, the series n¼ÀN fn is absolutely convergent in LNðRÞ: From this the result of the lemma readily follows. &

It is clear that in the representation (36) the constants cn and the constant term of P are not uniquely determined by f : However, without loss of generality we may assume that the constant term in P is zero, in which case this polynomial P is unique. A ’1 From now on we will consider only functions f BN;1ðRÞ for which the polynomial P in (36) is of degree at most one. These functions f can be written as

XN À1 f ¼ fF ðcnFf ÞÀcn1gþAf x; ð37Þ n¼ÀN with cnAC ðnAZÞ and uniquely determined Af AC: For convenience we will denote e1 this linear space of functions by BN;1ðRÞ; i.e.,

e1 A ’1 BN;1ðRÞ¼ff BN;1ðRÞ : f can be represented as in ð37Þg:

A e1 For f BN;1ðRÞ we deﬁne

jj f jje ¼jjf jj ’1 þjAj: B1 BN N;1 ;1

Clearly, jj Á jj is a seminorm on Be1 ðRÞ and jj f jj ¼ 0 if and only if f is a Be1 N;1 Be1 N;1 N;1 constant. Furthermore we observe that it follows from the proof of Lemma 7.1 that A e1 every f BN;1ðRÞ has a bounded derivative, so f has at most linear growth. The following result is essentially contained in [34], to which we have to refer the reader for the proof.

e1 À1 Proposition 7.2 (V. Peller). BN;1ðRÞDC ðC0Þ and there exists a constant C40 such e that jjc jj pCjj f jj for all f AB1 ðRÞ: f A0 Be1 N;1 N;1 ARTICLE IN PRESS

B. de Pagter, F.A. Sukochev / Journal of Functional Analysis 212 (2004) 28–75 69

The following corollary is now an immediate consequence of Theorem 5.16.

Corollary 7.3. Let ðM; tÞ be a semiﬁnite von Neumann algebra, let E be a separable A e1 symmetric Banach function space on ð0; NÞ and E ¼ EðM; tÞ: If f BN;1ðRÞ; then f is f Gaˆteaux operator differentiable along Eh it every point aAMh: Moreover, there exists e1 f a constant C40 such that jjD f ðaÞjjpCjj f jj for all f AB ðRÞ and all aAMh: E Be1 N;1 N;1

The following proposition yields a large class of functions to which the above corollary applies. We denote by MbðRÞ the space of all bounded Borel measures on R: For mAMbðRÞ we denote the total variation measure by jmj: The norm in MbðRÞ is deﬁned by jjmjj1 ¼jmjðRÞ for all mAMbðRÞ:

A 0 0A A e1 Proposition 7.4. If f S ðRÞ such that Ff MbðRÞ; then f BN;1ðRÞ:

N \ Proof. Let fcngn¼ÀN be a resolution of the identity on R f0g as introduced at the beginning of the present section. For convenience we put m ¼ Ff 0: For nAZ we N À1 a deﬁne the function ZnACc ðRÞ by ZnðxÞ¼ðixÞ cnðxÞ whenever x 0andZnð0Þ¼0: nÀ1 nþ1 Since the support of cn is contained in fxAR :2 pjxjp2 g; it is clear that

ÀnÀ1 Ànþ1 2 cnðxÞpjZnðxÞjp2 cnðxÞð38Þ for all xAR: Using that Ff 0 ¼ðixÞFf we ﬁnd that

Znm ¼ ZnðixÞFf ¼ cnFf for all nAZ; where Znm is the Borel measure with Radon–Nikodym derivative Zn with À1 respect to m: Consequently, F ðcnFf Þ is a function, given by Z À1 1ffiffiffiffiffiffi ixx F ðcnFf ÞðxÞ¼p ZnðxÞe dmðxÞ 2p R for all xAR: In combination with (38), this implies that Z n À1 2ffiffiffiffiffiffi 2 jjF ðcnFf ÞjjNpp cn djmj 2p R for all nAZ and so

XN XN Z n À1 2ffiffiffiffiffiffi 2 jjF ðc Ff ÞjjNp p c djmj n 2p n n¼ÀN n¼À N R ! Z XN 2ffiffiffiffiffiffi 2ffiffiffiffiffiffi ¼ p cn djmjpp jjmjj1oN; 2p R n¼ÀN 2p ARTICLE IN PRESS

70 B. de Pagter, F.A. Sukochev / Journal of Functional Analysis 212 (2004) 28–75

A ’1 0A 0 which shows that f BN;1ðRÞ: Finally, Ff MbðRÞ implies that f is a bounded A e1 function and so we may conclude that f BN;1ðRÞ: &

A e1 Remark 7.5. One might think that every f BN;1ðRÞ can be written as the sum of a bounded function and a linear function. The following example shows that this is not the case. Deﬁne f : R-R by

XN x f ðxÞ¼ sin2 : k2 k¼1

Note that this series is uniformly convergent on compact subsets of R: A simple computation shows that ffiffiffiffiffiffi p XN 0 2p 1 Ff ¼ 2 d 2 À d 2 2i k 2 À 2 k¼1 k k

0A A e1 and hence Ff MbðRÞ: Therefore, it follows from Proposition 7.4 that f BN;1ðRÞ: On the other hand, it is an easy exercise to show that there exist two constants c1; c240 such that pffiffiffi pffiffiffi c1 xpf ðxÞpc2 x whenever xXa for some 0oaAR:

Another important class of functions to which the above results apply is C1þeðRÞ with e40: Here C1þeðRÞ denotes the space of all bounded continuously differentiable e functions on R for which f 0 is bounded and satisﬁes j f 0ðxÞÀf 0ðyÞjpCjx À yj for all x; yAR and some constant C40: From the results in [40] it follows that 1þe D e1 C ðRÞ BN;1ðRÞ: We collect these results in the following corollary, extending those from [4, Theorem 10; 6, Proposition 7.8; 13 (Introduction); 42].

Corollary 7.6. Let ðM; tÞ be a semiﬁnite von Neumann algebra, let E be a separable symmetric Banach function space on ð0; NÞ and E ¼ EðM; tÞ: Suppose that f AS0ðRÞ satisﬁes one of the following two conditions:

0 (i) Ff AMbðRÞ; (ii) f AC1þeðRÞ for some e40: f Then f is Gaˆteaux operator differentiable along Eh at every point aAMh:

À1 Now we will exhibit an example of a function f : R-R which belongs to C ðC1Þ À1 À1 but not to C ðC0Þ: Actually we will construct a function f AC ðC1Þ which does not satisfy the conclusion of Corollary 5.3 for E ¼ L1ðR; tÞ; where R is the hyperﬁnite ARTICLE IN PRESS

B. de Pagter, F.A. Sukochev / Journal of Functional Analysis 212 (2004) 28–75 71

II1-factor with tracial state t: For sake of reference we formulate the following lemma.

Lemma 7.7. Let R be the hyperﬁnite II1-factor equipped with tracial state t: Then N N there exist two sequences fangn¼1 and fbngn¼1 in Rh such that n jjjanjÀjbnjjj1X2 njjan À bnjj1 for n ¼ 1; 2; y and jjanjjN; jjbnjjNp1 for all n:

Proof. For k ¼ 1; 2; y we denote by Mk the von Neumann algebra of all k Â k- matrices equipped with normalized trace trk: For each k; the algebra Mk is - isomorphic to a subalgebra Rk of R and this -isomorphism induces an isometry between L1ðMk; trkÞ and L1ðRk; tÞ: Clearly, L1ðRk; tÞ can be identiﬁed with a subspace of L1ðR; tÞ: In [13] it is shown that for all k ¼ 1; 2; y there exist self- adjoint Ak; BkAMk such that

jjjA jÀjB jjj XC ln kjjA À B jj ; k k L1ðMk;trkÞ k k L1ðMk;trkÞ N where C40 is a universal constant. Consequently, there exist sequences fakgk¼1 and N fbkgk¼1 in Rh such that

jjjakjÀjbkjjj1XC ln kjjak À bkjj1 for all k: Without loss of generality we can assume moreover that jjakjjN; jjbkjjNp1 for all k: Passing to a subsequence gives the desired result. &

- Now we start the construction of the function f : First we deﬁne c0 : R R by 8 <> t if 0ptp1; c ðtÞ¼ 3 À 1 t if 1ptp3; 0 :> 2 2 0 otherwise and let XN Àn cðtÞ¼ 2 c0ðt À 3nÞ: n¼0 Observe that

cðtÞ¼2Ànjt À 3nj whenever 3n À 2ptp3n þ 1 ð39Þ A N N and 1pn N: Let fangn¼1 and fbngn¼1 be two sequences in Rh as in Lemma 7.7 and put n d ¼ jja À b jj: ð40Þ n 4 n n ARTICLE IN PRESS

72 B. de Pagter, F.A. Sukochev / Journal of Functional Analysis 212 (2004) 28–75

Now let f ACNðRÞ be a real valued function such that suppð f ÞD½À1; NÞ with the properties: ÀÁÀÁÀÁÀÁ (i) y 1 1 0 1 0 1 for all n ¼ 0; 1; we have f 3n þ 2 ¼ c 3n þ 2 ; f 3n þ 2 ¼ c 3n þ 2 ; 0 0 f ð3n þ 2Þ¼cð3n þÂÃ2Þ and f ð3n þ 2Þ¼c ð3n þ 2Þ; (ii) 1 y ÂÃf is concave on 3n þ 2; 3n þ 2 for all n ¼ 0; 1; and f is convex on 1 y 3n À 1; 3n þ 2 for all n ¼ 1; 2; ; (iii) j f ðtÞÀcðtÞjpdn whenever 3n À 1ptp3n þ 1 for all n ¼ 1; 2; y .

0 0 It is easy to construct such a function f : We claim that f AV1ðRÞ; i.e., that f is of bounded variation on R (and right-continuous, which is obvious inÂÁ this case). 0 1 Indeed, it is clearly enough to show that f is of bounded variation on 2; 2 : From (i), (ii) and (iii) it follows that 1 1 Var f 0; 3n þ ; 3n þ 2 ¼ f 0 3n þ À f 0ð3n þ 2Þ 2 2 1 ¼ c0 3n þ À c0ð3n þ 2Þ 2 ¼ 2Àn þ 2ÀnÀ1 for all n ¼ 0; 1; y and 1 1 Var f 0; 3n À 1; 3n þ ¼ f 0 3n þ À f 0ð3n À 1Þ 2 2 1 ¼ c0 3n þ À c0ð3n À 1Þ 2 ¼ 2Àn þ 2Àn ÀÁÂÁ y 0 1 N N forÀÁ all nÀÃ¼ 1; 2; . This implies that Var f ; 2; o : Since it is clear that 0 N 1 N Var f ; À ; 2 o ; our claim follows. As observed in [28, Proposition 8.5], this A m A implies that cf LNðV1 Þ and hence cf A1 (where we use the notation introduced in Section 6 of the present paper). Moreover, since f is a C1-function, it is clear that 2 À1 cf ACbðR Þ; therefore cf AC1; i.e., f AC ðC1Þ: À1 Next we will show that f eC ðC0Þ: This will follow from the following lemma.

Lemma 7.8. Let R be the hyperﬁnite II1-factor equipped with tracial state t: The operator function f is not Lipschitz in the space L1ðR; tÞ; i.e., there does not exist a constant C40 such that jj f ðaÞÀf ðbÞjj1pCjja À bjj1 for all self-adjoint a; bAL1ðR; tÞ:

N N Proof. Let fangn¼1 and fbngn¼1 be the sequences in Rh from Lemma 7.7. We deﬁne e aen ¼ an þ 3n1 and bn ¼ bn þ 3n1 for all n ¼ 1; 2; y . Since sðaenÞD½3n À 1; 3n þ 1; it ARTICLE IN PRESS

B. de Pagter, F.A. Sukochev / Journal of Functional Analysis 212 (2004) 28–75 73 follows from (iii) above that e e jj f ðanÞÀcðanÞjjNpdn:

Moreover, since tð1Þ¼1; we have jjxjj1pjjxjjN for all xAR: Consequently, e e jj f ðanÞÀcðanÞjj1pdn and similarly we ﬁnd that e e jj f ðbnÞÀcðbnÞjj1pdn for all n: It follows from (39) that

Àn Àn cðaenÞ¼2 jaen À 3n1j¼2 janj

e Àn and similarly cðbnÞ¼2 jbnj: Collecting the above, using the properties of an; bn and the deﬁnition (40) of dn; it follows that

e e e e jj f ðanÞÀf ðbnÞjj1X jjcðanÞÀcðbnÞjj1 À 2dn

Àn ¼ 2 jjjanjÀjbnjjj1 À 2dn n X njja À b jj À 2d ¼ jja À b jj n n 1 n 2 n n 1 n e ¼ jjae À b jj 2 n n 1 for all n ¼ 1; 2; y; by which the lemma is proved. &

À1 Corollary 7.9. The function f : R-R as deﬁned above satisﬁes f AC ðC1Þ but À1 f eC ðC0Þ:

À1 À1 Proof. We have already shown above that f AC ðC1Þ: If f AC ðC0Þ; then it would follow from Corollary 5.3 that the operator function f is Lipschitz in the space L1ðR; tÞ; which contradicts the result of the above lemma. Hence we may conclude À1 that f eC ðC0Þ: &

References

[1] J. Arazy, T.J. Barton, Y. Friedman, Operator differentiable functions, Integral Equations Operator Theory 13 (1990) 461–487. [2] C. Bennett, R. Sharpley, Interpolation of Operators, Academic Press, Orlando, 1988. [3] E. Berkson, T.A. Gillespie, P.S. Muhly, Abstract spectral decomposition guaranteed by the Hilbert transform, Proc. London Math. Soc. 53 (1986) 489–517. ARTICLE IN PRESS

74 B. de Pagter, F.A. Sukochev / Journal of Functional Analysis 212 (2004) 28–75

[4] M.S. Birman, M.Z. Solomyak, Double Stieltjes operator integrals (Russian), Probl. Math. Phys. Leningrad Univ. 1 (1966) 33–67. [5] M.S. Birman, M.Z. Solomyak, Double Stieltjes operator integrals II (Russian), Probl. Math. Phys. Leningrad Univ. 2 (1967) 26–60. [6] M.S. Birman, M.Z. Solomyak, Double Stieltjes operator integrals III (Russian), Probl. Math. Phys. Leningrad Univ. 6 (1973) 27–53. [7] M.S. Birman, M.Z. Solomyak, Spectral Theory and Self-Adjoint Operators in Hilbert Space, Reidel Publishing Company, Dordrecht, 1987. [8] R.P. Boas, Entire Functions, Academic Press, New York, 1954. [9] V.I. Chilin, F.A. Sukochev, Weak convergence in non-commutative symmetric spaces, J. Operator Theory 31 (1994) 35–65. [10] Ph. Cle´ ment, B. de Pagter, F.A. Sukochev, H. Witvliet, Schauder decompositions and multiplier theorems, Studia Math. 138 (2000) 135–163. [11] Yu.L. Daletskii, S.G. Krein, Formulas of differentiation according to a parameter of functions of Hermitian operators, Dokl. Akad. Nauk SSSR (NS) 76 (1951) 13–16. [12] Yu.L. Daletskii, S.G. Krein, Integration and differentiation of functions of Hermitian operators and applications to the theory of perturbations, Voronezh. Gos. Univ. Trudy Sem. Funk. Anal. 1 (1956) 81–105. [13] E.B. Davies, Lipschitz continuity of functions of operators in the Schatten classes, J. London Math. Soc. 37 (1988) 148–157. [14] J. Diestel, J.J. Uhl, Vector Measures, American Mathematical Society, Providence, RI, 1977. [15] J. Dixmier, Von Neumann algebras, in: North-Holland Mathematical Library, Vol. 27, North- Holland, Amsterdam, 1981. [16] P.G. Dodds, T.K. Dodds, B. de Pagter, Non-commutative Banach function spaces, Math. Z. 201 (1989) 583–597. [17] P.G. Dodds, T.K. Dodds, B. de Pagter, Noncommutative Ko¨ the duality, Trans. Amer. Math. Soc. 339 (1993) 717–750. [18] P.G. Dodds, T.K. Dodds, B. de Pagter, F.A. Sukochev, Lipschitz continuity of the absolute value and Riesz projections in symmetric operator spaces, J. Funct. Anal. 148 (1997) 28–69. [19] N. Dunford, J.T. Schwartz, Linear Operators II: Spectral Theory, Interscience Publishers, New York, 1963. [20] Th. Fack, H. Kosaki, Generalized s-numbers of t-measurable operators, Paciﬁc. J. Math. 123 (1986) 269–300. [21] E. Hewitt, K. Stromberg, Real and Abstract Analysis, Springer, Berlin, Heidelberg, New York, 1965. [22] R.V. Kadison, J.R. Ringrose, Fundamentals of the Theory of Operator Algebras, Vol. I, Academic Press, New York, 1983. [23] R.V. Kadison, J.R. Ringrose, Fundamentals of the Theory of Operator Algebras, Vol. II, Academic Press, Orlando, 1986. [24] S.G. Krein, Ju.I. Petunin, E.M. Semenov, Interpolation of Linear Operators, in: Translations of Mathematical Monographs, Vol. 54, American Mathematical Society, Providence, RI, 1982. [25] J. Lindenstrauss, L. Tzafriri, Classical Banach Spaces II, Springer, Berlin, 1979. [26] E. Nelson, Notes on non-commutative integration, J. Funct. Anal. 15 (1974) 103–116. [27] B. de Pagter, F.A. Sukochev, H. Witvliet, Unconditional decompositions and Schur multipliers, in: A. Dijksma, et al. (Eds.), Operator Theory: Advances and Applications, Vol. 124, Birkha¨ user Verlag, Basel, 2001, pp. 505–525. [28] B. de Pagter, F.A. Sukochev, H. Witvliet, Double operator integrals, J. Funct. Anal. 192 (2002) 52–111. [29] J. Peetre, New Thoughts on Besov spaces, Duke University Mathematical Series I, Durham, 1976. [30] V. Peller, Hankel operators in the perturbation theory of unitary and self-adjoint operators, Funct. Anal. Appl. 19 (1985) 111–123 (Russian). [31] V. Peller, Hankel operators in the perturbation theory of unbounded selfadjoint operators, in: C. Sadosky (Ed.), Analysis and Partial Differential Equations, Lecture Notes in Pure and Applied Mathematicas, Vol. 122, Marcel Dekker, New York, 1990, pp. 529–544. ARTICLE IN PRESS

B. de Pagter, F.A. Sukochev / Journal of Functional Analysis 212 (2004) 28–75 75

[32] M. Reed, B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis, Academic Press, San Diego, 1980. [33] J. Rubio de Francia, Martingale and integral transforms of Banach space valued functions, in: J. Bastero, M. San Miguel (Eds.), Probability in Banach Spaces, Vol. 9, Lecture Notes in Mathematics, Vol. 1221, Springer, New York, 1994. [34] L. Schwartz, The´ orie des Distributions, Hermann, Paris, 1973. [35] S. Stratila, L. Zsido´ , Lectures on von Neumann Algebras, Abacus Press, Tungbridge Wells, Kent, England, 1979. [36] F.A. Sukochev, V.I. Chilin, Symmetric spaces over semiﬁnite von Neumann algebras, Dokl. Akad. Nauk SSSR 313 (1990) 811–815 (Russian); translation in Sov. Math. Dokl. 42 (1991) 97–101. [37] M. Takesaki, Theory of Operator Algebras I, Springer, New York, 1979. [38] M. Terp, Lp spaces associated with von Neumann algebras, Notes, Copenhagen University, 1981. [39] O.E. Tikhonov, Continuity of operator functions in topologies connected with a trace on a von Neumann algebra, Izv. Vyssh. Uchebn. Zaved. Mat. 1 (1987) 77–79 (Russian); translation in Soviet Math. (Iz. VUZ) 31 (1987) 110–114. [40] H. Triebel, Theory of Function Spaces, Birkha¨ user Verlag, Basel, 1983. [41] H. Triebel, Theory of Function Spaces II, Birkha¨ user Verlag, Basel, 1992. [42] H. Widom, When are differentiable functions differentiable? in: Linear and Complex Analysis Problem Book (199 Research Problems), Lecture Notes in Mathematics, Vol. 1043, Springer, Berlin, 1984, pp. 184–188.