Connections of unbounded operators and some related topics: von Neumann algebra case
aa r X i v : . [ m a t h . OA ] F e b Connections of unbounded operators and some related topics:von Neumann algebra case
Fumio Hiai , and Hideki Kosaki Graduate School of Information Sciences, Tohoku University,Aoba-ku, Sendai 980-8579, Japan Graduate School of Mathematics, Kyushu University,Nishi-ku, Fukuoka 819-0395, Japan
Abstract
The Kubo-Ando theory deals with connections for positive bounded opera-tors. On the other hand, in various analysis related to von Neumann algebrasit is impossible to avoid unbounded operators. In this article we try to extenda notion of connections to cover various classes of positive unbounded operators(or unbounded objects such as positive forms and weights) appearing naturallyin the setting of von Neumann algebras, and we must keep all the expected prop-erties maintained. This generalization is carried out for the following classes: (i)positive τ -measurable operators (affiliated with a semi-finite von Neumann alge-bra equipped with a trace τ ), (ii) positive elements in Haagerup’s L p -spaces, (iii)semi-finite normal weights on a von Neumann algebra. Investigation on thesegeneralizations requires some analysis (such as certain upper semi-continuity)on decreasing sequences in various classes. Several results in this direction areproved, which may be of independent interest. Ando studied Lebesgue decompo-sition for positive bounded operators by making use of parallel sums. Here, suchdecomposition is obtained in the setting of non-commutative (Hilsum) L p -spaces. Key words and phrases: closable operator, connection, Connes’ spatial theory, extended positive self-adjoint operator, form sum, γ -homogeneous operator, generalized s -number,geometric mean, graph analysis, Haagerup L p -space, Hilsum L p -space, Kubo-Ando theory, Lebesgue decomposition, measure topology, modular automor-phism group, operator monotone function, operator valued weight, parallel sum,positive form, positive self-adjoint operator, Radon-Nikodym cocycle, relativemodular operator, spatial derivative, strong resolvent convergence, τ -measurableoperator, trace, von Neumann algebra, weight E-mail: [email protected] E-mail: [email protected] ontents τ -measurable operators 12 τ -measurability and decreasing sequences . . . . . . . . . . . . . . . . . 133.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.4 Connections on L ( M , τ ) + M . . . . . . . . . . . . . . . . . . . . . . 313.5 Comparison between a α b and | a − α b α | . . . . . . . . . . . . . . . . . . 35 L p -spaces 375 Parallel sums of weights 42 L p -spaces 55 A Haagerup’s L p -spaces 69B Connes’ spatial theory 71 B.1 Spatial derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71B.2 Hilsum’s L p -spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75B.3 Canonical correspondence of P ( M , C ) and P ( B ( H ) , M ′ ) . . . . . . . . 76 C Infima of decreasing sequences of weights 79References 85
Operator means in various settings have been studied in operator theory and operatoralgebras, and they sometimes play unexpectedly important roles. Studies of parallelsums in [1, 2] and geometric means in [59] are such examples (and surely many others2hat the authors are unaware of). Also many properties of typical operator meanssuch as parallel sums (twice of harmonic means) and geometric means were completelyclarified in [5]. Motivated by these pioneering works, the so-called Kubo-Ando theorywas established in [52]. In this theory a notion of operator means (or more generallyconnections) of positive operators was introduced in an axiomatic fashion. They arein a one-to-one correspondence with the set of all non-negative operator monotonefunctions on R + (see the beginning of § L p -spaces. The main purpose of the article is to make a satisfactory theory of con-nections for various classes of unbounded operators (or unbounded objects such asweights) appearing naturally in analysis related to von Neumann algebras. Generallyunbounded operators are much more difficult to handle compared with bounded onessince delicate problems such as domain questions and closability problems (under alge-braic operations) have to be taken care of. For a semi-finite von Neumann algebra M with a trace τ one can introduce the class of τ -measurable operators (see the beginningof § L p -spaces (see [16, 53, 73] for earlydevelopment of the theory). The notion of τ -measurability was originally given in [62](see also [66] for related topics) in a slightly different form, but the approach in [56] ismore convenient to our purpose. Thus, the latter approach is employed, and we will use[56, 71] as our standard references on the subject. When a von Neumann algebra is fi-nite, all unbounded closed operators (affiliated with M ) is automatically τ -measurable.This means that manipulations of unbounded operators in this case are completely un-der control, which was already shown by Murray-von Neumann since the beginning ofthe subject. The recent articles [17, 18] have somewhat similar nature in the sense thatcertain means for τ -measurable operators are investigated. However, what are studiedthere are means in the sense of [35, 36] (which are different from Kubo-Ando means[52] considered here). Hence there is no direct relation between [17, 18] and our presentwork, but some of standard techniques for τ -measurable operators (such as devices in[23]) are in common use.We explain contents of later sections briefly, and more detailed description is givenat the beginning of each section. The most general situation is treated in §
2. Namely,we consider positive forms here, which are roughly positive self-adjoint operators with3ossibly non-dense domains, or equivalently, elements in the extended positive part \ B ( H ) + (see [29]). A general theory for parallel sums for positive forms was worked outin [50]. Thus, by mimicking an integral expression of a connection (in terms of parallelsums) one can define connections of positive forms. Some properties of connectionsbased on this naive definition are discussed in § § M with a trace τ , and studyconnections of τ -measurable operators. The definition of a τ -measurable operatorand some related topics are explained at the beginning of § § § τ -measurable operators in this section, and we need some general results on these matters.They are collected in § § L p -spaces L p ( M ) ([30]). We study connections in these L p -spaces. (A similar studythough specialized to connections in L ( M ) is also given in [34, Appndix D].) Thecrossed product M ⋊ σ R relative to a modular action σ is a semi-finite algebra equippedwith a so-called canonical trace τ (which is a standard fact in structure analysis of typeIII von Neumann algebras). The Haagerup L p -spaces are constructed by making useof this semi-finite algebra. Since τ -measurability and the measure topology there are(technically) important ingredients in Haagerup’s theory (see Appendix A for details),some of arguments in this section depend heavily upon those in §
3. Many interestingproperties of connections were known in [52]. We show (besides some special topics in § § § § L ( M ) can be identified with the predual M ∗ , meaning that a rea-sonable theory for normal positive linear functionals on M is at our disposal. Moregenerally, in § M . To this end, we use Connes’ spatial theory [13] and it is achieved by consid-ering parallel sums (in the sense of [50]) of spatial derivatives arising from weights inquestion. In this study a notion of ( − § § § L p -spaces. Since a reasonable notion of parallel sumshas been prepared in [50] and § §
4, we can play the same game by modifying argu-ments in [3]. However, we take a different approach akin to that in [46, 47]. Namely,4ased on graph analysis for relevant relative modular operators we express Lebesgue de-composition in a more explicit manner. For this purpose Hilsum’s L p -spaces (which areisometrically isomorphic to L p ( M )) are more convenient so that we use his L p -spacesin §
6. In fact, for M in its standard representation the positive parts of these L p -spacesconsist of powers of relative modular operators. Results in this section generalize thosein [47] (dealing with the predual M ∗ ∼ = L ( M )).We have three appendices for the reader’s convenience. In Appendix A basic facts onthe Haagerup L p -spaces (used in §
4) are reviewed. Appendix B explains several materi-als related to Connes’ spatial theory [13]. At first Connes’ notion of spatial derivativesis explained in § B.1 (via a slightly different approach presented in [71]) and then § B.2explains Hilsum’s L p -spaces. Materials in these two subsections are needed in § § M (acting on a Hilbert space H ) and operator-valued weights from B ( H ) to the commutant M ′ . We review this correspondence in § B.3. Parallel sumsof weights in § τ -measurable operators and so on.In the last appendix (Appendix C) we study decreasing sequences of (normal) weights.Probably results here have never been recorded in literature. We will study connections for various classes of unbounded operators (such as τ -measurable positive operators and so on) appearing naturally in analysis with vonNeumann algebras. Therefore, we begin with the most general situation, i.e., (un-bounded) positive self-adjoint operators. Actually, the notion of positive forms in [50]is more convenient for our purpose. In § R + (see the beginning of § § .1 Positive forms This section is a brief summary of [50] to fix ideas and notations for later use. Ourstandard references for basic facts on unbounded operators are [41, 61, 67]. Let H bea Hilbert space (assumed infinite-dimensional). By a positive form q on H we mean afunction q : H → [0 , + ∞ ] satisfying(i) q ( λξ ) = | λ | q ( ξ ) for all ξ ∈ H and λ ∈ C (with the convention 0 · ∞ = 0),(ii) q ( ξ + ξ ) + q ( ξ − ξ ) = 2 q ( ξ ) + 2 q ( ξ ) for all ξ , ξ ∈ H ,(iii) q is lower semi-continuous on H (i.e., q ( ξ ) ≤ lim inf n →∞ q ( ξ n ) whenever ξ n → ξ ).The domain D ( q ) of q is given by D ( q ) := { ξ ∈ H ; q ( ξ ) < ∞} , which is obviouslya linear subspace of H . We note that a positive form q on H bijectively correspondsto an extended positive self-adjoint operator h in the sense of Haagerup [29] or Kato[40], i.e., a positive self-adjoint operator h on some (closed) subspace K of H . We write \ B ( H ) + , following [29], for the set of such extended positive self-adjoint operators on H .The correspondence q ↔ h is given in such a way that D ( q ) = D ( h / ) (so D ( q ) = K )and q ( ξ ) = k h / ξ k for ξ ∈ D ( q ). In terms of the spectral decomposition h = R ∞ λ de λ on K (so e ∞ := lim λ →∞ e λ is the projection onto K ), we can also write q ( ξ ) = Z ∞ λ d k e λ ξ k + ∞k e ⊥∞ ξ k , ξ ∈ H (2.1)(with the convention ∞ · q = q h in this case. For each h ∈ \ B ( H ) + wecan define the resolvent (1 + h ) − , which is understood to be 0 on K ⊥ = D ( q ) ⊥ .For positive forms q , q corresponding to h , h ∈ \ B ( H ) + respectively, we note that q ≤ q , i.e., q ( ξ ) ≤ q ( ξ ) for all ξ ∈ H if and only if h ≤ h in the form sense,i.e., D ( h / ) ⊆ D ( h / ) and k h / ξ k ≤ k h / ξ k for all ξ ∈ D ( h / ). Moreover, it iswell-known that h ≤ h in the form sense if and only if (1 + h ) − ≥ (1 + h ) − . The(point-wise) sum q + q is a positive form, which corresponds to the so-called formsum h ˙+ h [41]. Let q be a positive form corresponding to h = R ∞ λ de λ (on K ). Let K be the null space of h or K = { ξ ∈ H ; q ( ξ ) = 0 } . We can define k ∈ \ B ( H ) + by k := R (0 , ∞ ) λ − de λ + 0 e ⊥∞ (on K ⊥ ). Then the inverse q − of q is defined as the positiveform corresponding to k , that is, q − ( ξ ) = Z [0 , ∞ ) λ − d k e λ ξ k (cid:18) = Z (0 , ∞ ) λ − d k e λ ξ k + ∞k e ξ k (cid:19) , ξ ∈ H . The expression of q − in terms of the pure positive form notation was shown in [50,Lemma 2] as q − ( ξ ) = sup ζ ∈H | ( ξ, ζ ) | q ( ζ ) (2.2)with the convention 0 / α/ ∞ ( α > q q − is order-reversingand ( q − ) − = q .The parallel sum of two positive forms was introduced in [50].6 efinition 2.1. For positive forms φ, ψ define the parallel sum φ : ψ by φ : ψ := ( φ − + ψ − ) − . Theorem 2.2 ([50]) . The parallel sum φ : ψ is the maximum of all the positive forms q satisfying q ( ξ + ξ ) ≤ φ ( ξ ) + ψ ( ξ ) , ξ , ξ ∈ H . In fact, we set ρ ( ξ ) := inf { φ ( ξ ) + ψ ( ξ ); ξ = ξ + ξ } . (2.3)Then ρ is quadratic , that is, ρ satisfies (i) and (ii) of the definition of positive forms.The above theorem says that φ : ψ is the maximum of all the positive forms q satisfying q ( ξ ) ≤ ρ ( ξ ) for all ξ ∈ H . Hence we can write [63, 64]( φ : ψ )( ξ ) = inf ξ n → ξ lim inf n →∞ ρ ( ξ n ) . (2.4)If { q n } is an increasing sequence of positive forms, then the point-wise supremum q =sup n q n is a positive form. On the other hand, when { q n } is decreasing, the point-wiseinfimum inf n q n is quadratic but not necessarily lower semi-continuous. We have themaximum of all the positive forms q satisfying q ( ξ ) ≤ inf n q n ( ξ ) for all ξ ∈ H [63, 64],which is denoted by Inf n q n . Note that { q − n } is increasing (in the decreasing case) andconsequently the point-wise supremum sup n q − n is a positive form as mentioned above.We actually have Inf n q n = (cid:18) sup n q − n (cid:19) − (2.5)(see [50, Lemma 16]). The following was given in [50, Corollary 17]: Theorem 2.3. If { φ n } and { ψ n } are decreasing sequences of positive forms, then Inf n ( φ n : ψ n ) = (cid:16) Inf n φ n (cid:17) : (cid:16) Inf n ψ n (cid:17) . (2.6)This result is obtained by repeated use of (2.5). Indeed, we observe (cid:16) Inf n φ n (cid:17) − + (cid:16) Inf n ψ n (cid:17) − = sup n φ − n + sup n ψ − n = sup n (cid:0) φ − n + ψ − n (cid:1) = sup n ( φ n : ψ n ) − with the decreasing sequence { φ n : ψ n } of parallel sums so that one can just take theinverses of the both sides.The obvious estimate Inf n φ n + Inf n ≤ inf n φ n + inf n ψ n = inf n ( φ n + ψ n ) and themaximality of Inf n ( φ n + ψ n ) stated above yieldInf n φ n + Inf n ψ n ≤ Inf n ( φ n + ψ n ) , but the equality generally fails to hold here. Related results (for other classes ofunbounded objects) will be mentioned in Theorem 3.10, Remark 3.11 and Proposition3.27 (see also the paragraph before that proposition).7et { q n } be a sequence of positive forms corresponding to { h n } in \ B ( H ) + . We saythat a sequence { h n } in \ B ( H ) + converges in the strong resolvent sense to h ∈ \ B ( H ) + if (1 + h n ) − → (1 + h ) − strongly [41, 60]. The following is from [50, Lemma 18]: Lemma 2.4.
Let q n ( n ∈ N ) and q be positive forms corresponding to h n and h in \ B ( H ) + . Assume that q n is decreasing. Then q = Inf n q n if and only if h n → h in thestrong resolvent sense, that is, (1 + h n ) − ր (1 + h ) − strongly. ( Similarly, when q n isincreasing, q = sup n q n if and only if (1 + h n ) − ց (1 + h ) − strongly. )From this lemma we also write h = Inf n h n when h n is decreasing and h n → h inthe strong resolvent sense. At first recall the notion of connections for bounded positive operators on H studiedin Kubo and Ando [52]. A connection σ is a binary operation σ : B ( H ) + × B ( H ) + → B ( H ) + satisfying, for every A, B, C, D ∈ B ( H ) + ,(I) if A ≤ C and B ≤ D , then AσB ≤ CσD ,(II) C ( AσB ) C ≤ ( CAC ) σ ( CBC ),(III) if A n ց A and B n ց B strongly, then A n σB n ց AσB strongly.A main result of [52] says that there is a bijective correspondence between the con-nections σ and the non-negative operator monotone functions f on (0 , ∞ ), determinedin such a way that AσB = A / f ( A − / BA − / ) A / for A, B ∈ B ( H ) + with A invertible. (A connection σ is called an operator mean if IσI = I or f (1) = 1.) The function f , called the representing function of σ , has theintegral expression f ( s ) = α + βs + Z (0 , ∞ ) (1 + t ) ss + t dµ ( t ) , s ∈ (0 , ∞ ) , (2.7)with unique α, β ≥ µ on (0 , ∞ ), see, e.g., [9].Based on (2.7) it was shown in [52] that the connection σ has the expression AσB = αA + βB + Z (0 , ∞ ) tt (( tA ) : B ) dµ ( t ) , A, B ∈ B ( H ) + . (2.8)For each connection σ on B ( H ) + the connection σ of positive forms was introducedin [51] by extending the above expression as follows:8 efinition 2.5. For a connection σ given in (2.8) the connection σ of positive forms φ, ψ is defined by( φσψ )( ξ ) := αφ ( ξ ) + βψ ( ξ ) + Z (0 , ∞ ) tt (( tφ ) : ψ )( ξ ) dµ ( t ) , ξ ∈ H , (2.9)where the lower semi-continuity of ξ ( φσψ )( ξ ) on H is easily seen from that of(( tφ ) : ψ )( ξ ) and Fatou’s lemma.By definition it is obvious that the connection φσψ for positive forms is a general-ization of AσB for
A, B ∈ B ( H ) + , that is, φσψ = q AσB if φ ( ξ ) = q A ( ξ ) = k A / ξ k and ψ ( ξ ) = q B ( ξ ) = k B / ξ k for ξ ∈ H .The notions of transpose and adjoint play an important part in theory of connections(and operator means) on B ( H ) + . Corresponding to the transpose ˜ f ( t ) := tf ( t − ) andthe adjoint f ∗ ( t ) := f ( t − ) − for operator monotone functions f > , ∞ ) we havethe transpose ˜ σ and the adjoint σ ∗ of a connection σ on B ( H ) + [52]. In the rest of thesection we examine ˜ σ and σ ∗ for connections of positive forms. Proposition 2.6.
For any connection σ and positive forms φ, ψ we have φ ˜ σψ = ψσφ. Proof.
For the representing function f of σ with the expression (2.7) we write˜ f ( s ) = αs + β + Z (0 , ∞ ) (1 + t ) s ts dµ ( t )= β + αs + Z (0 , ∞ ) (1 + t ) ss + t d ˜ µ ( t ) , where d ˜ µ ( t ) := dµ ( t − ) for t ∈ (0 , ∞ ). Hence it follows that for every ξ ∈ H ,( φ ˜ σψ )( ξ ) = βφ ( t ) + αψ ( ξ ) + Z (0 , ∞ ) tt (( tφ ) : ψ )( ξ ) d ˜ µ ( t )= αψ ( ξ ) + βφ ( ξ ) + Z (0 , ∞ ) tt t (( t − φ ) : ψ )( ξ ) dµ ( t ) . Since t (( t − φ ) : ψ )( ξ ) = ( φ : ( tψ ))( ξ ) = (( tψ ) : φ )( ξ ), we have ( φ ˜ σψ )( ξ ) = ( ψσφ )( ξ ).The parallel sum is the adjoint of the sum, and φ : ψ = ( φ − + ψ − ) − holds asthe definition itself (Definition 2.1). But this is not true for general connection σ (seeRemark 2.10 below). Proposition 2.7.
Let σ be a connection. Either if h, k ∈ B ( H ) + , or if h, k ∈ \ B ( H ) + are bounded from below, i.e., h ≥ ε , k ≥ ε for some ε > , then q h σ ∗ q k = ( q − h σq − k ) − , i.e., ( q h σ ∗ q k ) − = q − h σq − k . roof. Assume h, k ∈ B ( H ) + first. From the decreasing convergence (in (III) above)of the connection σ ∗ on B ( H ) + we have q h σ ∗ q k = inf ε> ( q h + ε σ ∗ q k + ε ) = Inf ε> ( q h + ε σ ∗ q k + ε )so that ( q h σ ∗ q k ) − = sup ε> ( q h + ε σ ∗ q k + ε ) − by [50, Lemma 16,(ii)]. Since (( h + ε σ ∗ ( k + ε − = ( h + ε − σ ( k + ε − in thecase of invertible bounded positive operators, it follows that( q h + ε σ ∗ q k + ε ) − = q − h + ε σq − k + ε . Hence it suffices to show that q − h σq − k = sup ε> ( q − h + ε σq − k + ε ) . (2.10)For every ξ ∈ H , in terms of the expression (2.9) we write( q − h + ε σq − k + ε )( ξ ) = α (( h + ε − ξ, ξ ) + β (( k + ε − ξ, ξ )+ Z (0 , ∞ ) tt ( { ( t ( h + ε − ) : ( k + ε − } ξ, ξ ) dµ ( t )= α (( h + ε − ξ, ξ ) + β (( k + ε − ξ, ξ )+ Z (0 , ∞ ) tt ( { ( t − h + k ) + ε ( t − + 1) } − ξ, ξ ) dµ ( t ) . (2.11)With the spectral decomposition h = R k h k λ de λ one can compute(( h + ε − ξ, ξ ) = Z k h k λ + ε d k e λ ξ k ր Z (0 , k h k ] λ − d k e λ ξ k + ∞k e ξ k = q − h ( ξ )as ε ց
0. Similarly, (( k + ε − ξ, ξ ) ր q − k ( ξ ) and ( { ( t − h + k ) + ε ( t − + 1) } − ξ, ξ ) ր q − t − h + k ( ξ ) as ε ց
0. Since h, k are bounded, one furthermore has q t − h + k = t − q h + q k so that q − t − h + k ( ξ ) = ( t − q h + q k ) − ( ξ ) = (( tq − h ) : q − k )( ξ ). Hence from the monotoneconvergence theorem applied to (2.11) it follows thatsup ε> ( q − h + ε σq − k + ε )( ξ ) = αq − h ( ξ ) + βq − k ( ξ ) + Z (0 , ∞ ) tt (( tq − h ) : q − k )( ξ ) dµ ( t )= ( q − h σq − k )( ξ ) , showing (2.10).Next, assume that h, k are bounded from below. Then we can apply the above case to q h − = q − h , q k − = q − k and σ ∗ in place of q h , q k and σ to have ( q − h σq − k ) − = q h σ ∗ q k .10 roposition 2.8. For any connection σ and positive forms φ, ψ we have φσ ∗ ψ ≤ ( φ − σψ − ) − , i.e., ( φσ ∗ ψ ) − ≥ φ − σψ − . Proof.
Write φ = q h and ψ = q k with h, k ∈ \ B ( H ) + and take the spectral decomposi-tions h = Z ∞ λ de λ + ∞ e ⊥∞ , k = Z ∞ λ df λ + ∞ f ⊥∞ . For each n ∈ N set h n := (1 /n ) e /n + Z (1 /n, ∞ ) λ de λ + ∞ e ⊥∞ ,k n := (1 /n ) f /n + Z (1 /n, ∞ ) λ df λ + ∞ f ⊥∞ . Since q h n ( ξ ) = 1 n k e /n ξ k + Z (1 /n, ∞ ) λ k e λ ξ k + ∞k e ⊥∞ ξ k = Z [0 , /n ] (1 /n − t ) d k e λ ξ k + q h ( ξ ) ց q h ( ξ )as n → ∞ for every ξ ∈ H , one has q h = inf n q h n = Inf n q h n so that q − h = sup n q − h n by [50, Lemma 16,(ii)]. The same holds for q k as well. Since t − q h n ( ξ ) + q k n ( ξ ) ց t − q h ( ξ ) + q k ( ξ ), it follows that t − q h + q k = Inf n ( t − q h n + q k n ) so that( tq − h : q − k ) = ( t − q h + q k ) − = sup n ( t − q h n + q k n ) − = sup n (( tq − h n ) : q − k n ) , t > . From these and the monotone convergence theorem with the expression (2.9) we havesup n ( q − h n σq − k n )( ξ ) = sup n (cid:20) αq − h n ( ξ ) + βq − k n ( ξ ) + Z (0 , ∞ ) tt (( tq − h n ) : q − k n )( ξ ) dµ ( t ) (cid:21) = αq − h ( ξ ) + βq − k ( ξ ) + Z (0 , ∞ ) tt (( tq − h ) : q − k )( ξ ) dµ ( t )= ( q − h σq − k )( ξ ) , ξ ∈ H . Moreover, since h n , k n are bounded from below, Proposition 2.7 implies that q − h n σq − k n =( q h n σ ∗ q k n ) − ≤ ( q h σ ∗ q k ) − for all n . Therefore, q − h σq − k ≤ ( q h σ ∗ q k ) − follows. Example 2.9.
The most studied family of operator means (for bounded positive op-erators) is the weighted geometric means α (0 ≤ α ≤ geometricmean / was first introduced by Pusz and Woronowicz [59] and developed in[4, 5]. The representing function of α (0 < α <
1) is s α = sin αππ Z ∞ ss + t dtt − α , s > . φ, ψ we write( φ α ψ )( ξ ) = sin αππ Z ∞ (( tφ ) : ψ )( ξ ) dtt − α , < α < , (2.12)and φ ψ = φ , φ ψ = ψ . Since ( α )˜= − α and ( α ) ∗ = α obviously, Proposi-tions 2.6 and 2.8 imply that φ α ψ = ψ − α φ, ( φ α ψ ) − ≥ φ − α ψ − for any positive forms φ, ψ . We here give the homogeneity property of α :( r φ ) α ( r ψ ) = r − α r α ( φ α ψ ) , r , r ≥ . (2.13)Indeed, this is clear when r = 0 or r = 0 or α = 0 ,
1. So assume r , r > < α <
1. Then one can easily compute(( r φ ) α ( r ψ ))( ξ ) = sin αππ Z ∞ (( r tφ ) : ( r ψ ))( ξ ) dtt − α = sin αππ Z ∞ r (( r r − tφ ) : ψ )( ξ ) dtt − α = sin αππ Z ∞ r ( tφ : ψ )( ξ ) r − r ( r − r t ) − α dt = r − α r α ( φ α ψ )( ξ ) . Remark 2.10.
It is known [49] that there exists a pair (
A, B ) of non-singular positiveself-adjoint operators such that all of A , B , A − and B − have dense domains and D ( A ) ∩ D ( B ) = D ( A − ) ∩ D ( B − ) = { } . For such
A, B consider h := A and k := B . Then for every t > tq h ) : q k = ( tq − h ) : q − k = 0 for all t >
0. Consider the weighted geometric mean α with 0 < α <
1. Then by (2.12) we have q h α q k = q − h α q − k = 0, and hence( q h α q k ) − = ∞ > q − h α q − k . This shows that ( φσ ∗ ψ ) − = φ − σψ − does nothold for general positive forms when σ = α (0 < α < τ -measurable operators In this section we consider a semi-finite von Neumann algebra M with a trace τ ,and connections for τ -measurable operators will be studied. In § τ -measurable operators needed in later subsections, and among otherthings various behaviors of decreasing sequences of such operators are studied (seeTheorems 3.8 and 3.10 for instance). Then, in § § L + M (whichis a natural ambient space in non-commutative interpolation theory for instance) isconsidered in § § .1 τ -measurability and decreasing sequences We begin with brief explanation on τ -measurable operators together with their general-ized s -numbers. Throughout let M be a semi-finite von Neumann algebra on a Hilbertspace H with a faithful, semi-finite and normal trace τ . A densely defined closed op-erator a affiliated with M is said to be τ -measurable if, for any δ >
0, there exists aprojection e ∈ M such that e H ⊆ D ( a ) (so k ae k < ∞ by the closed graph theorem)and τ ( e ⊥ ) ≤ δ . It is known that a is τ -measurable if and only if τ ( e ( s, ∞ ) ( | a | )) < ∞ for some s >
0, where | a | is the positive part of the polar decomposition. This notionwas introduced in [56]. Let M be the space of τ -measurable operators affiliated with M , and M + be the positive part of M . As usual we simply write a + b and ab forthe strong sum and the strong product. See [56, 71] for more details on τ -measurableoperators.For each a ∈ M we write e I ( | a | ) for the spectral projection of | a | corresponding toan interval I ⊆ [0 , ∞ ). The generalized s -number µ t ( a ) ( t >
0) was introduced in [23]as µ t ( a ) := inf {k ae k ; e is a projection in M with τ (1 − e ) ≤ t } = inf { s ≥ τ ( e ( s, ∞ ) ( | a | )) ≤ t } . This notion is useful in analysis of τ -measurable operators. Note that M is a completemetrizable topological *-algebra whose neighborhood basis of 0 is V ( ε, δ ) := { a ∈ M ; there is a projection e in M such that k ae k ≤ ε, τ (1 − e ) ≤ δ } , ε, δ > , and furthermore (see the proof of [23, Lemma 3.1]), a ∈ V ( ε, δ ) ⇐⇒ µ δ ( a ) ≤ ε. (3.1)We write S for the set of a ∈ M such that τ ( e ( ε, ∞ ) ( | a | )) < ∞ for all ε > t →∞ µ t ( a ) = 0. The space S is a closed (in the measure topology)two-sided ideal of M and contains L p ( M , τ ) for all p ∈ (0 , ∞ ), where L p ( M , τ ) is thenon-commutative L p -space associated with ( M , τ ). These L p -spaces were studied in[16, 73]. The latter directly deals with unbounded operators whereas in the former the L p -space is introduced as the Banach space completion of a certain space of boundedoperators equipped with the p -norm. We recall that the L p -norm is given by k a k p = τ ( | a | p ) /p = (cid:18)Z ∞ µ t ( a ) p dt (cid:19) /p . Decreasingness of a generalized s -number yields the obvious estimate k a k pp ≥ sµ s ( a ) p for each s > µ s ( a ) ≤ k a k p s − /p , s > . (3.2)13ote that when M = B ( H ) with the usual trace, M = B ( H ) and S is the set ofcompact operators on H . Hence, operators in S are called τ -compact operators insome literature.In this subsection we prepare a few basic facts on τ -measurable operators, in par-ticular, on convergence of decreasing sequences in M + , which will be used later inthis section. For each a ∈ M + the corresponding positive form q a has the domain D ( q a ) = D ( a / ) dense in H since a is densely defined.The first four lemmas might be known to experts while we give the proofs forcompleteness. Lemma 3.1.
Let a, b be positive self-adjoint operators affiliated with M . Assume that a ≤ b in the form sense, or equivalently (1 + a ) − ≥ (1 + b ) − . Then b ∈ M + implies a ∈ M + .Proof. Set x := (1 + a ) − and y := (1 + b ) − ; then x, y ∈ M + and x ≥ y . From astandard argument as in the proof of [23, Proposition 2.2] it follows that τ ( e [0 ,α ) ( x )) ≤ τ ( e [0 ,α ) ( y )) for all α >
0. Since e ( s, ∞ ) ( a ) = e [0 , (1+ s ) − ) ( x ) , e ( s, ∞ ) ( b ) = e [0 , (1+ s ) − ) ( y ) , s ≥ , we find that τ ( e ( s, ∞ ) ( a )) ≤ τ ( e ( s, ∞ ) ( b )) for all s ≥
0, showing the assertion.
Lemma 3.2.
Let a, b ∈ M + . The order a ≤ b in M + , i.e., b − a ∈ M + is equivalentto a ≤ b in the form sense.Proof. The proof is standard as follows: a ≤ b in M + ⇐⇒ a ≤ b in M + ⇐⇒ (1 + b ) − / (1 + a )(1 + b ) − / ≤ ⇐⇒ (1 + a ) / (1 + b ) − (1 + a ) / ≤ ⇐⇒ (1 + b ) − ≤ (1 + a ) − , showing the assertion.The following factorization technique is the τ -measurable operator version of [21]and will be repeatedly used throughout the section: Lemma 3.3. If a, b ∈ M + and a ≤ λb for some λ > , then there exists a unique x ∈ s ( b ) M s ( b ) , where s ( b ) is the support projection of b , such that a / = xb / .Moreover, we have k x k ≤ λ / .Proof. Assume that a ≤ λb with λ >
0. From Lemma 3.2 it follows that k a / ξ k ≤ λ k b / ξ k for all ξ ∈ D ( b / ). Since the range of b / is dense in s ( b ) H , there is aunique operator x on H such that xs ( b ) ⊥ = 0 and a / ξ = xb / ξ for all ξ ∈ D ( b / ).Moreover, we have k x k ≤ λ / . From uniqueness we easily see that x ∈ M and hence x = s ( a ) xs ( b ) ∈ s ( b ) M s ( b ). Since D ( b / ) is τ -dense, we have a / = xb / by [71,Chap. I, Proposition 12]. 14 emma 3.4. For every a, b ∈ M + the strong sum a + b in M + coincides with theform sum a ˙+ b , that is, q a + b = q a + q b .Proof. Let h := a + b in M + . By Lemma 3.3 there exist x, y ∈ s ( h ) M s ( h ) such that a / = xh / and b / = yh / , so a / = h / x ∗ and b / = h / y ∗ as well. We have h = h / x ∗ xh / + h / y ∗ yh / = h / ( x ∗ x + y ∗ y ) h / , from which x ∗ x + y ∗ y = s ( h ) follows immediately. Clearly, D ( h / ) ⊆ D ( a / ) ∩ D ( b / ).Moreover, since h / = ( x ∗ x + y ∗ y ) h / = x ∗ a / + y ∗ b / , we have D ( a / ) ∩ D ( b / ) ⊆D ( h / ). Therefore, D ( h / ) = D ( a / ) ∩D ( b / ) = D (( a ˙+ b ) / ). For every ξ ∈ D ( h / ), k a / ξ k + k b / ξ k = k xh / ξ k + k yh / ξ k = (( x ∗ x + y ∗ y ) h / ξ, h / ξ ) = k h / ξ k . Hence q a + q b = q h , that is, a ˙+ b = h follows. Definition 3.5.
Assume that a, b ∈ M + satisfies a ≤ λb for some λ >
0. By Lemma3.3 we have a unique x ∈ s ( b ) M s ( b ) such that a / = xb / . It is convenient for lateruse to introduce the notation T a/b := x ∗ x , which is a unique T ∈ ( s ( b ) M s ( b )) + suchthat a = b / T b / . Moreover, T a/b ≤ λs ( b ) holds. From the proof of Lemma 3.4 wealso note that when a, b ∈ M + and h := a + b , we have T a/h + T b/h = s ( h ). Lemma 3.6.
Let a, a n ∈ M ( n ∈ N ) and assume that sup n k a n k < ∞ . If a n → a inthe measure topology, then a n → a in the strong operator topology.Proof. We may and do assume a = 0. First, let us show the result when M isstandardly represented (by left multiplication) on the Hilbert space L ( M , τ ). Let α := sup n k a n k < + ∞ . For every z ∈ L ( M , τ ), by [23, Lemma 2.5,(vii)] we have k a n z k = Z ∞ µ t ( a n z ) dt ≤ Z ∞ µ t/ ( a n ) µ t/ ( z ) dt. From the assumption a n → µ t/ ( a n ) → , ∞ ) , dt ). Moreover, note that µ t/ ( a n ) µ t/ ( z ) ≤ α µ t/ ( z ) ∈ L ((0 , ∞ ) , dt ). Hence Lebesgue’s dominated convergence theorem can be applied tosee k a n z k → a n → M . To be more precise, we may show that if a n → M with sup n k a n k < ∞ and π : M → M is an isomorphism, then π ( a n ) → M . For this, we may consider the cases of π being an amplification, aninjective induction, and a spatial isomorphism, separately (see [69, Chap. IV, Theorem5.5]). The assertion is obvious for the latter two cases. For the first case, let π be theamplification of M onto M ⊗
H ⊗ K . Then it is obvious that π ( a n )( ξ ⊗ η ) → ξ ∈ H and η ∈ K . The assertion is immediate since { ξ ⊗ η : ξ ∈ H , η ∈ K} istotal in H ⊗ K (and { a n } is uniformly bounded).15he next proposition will be useful in our discussions below, which is probablyknown to experts but we cannot find a suitable reference. Proposition 3.7.
Let a n ∈ M + ( n ∈ N ) . If a n → a in the measure topology for some a ∈ M + , then a n → a in the strong resolvent sense.Proof. Since (1 + a ) − − (1 + a n ) − = (1 + a ) − ( a n − a )(1 + a n ) − , we have for every ε > µ ε ((1 + a ) − − (1 + a n ) − ) ≤ µ ε ( a n − a ) −→ n → ∞ , which means that (1 + a n ) − → (1 + a ) − in the measure topology. HenceLemma 3.6 implies that (1 + a n ) − → (1 + a ) − strongly, that is, a n → a in the strongresolvent sense.Let a n ∈ M + ( n ∈ N ) and assume that a ≥ a ≥ · · · . Then there is a positiveself-adjoint operator a on H such that a n → a in the strong resolvent sense, i.e.,(1 + a n ) − ր (1 + a ) − (see Lemma 2.4). Note that Lemma 3.1 guarantees a ∈ M + .Let us agree to express this situation as a n ց a in the strong resolvent sense. Theorem 3.8.
Let a n ∈ M + ( n ∈ N ) and assume that a n ց a in the strong resolventsense. If a ∈ S + , then a n → a in the measure topology. If < p < ∞ and a ∈ L p ( M , τ ) + , then a ∈ L p ( M , τ ) + and k a n − a k p → .Proof. First, assume a ∈ L ( M , τ ) + . Define ϕ ∈ M ∗ + by ϕ ( x ) := lim n →∞ τ ( a n x ) for x ∈ M . Since 0 ≤ ϕ ≤ τ ( a · ), ϕ is normal, i.e., ϕ ∈ M + ∗ . Hence ϕ = τ ( b · ) for some b ∈ L ( M , τ ) + . Since a n ≥ b for all n , k a n − b k = τ (( a n − b )1) = τ ( a n − ϕ (1) → a = b . Hence a ∈ L ( M , τ ) + and k a n − a k → a ∈ S + and take the spectral decomposition a = R ∞ λ de λ . Letus show that { a n } converges in M . In vew of (3.1) it suffices to prove that for any ε > n such that δ ε ( a n − a m ) ≤ ε for all n, m ≥ n . To do so, let ε > a n into sum of several small pieces and an element of L ( M , τ ) + . Choose r > δ > δµ ε ( a ) < ε and τ ( e ⊥ r ) < ε . Let f := e δ , f := e ⊥ r and f := 1 − ( f + f ). Then τ ( f ) ≤ τ ( f ⊥ ) < ∞ (thanks to a ∈ S ) and τ ( f ) < ε . Write a n = a n f + a n f + a n f = a n f + a n f + f a n f + f a n f + f a n f. Since τ ( f ) < ∞ and k f a f k ≤ r , one has f a f ∈ L ( M , τ ) + . From the case provedabove it follows that k f a n f − b k → b ∈ L ( M , τ ) + . Hence there is an n ∈ N such that µ ε ( f a n f − f a m f ) < ε, n, m ≥ n . (3.3)16ince τ ( f ) < ε , one has µ ε ( f a n f ) ≤ µ ε ( f a n ) = µ ε ( a n f ) = µ / ε ( f a n f ) = 0 , n ∈ N . (3.4)From [23, Lemma 2.5,(vi)] one moreover finds that µ ε ( f a n f ) ≤ µ ε ( f a n ) = µ ε ( a n f ) = µ ε ( f a n f ) ≤ k f a / n k µ ε ( a n ) k a / n f k = k f a n f k / µ ε ( a n ) k f a n f k / ≤ k f a f k µ ε ( a ) ≤ δµ ε ( a ) < ε so that µ ε ( f a n f ) ≤ µ ε ( a n f ) ≤ ε, n ∈ N . (3.5)By [23, Lemma 2.5,(v)] and (3.3)–(3.5) we find that µ ε ( a n − a m ) ≤ µ ε ( a n f ) + µ ε ( a m f ) + µ ε ( f a n f ) + µ ε ( f a m f )+ µ ε ( a n f ) + µ ε ( a m f ) + µ ε ( f a n f ) + µ ε ( f a m f )+ µ ε ( f a n f − f a m f ) ≤ ε for all n, m ≥ n . Since ε > { a n } converges to some b ∈ M + in the measure topology. Since b = a by Proposition 3.7, the first assertionholds. The second follows from the first and [23, Theorem 3.6].The assumption of a ∈ S (or at least a n ∈ S for some n ) is essential for the firstassertion of Theorem 3.8. For this, we may consider a sequence a n in B ( H ) + such that a n ց a but k a n − a k 6→
0. Note that the measure topology on B ( H ) coincides withthe operator norm topology.The next lemma gives a convenient description of the limit in the strong resolventsense of a decreasing sequence in M + . Lemma 3.9.
Let a n , b ∈ M + ( n ∈ N ) be such that a ≥ a ≥ · · · and a ≤ λb forsome λ > . Then T a n /b ( see Definition 3.5 ) is decreasing, and if T := lim n T a n /b ( thestrong limit ) then a n ց b / T b / in the strong resolvent sense.Proof. Since k a / n ξ k = ( T a n /b b / ξ, b / ξ ) is decreasing for every ξ ∈ D ( b / ), we seethat T a n /b is decreasing so that the strong limit T := lim n T a n /b ( ∈ M + ) exists. Now,let a := b / T b / and a := lim n a n in the strong resolvent sense, both of which arein M + (by Lemma 3.1). Since (1 + a ) − ≥ (1 + a n ) − for all n (by Lemma 3.2) and(1 + a n ) − ր (1 + a ) − , one has (1 + a ) − ≥ (1 + a ) − so that a ≤ a in M + by Lemma3.2 again. On the other hand, for every ξ ∈ D ( b / ),( T a/b b / ξ, b / ξ ) = k a / ξ k ≤ k a / n ξ k = ( T a n /b b / ξ, b / ξ ) , which implies T a/b ≤ T a n /b for all n so that T a/b ≤ T . Hence a ≤ a in M + as well.17 heorem 3.10. Let a n , b n ∈ M + and assume that a n ց a and b n ց b in the strongresolvent sense. Then a n + b n ց a + b in the strong resolvent sense. Moreover, if a , b ∈ S + , then a n + b n → a + b in the measure topology. If a , b ∈ L p ( M , τ ) + forsome p ∈ (0 , ∞ ) , then a n + b n → a + b in the ( quasi- ) norm k · k p .Proof. Let h := a + b . Set T n := T a n /h , T := T a/h , S n := T b n /h and S := T b/h ,which are positive operators in s ( h ) M s ( h ). By Lemma 3.9 we have T n ց T and S n ց S strongly, and hence T n + S n ց T + S strongly. It is immediate to see that T ( a n + b n ) /h = T n + S n and T ( a + b ) /h = T + S . Hence the conclusion of the first assertionfollows from Lemma 3.9 again. The latter assertions are immediate from Theorem3.8. Remark 3.11.
Recall (Lemma 2.4) that a n ց a in the strong resolvent sense isequivalently written as q a = Inf n q a n in terms of symbol Inf. Theorem 3.10 says thatfor every decreasing sequences a n and b n in M + ,Inf n ( q a n + q b n ) = (cid:16) Inf n q a n (cid:17) + (cid:16) Inf n q b n (cid:17) . (3.6)Indeed, by Lemma 3.4 and Theorem 3.10,Inf n ( q a n + q b n ) = Inf n q a n + b n = q a + b = q a + q b = (cid:16) Inf n q a n (cid:17) + (cid:16) Inf n q b n (cid:17) . Note that this is not true for general (not τ -measurable) positive self-adjoint operators.See [63, 50] and [65, § § Lemma 3.12.
Let a ∈ S . Let c, c n ∈ M ( n ∈ N ) be such that c n → c in the strong*topology. Then ac n → ac and c n a → ca in the measure topology.Proof. Note that α := sup n k c n k < + ∞ by the uniform boundedness principle. Itsuffices to show only the first convergence. Taking the polar decomposition of a wemay assume a ≥
0. Take the spectral decomposition a = R ∞ λ de λ . For any ε > r > ε such that τ ( e ⊥ r ) < ε , and decompose a as a = a + a + a with a := Z [0 ,ε ] λ de λ , a := Z ( ε,r ] λ de λ , a := Z ( r, ∞ ) λ de λ . Then it is clear that µ ε ( a ) ≤ ε and µ ε ( a ) = 0. Since a ∈ S + , note that τ ( e ( ε,r ] ) < ∞ so that τ ( a ) ≤ r τ ( e ( ε,r ] ) < ∞ . We have µ ε ( a ( c n − c )) ≤ µ ε ( a ( c n − c )) + µ ε ( a ( c n − c )) + µ ε ( a ( c n − c )) ≤ αµ ε ( a ) + µ ε ( a ( c n − c )) + 2 αµ ε ( a ) ≤ αε + µ ε ( a ( c n − c )) . τ ( a ) < ∞ and ( c n − c ) ∗ ( c n − c ) → σ -weakly thanks to k c n k ≤ α ), we have k a ( c n − c ) k = τ ( a ( c n − c ) ∗ ( c n − c )) −→ a ( c n − c ) → n →∞ µ ε ( a − e n ae n ) ≤ αε. Since ε > ac n → ac in the measure topology. As in the previous subsection let M be a semi-finite von Neumann algebra with afaithful semi-finite normal trace τ . Throughout this subsection let σ be any connectionin the sense of Kubo and Ando [52] having the expression in (2.8). We define theconnection σ for positive τ -measurable operators in two different ways and prove thatthey coincide.The first definition is the restriction of Definition 2.5 to positive forms correspondingto τ -measurable operators. For this we first give a lemma. Lemma 3.13.
There exist a constant λ > , depending on σ only, such that φσψ ≤ λ ( φ + ψ ) for all positive forms φ, ψ .Proof. Let φ, ψ be arbitrary positive forms. Theorem 2.2 in particular implies that φ : ψ ≤ φ and φ : ψ ≤ ψ . Therefore, from (2.9) we observe φσψ ≤ αφ + βψ + Z (0 , tt tφ dµ ( t ) + Z (1 , ∞ ) tt ψ dµ ( t ) ≤ αφ + βψ + 2 µ ((0 , φ + 2 µ ((1 , ∞ )) ψ. Hence φσψ ≤ λ ( φ + ψ ) with λ := max { α + 2 µ ((0 , , β + 2 µ ((1 , ∞ )) } . Proposition 3.14.
For every a, b ∈ M + there exist a unique h ∈ M + such that q a σq b = q h . Proof.
By Lemmas 3.13 and 3.4 we have q a σq b ≤ λ ( q a + q b ) = λq a + b . This implies that the domain of q a σq b is dense in H . Hence there is a positive self-adjoint operator h such that q a σq b = q h . For every unitary u ′ ∈ M ′ it follows from thedefinition (2.9) and [50, Corollary 7] that u ′∗ ( q a σq b ) u ′ = αu ′∗ q a u ′ + βu ′∗ q b u ′ + Z (0 , ∞ ) tt (( tu ′∗ q a u ′ ) : ( u ′∗ q b u ′ )) dµ ( t ) . u ′∗ q a u ′ = q u ′∗ au ′ = q a and similarly for q b , we have u ′∗ ( q a σq b ) u ′ = a a σq b , i.e., u ′∗ q h u ′ = q h , which means that u ′∗ hu ′ = h . Therefore, h is affiliated with M . Moreover,since q h ≤ λq a + b , we have h ≤ λ ( a + b ) in the form sense so that h ∈ M + by Lemma3.1. Finally, the uniqueness of h is obvious. Definition 3.15 (First) . Proposition 3.14 says that for each a, b ∈ M + we can define aσb ∈ M + by the condition q aσb = q a σq b . Thus, we have the connection σ : M + ×M + → M + . When M = B ( H ) = M , this connection σ reduces to the connection σ in [52] on B ( H ) + × B ( H ) + , see [50, § Definition 3.16 (Second) . For every a, b ∈ M + choose an h ∈ M + such that a + b ≤ λh for some λ >
0. With T a/h and T b/h (see Definition 3.5) we define the connection a [ σ ] b ∈ M + by a [ σ ] b := h / ( T a/h σT b/h ) h / , (3.7)where T a/h σT b/h is the connection [52] for bounded positive operators. The definitionis justified by Lemma 3.18 below, so we have the connection [ σ ] : M + × M + → M + .We use the symbol [ σ ] to distinguish it from σ in the previous definition while theircoincidence will shortly be proved.Recall [52] that the transformer inequality for connections σ on B ( H ) + holds truein the following slightly more general situation than (II) of § C ∗ ( AσB ) C ≤ ( C ∗ AC ) σ ( C ∗ BC ) , A, B ∈ B ( H ) + , C ∈ B ( H ) . Moreover, concerning the equality case of the transformer inequality, the followingresult was shown in [25, Theorem 3], which we state as a lemma to use in provingLemma 3.18 and Theorem 3.31:
Lemma 3.17.
Let
A, B ∈ B ( H ) + and C ∈ B ( H ) be such that s ( A + B ) ≤ s ( CC ∗ ) ,where s ( · ) denotes the support projection. Then for any connection σ , C ∗ ( AσB ) C = ( C ∗ AC ) σ ( C ∗ BC ) . Lemma 3.18.
For every a, b ∈ M + the a [ σ ] b given in (3.7) is an element of M + determined independently of the choice of h as in Definition 3.16.Proof. For any h ∈ M + with a + b ≤ λh for some λ >
0, since T a/h , T b/h ∈ M + , wehave for every unitary u ′ ∈ M ′ , u ′∗ ( T a/h σT b/h ) u ′ = ( u ′∗ T a/h u ′ ) σ ( u ′∗ T b/h u ′ ) = T a/h σT b/h . Hence T a/h σT b/h ∈ M + so that the right hand side of (3.7) is indeed in M + . To provethe independence of the choice of h , let k ∈ M + be another choice such as h . We may20ssume that h ≤ k . In fact, we may consider the cases of h, h + k and of k, h + k . Thenby Lemma 3.3 we have an x ∈ s ( h ) M s ( k ) such that h / = xk / . Since h / T a/h h / = a = k / T a/k k / = h / x ∗ T a/k xh / , one has T a/h = x ∗ T a/k x and similarly T b/h = x ∗ T b/k x . Since s ( a + b ) ≤ s ( h ) = s ( xx ∗ ),it follows from Lemma 3.17 that k / ( T a/k σT b/k ) k / = h / x ∗ ( T a/k σT b/k ) xh / = h / (( x ∗ T a/k x ) σ ( x ∗ T b/k x )) h / = h / ( T a/h σT b/h ) h / . In particular, when b ≤ λa for some λ >
0, taking h = a in (3.7) we write a [ σ ] b = a / f ( T b/a ) a / . (3.8)Note that T B/A = A − / BA − / when A, B ∈ B ( H ) + with A invertible. Therefore, (3.8)is an extended version of the familiar formula (for Kubo-Ando connections) AσB = A / f ( A − / BA − / ) A / for A invertible.In the rest of the subsection we will prove that Definitions 3.15 and 3.16 areequivalent. The next lemma is similar to the familiar convergence formula AσB =lim ε ց ( A + εI ) σ ( B + εI ) for A, B ∈ B ( H ) + . Lemma 3.19.
Let a, b, h ∈ M + be such that a + b ≤ λh for some λ > . Then wehave a [ σ ] b = lim ε ց ( a + εh )[ σ ]( b + εh ) , (3.9) a decreasing limit in the strong resolvent sense.Proof. For any ε > a + εh )[ σ ]( b + εh ) = h / ( T ( a + εh ) /h σT ( b + εh ) /h ) h / = h / (( T a/h + εs ( h )) σ ( T b/h + εs ( h ))) h / . Since T a/h + εs ( h ) ց T a/h and T b/h + εs ( h ) ց T b/h strongly (even in the operator norm)as ε ց
0, we have ( T a/h + εs ( h )) σ ( T b/h + εs ( h )) ց T a/h σT b/h strongly as ε ց § Remark 3.20.
A typical choice of h in the above lemma is h = a + b . In this case,when both of a, b are in S + , it follows from Theorem 3.8 that (3.9) becomes the limit inthe measure topology. Moreover, when a, b ∈ L p ( M, τ ) + with p ∈ (0 , ∞ ) and h = a + b ,(3.9) becomes the limit in the (quasi-)norm k · k p .21 emma 3.21. Let a, b ∈ M + . For every ξ ∈ D ( a / ) ∩ D ( b / ) we have q a [:] b ( ξ ) = inf { q a ( η ) + q b ( ζ ); η, ζ ∈ H , η + ζ = ξ } , (3.10) where a [:] b is given in Definition 3.16 for parallel sum.Proof. Firstly we recall the well-known variational formula of A : B for A, B ∈ B ( H ) + ,that is, for every ξ ∈ H ,(( A : B ) ξ, ξ ) = inf { ( Aη, η ) + (
Bζ , ζ ); η, ζ ∈ H , η + ζ = ξ } (3.11)(see [2, Theorem 9] and also [33, Lemma 3.1.5]). For any ξ ∈ H write q ( ξ ) for the righthand side of (3.10). Let h := a + b so that D ( h / ) = D ( a / ) ∩ D ( b / ) (see Lemma3.4). Assume ξ ∈ D ( a / ) ∩ D ( b / ). From (3.7) for parallel sum we have q a [:] b ( ξ ) = k ( a [:] b ) / ξ k = k ( T a/h : T b/h ) / h / ξ k = inf {k T / a/h η k + k T / b/h ζ k ; η, ζ ∈ H , η + ζ = h / ξ } thanks to (3.11). For every η, ζ ∈ H with η + ζ = h / ξ choose a sequence η n ∈ D ( h / )such that h / η n → s ( h ) η . Let ζ n := ξ − η n ∈ D ( h / ); then η n + ζ n = ξ and h / ζ n = h / ξ − h / η n −→ h / ξ − s ( h ) η = s ( h )( h / ξ − η ) = s ( h ) ζ . We hence find that k T / a/h η k + k T / b/h ζ k = k T / a/h s ( h ) η k + k T / b/h s ( h ) ζ k = lim n →∞ (cid:0) k T / a/h h / η n k + k T / b/h h / ζ n k (cid:1) = lim n →∞ ( q a ( η n ) + q b ( ζ n )) ≥ q ( ξ ) , which implies q a [:] b ( ξ ) ≥ q ( ξ ). Conversely, let η, ζ ∈ H with η + ζ = ξ . If η
6∈ D ( a / )or ζ
6∈ D ( b / ), then q a ( η ) + q b ( ζ ) = ∞ . So we may assume that η ∈ D ( a / ) and ζ ∈ D ( b / ). Then, since ξ ∈ D ( a / ) ∩ D ( b / ), we must have η, ζ ∈ D ( a / ) ∩ D ( b / ) = D ( h / ) and therefore q a ( η ) + q b ( ζ ) = k T / a/h h / η k + k T / b/h h / ζ k ≥ k ( T a/h : T b/h ) / h / ξ k = k ( a [:] b ) / ξ k . Hence q ( ξ ) ≥ q a [:] b ( ξ ) follows and the assertion has been shown. Lemma 3.22.
Let a, b ∈ M + . Then q a [:] b coincides with q a : q b given in Definition 2.1.Hence we have a [:] b = a : b .Proof. First, assume that λ − b ≤ a ≤ λb for some λ >
0. Then, from (3.8) it isimmediately verified that (1 : λ − ) a ≤ a [:] b ≤ (1 : λ ) a , so we have D ( a / ) = D ( b / ) = D (( a [:] b ) / ). Hence both sides of (3.10) are ∞ when ξ ∈ H \ D ( a / ), so the equality(3.10) holds for all ξ ∈ H . This means that when λ − b ≤ a ≤ λb , the quadratic form22efined as in (2.3) (for q a , q b ), i.e., the right hand side of (3.10) for all ξ ∈ H is equalto q a [:] b . Since the latter is lower semi-continuous on H , the procedure of inf ξ n → ξ lim inf n →∞ in (2.4) is not needed, and by Theorem 2.2 we have q a [:] b = q a : q b in this case.For general a, b ∈ M + let h := a + b . The convergence in (3.9) for parallel sum isrewritten as q a [:] b = Inf ε> q ( a + εh )[:]( b + εh ) , as well as q a = Inf ε> q a + εh and q b = Inf ε> q b + εh . For each ε >
0, since λ − ( b + εh ) ≤ a + εh ≤ λ ( b + εh ) for some λ >
0, it follows from the above case that q ( a + εh )[:]( b + εh ) = q a + εh : q b + εh . Hence by (2.6) we obtain q a [:] b = Inf ε> ( q a + εh : q b + εh ) = (cid:16) Inf ε> q a + εh (cid:17) : (cid:16) Inf ε> q b + εh (cid:17) = q a : q b , and the assertion has been shown.Our main result of the subsection is the following: Theorem 3.23.
For every a, b ∈ M + we have q a [ σ ] b = q a σq b and hence a [ σ ] b = aσb. Proof.
Consider the integral expressions in (2.8) and (2.9) for σ . First assume α, β > h := a + b . For every ξ ∈ D ( h / ) = D ( a / ) ∩ D ( b / ) we have q a [ σ ] b ( ξ ) = k ( a [ σ ] b ) / ξ k = (cid:0) ( T a/h σT b/h ) h / ξ, h / ξ (cid:1) = α k T / a/h h / ξ k + β k T / b/h h / ξ k + Z (0 , ∞ ) tt k (( tT a/h ) : T b/h ) / h / ξ k dµ ( t )= αq a ( ξ ) + βq b ( ξ ) + Z (0 , ∞ ) tt q ( ta )[:] b ( ξ ) dµ ( t )= αq a ( ξ ) + βq b ( ξ ) + Z (0 , ∞ ) tt (( tq a ) : q b )( ξ ) dµ ( t )= ( q a σq b )( ξ ) . In the above we have used q ( ta )[:] b = ( tq a ) : q b due to Lemma 3.22. Since a [ σ ] b ≥ h / ( αT a/h + βT b/h ) h / = αa + βb , note that q a [ σ ] b ( ξ ) = ∞ for all ξ ∈ H \ D ( h / ).Since ( q a σq b )( ξ ) ≥ αq a ( ξ )+ βq b ( ξ ), note also that ( q a σq b )( ξ ) = ∞ for all ξ ∈ H\D ( h / ).Therefore, q a [ σ ] b ( ξ ) = ( q a σq b )( ξ ) for all ξ ∈ H .Next, for general σ with the representing operator monotone function f on (0 , ∞ )we consider the connection σ with representing function f ( t ) := f ( t ) + 1 + t . Theabove proved case gives q a [ σ ] b = q a σ q b . (3.12)23e moreover have q a σ q b = q a σq b + q a + q b = q aσb + q a + q b = q aσb + a + b , (3.13)where the second equality is Definition 3.15 and the last equality is due to Lemma 3.4.On the other hand, letting h := a + b , by the definition (3.7) we have( a + εh )[ σ ]( b + εh ) = h / ( T ( a + εh ) /h σ T ( b + εh ) /h ) h / = h / ( T ( a + εh ) /h σT ( b + εh ) /h + T ( a + εh ) /h + T ( b + εh ) /h ) h / = ( a + εh )[ σ ]( b + εh ) + ( a + εh ) + ( b + εh )for every ε >
0. Hence by Lemma 3.19 and Theorem 3.10, letting ε ց a [ σ ] b = a [ σ ] b + a + b , that is, q a [ σ ] b = q a [ σ ] b + a + b . (3.14)Combining (3.12)–(3.14) implies that a [ σ ] b + a + b = aσb + a + b (in M + ), from which a [ σ ] b = aσb follows.From now on we will use the same symbol σ for both definitions of Definitions 3.15and 3.16. From the definition (3.7) the following is clear: Proposition 3.24.
Let σ , σ be connections with the representing operator monotonefunctions f , f on (0 , ∞ ) respectively. If f ( t ) ≤ f ( t ) for all t ∈ (0 , ∞ ) , then aσ b ≤ aσ b for all a, b ∈ M + . In particular, for every symmetric (i.e., ˜ σ = σ ) operator mean σ we have2( a : b ) ≤ aσb ≤ a + b , a, b ∈ M + . (3.15)Furthermore, we note that (3.15) holds for general positive forms φ, ψ , that is, 2( φ : ψ ) ≤ φσψ ≤ ( φ + ψ ) /
2. (The proof of this is not difficult from the definition (2.9) andleft to the reader.) But we do not know whether Proposition 3.24 holds for positiveforms in general.
In this subsection we extend properties of connections for bounded positive operatorsto those for τ -measurable operators.Let σ be any connection corresponding to an operator monotone function f > , ∞ ), originally defined on B ( H ) + in [52] and extended to M + in the previoussubsection. Theorem 3.25.
Let a, b, a i , b i ∈ M + . (1) (Monotonicity). If a ≤ a and b ≤ b , then a σb ≤ a σb . If a n , b n ∈ M + ( n ∈ N ) , a n ց a and b n ց b in thestrong resolvent sense, then a n σb n ց aσb in the strong resolvent sense and hence Inf n ( a n σb n ) = (cid:16) Inf n a n (cid:17) σ (cid:16) Inf n b n (cid:17) . (3.16)(3) (Concavity). We have ( a + a ) σ ( b + b ) ≥ a σb + a σb . (4) (Transpose). We have a ˜ σb = bσa .Proof. All of (1)–(4) are immediately seen by applying the same properties for boundedpositive operators to T a/h σT b/h in the definition (3.7). The properties (1), (3) and (4)are also obvious by Definition 3.15 since connections of positive forms satisfy those(see [50] and Proposition 2.6). We here prove (2) for instance. Let h := a + b .Since T a n /h ց T a/h and T b n /h ց T b/h strongly by Lemma 3.9, one has T a n /h σT b n /h ց T a/h σT b/h strongly. Therefore, by Lemma 3.9 again one has a n σb n = h / ( T a n /h σT b n /h ) h / ց h / ( T a/h σT b/h ) h / = aσb in the strong resolvent sense, which is also written as (3.16) due to Lemma 2.4.Let M ++ := { a ∈ M + ; a − ∈ M + } . The next proposition extends the formula Aσ ∗ B = ( A − σB − ) − for A, B ∈ B ( H ) ++ to τ -measurable operators. Proposition 3.26 (Adjoint) . Let σ ∗ be the adjoint of σ . For every a, b ∈ M + we have ( q a σ ∗ q b ) − = q − a σq − b . Moreover, if a, b ∈ M ++ , then aσ ∗ b, a − σb − ∈ M ++ and aσ ∗ b = ( a − σb − ) − . Proof.
Let a, b ∈ M + . With the spectral decomposition a = R ∞ λ de λ set a n :=(1 /n ) e n + R (1 /n, ∞ ) λ de λ and similarly b n for n ∈ N . Since a n ց a and b n ց b in thestrong resolvent sense (even in the measure topology), it follows from Theorem 3.25,(2)that a n σ ∗ b n ց aσ ∗ b in the strong resolvent sense, that is, q a σ ∗ q b = Inf n ( q a n σ ∗ q b n ).Therefore, ( q a σ ∗ q b ) − = sup n ( q a n σ ∗ q b n ) − = sup n ( q − a n σq − b n ) ≤ q − a σq − b , where we have used [50, Lemma 16,(ii)] for the first equality and Proposition 2.7 for thesecond. Since Proposition 2.8 gives the reverse inequality, the first assertion follows.Next, assume a, b ∈ M ++ . Then aσ ∗ b, a − σb − ∈ M + . In this case, the firstassertion means that q ( aσ ∗ b ) − = q a − σb − and so ( aσ ∗ b ) − = a − σb − . This implies thelatter assertion.We note that the convergence a n σb n ց aσb in the measure topology does not hold ingeneral even when a n ց a and b n ց b in the measure topology. In fact, it is known [32,Example 4.5] that there are A, B ∈ B ( H ) + such that lim ε ց k ( A + εI ) B k > k A B k .The situation being so, the next result seems rather best possible for the convergencein the measure topology. 25 roposition 3.27. Let a n , b n ∈ M + be such that a n ց a and b n ց b in the strongresolvent sense. Assume that one of the following conditions is satisfied: (1) a , b ∈ S , (2) a ∈ S and lim t →∞ f ( t ) /t = 0 , (3) b ∈ S and f (0 + ) = 0 .Then a n σb n ∈ S for all n and a n σb n ց aσb in the measure topology.Proof. By Theorem 3.25,(2) we have a n σb n ց aσb in the strong resolvent sense. Once a σb ∈ S is shown, by Theorem 3.8 we have a n σb n ց aσb in the measure topology.The case (1) is obvious since a σb ≤ λ ( a + b ) for some λ > h ∈ M + such that h ≥ a + b ≤ h . We write a σb = h / ( T a /h σT b /h ) h / . Since b = h / T b /h h / is in S and h ≥
1, it followsthat T b /h ∈ M + is in S as well. Note that T a /h σT b /h ≤ σT b /h = f ( T b /h ) . (3.17)Since f (0 + ) = 0 and f is strictly increasing on (0 , ∞ ), by [23, Lemma 2.5,(iv)] we have µ t ( f ( T b /h )) = f ( µ t ( T b /h )) for all t >
0, which implies that f ( T b /h ) ∈ S and hence T a /h σT b /h ∈ S thanks to (3.17). Therefore, a σb ∈ S follows as well. The case (2)is shown immediately from (3) since a σb = b ˜ σa and ˜ f (0 + ) = lim t →∞ f ( t ) /t = 0,where ˜ f is the transpose of f . Theorem 3.28 (Transformer inequality) . For every a, b ∈ M + and c ∈ M we have c ∗ ( aσb ) c ≤ ( c ∗ ac ) σ ( c ∗ bc ) . (3.18)We first prove the case of parallel sum as a lemma. Lemma 3.29.
For every a, b ∈ M + and c ∈ M we have c ∗ ( a : b ) c ≤ ( c ∗ ac ) : ( c ∗ bc ) .Proof. By Theorem 2.2 it suffices to prove that, for every η, ζ ∈ H , q c ∗ ( a : b ) c ( η + ζ ) ≤ q c ∗ ac ( η ) + q c ∗ bc ( ζ ) . Let the above right hand side be finite, that is, η ∈ D (( c ∗ ac ) / ) and ζ ∈ D (( c ∗ bc ) / ).Take the polar decomposition a / c = v | a / c | . Since ( c ∗ ac ) / = | a / c | = v ∗ a / c (thestrong product), note that { ξ ∈ H ; ξ ∈ D ( c ) , cξ ∈ D ( a / ) } is a core of ( c ∗ ac ) / .Hence one can choose a sequence η n such that η n → η , η n ∈ D ( c ), cη n ∈ D ( a / ), and q c ∗ ac ( η n ) → q c ∗ ac ( η ). For each η n one has also q c ∗ ac ( η n ) = k v ∗ a / cη n k = k a / cη n k = q a ( cη n ) . (3.19)Similarly, one can choose a sequence ζ n such that ζ n → ζ , ζ n ∈ D ( c ), cζ n ∈ D ( b / ), q c ∗ bc ( ζ n ) → q c ∗ bc ( ζ ), and q c ∗ bc ( ζ n ) = q b ( cζ n ). Since a : b ≤ a, b , note that cη n , cζ n ∈ (( a : b ) / ) and a similar argument as in (3.19) gives q c ∗ ( a : b ) c ( η n + ζ n ) = q a : b ( cη n + cζ n ).From the lower semi-continuity of q c ∗ ( a : b ) c it follows that q c ∗ ( a : b ) c ( η + ζ ) ≤ lim inf n →∞ q c ∗ ( a : b ) c ( η n + ζ n ) = lim inf n →∞ q a : b ( cη n + cζ n ) ≤ lim inf n →∞ ( q a ( cη n ) + q b ( cζ n )) = q c ∗ ac ( η ) + q c ∗ bc ( ζ )due to Theorem 2.2 for the latter inequality. Proof of Theorem 3.28.
To prove this, we use the definition (2.9) and Definition 3.15.Assume first that α, β > q c ∗ ( aσb ) c ( ξ ) ≤ q ( c ∗ ac ) σ ( c ∗ bc ) ( ξ ) , ξ ∈ H . (3.20)Let the right hand side be finite. Then by the assumption α, β > q c ∗ ac ( ξ ) < ∞ and q c ∗ bc ( ξ ) < ∞ so that q c ∗ ( a + b ) c ( ξ ) < ∞ by Lemma 3.4. This implies that ξ ∈ D (( a + b ) / c ). Hence one can choose a sequence ξ n such that ξ n ∈ D ( c ), cξ n ∈ D (( a + b ) / )and k ( a + b ) / c ( ξ n − ξ ) k →
0. By Lemmas 3.13 and 3.4 one has q ( c ∗ ac ) σ ( c ∗ bc ) ( ξ n − ξ ) ≤ λ ( q c ∗ ac + q c ∗ bc )( ξ n − ξ ) = λq c ∗ ( a + b ) c ( ξ n − ξ )= λ k ( a + b ) / c ( ξ n − ξ ) k −→ . Moreover, since q aσb ≤ λq a + b means that aσb ≤ λ ( a + b ) (in the form sense), one has q c ∗ ( aσb ) c ( ξ n − ξ ) = k ( aσb ) / c ( ξ n − ξ ) k ≤ λ k ( a + b ) / c ( ξ n − ξ ) k −→ . These imply that q ( c ∗ ac ) σ ( c ∗ bc ) ( ξ ) = lim n →∞ q ( c ∗ ac ) σ ( c ∗ bc ) ( ξ n ) , q c ∗ ( aσb ) c ( ξ ) = lim n →∞ q c ∗ ( aσb ) c ( ξ n ) . Hence we may prove (3.20) for ξ such that ξ ∈ D ( c ) and cξ ∈ D (( a + b ) / ). For eachsuch ξ we have by (2.9) q c ∗ ( aσb ) c ( ξ ) = k ( aσb ) / cξ k = q aσb ( cξ )= αq a ( cξ ) + βq b ( cξ ) + Z (0 , ∞ ) tt q ( ta ): b ( cξ ) dµ ( t )= αq c ∗ ac ( ξ ) + βq c ∗ bc ( ξ ) + Z (0 , ∞ ) tt q c ∗ (( ta ): b ) c ( ξ ) dµ ( t ) ≤ αq c ∗ ac ( ξ ) + βq c ∗ bc ( ξ ) + Z (0 , ∞ ) tt q ( c ∗ ( ta ) c ):( c ∗ bc ) ( ξ ) dµ ( t )= q ( c ∗ ac ) σ ( c ∗ bc ) ( ξ ) , where Lemma 3.29 has been used for the above inequality. Therefore, (3.20) has beenshown so that (3.18) follows.For general σ we use the same trick as in the proof of Theorem 3.23. Let σ bethe connection corresponding to f ( t ) + 1 + t . The above proved case gives c ∗ ( aσ b ) c ≤ ( c ∗ ac ) σ ( c ∗ bc ). Since c ∗ ( aσ b ) c = c ∗ ( aσb + a + b ) c and ( c ∗ ac ) σ ( c ∗ bc ) = ( c ∗ ac ) σ ( c ∗ bc ) + c ∗ ac + c ∗ bc as in (3.13), we obtain (3.18) for general σ .27 emark 3.30. When φ, ψ are general positive forms and c is a bounded operator, thetransformer inequality c ∗ ( φ : ψ ) c ≤ ( c ∗ φc ) : ( c ∗ ψc )was shown in [50, Corollary 7] with the definition ( c ∗ φc )( ξ ) := φ ( cξ ), ξ ∈ H . Thistransformer inequality immediately extends to φσψ from the definition (2.9). Notethat if a ∈ M + and c ∈ M , then q a ( cξ ) = q c ∗ ac ( ξ ) for all ξ ∈ H since D ( a / c ) = { ξ ∈H ; cξ ∈ D ( a / ) } (i.e., a / c is closed without taking closure) in this case. Therefore,Theorem 3.28 is included in [50] when c ∈ M .The next theorem shows the equality case in the transformer inequality (3.18). Theorem 3.31.
Let a, b ∈ M + and c ∈ M be such that s ( a + b ) ≤ s ( cc ∗ ) . Then c ∗ ( aσb ) c = ( c ∗ ac ) σ ( c ∗ bc ) for any connection σ .Proof. Let h := a + b . For any connection σ , by Definition 3.16 we write c ∗ ( aσb ) c = c ∗ h / ( T a/h σT b/h ) h / c. Let c ∗ h / = vk / be the polar decomposition with k / := | c ∗ h / | so that k = h / cc ∗ h / and s ( k ) = s ( h ) since s ( h ) ≤ s ( cc ∗ ). Furthermore, set ˜ a := k / T a/h k / and ˜ b := k / T b/h k / . We then have ˜ a + ˜ b = k / ( T a/h + T b/h ) k / = k / s ( h ) k / = k , T ˜ a/k = T a/h and T ˜ b/k = T b/h since s ( T a/h ) , s ( T b/h ) ≤ s ( h ) = s ( k ) (see Definition 3.5).Therefore,˜ aσ ˜ b = k / ( T ˜ a/k σT ˜ b/k ) k / = v ∗ c ∗ h / ( T a/h σT b/h ) h / cv = v ∗ c ∗ ( aσb ) cv so that c ∗ ( aσb ) c = v (˜ aσ ˜ b ) v ∗ . Since v ˜ av ∗ = vk / T a/h k / v ∗ = c ∗ h / T a/h h / c = c ∗ ac and similarly v ˜ bv ∗ = c ∗ bc , it suffices to show that v (˜ aσ ˜ b ) v ∗ = ( v ˜ av ∗ ) σ ( v ˜ bv ∗ ) . (3.21)Note that v ˜ av ∗ + v ˜ bv ∗ = c ∗ hc = vkv ∗ and ( vkv ∗ ) / = vk / v ∗ , and it is immediate tosee that T v ˜ av ∗ /vkv ∗ = vT ˜ a/k v ∗ and T v ˜ bv ∗ /vkv ∗ = vT ˜ b/k v ∗ . Hence (3.21) follows since( v ˜ av ∗ ) σ ( v ˜ bv ∗ ) = vk / v ∗ (( vT ˜ a/k v ∗ ) σ ( vT ˜ b/k v ∗ )) vk / v ∗ = vk / v ∗ v ( T ˜ a/k σT ˜ b/k ) v ∗ vk / v ∗ = vk / ( T ˜ a/k σT ˜ b/k ) k / v ∗ = v (˜ aσ ˜ b ) v ∗ , where the second equality in the above is due to Lemma 3.17 since s ( T ˜ a/k + T ˜ b/k ) = s (˜ a + ˜ b ) = s ( k ) = s ( v ∗ v ). 28n particular, when a ∈ M ++ , Theorem 3.31 shows aσb = a / f ( a − / ba − / ) a / , (3.22)which is the generalization of the familiar expression of AσB for
A, B ∈ B ( H ) + with A invertible.Now, we consider M ⊗ M , the tensor product of M and the 2 × M = M ( C ), equipped with the faithful semi-finite normal trace τ ⊗ Tr, where Tr isthe usual trace on M . The space M ⊗ M of τ ⊗ Tr-measurable operators is identifiedwith
M ⊗ M in the natural way.The following result is the τ -measurable operator version of [4, Theorem I.1]: Lemma 3.32.
For every a, b, c ∈ M + , (cid:20) a cc ∗ b (cid:21) ≥ in M ⊗ M if and only if a, b ∈M + and c = a / zb / for some contraction z ∈ M .Proof. Assume that a, b ≥ c = a / zb / with z ∈ M , k z k ≤
1. Then (cid:20) a cc ∗ b (cid:21) = (cid:20) a a / zb / b / z ∗ a / b (cid:21) = (cid:20) a / b / (cid:21) (cid:20) zz ∗ (cid:21) (cid:20) a / b / (cid:21) ≥ . Conversely, assume that (cid:20) a cc ∗ b (cid:21) ≥
0. Let e := s ( a ) and f := s ( b ). Setting (cid:20) h mm ∗ k (cid:21) = (cid:20) a cc ∗ b (cid:21) / we write (cid:20) a cc ∗ b (cid:21) = (cid:20) h mm ∗ k (cid:21) = (cid:20) h + mm ∗ hm + mkm ∗ h + km ∗ m ∗ m + k (cid:21) , so that a = h + mm ∗ , b = m ∗ m + k , c = hm + mk. Since h , mm ∗ ≤ a , there are x, x ∈ e M e such that h = xa / and | m ∗ | = x a / .Since k , m ∗ m ≤ b , there are y, y ∈ f M f such that k = yb / and | m | = y b / . Withthe polar decompositions m = v | m | and m ∗ = w | m ∗ | we write c = hv | m | + | m ∗ | w ∗ k = a / x ∗ vy b / + a / x ∗ w ∗ yb / = a / zb / with z := x ∗ vy + x ∗ w ∗ y ∈ e M f . Therefore,0 ≤ (cid:20) a cc ∗ b (cid:21) = (cid:20) a a / zb / b / z ∗ a / b (cid:21) = (cid:20) a / b / (cid:21) (cid:20) e zz ∗ f (cid:21) (cid:20) a / b / (cid:21) , and it remains to show k z k ≤
1. For every ε >
0, multiplying (cid:20) ( ε + a / ) −
00 ( ε + b / ) − (cid:21) to the above both sides, we have0 ≤ " a / ε + a / b / ε + b / e zz ∗ f (cid:21) " a / ε + a / b / ε + b / . ε ց ≤ (cid:20) e f (cid:21) (cid:20) e zz ∗ f (cid:21) (cid:20) e f (cid:21) = (cid:20) e zz ∗ f (cid:21) thanks to z ∈ e M f . This implies (cid:20) zz ∗ (cid:21) ≥
0, which is equivalent to k z k ≤ × τ -measurable operators: Proposition 3.33.
For every a, b ∈ M + we have a b = max (cid:26) c ∈ M + ; (cid:20) a cc b (cid:21) ≥ in M ⊗ M (cid:27) . Proof.
Let h := a + b and a / = xh / , b / = yh / with x, y ∈ e M e , where e := s ( h ).Since a b = h / ( T a/h T b/h ) h / , one has (cid:20) a a ba b b (cid:21) = (cid:20) h / T a/h h / h / ( T a/h T b/h ) h / h / ( T a/h T b/h ) h / h / T b/h h / (cid:21) = (cid:20) h / h / (cid:21) (cid:20) T a/h T a/h T b/h T a/h T b/h T b/h (cid:21) (cid:20) h / h / (cid:21) ≥ . Let c ∈ M + be such that (cid:20) a cc b (cid:21) ≥
0. By Lemma 3.32 there is a contraction z ∈ M such that c = a / zb / and so c = h / x ∗ zyh / = h / ˆ zh / , where ˆ z := x ∗ zy ∈ e M e .Since h / ε + h / ˆ z h / ε + h / = ( ε + h / ) − c ( ε + h / ) − ≥ , ε > , one has ˆ z ≥ ε ց
0. Furthermore, since0 ≤ (cid:20) a cc b (cid:21) = (cid:20) h / T a/h h / h / ˆ zh / h / ˆ zh / h / T b/h h / (cid:21) = (cid:20) h / h / (cid:21) (cid:20) T a/h ˆ z ˆ z T b/h (cid:21) (cid:20) h / h / (cid:21) and (cid:20) T a/h ˆ z ˆ z T b/h (cid:21) ∈ (cid:20) e e (cid:21) ( M ⊗ M ) (cid:20) e e (cid:21) , it follows as above that (cid:20) T a/h ˆ z ˆ z T b/h (cid:21) ≥ z ≤ T a/h T b/h . This gives c ≤ h / ( T a/h T b/h ) h / = a b .The following is the extension of a similar characterization of the parallel sum (see[4]) to the setting of τ -measurable operators: Proposition 3.34.
For every a, b ∈ M + , a : b = max (cid:26) c ∈ M + ; (cid:20) a b (cid:21) ≥ (cid:20) c cc c (cid:21) in M ⊗ M (cid:27) . roof. Let h := (cid:20) a b (cid:21) and k := (cid:20) a : b a : ba : b a : b (cid:21) . Then h, k ≥ M ⊗ M and h / = (cid:20) a / b / (cid:21) , k / = 1 √ (cid:20) ( a : b ) / ( a : b ) / ( a : b ) / ( a : b ) / (cid:21) . Hence D ( h / ) = D ( a / ) ⊕ D ( b / ) and D ( k / ) = D (( a : b ) / ) ⊕ D (( a : b ) / ). Notethat D ( a / ) ⊆ D (( a : b ) / ) and D ( b / ) ⊆ D (( a : b ) / ) so that D ( h / ) ⊆ D ( k / ).For every η ∈ D ( a / ) and ζ ∈ D ( b / ) we have k h / ( η ⊕ ζ ) k = k a / η k + k b / ζ k = q a ( η ) + q b ( ζ ) ≥ q a : b ( η + ζ ) = k k / ( η ⊕ ζ ) k . In the above we have used Theorem 2.2 together with Definition 3.15. Therefore, h ≥ k follows (see Lemma 3.2).Let c ∈ M + be such that (cid:20) a b (cid:21) ≥ (cid:20) c cc c (cid:21) . Let z := (cid:20) c cc c (cid:21) . Note that z / = √ (cid:20) c / c / c / c / (cid:21) and D ( z / ) = D ( c / ) ⊕ D ( c / ). Hence h ≥ z means that D ( a / ) ⊆D ( c / ), D ( b / ) ⊆ D ( c / ) and for every η ∈ D ( a / ) and ζ ∈ D ( b / ), k a / η k + k b / ζ k = k h / ( η ⊕ ζ ) k ≥ k z / ( η ⊕ ζ ) k = k c / ( η + ζ ) k . This implies that q c ( η + ζ ) ≤ q a ( η ) + q b ( ζ ) for all η, ζ ∈ H . Hence Theorem 2.2 gives q c ≤ q a : q b , that is, c ≤ a : b . L ( M , τ ) + M The subspace L ( M , τ ) + M of M appears in several situations related to ( M , τ ),for instance, in interpolation theory (e.g., [44, 72]), majorization theory (e.g., [31],[55, Chap. 3]), symmetric operator ideals (e.g., [55, Chap. 2]) and so on. Indeed,the integral R s µ t ( a ) dt in (ii) below is of fundamental importance in a twofold sense.Firstly it is a natural continuous analogue of the Ky Fan norm (see [9]) for compactoperators (i.e., the sum of the first s largest eigenvalues), and secondly it is nothingbut the K -functional in real interpolation theory (see [8] and [23, p. 289]):inf {k b k + t k c k ; a = b + c } (cid:0) = K ( t, a ; L ( M , τ ) , M ) (cid:1) . In this subsection we present further results on connections aσb when a, b ∈ M + arein L ( M ) + M .It is well-known [31, Proposition 1.2] that the following conditions for a ∈ M areequivalent:(i) a ∈ L ( M , τ ) + M ;(ii) R t µ s ( a ) ds < + ∞ for some (equivalently, for all) t > | a | e ( r, ∞ ) ( | a | ) ∈ L ( M , τ ) for some r ≥ L p for L p ( M , τ ) and( L + M ) + for ( L + M ) ∩ M + . Note that L ∩ M ⊆ L p ⊆ L + M for all p ∈ [1 , ∞ ].Since ax, xa ∈ L for any a ∈ L + M and x ∈ L ∩ M , one can define a duality pairingbetween L + M and L ∩ M by h a, x i τ := τ ( ax ) (= τ ( xa )) , a ∈ L + M , x ∈ L ∩ M . By the condition (iii) it is clear that ( L + M ) + = L + M + and so h a, x i τ ≥ x ∈ ( L + M ) + and x ∈ L ∩ M + . In particular, when τ (1) < + ∞ , of course L + M = L and L ∩ M = M so that the above duality pairing is the usual dualitybetween L ∼ = M ∗ and M .The first lemma might be well-known to experts while we give a proof for complete-ness. Lemma 3.35.
Let a, b ∈ ( L + M ) + . If h a, x i τ ≤ h b, x i τ for all x ∈ L ∩ M + , then a ≤ b . Hence, if h a, x i τ = h b, x i τ for all x ∈ L ∩ M + , then a = b .Proof. Take the Jordan decomposition b − a = ( b − a ) + − ( b − a ) − . Let h := ( b − a ) − and e := s ( h ). For every x ∈ L ∩ M + we note that0 ≤ h h, x i τ = h e ( a − b ) e, x i τ = h a − b, exe i τ ≤ , where the inequality follows from the assumption since e ( L ∩ M + ) e ⊆ L ∩ M + obviously. Hence h h, x i τ = 0 for all x ∈ L ∩ M + . Now let us show h = 0 (whichmeans a ≤ b ).By the condition (iii) above we can choose an r ≥ f := e ( r, ∞ ) ( h ) satisfies τ ( f ) < + ∞ , hf ∈ L and hf ⊥ ∈ M + . Since f M + f ⊂ L ∩ M + , one has τ ( hf x ) = τ ( hf xf ) = 0 for all x ∈ M + so that hf = 0. One also has τ ( hf ⊥ x ) = τ ( hf ⊥ xf ⊥ ) = 0for all x ∈ L ∩ M + . Since L ∩ M + is dense in L , it follows that τ ( hf ⊥ x ) = 0 forall x ∈ L and hence hf ⊥ = 0. Therefore, h = hf + hf ⊥ = 0.The following two lemmas will be needed in proving subsequent theorems: Lemma 3.36.
Let a ∈ ( L + M ) + , x , x ∈ L ∩ M and k , k ∈ M . Then x ∗ x a / k k ∗ a / ∈ L , x a / k , x a / k ∈ L and τ ( x ∗ x a / k k ∗ a / ) = τ ( k ∗ a / x ∗ x a / k ) = ( x a / k , x a / k ) with the inner product ( · , · ) in L .Proof. Choose an r ≥ af ∈ L and af ⊥ ∈ M + with f := e ( r, ∞ ) ( a ).Then a / = a / f + a / f ⊥ ∈ L + M so that a / k , k ∗ a / ∈ L + M . Since x ∗ x ∈ L ∩ M ⊆ L obviously, one has x ∗ x a / k ∈ L ∩ L (noting L · L ⊆ L , L · M ⊆ L , M · L ⊆ L and L · M ⊆ L ). Therefore, x ∗ x a / k k ∗ a / ∈ L and τ ( x ∗ x a / k k ∗ a / ) = τ ( k ∗ a / x ∗ x a / k ) . Moreover, one has x a / k , x a / k ∈ L so that the above quantity is written as( x a / k , x a / k ). 32 emma 3.37. Let a, a n ∈ ( L + M ) + for n ∈ N . If a n ց a in the strong resolventsense, then h a n , x i τ ց h a, x i τ for every x ∈ L ∩ M + .Proof. By the assumption it is clear that h a n , x i τ decreases and h a n , x i τ ≥ h a, x i τ . Soit suffices to show that inf n h a n , x i τ = h a, x i τ . For each δ > a δ := (1 + δa ) − a (= δ − (1 − (1 + δa ) − )) and a n,δ := (1 + δa n ) − a n . Note that h a n , x i τ = h a n − a n,δ , x i τ + h a n,δ , x i τ = h δ (1 + δa n ) − a n , x i τ + h a n,δ , x i τ . (3.23)Moreover, by the assumption note that for each δ > a n,δ ց a δ strongly in M + so that h a n,δ , x i τ = τ ( a n,δ x ) ց τ ( a δ x ) = h a δ , x i τ (3.24)as n → ∞ . Since s ≥ δ (1 + δs ) − s is a continuous increasing function, by [23,Lemma 2.5,(iv), (iii)] we have µ t ( δ (1 + δa n ) − a n ) = δµ t ( a n ) δµ t ( a n ) ≤ δµ t ( a ) δµ t ( a ) , t > . Therefore, h δ (1 + δa n ) − a n , x i τ ≤ τ (cid:0)(cid:12)(cid:12) δ (1 + δa n ) − a n x (cid:12)(cid:12)(cid:1) = Z ∞ µ t (cid:0) δ (1 + δa n ) − a n x (cid:1) dt (by [23, Proposition 2.7]) ≤ Z ∞ µ t/ (cid:0) δ (1 + δa n ) − a n (cid:1) µ t/ ( x ) dt (by [23, Lemma 2.5,(vii)])= 2 Z ∞ µ t (cid:0) δ (1 + δa n ) − a n (cid:1) µ t ( x ) dt ≤ Z ∞ δµ t ( a ) δµ t ( a ) µ t ( x ) dt ≤ k x k ∞ Z r µ t ( a ) dt + 2 δµ r ( a ) Z ∞ r µ t ( x ) dt ≤ k x k ∞ Z r µ t ( a ) dt + 2 δµ r ( a ) k x k (3.25)for any r >
0. Since lim r ց R r µ t ( a ) dt = 0 (due to the condition (ii) at the beginningof this subsection), for each ε > r > k x k ∞ R r µ t ( a ) dt ≤ ε/ δ > δµ r ( a ) k x k ≤ ε/
2. For such a δ > h a n , x i τ ≤ ε + h a n,δ , x i τ so that inf n h a n , x i τ ≤ ε + h a δ , x i τ ≤ ε + h a, x i τ thanks to (3.24), which implies inf n h a n , x i τ = h a, x i τ as desired.33ote that if a, b ∈ ( L + M ) + then aσb ∈ ( L + M ) + for any connection σ (seeLemma 3.13). The next theorem is an extension of the variational formula (3.11) (forbounded positive operators) to a : b for a, b ∈ ( L + M ) + . Theorem 3.38.
For every a, b ∈ ( L + M ) + and x ∈ L ∩ M we have h a : b, x ∗ x i τ = inf {h a, y ∗ y i τ + h b, z ∗ z i τ ; y, z ∈ L ∩ M , y + z = x } . (3.26) Proof.
Let h := a + b . By Theorem 3.25,(2) and Lemma 3.37 note that, with x, y, z ∈ L ∩ M , h a : b, x ∗ x i τ = inf ε> h ( a + εh ) : ( b + εh ) , x ∗ x i τ and inf y + z = x {h a, y ∗ y i τ + h b, z ∗ z i τ } = inf y + z = x inf ε> {h a + εh, y ∗ y i τ + h b + εh, z ∗ z i τ } = inf ε> inf y + z = x {h a + εh, y ∗ y i τ + h b + εh, z ∗ z i τ } . So it suffices to prove (3.26) when λ − b ≤ a ≤ λb for some λ >
0. In this case, let e := s ( a ) = s ( b ) ∈ M and take a k ∈ e M e with a / = kh / and T a/h = k ∗ k . Asimmediately seen, note that k is invertible in e M e . Since T a/h + T b/h = e (see Definition3.5), we have a : b = h / ( T a/h : T b/h ) h / = h / T a/h ( T a/h + T b/h ) − T b/h h / = h / k ∗ k ( e − k ∗ k ) h / = h / k ∗ ( e − kk ∗ ) kh / = a / ( e − kk ∗ ) a / . Now let x, y, z ∈ L ∩ M with y + z = x , and using Lemma 3.36 we compute h a, y ∗ y i τ + h b, z ∗ z i τ − h a : b, x ∗ x i τ = τ (( x − z ) ∗ ( x − z ) a ) + τ ( z ∗ zb ) − τ ( x ∗ x ( a − a / kk ∗ a / ))= τ ( z ∗ za ) + τ ( z ∗ zb ) − τ ( x ∗ za ) − τ ( z ∗ xa ) + τ ( x ∗ xa / kk ∗ a / )= τ ( z ∗ zh ) + τ ( k ∗ a / x ∗ xa / k ) − τ ( x ∗ za / k ∗− k ∗ a / )= k zh / k + k xa / k k − za / k ∗− , xa / k )= k zh / k + k xa / k k − zh / , xa / k ) (3.27)thanks to h / = k − a / = a / k ∗− with inverses k − , k ∗− in e M e . The Schwarzinequality gives h a, y ∗ y i τ + h b, z ∗ z i τ − h a : b, x ∗ x i τ ≥ . We note that { zh / : z ∈ L ∩ M} ⊆ L e . Let us show that the left hand side hereis dense in L e in the norm k · k . For this it suffices to show that if w ∈ L annihilatesthe above left hand side (i.e., τ ( wzh / ) = 0 for all z ∈ L ∩M ), then w annihilates L e ,equivalently, ew = 0. So assume that w ∈ L satisfies τ ( wzh / ) = 0 for all z ∈ L ∩M .Take the spectral decomposition h = R ∞ λ de λ . For each n ∈ N let f n := e n − e /n .34hen, for every z ∈ L ∩ M , from zf n ∈ L ∩ M we have τ ( wzf n h / ) = 0. Note that f n h / is bounded and invertible in f n M f n . Hence f n h / w ∈ L and τ ( f n h / wz ) = 0for all z ∈ L ∩ M . Since L ∩ M is dense in L , it follows that f n h / w = 0 and hence f n w = 0, so τ ( f n ww ∗ ) = 0. Since f n ր e , we have τ ( eww ∗ ) = 0, i.e., ew = 0.Since xa / k ∈ L e by Lemma 3.36, from what we have just shown one can choosea sequence z n ∈ L ∩ M such that k z n h / − xa / k k →
0. Letting y n := x − z n andputting y = y n , z = z n in (3.27) one has h a, y ∗ n y n i τ + h b, z ∗ n z n i τ − h a : b, x ∗ x i τ = k z n h / k + k xa / k k − z n h / , xa / k ) −→ . Thus, the expression (3.26) has been obtained.Note by Lemma 3.35 that a : b for a, b ∈ ( L + M ) + can be determined by thevariational expression (3.26). a α b and | a − α b α | Consider the following conditions for a ∈ M :(i) R δ log + µ s ( a ) ds < ∞ for some (equivalently, any) δ >
0, where log + t stands formax { log t, } for t ≥ R δ µ s ( a ) q ds < ∞ for some (equivalently, any) δ > q > ⇒ (i) is immediately seen. If µ s ( a ) ≤ Cs − β (0 < s < δ ) forsome (equivalently, any) δ > C, β > a ∈ L p ( M , τ ) forsome p ∈ (0 , ∞ ], see (3.2)), then (ii) is satisfied. Whenever a ∈ M satisfies (i), wedefine Λ t ( a ) ∈ [0 , ∞ ) for t > t ( a ) := exp Z t log µ s ( a ) ds. This is a generalization of the
Fuglede-Kadison determinant ∆( a ) [24] for a ∈ M . Infact, we have ∆ τ (1) ( a ) = ∆( a ) when M is finite and τ (1) < ∞ . The majorization prop-erties in terms of Λ t ( a ) ( t >
0) are useful [22, 23] to derive trace and norm inequalitiesfor τ -measurable operators. We remark that the Fuglede-Kadison determinant in sucha generalized setting and certain related estimates (in logarithmic submajorization) aresubjects in the recent article [19]. Lemma 3.39. If a, b ∈ M satisfy (2 ◦ ) , then a + b , ab and a α ( α > satisfy the same.Proof. Assume that R δ µ s ( a ) p ds < ∞ and R δ µ s ( b ) q ds < ∞ for some δ, p, q >
0. By[23, Lemma 2.5] note that µ s ( a + b ) ≤ µ s/ ( a ) + µ s/ ( b ) and µ s ( ab ) ≤ µ s/ ( a ) µ s/ ( b ) for35ll s >
0. With 1 /r = 1 /p + 1 /q H¨older’s inequality gives Z δ µ s ( ab ) r ds ≤ (cid:18)Z δ µ s/ ( a ) p ds (cid:19) r/p (cid:18)Z δ µ s/ ( b ) q ds (cid:19) r/q = (cid:18) Z δ/ µ s ( a ) p ds (cid:19) r/p (cid:18) Z δ/ µ s ( b ) q ds (cid:19) r/q < ∞ . With r := min { p, q } one has Z δ µ s ( a + b ) r ds ≤ r Z δ ( µ s/ ( a ) r + µ s/ ( b ) r ) ds ≤ r (cid:18)Z δ (1 + µ s/ ( a ) p ) ds + Z δ (1 + µ s/ ( b ) q ) ds (cid:19) < ∞ . The assertion for a α is clear since µ s ( a α ) = µ s ( a ) α .The next theorem is the main result of [48] while it was shown in the case a, b ∈ S + ,but the whole proof is valid in the present setting. The proof in [48] starts with theinequality Λ t ( ab ) ≤ Λ t ( a )Λ t ( b ) , t > , for a, b ∈ S (:= M ∩ S ) in [22], which indeed holds true for a, b ∈ M satisfying (ii) asshown in [23, pp. 287–288]. In the proofs in [23, 48] the equalitylim p ց (cid:18) Z t φ ( s ) p dst (cid:19) /p = exp Z t log φ ( s ) ds if Z t φ ( s ) q ds < ∞ for some q > φ ( s ) = µ s ( ab ), µ s ( | ab | r ), etc., where the condition (ii) is essential. Theorem 3.40 ([48]) . If a, b ∈ M + satisfy ( ii ) , then Λ t ( | ab | r ) ≤ Λ t ( a r b r ) , t > , r ≥ , (3.28) or equivalently, Λ t ( a r b r ) ≤ Λ t ( | ab | r ) , t > , < r ≤ . (3.29)The aim of this subsection is to compare a α b (see Example 2.9) and | a − α b α | for0 ≤ α ≤ Theorem 3.41. If a, b ∈ M + satisfy (2 ◦ ) , then Λ t ( a α b ) ≤ Λ t ( a − α b α ) , t > , ≤ α ≤ . (3.30)36 roof. The case α = 0 or 1 is trivial, so assume 0 < α <
1. Since Λ t ( a α b ) =Λ t ( b − α a ) and Λ t ( a − α b α ) = Λ t ( b α a − α ), we may assume 1 / ≤ α <
1. For each n ∈ N let a n := (1 /n ) + a . By Lemma 3.39 note that a n and a − / n ba − / n satisfy (ii).We have Λ t ( a α b ) ≤ Λ t ( a n α b )= Λ t ( a / n ( a − / n ba / n ) α a / n ) (by (3.22))= Λ t (( a − / n ba − / n ) α/ a / n ) ≤ Λ t ( | ( a − / n ba − / n ) / a / αn | α ) (by (3.29))= Λ t ( a / αn ( a − / n ba − / n ) a / αn ) α = Λ t ( | b / a − α α n | α ) ≤ Λ t ( b α a − αn ) (by (3.28) since 2 α ≥ t ( a − αn b α ) . Now, to prove (3.30), it suffices to show thatΛ t ( a − α b α ) = lim n →∞ Λ t ( a − αn b α ) . (3.31)Recalling continuity of an operator function in the measure topology [68] (as long as afunction in question is continuous), we have | a − αn b α | = b α a − α ) n b α ց b α a − α ) b α = | a − α b | in the measure topology as n → ∞ . Hence by [23, Lemma 3.4,(ii)], µ s ( a − αn b α ) ց µ s ( a − α b α ) as n → ∞ for a.e. s ∈ (0 , t ). Since R t log + µ s ( a − α b α ) ds < ∞ from (ii), onecan apply the monotone convergence theorem to log + µ s ( a − α b α ) − log µ s ( a − αn b α ) ≥ Remark 3.42.
Let a, b be as stated in Theorems 3.40 and 3.41. Theorem 3.40 saysthat r > Λ t ( | a r b r | /r ) is monotone increasing for each t >
0. Hence Theorem 3.41furthermore implies that for 0 < r ≤ q ,Λ t (( a r α b r ) /r ) ≤ Λ t ( | a q (1 − α ) b qα | /q ) , t > , ≤ α ≤ . (3.32) Problem 3.43.
For any a, b as above we conjecture that r > Λ t (( a r α b r ) /r ) ismonotone decreasing for each t >
0. This is well-known for the B ( H ) case (in par-ticular, for matrices) as log-majorization, whose proof is based on the anti-symmetrictensor power technique, see, e.g., [32, 33]. The technique is not at our disposal in thevon Neumann algebra setting. If the conjecture is true, then (3.32) holds for all r, q > L p -spaces In this section let M be a general von Neumann algebra on a Hilbert space H , and wewill discuss connections on Haagerup’s L p -spaces. A brief description of Haagerup’s L p -spaces L p ( M ) for 0 < p ≤ ∞ is given in Appendix A for the reader’s convenience. The37asis of Haagerup’s L p ( M ) is the crossed product von Neumann algebra R := M ⋊ σ R with respect to the modular automorphism group σ t for a faithful normal semi-finiteweight ϕ on M . Note that R is semi-finite with the canonical trace τ , and consider thespace R of τ -measurable operators affiliated with R . Then L p ( M )’s are constructedinside R by (A.4) in terms of the dual action θ s (see Appendix A).In particular, when M is semi-finite with a faithful semi-finite normal trace τ , wenote that all the results in this section hold true in the setting of the conventional L p -spaces L p ( M , τ ) ([16, 73]) with respect to τ . Note that Haagerup’s L p ( M ) in thissituation is identified (up to an isometric isomorphism) with L p ( M , τ ), and all theresults below reduce to those for L p ( M , τ ). In fact, some of them are included in § L p ( M , τ ) setting.Now, let σ be any connection. As defined in the previous section, for every a, b ∈ R + we have aσb ∈ R + . The next lemma says that we have the connection σ : L p ( M ) + × L p ( M ) + → L p ( M ) + by restricting σ to L p ( M ) + for any p ∈ (0 , ∞ ]. Lemma 4.1.
Let < p ≤ ∞ . If a, b ∈ L p ( M ) + , then aσb ∈ L p ( M ) + .Proof. Let h := a + b ∈ L p ( M ) + . Let x, y ∈ s ( h ) R s ( h ) such that a / = xh / and b / = yh / as in Definition 3.5. Here note that the support projection s ( h ) is in M .Applying θ s to a / = xh / gives θ s ( a ) / = θ s ( x ) θ s ( h ) / . Since θ s ( a ) = e − s/p a and θ s ( h ) = e − s/p h for all s ∈ R , one has a / = θ s ( x ) h / . Hence from the uniqueness of x it follows that θ s ( x ) = x for all s ∈ R so that T a/h = x ∗ x ∈ M + , and similarly T b/h = y ∗ y ∈ M + . Therefore, one has aσb = h / ( T a/h σT b/h ) h / ∈ L p ( M ) + immediately.The connection aσb for a, b ∈ L p ( M ) + does not depend upon the choice of a faithfulsemi-finite normal weight ϕ on M . In fact, for another faithful semi-finite normalweight ϕ on M we have the (canonical) isomorphism κ : R → R := M ⋊ σ ϕ R .The κ induces an isometric isomorphism from L p ( M ) in R and that in R (see theremark at the end of Appendix A), for which we have κ ( aσb ) = κ ( a ) σκ ( b ) for any a, b ∈ L p ( M ) + .By restricting the properties of Theorem 3.25 (for R + ) to L p ( M ) + for any p ∈ (0 , ∞ ],it is clear that all of them hold for a, b, a i , b i ∈ L p ( M ) + . But the decreasing convergenceholds more strongly in the norm k · k p as follows. For this it is worthwhile to first givethe variant of Theorem 3.8 for a decreasing sequence in L p ( M ) + . Proposition 4.2.
Let < p < ∞ . If a n ∈ L p ( M ) + ( n ∈ N ) and a n ց a in the strongresolvent sense, then a ∈ L p ( M ) + and k a n − a k p → .Proof. Since L p ( M ) ⊆ S (defined inside R ) by [45, Lemma B], Theorem 3.8 impliesthat a n ց a in the measure topology. By [71, Chap. II, Proposition 26] or [45, LemmaB] note that the k · k p -norm topology on L p ( M ) coincides with the relative topologyinduced from the measure topology on R . (More precisely we have µ t ( x ) = t − /p k x k p for x ∈ L p ( M ).) Hence we have the conclusion.Now, the next theorem follows from Propositions 3.27 and 4.2.38 heorem 4.3. Let < p < ∞ . If a n , b n ∈ L p ( M ) + ( n ∈ N ) , a n ց a and b n ց b inthe strong resolvent sense, then a, b ∈ L p ( M ) + and k a n σb n − aσb k p → . For a, b ∈ L p ( M ) + and c ∈ L q ( M ) with 0 < p, q ≤ ∞ we have the transformerinequality c ∗ ( aσb ) c ≤ ( c ∗ ac ) σ ( c ∗ bc ) in L r ( M ) (1 /p + 2 /q = 1 /r ) by Theorem 3.28 (andH¨older’s inequality for Haagerup’s L p -spaces). Furthermore, Theorem 3.31 shows that c ∗ ( aσb ) c = ( c ∗ ac ) σ ( c ∗ bc ) in L r ( M ) whenever s ( a + b ) ≤ s ( cc ∗ ). In particular, when p = 1 and r = ∞ , this says that c ∗ ( φσψ ) c = ( c ∗ φc ) σ ( c ∗ ψc ) (4.1)for every φ, ψ ∈ M + ∗ and c ∈ M with s ( φ + ψ ) ≤ s ( cc ∗ ).The following proposition is the variational expression of the parallel sum of a, b ∈ L p ( M ), 1 ≤ p < ∞ , in terms of the L p - L q -duality. When p = 1, this reduces to theexpression of φ : ψ for φ, ψ ∈ M + ∗ as( φ : ψ )( x ∗ x ) = inf { φ ( y ∗ y ) + ψ ( z ∗ z ); y, z ∈ M , y + z = x } , x ∈ M . (4.2) Proposition 4.4.
Let ≤ p ≤ ∞ and /p + 1 /q = 1 . For every a, b ∈ L p ( M ) + wehave h a : b, x ∗ x i p,q = inf {h a, y ∗ y i p,q + h b, z ∗ z i p,q ; y, z ∈ L q ( M ) , y + z = x } for every x ∈ L q ( M ) , where h a, c i p,q = tr( ac ) for a ∈ L p ( M ) and c ∈ L q ( M ) ( see (A.5)) .Proof. The case p = ∞ (hence q = 1) is the well-known variational expression (see(3.11)) since M = L ∞ ( M ) standardly acts on L ( M ) (by left mulitplication) and h a, x ∗ x i ∞ , = tr( ax ∗ x ) = ( ax ∗ , x ∗ ) for x ∈ L ( M ).Assume 1 ≤ p < ∞ . Let h := a + b and e := s ( h ) ∈ M . The proof below is similarto that of Theorem 3.38 (with h· , ·i p,q in place of h· , ·i τ ), so we only sketch it. As before(using Theorem 4.3 in the present case) we may assume that λ − b ≤ a ≤ λb for some λ >
0, and take a k ∈ e M e , invertible in e M e , such that a / = kh / and T a/h = k ∗ k (see the proof of Lemma 4.1). As in the previous proof we have a : b = a / ( e − kk ∗ ) a / and for y, z ∈ L q ( M ) with y + z = x , h a, y ∗ y i p,q + h b, z ∗ z i p,q − h a : b, x ∗ x i p,q = k zh / k + k xa / k k − zh / , xa / k ) . The remaining proof is the same as in the proof of Theorem 3.38 since L q ( M ) h / isdense in L ( M ) e and xa / k ∈ L ( M ) e .The integral in the definition (2.9) of φσψ is given in the weak sense (i.e., in theevaluation at each ξ ∈ H ). In the following we give the integral expression of aσb for a, b ∈ L p ( M ), 1 ≤ p < ∞ , in the strong sense of Bochner integral (see, e.g., [15]):39 roposition 4.5. Let ≤ p < ∞ and a, b ∈ L p ( M ) . Then the L p ( M ) -valued function t tt (( ta ) : b ) is Bochner integrable on (0 , ∞ ) with respect to the measure µ and aσb = αa + βb + Z (0 , ∞ ) tt (( ta ) : b ) dµ ( t ) . Proof.
Let h := a + b . From the expression (2.8) applied to A = T a/h and B = T b/h we have the integral expression of aσb in the weak sense in terms of the L p - L q -duality.That is, for every c ∈ L q ( M ), 1 /p + 1 /q = 1, since h / ch / ∈ L ( M ) ( ∼ = M ∗ ), wehave tr c ( a : b ) = tr ( h / ch / )( T a/h σT b/h )= α tr ( h / ch / ) T a/h + β tr ( h / ch / ) T b/h + Z (0 , ∞ ) tt tr ( h / ch / )(( tT a/h ) : T b/h ) dµ ( t )= α tr ca + β tr cb + Z (0 , ∞ ) tt tr c (( ta ) : b ) dµ ( t ) . Let us show that the integral R (0 , ∞ ) 1+ tt (( ta ) : b ) dµ ( t ) exists as a Bochner integral.When λ − ≤ a ≤ λb for some λ >
0, since ( ta ) : b = b / (( tT a/b ) : e ) b / (where e := s ( b )) and t ( tT a/b ) : e is continuous in the operator norm, it follows that t ( ta ) : b is continuous in the norm k · k p . For general a, b ∈ L p ( M ) + , since( ta + εh ) : ( b + εh ) → ( ta ) : b in the norm k · k p as ε ց t ( ta ) : b is strongly measurable on (0 , ∞ ) in the norm k · k p . Furthermore, since k ( ta ) : b k p ≤ k ( t ( a + b )) : ( a + b ) k p = t t k a + b k p , we have R (0 , ∞ ) 1+ tt k ( ta ) : b k p dµ ( t ) < ∞ . Hence the result follows.Consider the tensor products M ⊗ M with ϕ ⊗ Tr, where ϕ is a faithful semi-finitenormal weight on M and Tr is the usual trace on M = M ( C ). Since σ ϕ ⊗ Tr t = σ ϕ t ⊗ id with the identity map id on M , we have( M ⊗ M ) ⋊ σ ϕ ⊗ id R = R ⊗ M , and the dual action on R ⊗ M is θ s ⊗ id , where θ s is the dual action on R . Hence L p ( M ⊗ M ) is identified with L p ( M ) ⊗ M and its positive part is ( L p ( M ) ⊗ M ) ∩ ( R ⊗ M ) + . In this setting we can write the expressions of Propositions 3.33 and 3.34restricted to a, b ∈ L p ( M ) + , 0 < p ≤ ∞ , as follows: a b = max (cid:26) c ∈ L p ( M ) + ; (cid:20) a cc b (cid:21) ≥ L p ( M ) ⊗ M (cid:27) ,a : b = max (cid:26) c ∈ L p ( M ) + ; (cid:20) a b (cid:21) ≥ (cid:20) c cc c (cid:21) in L p ( M ) ⊗ M (cid:27) . L p -norm inequality for the weighted geo-metric mean, which is a consequence of Theorem 3.41. Similarly to [48, Theorem 4] akey idea of the proof is to use the following formulas (see [45, 23]) for every a ∈ L p ( M )where 0 < p ≤ ∞ : µ t ( a ) = t − /p k a k p , Λ t ( a ) = (( et − ) /p k a k p ) t , t > , (4.3)as already mentioned in the proof of Proposition 4.2. Note that they hold for the case p = ∞ as well. Indeed, if a ∈ L ∞ ( M ) = M , then any spectral projection of | a | is θ -invariant, so for any s < k a k ∞ we have τ ( e ( s, k a k ∞ ] ( | a | ) = ∞ . Hence µ t ( a ) = k a k ∞ forall t >
0, showing the formulas in (4.3) for p = ∞ .Another fact we need in the proof below is that for any connection σ , θ s ( xσy ) = θ s ( x ) σθ s ( y ) , s ∈ R , x, y ∈ R + , (4.4)which is easily verified as follows: With h := x + y we have θ s ( T x/h σT y/h ) = θ s ( T x/h ) σθ s ( T y/h ) = T θ s ( x ) /θ s ( h ) σT θ s ( y ) /θ s ( h ) so that θ s ( xσy ) = θ s ( h / ( T x/h σT y/h ) h / )= θ s ( h ) / ( T θ s ( x ) /θ s ( h ) σT θ s ( y ) /θ s ( h ) ) θ s ( h ) / = θ s ( x ) σθ s ( y ) . (Note that Lemma 4.1 is also immediate from the fact (4.4).) Theorem 4.6.
Let p , p , p ∈ (0 , ∞ ] and ≤ α ≤ be such that /p = (1 − α ) /p + α/p . Assume a ∈ L p ( M ) + and b ∈ L p ( M ) + . Then we have a α b , | a − α b α | ∈ L p ( M ) + and k a α b k p ≤ k a − α b α k p ≤ k a k − αp k b k αp . (4.5) Proof.
Since the case α = 0 or 1 is trivial, we may assume 0 < α <
1. From (4.3) notethat x ∈ R satisfies the condition (ii) of § x ∈ L p ( M ) for some p ∈ (0 , ∞ ].From (4.4) and (2.13) we have θ s ( a α b ) = ( e − s/p a ) α ( e − s/p b )= e − s ((1 − α ) /p + α/p ) ( a α b ) = e − s/p ( a α b ) , s ∈ R , showing a α b ∈ L p ( M ). On the other hand, from H¨older’s inequality for Haagerup’s L p -spaces, we find that a − α b α ∈ L p ( M ) and k a − α b α k p ≤ k a − α k p / (1 − α ) k b α k p /α = k a k − αp k b k αp , which is the second inequality of (4.5). By (3.30) (for a, b ∈ R + ) and (4.3) we furtherhave e /p k a α b k p = Λ ( a α b ) ≤ Λ ( a − α b α ) = e /p k a − α b α k p , which gives the first inequality of (4.5). 41 emark 4.7. In the situation of Theorem 4.6, for any r > k ( a r α b r ) /r k p ≤ k | a r (1 − α ) b rα | /r k p ≤ k a k − αp k b k αp by applying (4.5) to a r , b r in view of 1 / ( p/r ) = (1 − α ) / ( p /r ) + α/ ( p /r ). Since r >
7→ k | a r (1 − α ) b rα | /r k p is monotone increasing [48, Theorem 4], we have k ( a r α b r ) /r k p ≤ k | a q (1 − α ) b qα | /q k p , < r ≤ q. If the conjecture of Problem 3.43 is true, then it follows that r >
7→ k ( a r α b r ) /r k p is monotone decreasing and the above inequality holds for all independent r, q > We have studied connections for various classes of unbounded objects. In every caseparallel sums are building blocks for connections (see Definition 2.5, § § § § M be a von Neumann algebra on a Hilbert space H .We fix a faithful semi-finite normal weight χ on the commutant M ′ and use Connes’spatial derivatives (see Appendix § B.1).
For notational convenience we set P ( M , C ) := the set of all semi-finite normal weights on M . A (densely defined) positive self-adjoint operator T acting on H is said to be ( − -homogeneous (relative to χ ) if T it y ′ = σ ′− t ( y ′ ) T it for each t ∈ R and y ′ ∈ M ′ . Here, { σ ′ t } t ∈ R is the modular automorphism group on M ′ induced by χ , and T it is understood to be defined on the support of T . We take aweight φ ∈ P ( M , C ) and the spatial derivative dφ/dχ will be considered (see Appendix § B.1). When φ is faithful, we have σ ′ t ( y ′ ) = ( dφ/dχ ) − it y ′ ( dφ/dχ ) it for y ′ ∈ M ′ ([13, Theorem 9], and also see (i), (ii) at the beginning of § B.3) and hence( dφ/dχ ) it y ′ = ( dφ/dχ ) it y ′ ( dφ/dχ ) − it ( dφ/dχ ) it = σ ′− t ( y ′ )( dφ/dχ ) it , dφ/dχ is ( − − T of the form T = dφ/dχ for some φ ∈ P ( M , C ) (see [13, Theorem 13]).We set M + − := the set of all ( − H . From the explanation so far φ ∈ P ( M , C ) ←→ dφ/dχ ∈ M + − (5.1)is an order preserving one-to-one correspondence.Here is our strategy (for investigating a notion of parallel sums of weights): Areasonable theory on parallel sums of positive forms was worked out in [50]. A positiveform means a lower semi-continuous positive quadratic form defined everywhere in H with the value + ∞ allowed, i.e., an element in the extended positive part \ B ( H ) + (see § φ, ψ ∈ P ( M , C ) so that we have(densely defined) positive self-adjoint operators dφ/dχ and dψ/dχ . Regard them aspositive forms, which means that dφ/dχ for instance is identified with q dφ/dχ ( ξ ) = ( k ( dφ/dχ ) / ξ k when ξ ∈ D (( dφ/dχ ) / ) , + ∞ when ξ
6∈ D (( dφ/dχ ) / ) . As positive forms we define their parallel sum, i.e.,( dφ/dχ ) : ( dψ/dχ ) = (cid:0) ( dφ/dχ ) − + ( dψ/dχ ) − (cid:1) − . Since ( dφ/dχ ) : ( dψ/dχ ) is majorized by dφ/dχ and dψ/dχ , this parallel sum (definedas a positive form) corresponds to a (densely defined) positive self-adjoint operator.Recall that spatial derivatives dφ/dχ and dψ/dχ are ( − dφ/dχ ) : ( dψ/dχ ) is proved to be ( − it -power (cid:0) ( dφ/dχ ) : ( dψ/dχ ) (cid:1) it seems impossible so that checking( − L p -spaces).Here, ϕ is a faithful normal semi-finite weight on M , and let R := M ⋊ σ R bethe crossed product acting on L ( R , H ) = H ⊗ L ( R , dt ) with respect to the modularautomorphism group σ t = σ ϕ t . Recall that R is semi-finite with the canonical trace τ and the dual action θ s is given on R . These materials are essential to define Haagerup’s L p -spaces as explained in Appendix A. We set H = H ( R , θ ) := the set of all positive self-adjoint operators h affiliated with R satisfying θ s ( h ) = e − s h for each s ∈ R . h φ ∈ H ←→ dφ/dχ ∈ M + − . (5.2)This correspondence admits the following explicit description (see [38, Theorem 2] andalso [71, Chap. IV, Proposition 4]): Lemma 5.1.
With the unitary operator U on L ( R , H ) defined by ( U ξ )( t ) = ( dϕ /dχ ) it ξ ( t ) ( ξ ∈ L ( R , H )) we have U h φ U ∗ = ( dφ/dχ ) ⊗ H for each φ ∈ P ( M , C ) , where H is the generator of the translations λ ( t ) ξ = ξ ( · − t ) , ξ ∈ L ( R , H ) , i.e., λ ( t ) = H it ( see Appendix A ) . The images (under this transformation U · U ∗ ) of the standard generators in thecrossed product R = M ⋊ σ R can be also explicitly written down (see [71, Chap. IV,Proposition 3]) although they are not needed here.So far (also in Appendix A) we have used the identification L ( R , H ) ∼ = H ⊗ L ( R , dt ). In the proof of the next lemma we will use L ( R , H ) ∼ = L ( R , dt ) ⊗ H in-stead, which seems more fitting to present arguments. We take the Fourier transformon L ( R , dt ) and arrive at U h φ U ∗ = e t ⊗ dφ/dχ, (5.3)where e t means the multiplication operator m e · on L ( R , dt ) induced by the exponentialfunction t e t . We note that the action of e t ⊗ dφ/dχ to a vector ξ ∈ L ( R , H ) lookslike (cid:0)(cid:0) e t ⊗ dφ/dχ (cid:1) ξ (cid:1) ( t ) = e t ( dφ/dχ ) ξ ( t ) . (5.4)When one identifies L ( R , dt ) ⊗ H with the disintegration Z ⊕ R H dt , this means U h φ U ∗ = Z ⊕ R e t ( dφ/dχ ) dt (see [54, 57] for instance). However, for computations of parallel sums we will have toregard the positive self-adjoint operator e t ⊗ dφ/dχ as a positive form, and probablyreduction theory for positive forms has not been properly formulated. Thus, reductionpicture for positive forms will not be used in the poof of the next lemma. However, wewill use this picture to express spectral projections of relevant positive forms, which isperfectly legitimate. Lemma 5.2.
For each φ and ψ we have (cid:0) e t ⊗ ( dφ/dχ ) (cid:1) : (cid:0) e t ⊗ ( dψ/dχ ) (cid:1) = e t ⊗ (( dφ/dχ ) : ( dψ/dχ )) . (5.5)44 roof. Our strategy is to compare spectral projections E I ( e t ⊗ ( dφ/dχ )) and e I ( dφ/dχ )(with I ⊆ [0 , + ∞ ]). Since e t ⊗ ( dφ/dχ ) is actually a (densely defined) positive self-adjoint operator on L ( R , dt ) ⊗ H , we have E { + ∞} ( e t ⊗ ( dφ/dχ )) = 0 . Also, since the support of dφ/dχ is s ( φ ) (i.e., the support as a weight), we have E { } ( e t ⊗ ( dφ/dχ )) = 1 L ( R ,dt ) ⊗ (1 − s ( φ )) . For each λ >
0, the expression (5.4) shows E [ λ, ∞ ) ( e t ⊗ ( dφ/dχ )) = Z ⊕ R e [ e − t λ, ∞ ) ( dφ/dχ ) dt, (5.6)where a disintegration symbol is used to express the projection E [ λ, ∞ ) ( e t ⊗ ( dφ/dχ )).Thus, spectral projections E I (( e t ⊗ ( dφ/dχ )) − ) for the positive form ( e t ⊗ ( dφ/dχ )) − (on L ( R , dt ) ⊗ H ) are given by E { + ∞} (( e t ⊗ ( dφ/dχ )) − ) = E { } ( e t ⊗ ( dφ/dχ )) = 1 L ( R ,dt ) ⊗ (1 − s ( φ )) ,E { } (( e t ⊗ ( dφ/dχ )) − ) = E { + ∞} ( e t ⊗ ( dφ/dχ )) = 0 ,E (0 ,λ ] (( e t ⊗ ( dφ/dχ )) − ) = E [1 /λ, ∞ ) ( e t ⊗ ( dφ/dχ ))= Z ⊕ R e [ e − t /λ, ∞ ) ( dφ/dχ ) dt = Z ⊕ R e (0 ,e t λ ] (( dφ/dχ ) − ) dt. Note that (5.6) was used, and e I (( dφ/dχ ) − ) of course means a spectral projection for( dφ/dχ ) − . The above E (0 ,λ ] (( e t ⊗ ( dφ/dχ )) − ) becomes the spectral projection of thetensor product of the multiplication operator e − t (= m e −· ) and the inverse of dφ/dχ (considered as a non-singular operator on s ( φ ) H ). From the above computations it iseasy to see that the positive form ( e t ⊗ ( dφ/dχ )) − (on L ( R , dt ) ⊗ H ) is given by (cid:0) ( e t ⊗ ( dφ/dχ )) − ξ, ξ (cid:1) = Z R e − t (( dφ/dχ ) − ξ ( t ) , ξ ( t )) dt for ξ ∈ L ( R , dt ) ⊗ H . Here, ( dφ/dχ ) − should be understood as a positive form (with e { + ∞} (( dφ/dχ ) − ) = e { } ( dφ/dχ ) = 1 − s ( φ )).We obviously have the same formula as above for dψ/dχ so that the (form) sum( e t ⊗ ( dφ/dχ )) − + ( e t ⊗ ( dψ/dχ )) − (on L ( R , dt ) ⊗ H ) is given by (cid:0) (( e t ⊗ ( dφ/dχ )) − + ( e t ⊗ ( dψ/dχ )) − ) ξ, ξ (cid:1) = Z R e − t (( dφ/dχ ) − ξ ( t ) , ξ ( t )) dt + Z R e − t (( dψ/dχ ) − ξ ( t ) , ξ ( t )) dt = Z R e − t ((( dφ/dχ ) − + ( dψ/dχ ) − ) ξ ( t ) , ξ ( t )) dt with the form sum ( dφ/dχ ) − + ( dψ/dχ ) − (on H ). Thus, spectral projections for( e t ⊗ ( dφ/dχ )) − + ( e t ⊗ ( dψ/dχ )) − and those for ( dφ/dχ ) − + ( dψ/dχ ) − are relatedin the following fashion: E (0 ,λ ] (( e t ⊗ ( dφ/dχ )) − + ( e t ⊗ ( dψ/dχ )) − ) = Z ⊕ R e (0 ,e t λ ] (( dφ/dχ ) − + ( dψ/dχ ) − ) dt E { } (( e t ⊗ ( dφ/dχ )) − + ( e t ⊗ ( dψ/dχ )) − ) = 0 ,E { + ∞} (( e t ⊗ ( dφ/dχ )) − + ( e t ⊗ ( dψ/dχ )) − )= 1 L ( R ,dt ) ⊗ e { + ∞} (( dφ/dχ ) − + ( dψ/dχ ) − ) . Hence, spectral projections for the parallel sum in question are given by E [ λ, ∞ ) ((( e t ⊗ ( dφ/dχ )) − + ( e t ⊗ ( dψ/dχ )) − ) − )= E (0 , /λ ] (( e t ⊗ ( dφ/dχ )) − + ( e t ⊗ ( dψ/dχ )) − )= Z ⊕ R e (0 ,e t /λ ] (( dφ/dχ ) − + ( dψ/dχ ) − ) dt = Z ⊕ R e [ e − t λ, ∞ ) ((( dφ/dχ ) − + ( dψ/dχ ) − ) − ) dt. We also have E { + ∞} ((( e t ⊗ ( dφ/dχ )) − + ( e t ⊗ ( dψ/dχ )) − ) − ) = 0 , (5.7) E { } ((( e t ⊗ ( dφ/dχ )) − + ( e t ⊗ ( dψ/dχ )) − ) − )= 1 L ( R ,dt ) ⊗ e { + ∞} (( dφ/dχ ) − + ( dψ/dχ ) − )= 1 L ( R ,dt ) ⊗ e { } ((( dφ/dχ ) − + ( dψ/dχ ) − ) − ) . (5.8)Thus, we conclude (cid:16)(cid:0) e t ⊗ ( dφ/dχ ) (cid:1) − + (cid:0) e t ⊗ ( dψ/dχ ) (cid:1) − (cid:17) − = e t ⊗ (cid:0) ( dφ/dχ ) − + ( dψ/dχ ) − (cid:1) − , which is exactly (5.5).The relation (5.7) corresponds to the fact that the parallel sum is a densely definedpositive self-adjoint operator while (5.8) indicates possibility of a non-trivial kernel.We are now ready to prove the next result. Proposition 5.3.
For φ , ψ in P ( M , C ) the parallel sum ( dφ/dχ ) : ( dψ/dχ ) = (cid:0) ( dφ/dχ ) − + ( dψ/dχ ) − (cid:1) − ( which is defined in § ) is ( − -homogeneous relative to χ .Proof. From (5.3) and (5.5) we have U ( h φ : h ψ ) U ∗ = e t ⊗ (( dφ/dχ ) : ( dψ/dχ )) . (5.9)Thus, it remains to show that the parallel sum h φ : h ψ belongs to H (see (5.2)). Let q φ , q ψ be positive forms corresponding to positive self-adjoint operators h φ , h ψ respectively.46ince h φ is affiliated with R and has the scaling property θ s ( h φ ) = e − s h φ , the associatedform q φ satisfies q φ ( u ′ ξ ) = q φ ( ξ ) and q φ ( µ ( s ) ∗ ξ ) = e − s q φ ( ξ )for any unitary u ′ ∈ R ′ and µ ( s ) (defined by (A.1)).By the variational expression for inverses (see (2.2) in § q − φ ( u ′ ξ ) = sup ζ | ( u ′ ξ, ζ ) | q φ ( ζ ) = sup ζ | ( ξ, u ′∗ ζ ) | q φ ( ζ )= sup ζ | ( ξ, ζ ) | q φ ( u ′ ζ ) = sup ζ | ( ξ, ζ ) | q φ ( ζ ) = q − φ ( ξ ) ,q − φ ( µ ( s ) ∗ ξ ) = sup ζ | ( µ ( s ) ∗ ξ, ζ ) | q φ ( ζ ) = sup ζ | ( ξ, µ ( s ) ζ ) | q φ ( ζ )= sup ζ | ( ξ, ζ ) | q φ ( µ ( s ) ∗ ζ ) = sup ζ | ( ξ, ζ ) | e − s q φ ( ζ ) = e s q − φ ( ξ )(where sup is taken over all vectors ζ ), and of course the same holds true for q ψ . It isthen trivial from the definition that the form sum q − φ + q − ψ has the same invariance,that is, (cid:0) q − φ + q − ψ (cid:1) ( u ′ ξ ) = (cid:0) q − φ + q − ψ (cid:1) ( ξ ) , (cid:0) q − φ + q − ψ (cid:1) ( µ ( s ) ∗ ξ ) = e s (cid:0) q − φ + q − ψ (cid:1) ( ξ ) . Thus, by repeating parallel arguments as above, we conclude (cid:0) q − φ + q − ψ (cid:1) − ( u ′ ξ ) = (cid:0) q − φ + q − ψ (cid:1) − ( ξ ) , (cid:0) q − φ + q − ψ (cid:1) − ( µ ( s ) ∗ ξ ) = e − s (cid:0) q − φ + q − ψ (cid:1) − ( ξ ) . This means that the positive self-adjoint operator h φ : h ψ (corresponding to the positiveform (cid:0) q − φ + q − ψ (cid:1) − ) is affiliated with R and θ s ( h φ : h ψ ) = e − s ( h φ : h ψ ) holds true, thatis, h φ : h ψ ∈ H . Definition 5.4.
We assume φ, ψ ∈ P ( M , C ). The weight in P ( M , C ) corresponding(via (5.1)) to the ( − dφ/dχ ) : ( dψ/dχ ) ∈ H (in Proposition5.3) is called the parallel sum of φ and ψ , and denoted by φ : ψ , that is, d ( φ : ψ ) /dχ = ( dφ/dχ ) : ( dψ/dχ ) (cid:16) ∈ M + − (cid:17) . Remark 5.5.
The parallel sum φ : ψ is independent of not only the choice of arepresenting Hilbert space H for M but also the choice of a faithful semi-finite normalweight χ on M ′ . Indeed, assume that κ : M (on H ) → M (on H ) is an isomorphism,and let ϕ be a faithful normal semi-finite weight on M and ϕ := ϕ ◦ κ − on M .Also, let χ and χ be faithful semi-finite normal weights on M ′ and M ′ respectively,for which U and U are defined as in Lemma 5.1. The κ : M → M extends to anisomorphism κ : R = M ⋊ σ ϕ R → R = M ⋊ σ ϕ R so that the canonical trace and47he dual action on R are τ = τ ◦ κ − and θ s := κ ◦ θ s ◦ κ − . Let φ, ψ ∈ P ( M , C )and set φ := φ ◦ κ − , ψ := ψ ◦ κ − ∈ P ( M , C ). Then it is standard to see that h φ = κ ( h φ ), h ψ = κ ( h ψ ) and h φ : h ψ = κ ( h φ : h ψ ), where κ gives rise to a map κ : H ( R , θ ) → H ( R , θ ) in a natural way. Since (5.9) and Definition 5.4 imply U h φ : ψ U ∗ = e t ⊗ ( d ( φ : ψ ) /dχ ) = U ( h φ : h ψ ) U ∗ ,U h φ : ψ U ∗ = e t ⊗ ( d ( φ : ψ ) /dχ ) = U ( h φ : h ψ ) U ∗ , it follows that h φ : ψ = h φ : h ψ and h φ : ψ = h φ : h ψ . Therefore, h φ : ψ = κ ( h φ : ψ ) sothat we have the desired identity φ : ψ = ( φ : ψ ) ◦ κ − . Note that the fact of independence shown above is included in Theorem 5.7 below whilethe proof of the latter is based on the former.
Lemma 5.6.
For φ, ψ ∈ P ( M , C ) their parallel sum φ : ψ ( ∈ P ( M , C )) satisfies ( φ : ψ )(( x + x ) ∗ ( x + x )) ≤ φ ( x ∗ x ) + ψ ( x ∗ x ) ( ≤ + ∞ ) for all x , x ∈ M .Proof. We make use of a family { ξ ι } ι ∈ I of vectors in D ( H , χ ) stated in Lemma B.4.Since P ι ∈ I θ χ ( ξ ι , ξ ι ) = 1, we have( x + x ) ∗ ( x + x ) = X ι ∈ I θ χ (( x + x ) ∗ ξ ι , ( x + x ) ∗ ξ ι ) ,x ∗ i x i = X ι ∈ I θ χ ( x ∗ i ξ ι , x ∗ i ξ ι ) (for i = 1 , φ : ψ )(( x + x ) ∗ ( x + x )) = X ι ∈ I (cid:0) φ : ψ )( θ χ (( x + x ) ∗ ξ ι , ( x + x ) ∗ ξ ι (cid:1) = X ι ∈ I q φ : ψ (( x + x ) ∗ ξ ι )= X ι ∈ I k ( d ( φ : ψ ) /dχ ) / ( x + x ) ∗ ξ ι k = X ι ∈ I k ( (cid:0) dφ/dχ ) : ( dψ/dχ ) (cid:1) / ( x + x ) ∗ ξ ι k ≤ X ι ∈ I k ( dφ/dχ ) / x ∗ ξ ι k + X ι ∈ I k ( dψ/dχ ) / x ∗ ξ ι k = φ ( x ∗ x ) + ψ ( x ∗ x ) . Here, the inequality appearing in the middle of computations follows from Theorem2.2, and also normality of involved weights, (B.4), Definition B.2 and Definition 5.4 arerepeatedly used. 48 heorem 5.7.
We assume φ, ψ ∈ P ( M , C ) . The parallel sum φ : ψ ( ∈ P ( M , C )) isthe maximum of all semi-finite normal weights ω satisfying ω (( x + x ) ∗ ( x + x )) ≤ φ ( x ∗ x ) + ψ ( x ∗ x ) ( ≤ + ∞ ) (5.10) for all x , x ∈ M .Proof. Normality and semi-finiteness (for weights) are expressed in terms of the σ -weak topology, and this topology is independent of the Hilbert space on which M acts.Therefore, in view of Remark 5.5 we may and do assume that the action of M on H is standard. We have shown Lemma 5.6, and it remains to show the maximality of theparallel sum φ : ψ . Let us assume that a (semi-finite) normal weight ω satisfies theinequality (5.10) (and ω ≤ φ : ψ is to be shown).We start from two vectors ζ ∈ D ( H , χ ) ∩ D (( dφ/dχ ) / ) and ζ ∈ D ( H , χ ) ∩ D (( dψ/dχ ) / ) . Both of R χ ( ζ i ) ( i = 1 ,
2) are bounded operators from H χ to H and satisfy y ′ R χ ( ζ i ) = R χ ( ζ i ) π χ ( y ′ ) for y ′ ∈ M ′ . Since the action of M on H is standard, there existsa surjective isometry I : H → H χ intertwining the respective two M ′ -actions, i.e., π χ ( y ′ ) I = Iy ′ . This means that the compositions R χ ( ζ i ) I are bounded operators on H and commute with y ′ ∈ M ′ , that is, R χ ( ζ i ) I ∈ M . We note θ χ ( ζ i , ζ i ) = R χ ( ζ i ) R χ ( ζ i ) ∗ = ( R χ ( ζ i ) I )( R χ ( ζ i ) I ) ∗ (for i = 1 , ,θ χ ( ζ + ζ , ζ + ζ ) = R χ ( ζ + ζ ) R χ ( ζ + ζ ) ∗ = ( R χ ( ζ ) + R χ ( ζ )) ( R χ ( ζ ) + R χ ( ζ )) ∗ = ( R χ ( ζ ) I + R χ ( ζ ) I ) ( R χ ( ζ ) I + R χ ( ζ ) I ) ∗ . We then compute k ( dω/dχ ) / ( ζ + ζ ) k = q ω ( ζ + ζ ) = ω (cid:0) θ χ ( ζ + ζ , ζ + ζ ) (cid:1) = ω (cid:0) ( R χ ( ζ ) I + R χ ( ζ ) I ) ( R χ ( ζ ) I + R χ ( ζ ) I ) ∗ (cid:1) ≤ φ (cid:0) ( R χ ( ζ ) I ) ( R χ ( ζ ) I ) ∗ (cid:1) + ψ (cid:0) ( R χ ( ζ ) I ) ( R χ ( ζ ) I ) ∗ (cid:1) = φ ( θ χ ( ζ , ζ )) + ψ ( θ χ ( ζ , ζ )) = q φ ( ζ ) + q ψ ( ζ )= k ( dφ/dχ ) / ζ k + k ( dψ/dχ ) / ζ k . (5.11)Here, the inequality comes from the assumption (5.10) (with R χ ( ζ i ) I ∈ M ) and ofcourse (B.4) and Definition B.2 are repeatedly used.To get the desired conclusion, it suffices to show that the inequality (5.11) remainsvalid for arbitrary vectors ζ , ζ ∈ H . Indeed, this would mean dω/dχ ≤ ( dφ/χ ) : ( dφ/dχ )thanks to the maximality of ( dφ/χ ) : ( dφ/dχ ) stated in Theorem 2.2. However, since( dφ/χ ) : ( dφ/dχ ) = d ( φ : ψ ) /dχ according to Definition 5.4, we have dω/dχ ≤ d ( φ : ψ ) /dχ , showing ω ≤ φ : ψ . 49o show (5.11) in full generality, we may and do assume ζ ∈ D (( dφ/χ ) / ) and ζ ∈D (( dψ/χ ) / ). Indeed, otherwise (5.11) trivially holds true, the right hand side being+ ∞ . Thanks to the core condition stated in Remark B.3,(iii) one can take sequences { ζ ,n } n ∈ N from D ( H , χ ) ∩ D (( dφ/dχ ) / ) and { ζ ,n } n ∈ N from D ( H , χ ) ∩ D (( dψ/dχ ) / )satisfying ζ ,n → ζ , ( dφ/dχ ) / ζ ,n → ( dφ/dχ ) / ζ ,ζ ,n → ζ , ( dψ/dχ ) / ζ ,n → ( dψ/dχ ) / ζ as n → ∞ . Since ζ ,n + ζ ,n → ζ + ζ as n → ∞ , lower semi-continuity of the positiveform k ( dω/dχ ) / · k and the first half of the proof (i.e., (5.11) under the additionalassumption) imply k ( dω/dχ ) / ( ζ + ζ ) k ≤ lim inf n →∞ k ( dω/dχ ) / ( ζ ,n + ζ ,n ) k ≤ lim inf n →∞ (cid:0) k ( dφ/dχ ) / ζ ,n k + k ( dψ/dχ ) / ζ ,n k (cid:1) = k ( dφ/dχ ) / ζ k + k ( dψ/dχ ) / ζ k as desired. Remark 5.8. (i) This theorem should be compared with the variational expression (4.2) for posi-tive functionals φ, ψ ∈ M + ∗ .(ii) All expected properties such as monotonicity, concavity and transformer inequal-ity (i.e., a ∗ ( φ : ψ ) a ≤ ( a ∗ φa ) : ( a ∗ ψa ) for all a ∈ M ) are valid since parallel sumsfor weights are defined by using corresponding spatial derivatives and the latterpossesses these properties (see [50]). It is also possible to see these propertiesfrom Theorem 5.7.(iii) We assume that { ω n } n ∈ N is a decreasing sequence of semi-finite normal weights.Let us denote the maximum of all semi-finite normal weights majorized by thepoint-wise infimum inf n ω n by Inf n ω n (see Remark C.5,(i) in Appendix C). Thenwe have d (cid:16) Inf n ω n (cid:17) /dχ = Inf n ( dω n /dχ ) ,dω n /dχ −→ d (cid:16) Inf n ω n (cid:17) /dχ in the strong resolvent sense(see Theorem C.4 and Remark C.5,(i)).From the above remark (iii) and Theorem 2.3 (see also [50, Theorem 19]) we havethe continuity property for decreasing sequences. Corollary 5.9. If { φ n } n ∈ N and { ψ n } n ∈ N are two decreasing sequences of semi-finitenormal weights, then Inf n ( φ n : ψ n ) = (cid:16) Inf n φ n (cid:17) : (cid:16) Inf n ψ n (cid:17) .
50e now recall Connes’ canonical order reversing correspondence between P ( M , C ) := the set of all faithful semi-finite normal weights on M ,P ( B ( H ) , M ′ ) := the set of all faithful semi-finite normaloperator valued weights from B ( H ) to M ′ , which is briefly explained in § B.3. We make use of this apparatus to our investigationon parallel sums for weights. For φ, ψ ∈ P ( M , C ) we have the canonical operatorvalued weights φ − , ψ − ∈ P ( B ( H ) , M ′ ) characterized by χ ◦ φ − = Tr(( dχ/dφ ) · ) and χ ◦ ψ − = Tr(( dχ/dψ ) · )(see (B.9)), and they sum up to χ ◦ ( φ − + ψ − ) = Tr(( dχ/dφ + dχ/dψ ) · )= Tr (cid:0)(cid:0) ( dφ/dχ ) − + ( dψ/dχ ) − (cid:1) · (cid:1) . (5.12)We assume that the sum φ − + ψ − is a semi-finite operator valued weight from B ( H )to M ′ (which means that dχ/dφ + dχ/dψ has a dense domain). Then the above leftside is nothing but Tr (cid:0)(cid:0) dχ/d ( φ − + ψ − ) − (cid:1) · (cid:1) with the inverse weight ( φ − + ψ − ) − ∈ P ( M , C ) (see (B.10)). Therefore, from (5.12)we get( dφ/dχ ) − + ( dψ/dχ ) − = dχ/d ( φ − + ψ − ) − (cid:18) = (cid:16) d ( φ − + ψ − ) − /dχ (cid:17) − (cid:19) and hence by taking the inverses of the both sides we conclude d ( φ − + ψ − ) − /dχ = (cid:0) ( dφ/dχ ) − + ( dψ/dχ ) − (cid:1) − = ( dφ/dχ ) : ( dψ/dχ ) . Since the far right side is d ( φ : ψ ) /dχ by the very definition (see Definition 5.4), wehave d ( φ − + ψ − ) − /dχ = d ( φ : ψ ) /dχ . Therefore, we arrive at the following mostnatural-looking expression: Theorem 5.10.
For two weights φ, ψ ∈ P ( M , C ) with the semi-finite sum φ − + ψ − of inverses we have φ : ψ = (cid:0) φ − + ψ − (cid:1) − . Here, the right hand side should be understood in terms of Connes’ canonical orderreversing correspondence between P ( M , C ) and P ( B ( H ) , M ′ ) . The semi-finiteness requirement for φ − + ψ − in the theorem is probably redundant.One can cut M ⊆ B ( H ) by relevant projections, and the correspondence P ( M , C ) ↔ P ( B ( H ) , M ′ ) may be enlarged to any (operator valued) weights without faithfulnessand/or semi-finiteness requirement. Then ( φ − + ψ − ) − might be justified in fullgenerality. 51 .3 Connections of weights Let us start from a connection σ in the Kubo-Ando sense. We assume that its repre-senting function f , a non-negative operator monotone function on (0 , ∞ ), is f ( s ) = α + βs + Z (0 , ∞ ) (1 + t ) ss + t dµ ( s )with α, β ≥ µ on (0 , ∞ ) (see (2.7) and (2.8)). For each φ, ψ ∈ P ( M , C ) we define a functional φσψ : M + → [0 , + ∞ ] by( φσψ )( x ) := αφ ( x ) + βψ ( x ) + Z (0 , ∞ ) tt (( tφ ) : ψ )( x ) dµ ( t ) , x ∈ M + . (5.13)We note that the function (( tφ ) : ψ )( x ) is non-decreasing in t (by Remark 5.8,(ii)) sothat it is a measurable function of t ∈ (0 , ∞ ).We have to check if φσψ defined by (5.13) is a normal weight. For an increasing net { x ι } ι ∈ I in M + with sup ι x ι = x we obviously have sup ι (( tφ ) : ψ )( x ι ) = (( tφ ) : ψ )( x )for each t >
0. However, in the proof of the next lemma one cannot use the monotoneconvergence theorem to show sup ι ( φσψ )( x ι ) = ( φσψ )( x ) since { x ι } ι ∈ I is not necessarilya sequence. Lemma 5.11.
The above φσψ ( defined by (5.13)) is a normal weight on M for any φ, ψ ∈ P ( M , C ) . Moreover, if φ + ψ is semi-finite, then φσψ is a semi-finite normalweight on M ( for any connection σ ) .Proof. The estimate in Lemma 3.13 deals with positive forms, but the same estimateclearly remains valid for weights (with identical arguments) so that semi-finiteness inthe second part is obvious. It remains to show normality.We take x ∈ M + and observe that ( (( tφ ) : ψ )( x ) ≤ ( φ : ψ )( x ) ≤ ( φ : ( t − ψ ))( x ) = t − (( tφ ) : ψ )( x ) for 0 < t < ,t − (( tφ ) : ψ )( x ) = ( φ : ( t − ψ ))( x ) ≤ ( φ : ψ )( x ) ≤ (( tφ ) : ψ )( x ) for t ≥ . Hence we have either (( tφ ) : ψ )( x ) < + ∞ for all t > tφ ) : ψ )( x ) = + ∞ for all t >
0. In the former case, since (( tφ ) : ψ )( x ) is concave in t > t >
0. For each n ∈ N we set t n,k := k/ n ( k = 1 , , . . . , n n )and define f n ( t ) := n n X k =1 [ t n,k ,t n,k +1 ) ( t ) 1 + t n,k +1 t n,k +1 (( t n,k φ ) : ψ )( x ) , t > . We claim that Z (0 , ∞ ) f n ( t ) dµ ( t ) = n n X k =1 λ n,k (( t n,k φ ) : ψ )( x ) with λ n,k := 1 + t n,k +1 t n,k +1 µ ([ t n,k , t n,k +1 ))52ncreases to Z (0 , ∞ ) tt (( tφ ) : ψ )( x ) dµ ( t )as n → ∞ .Indeed, from the fact noted above, we may and do assume that (( tφ ) : ψ )( x ) is finite(for all t > t (( tφ ) : ψ )( x ) is a non-decreasing continuous function on (0 , ∞ ) (as wasremarked above);(b) (1 + t ) /t is decreasing on (0 , ∞ );(c) The n th function f n takes a constant value t n,k +1 t n,k +1 (( t n,k φ ) : ψ )( x ) on each in-terval [ t n,k , t n,k +1 ) (of length 2 − n ), and here the values at the right and leftend-points (of the functions in (b), (a) respectively) are used (to get an in-creasing sequence). Also, this interval is further divided into two subintervals[ t n,k , t n,k +1 ) = [ t n +1 , k , t n +1 , k +1 ) ∪ [ t n +1 , k +1 , t n +1 , k +1) ) (of length 2 − ( n +1) ) whenone defines the next function f n +1 .Based on these we can immediately show f n ( t ) ր tt (( tφ ) : ψ )( x ) for all t > n → ∞ . Hence the claim is a consequence of the monotone convergence theorem.From the claim together with the definition (5.13) we observe( φσψ )( x ) = sup n (cid:20) αφ ( x ) + βψ ( x ) + n n X k =1 λ n,k (( t n,k φ ) : ψ )( x ) (cid:21) , x ∈ M + . (5.14)Since the functional inside the bracket here is obviously a normal weight, so is φσψ thanks to [26, Theorem 1.8].In view of the above lemma we define the following: Definition 5.12.
For every φ, ψ ∈ P ( M , C ) we call the normal weight φσψ (givenby (5.13)) on M the connection of φ, ψ . If φ + ψ is semi-finite, then we have φσψ ∈ P ( M , C ).From the definition (5.13) monotonicity, concavity and transformer inequality statedin Remark 5.8,(ii) immediately extend to connections φσψ for φ, ψ ∈ P ( M , C ), al-though it is unknown to us whether or not the transformer equality in (4.1) extends to φ, ψ ∈ P ( M , C ). Also, the transpose equality φ ˜ σψ = ψσφ holds as in Proposition 2.6.In the rest of this subsection we consider the situation where M is semi-finite.We begin with the special case M = B ( H ) with the usual trace Tr. Set χ on M ′ = C χ ( λ
1) = λ . Let a, b be positive self-adjoint operators on H . It is well-known that Tr( a · ) ∈ P ( M , C ) and d Tr( a · ) /dχ = a . Hence the connection aσb in53efinition 2.5 via positive forms becomes a special case of connections of semi-finitenormal weights (while Definition 5.4 is in turn based on Definition 2.5).Next we consider a general semi-finite von Neumann algebra M with a faithfulsemi-finite normal trace τ . For simplicity let us assume that M is represented in thestandard form h π ℓ ( M ) , L ( M , τ ) , J = ∗ , L ( M , τ ) + i ( π ℓ is the left multiplication). Set χ = τ ′ by τ ′ ( x ′ ) := τ ( J x ′∗ J ), x ′ ∈ M ′ + .Let a, b be positive self-adjoint operators a, b affiliated with M . It is well-knownthat τ a = τ ( a · ) ∈ P ( M , C ) [58] and dτ a /dτ ′ = a. (5.15)Indeed, since ( Dτ a : Dτ ) t = a it and dτ /dτ ′ = 1 in this case,( dτ a /dτ ′ ) it = ( Dτ a : Dτ ) t ( dτ /dτ ′ ) it = a it (see (c) of § B.3). The next result shows that consideration on connections of weightsin this subsection is quite consistent with what was done in § Proposition 5.13.
Let a, b be positive self-adjoint operators affiliated with M andwe assume that the connection aσb ( in the sense of Definition 2.5 ) is densely defined.Under these circumstances we have τ aσb = τ a στ b . Here, the right hand side is defined in the sense of Definition 5.12.Proof.
Definition 5.4 means d ( τ a : τ b ) /dτ ′ = ( dτ a /dτ ) : ( dτ b /dτ ) = a : b = dτ a : b /dτ ′ (due to (5.15)) so that τ a : b = τ a : τ b . Hence by the definition (5.13) we have for every x ∈ M + , ( τ a στ b )( x ) = ατ a ( x ) + βτ b ( x ) + Z (0 , ∞ ) (( tτ a ) : τ b )( x ) dµ ( t )= ατ ( ax ) + βτ ( bx ) + Z (0 , ∞ ) tt τ (( ta ) : b )( x ) dµ ( t ) . The formula (5.14) applied to τ a and τ b shows( τ a στ b )( x ) = sup n (cid:20) ατ ( ax ) + βτ ( bx ) + n n X k =1 λ n,k τ (( t n,k a ) : b ) x ) (cid:21) = sup n τ (cid:18)(cid:20) αa + βb + n n X k =1 λ n,k (( t n,k a ) : b ) (cid:21) x (cid:19) = sup n τ A n ( x ) (5.16)54ith the increasing sequence { A n } n ∈ N given by A n := αa + βb + n n X k =1 λ n,k (( t n,k a ) : b ) . Note that the connection aσb is defined via the positive form q aσb ( ξ ) = αq a ( ξ ) + βq b ( ξ ) + Z (0 , ∞ ) (( tq a ) : q b )( ξ ) dµ ( t ) , ξ ∈ H . Almost parallel arguments to those in the proof of Lemma 5.11 also show (5.14) forpositive forms and consequently we have q aσb ( ξ ) = sup n (cid:20) αq a ( ξ ) + βq b ( ξ ) + n n X k =1 λ n,k (( t n,k q a ) : q b )( ξ ) (cid:21) = sup n q A n ( ξ ) , which means that A n ր aσb in the strong resolvent sense (i.e., as positive forms). By[29, Theorem 1.12,(1)] this implies that τ A n ր τ aσb so that the desired result followsfrom (5.16).In particular, when a, b ∈ M + ( τ -measurable), τ aσb with the density aσb in § τ a στ b in Definition 5.12 (note that τ a + τ b = τ a + b is semi-finite in this case). Indeed, the definition of aσb in Definition 3.16 is independent ofthe representing Hilbert space H , and it agrees with that in the sense of Definition 2.5(or Definition 3.15). L p -spaces In this section we study Lebesgue decomposition in the setting of non-commutative L p -spaces. For bounded positive operators quite a satisfactory theory of Lebesguedecomposition was worked out by Ando [3]. For positive bounded operators a, b theincreasing sequence { a : ( nb ) } n ∈ N is bounded by a so that a [ b ] = sup n ( a : ( nb )) ( ≤ a ) (6.1)exists as the strong limit. It was proved in [3] that a = a [ b ] + ( a − a [ b ])is Lebesgue decomposition of a with respect to b , where a [ b ] and a − a [ b ] are b -absolutelycontinuous and b -singular respectively (see Definition 6.12 for absolute continuity andsingularity). This result can be shown by (ingenious) use of basic properties of parallelsums (see [3] for details). The above strong limit a [ b ] is known to be the largest b -absolutely continuous operator majorized by a , but uniqueness of decomposition (into55 -absolutely continuous and b -singular operators) generally fails to hold. Actuallydecomposition of each a ≥ b has a closed range. These results(together with some others) were obtained in [3]Recall that we have a reasonable notion of parallel sums (with all the expectedproperties) for positive elements in non-commutative L p -spaces (see § § a, b are replaced by positive elements in some L p -space ( p ∈ [1 , ∞ )), then one canplay the same game (see Remark 6.14). However, our approach in this section (which isakin to that in [47]) is somewhat different. Namely, by using relevant relative modularoperators and Radon-Nikodym cocycles, we try to express Lebesgue decomposition in amore explicit manner. Arguments involving relative modular operators are unavoidableso that L p -spaces consisting of powers of these operators seem fitting. Therefore, westart from a standard form and deal with Hilsum’s L p -spaces [38] with respect to afixed faithful state on the commutant (instead of Haagerup’s L p -spaces). These two L p -spaces are isometrically isomorphic, and a brief description of Hilsum’s L p -spaces isincluded in Appendix § B.2 for the reader’s convenience.
Let M be a von Neumann algebra with a standard form hM , H , J, Pi . Throughoutthis section we fix two faithful positive linear functionals ϕ , ψ in the predual M + ∗ ,whose unique implementing vectors in the natural cone P will be denoted by ξ ϕ , ξ ψ respectively (i.e., ϕ = ω ξ ϕ and ψ = ω ξ ψ ). The latter functional ψ (or more precisely ψ ′ to be explained shortly) will be needed just to define our L p -spaces. We note that ξ ϕ , ξ ψ are cyclic and separating vectors (due to faithfulness of ϕ and ψ ). For each ϕ ∈ M + ∗ we have the relative modular operator ∆ ϕψ as well as ∆ ϕ ψ (see the lastpart of Appendix § B.1). The positive self-adjoint operator ∆ ϕψ is exactly the spatialderivative dϕ/dψ ′ where ψ ′ ∈ M ′ + ∗ is defined by ψ ′ ( x ′ ) = ( x ′ ξ ψ , ξ ψ ) = ψ ( J x ′∗ J )for x ′ ∈ M ′ (see (B.6)).In this section Hilsum’s L p -spaces L p ( M , ψ ′ ) (1 ≤ p < ∞ ) will be used. We have,as explained in Appendix § B.2, L p ( M , ψ ′ ) + = { ∆ /pϕψ ; ϕ ∈ M + ∗ } , and ∆ /pϕψ (= ( dϕ/dψ ′ ) /p ) ∈ L p ( M , ψ ′ ) + corresponds to h /pϕ in the Haagerup L p -space L p ( M ) (see (B.7)).We will study various relations (such as absolute continuity and so on) between ∆ /pϕψ and ∆ /pϕ ψ in L p ( M , ψ ′ ) + . Definition 6.1.
For a functional ϕ ∈ M + ∗ (and a fixed faithful functional ϕ ∈ M + ∗ )we define the operator T ϕ with D ( T ϕ ) = M ′ ξ ϕ by T ϕ : J xJ ξ ϕ ∈ M ′ ξ ϕ J xJ ∆ / pϕϕ ξ ϕ ∈ H (with x ∈ M ) . ϕ ∈ M + ∗ let ( Dϕ : Dϕ ) t ( t ∈ R ) be Connes’ Radon-Nikodym cocycle [12],which is written in terms of relative modular operators as follows (see [13]):( Dϕ : Dϕ ) t = ∆ itϕϕ ∆ − itϕ (cid:0) = ( dϕ/dϕ ′ ) it ( dϕ /dϕ ′ ) − it (cid:1) = ∆ itϕψ ∆ − itϕ ψ (cid:0) = ( dϕ/dψ ′ ) it ( dϕ /dψ ′ ) − it (cid:1) ( t ∈ R ) (6.2)When ϕ ≤ ℓϕ (or more generally ∆ /pϕψ ≤ ℓ ∆ /pϕ ψ ) for some ℓ >
0, ( Dϕ : Dϕ ) − i/ p makes sense as an element in M . To be more precise, there is a σ -weakly M -valuedcontinuous function f ( z ) on the strip − / p ≤ Im z ≤ f ( t ) = ( Dϕ : Dϕ ) t ( t ∈ R ). Then ( Dϕ : Dϕ ) − i/ p is determinedas f ( − i/ p ), and we have J xJ ∆ / pϕϕ ξ ϕ = J xJ ( Dϕ : Dϕ ) − i/ p ξ ϕ = ( Dϕ : Dϕ ) − i/ p J xJ ξ ϕ , showing that T ϕ is just the restriction of ( Dϕ : Dϕ ) − i/ p to the dense subspace M ′ ξ ϕ in this situation.We will have to deal with increasing sequences in L p ( M , ψ ′ ) + quite often. Foran increasing sequence { A n } n ∈ N and ( A n ≤ ) A in L p ( M , ψ ′ ) + the notation A n ր A means A n → A in the σ ( L p , L q )-topology (see (B.8)), i.e., h A n , B i ր h A, B i for each B ∈ L q ( M , ψ ′ ) + where 1 /p + 1 /q = 1. Note that we have k A − A n k p → p = 1 the L -space L ( M , ψ ′ ) can be identified with the predual M ∗ andwe have k A − A n k = h A − A n , i →
0. On the other hand, for p ∈ (1 , ∞ ) the assertionfollows from the well-known uniform convexity of L p -spaces (see [23, §
5] for instance)together with lim n →∞ k A n k p = k A k p , which is a consequence of ( k A n k p ≤ k A k p (due to A n ≤ A ) , k A k p ≤ lim inf n →∞ k A n k p (due to lower semi-continuity of k · k p in σ ( L p , L q )) . The next proposition as well as its proof will be of fundamental importance in therest of the section. Indeed, the property (ii) below will be used as the definition of∆ /pϕ ψ -absolute continuity (for ∆ /pϕψ ) in § § Proposition 6.2.
We assume ≤ p < ∞ and /p + 1 /q = 1 . For ϕ ∈ M + ∗ thefollowing three properties are mutually equivalent :(i) the operator T ϕ ( Definition 6.1 ) is closable ;(ii) there exists an increasing sequence { A n } in L p ( M , ψ ′ ) + satisfying A n ր ∆ /pϕψ and A n ≤ ℓ n ∆ /pϕ ψ for some ℓ n > the map xξ ϕ ∈ M ξ ϕ ∆ /pϕψ , ∆ / qϕ ψ x ∗ x ∆ / qϕ ψ i ∈ R + is lower semi-continuous. roof. We will show (i) = ⇒ (ii), (ii) = ⇒ (iii), and (iii) = ⇒ (i). Firstly we assume (i).From the definition T ϕ satisfies u ′ T ϕ u ′∗ = T ϕ for an arbitrary unitary u ′ ∈ M ′ so thatthe closure T ϕ satisfies u ′ T ϕ u ′∗ = T ϕ , that is, T ϕ is affiliated with M (see also Remark6.9). Let T ∗ ϕ T ϕ = Z ∞ λ de λ be the spectral decomposition. We set h n := Z n λ de λ ∈ M + and A n := ∆ / pϕ ψ h n ∆ / pϕ ψ ∈ L p ( M , ψ ′ ) + . Clearly { A n } is an increasing sequence in L p ( M , ψ ′ ) + and A n ≤ n ∆ /pϕ ψ . For each x ∈ M we compute h A n , ∆ / qϕ ψ x ∗ x ∆ / qϕ ψ i = tr (cid:16) ∆ / pϕ ψ h n ∆ / pϕ ψ ∆ / qϕ ψ x ∗ x ∆ / qϕ ψ (cid:17) = tr (cid:16) h n ∆ / ϕ ψ x ∗ x ∆ / ϕ ψ (cid:17) = tr (cid:16)(cid:16) h n ∆ / ϕ ψ (cid:17) (cid:16) ∆ / ϕ ψ x ∗ x (cid:17) ∗ (cid:17) = ( h n ξ ϕ , J x ∗ xJ ξ ϕ ) = ( h n J xJ ξ ϕ , J xJ ξ ϕ ) . (6.3)Here we have used the fact that ∆ / ϕ ψ ∈ L ( M , ψ ′ ) + corresponds to ξ ϕ ∈ P . On theother hand, we have k T ϕ J xJ ξ ϕ k = k T ϕ J xJ ξ ϕ k = k J xJ ∆ / pϕϕ ξ ϕ k . (6.4)The vectors ∆ / pϕϕ ξ ϕ , J xJ ∆ / pϕϕ ξ ϕ here correspond to h / pϕ h / qϕ , h / pϕ h / qϕ x ∗ respec-tively in the Haargerup L -space L ( M ) (see [42, §
2] and [43, § / pϕψ ∆ / qϕ ψ , ∆ / pϕψ ∆ / qϕ ψ x ∗ respectively in our L ( M , ψ ′ ). Hence, the far right side of(6.4) is equal totr (cid:16)(cid:16) ∆ / pϕψ ∆ / qϕ ψ x ∗ (cid:17) (cid:16) ∆ / pϕψ ∆ / qϕ ψ x ∗ (cid:17) ∗ (cid:17) = tr (cid:16) ∆ / pϕψ ∆ / qϕ ψ x ∗ x ∆ / qϕ ψ ∆ / pϕψ (cid:17) = tr (cid:16) ∆ /pϕψ ∆ / qϕ ψ x ∗ x ∆ / qϕ ψ (cid:17) = h ∆ /pϕψ , ∆ / qϕ ψ x ∗ x ∆ / qϕ ψ i (6.5)in our L p - L q -duality notation (see (B.8)). Thus, we have shown k T ϕ J xJ ξ ϕ k = h ∆ /pϕψ , ∆ / qϕ ψ x ∗ x ∆ / qϕ ψ i . (6.6)Therefore, from (6.3) and (6.6) we observe h A n , ∆ / qϕ ψ x ∗ x ∆ / qϕ ψ i ≤ h ∆ /pϕψ , ∆ / qϕ ψ x ∗ x ∆ / qϕ ψ i , h A n , ∆ / qϕ ψ x ∗ x ∆ / qϕ ψ i ր h ∆ /pϕψ , ∆ / qϕ ψ x ∗ x ∆ / qϕ ψ i (as n → ∞ ) . The set of all elements of the form ∆ / qϕ ψ x ∗ x ∆ / qϕ ψ is a dense subset in L q ( M , ψ ′ ) + (aswill be seen in Lemma 6.3) so that we have A n ≤ ∆ /pϕψ and h A n , C i ր h ∆ /pϕψ , C i foreach C ∈ L q ( M , ψ ′ ) + , i.e., A n ր ∆ /pϕψ . Thus, (ii) is shown.58e next assume (ii). The assumption A n ≤ ℓ n ∆ /pϕ ψ guarantees A / n = u n ∆ / pϕ ψ forsome u n ∈ M with k u n k ≤ √ ℓ n so that we get h A n , ∆ / qϕ ψ x ∗ x ∆ / qϕ ψ i = h ∆ / pϕ ψ u ∗ n u n ∆ / pϕ ψ , ∆ / qϕ ψ x ∗ x ∆ / qϕ ψ i = k u n J xJ ξ ϕ k , where the second equality follows from the same computations as (6.3). Since A n ր ∆ /pϕψ , we have h ∆ /pϕψ , ∆ / qϕ ψ x ∗ x ∆ / qϕ ψ i = sup n h A n , ∆ / qϕ ψ x ∗ x ∆ / qϕ ψ i = sup n k u n J xJ ξ ϕ k . Being the supremum of continuous functions xξ ϕ ( u ∗ n u n J xξ ϕ , J xξ ϕ ) ( n = 1 , , · · · ),the above quantity is lower semi-continuous and (iii) is shown.Finally let us assume (iii). In the very first part of the proof the equation (6.6) wasshown from (6.4) and (6.5), but obviously (6.4), (6.5) always yield k T ϕ J xJ ξ ϕ k = h ∆ /pϕψ , ∆ / qϕ ψ x ∗ x ∆ / qϕ ψ i (since the second equality in (6.4) is just the definition of T ϕ in Definition 6.1). Thisshows the quadratic form associated with T ϕ defined on D ( T ϕ ) = M ′ ξ ϕ (i.e., the aboveleft side) is lower semi-continuous from the assumption (iii), and hence T ϕ is closableas desired. Indeed, let us assume that a sequence η n in D ( T ϕ ) satisfies η n → T ϕ η n → ζ . Since { T ϕ η n } is a Cauchy sequence, for each ε > N = N ε such that k T ϕ η n − T ϕ η m k ≤ ε for n, m ≥ N . We fix n ≥ N and let m → ∞ . Since η n − η m → η n , lower semi-continuity shows k T ϕ η n k ≤ lim inf m →∞ k T ϕ ( η n − η m ) k ≤ ε (for n ≥ N ) . Thus, we have lim n →∞ k T ϕ η n k = 0, i.e., ζ = 0, and hence Γ( T ϕ ) meets the “ y -axis”0 ⊕ H in the trivial way.The next result (required in the above proof) is certainly known to specialists.However, we present a proof for the reader’s convenience. Lemma 6.3.
The subset ∆ / qϕ ψ M + ∆ / qϕ ψ ( ⊆ L q ( M , ψ ′ ) + ) is dense in L q ( M , ψ ′ ) + .Proof. We may and do assume 1 ≤ q < ∞ . We note that A := { ω ∈ M + ∗ ; ω ≤ ℓ ϕ for some ℓ > } is dense in M + ∗ . In fact, ω x ′ ξ ϕ = ( · x ′ ξ ϕ , x ′ ξ ϕ ) with x ′ ∈ M ′ belongs to A due to ω x ′ ξ ϕ ≤ k x ′ k ϕ . Thus, the density follows from the following two facts:(i) M ′ ξ ϕ is dense in H ,(ii) k ω ξ − ω ξ k ≤ k ξ + ξ k · k ξ − ξ k for ξ i ∈ H .59hen, we observe that the set of all ∆ /qωψ , ω ∈ A , is dense in L q ( M , ψ ′ ) + (since ω ∈ M + ∗ ∆ /qωψ ∈ L q ( M , ψ ′ ) + is a continuous surjection due to the generalizedPowers-Sørmer inequality [37, Appendix] for instance). For ω ∈ A the Radon-Nikodymcocycle ( Dω : Dϕ ) t = ∆ itωψ ∆ − itϕ ψ (see (6.2)) extends to a bounded continuous functionon the strip − / ≤ Im z ≤ x = ( Dω : Dϕ ) − i/ q we have∆ /qωψ = ∆ / qϕ ψ x ∗ x ∆ / qϕ ψ ∈ ∆ / qϕ ψ M + ∆ / qϕ ψ thanks to ∆ / qωψ = x ∆ / qϕ ψ , and we are done. To investigate closability of T ϕ (mentioned in Proposition 6.2), it is natural to see thegraph of this operator. It requires some manipulations of relevant relative modularoperators and Radon-Nikodym cocycles.Let σ t (= σ ϕ t ) be the modular automorphism group associated with ϕ , and we set M := { x ∈ M ; t ∈ R σ t ( x ) ∈ M extends to an entire function } . The modular operator associated with ϕ ′ = ϕ ◦ j ∈ M ′ + ∗ (with j ( x ′ ) = J x ′∗ J for x ′ ∈ M ′ ) is J ∆ J = ∆ − . Therefore, J M J consists of all elements x ′ ∈ M ′ for which σ ′ t ( x ′ ) (cid:16) = σ ϕ ′ t ( x ′ ) (cid:17) extend to entire functions. We note J M J ξ ϕ = M ξ ϕ .Let us begin with the next general result on a core for relative modular operators. Lemma 6.4.
For each ω ∈ M + ∗ ( and a fixed faithful functional ϕ ∈ M + ∗ ) the subspace M ξ ϕ is a core for ∆ αωϕ with α ∈ [0 , / .Proof. Obviously D (∆ / ωϕ ) (cid:0) ⊆ D (∆ αωϕ ) (cid:1) is a core for ∆ αωϕ , that is, D (∆ / ωϕ ) is densein D (∆ αωϕ ) with respect to the graph norm of ∆ αωϕ . Thus, it suffices to show that M ξ ϕ is dense in D (∆ / ωϕ ) with respect to the graph norm of ∆ αωϕ , which follows fromthe following two facts:(i) M ξ ϕ is a core for ∆ / ωϕ ;(ii) the graph norm of ∆ αωϕ is majorized by a scalar multiple of that of ∆ / ωϕ on D (∆ / ωϕ ).From the definition M ξ ϕ is a core for ∆ / ωϕ . We note that M is a dense *-subalgebra of M in the σ -weak topology and hence in the strong* topology (thanks to the Kaplanskydensity theorem), that is, for x ∈ M we can take y ∈ M with arbitrarily small k ( x − y ) ξ ϕ k and k ( x ∗ − y ∗ ) ξ ω k (cid:16) = k ∆ / ωϕ ( x − y ) ξ ϕ k (cid:17) . Therefore, (i) holds true. On60he other hand, (ii) is a consequence of the following obvious estimate for η ∈ D (∆ / ωϕ ): k η k + k ∆ αωϕ η k = Z ∞ (1 + λ α ) d k e λ η k = Z (1 + λ α ) d k e λ η k + Z ∞ (1 + λ α ) d k e λ η k ≤ Z d k e λ η k + Z ∞ (1 + λ ) d k e λ η k ≤ Z ∞ (1 + λ ) d k e λ η k = 3 (cid:0) k η k + k ∆ / ωϕ η k (cid:1) , where ∆ ωϕ = R ∞ λ de λ is the spectral decomposition.We note ∆ /pϕϕ + ∆ /pϕ ∈ L p ( M , ϕ ′ ) + and it must be of the form∆ /pϕϕ + ∆ /pϕ = ∆ /pχϕ (6.7)with some χ ∈ M + ∗ . We observe that ∆ /pχϕ is non-singular (since so is ∆ /pϕ ), showingthat χ is faithful. When p = 1, we simply have χ = ϕ + ϕ (the usual sum asfunctionals), which is exactly the situation dealt with in [47]. When p = 2, χ is thefunctional coming from the vector sum ξ ϕ + ξ ϕ ∈ P , i.e., χ ( x ) = ( x ( ξ ϕ + ξ ϕ ) , ( ξ ϕ + ξ ϕ )) . We now go back to a general p , and for convenience we set ζ χ := ∆ / pχϕ ξ ϕ . (6.8)Majorization ∆ /pϕ , ∆ /pϕϕ ≤ ∆ /pχϕ guarantees that both of the Radon-Nikodym cocy-cles ( Dϕ : Dχ ) t and ( Dϕ : Dχ ) t admit bounded continuous extensions on the strip − / p ≤ Im z ≤ Definition 6.5.
With χ ∈ M + ∗ determined by (6.7) we set a = ( Dϕ : Dχ ) − i/ p and b = ( Dϕ : Dχ ) − i/ p , which are contractions in M . Lemma 6.6.
We have ξ ϕ = aζ χ and ∆ / pϕϕ ξ ϕ = bζ χ .Proof. We note ( Dϕ : Dχ ) t ∆ itχϕ ξ ϕ = ∆ itϕ ∆ − itχϕ ∆ itχϕ ξ ϕ = ∆ itϕ ξ ϕ = ξ ϕ , ( Dϕ : Dχ ) t ∆ itχϕ ξ ϕ = ∆ itϕϕ ∆ − itχϕ ∆ itχϕ ξ ϕ = ∆ itϕϕ ξ ϕ . We can substitute t = − i/ p here, which gives us the desired conclusion.61 heorem 6.7. The closure of the graph Γ( T ϕ ) ( ⊆ H ⊕ H ) of the operator T ϕ ( seeDefinition 6.1 ) is given by Γ( T ϕ ) = { ( aξ, bξ ) ∈ H ⊕ H ; ξ ∈ H} ( with the contractions a, b in Definition 6.5 ) .Proof. For x ′ ∈ M ′ Lemma 6.6 yields x ′ ξ ϕ = x ′ aζ χ = ax ′ ζ χ and x ′ ∆ / pϕϕ ξ ϕ = x ′ bζ χ = bx ′ ζ χ , and hence Definition 6.1 saysΓ( T ϕ ) = { ( ax ′ ζ χ , bx ′ ζ χ ); x ′ ∈ M ′ } . It is important to notice J M J ζ χ = H . (6.9)Indeed, we observe J M J ζ χ = J M J ∆ / pχϕ ξ ϕ = ∆ / pχϕ J M J ξ ϕ = R (∆ / pχϕ ) = H . Here, the first and fourth equalities follow from (6.8) and faithfulness of χ respectively.The third equality holds true due to the fact that M ξ ϕ = J M J ξ ϕ is a core for ∆ / pχϕ (thanks to Lemma 6.4) while the second is a consequence of ( − / p )-homogeneity of∆ / pχϕ relative to ϕ ′ (see [71, Chap. III, Corollary 34] or Lemma C.2 in § C), that is, x ′ ∆ / pχϕ ⊆ ∆ / pχϕ σ ′− i/ p ( x ′ ) (6.10)holds true for each x ′ ∈ J M J . Since M ′ ζ χ ( ⊇ J M J ζ χ ) is dense in H , we haveΓ( T ϕ ) ⊇ { ( aξ, bξ ); ξ ∈ H} ⊇ Γ( T ϕ ) , and it remains to show that the set { ( aξ, bξ ); ξ ∈ H} is closed.To show this closedness, we note a ∗ a + b ∗ b = 1 , (6.11)which is equivalent to(( a ∗ a + b ∗ b ) x ′ ζ χ , x ′ ζ χ ) = ( x ′ ζ χ , x ′ ζ χ ) ( x ′ ∈ J M J )thanks to (6.9). The left hand side here is equal to k ax ′ ζ χ k + k bx ′ ζ χ k = k x ′ aζ χ k + k x ′ bζ χ k = k x ′ ξ ϕ k + k x ′ ∆ / pϕϕ ξ ϕ k (by Lemma 6.6)= k ∆ / pϕ σ ′− i/ p ( x ′ ) ξ ϕ k + k ∆ / pϕϕ σ ′− i/ p ( x ′ ) ξ ϕ k = k ∆ / pχϕ σ ′− i/ p ( x ′ ) ξ ϕ k (by (6.7))= k x ′ ∆ / pχϕ ξ ϕ k = k x ′ ζ χ k − / p )-homogeneity of ∆ / pϕϕ and ∆ / pχϕ (see (6.10)) so that (6.11) has beenproved.To see the closedness in question, let us assume that ( ξ , ξ ) is in the closure of { ( aξ, bξ ); ξ ∈ H} . This means that there is a sequence { η n } such that ξ = lim n →∞ aη n and ξ = lim n →∞ bη n . Thanks to (6.11) we have η n = a ∗ aη n + b ∗ bη n −→ a ∗ ξ + b ∗ ξ (as n → ∞ ) . Therefore, with η = a ∗ ξ + b ∗ ξ we get ( ξ , ξ ) = ( aη, bη ) ∈ { ( aξ, bξ ); ξ ∈ H} , and weare done. Corollary 6.8.
The projection P from H ⊕ H onto Γ( T ϕ ) is given by P = (cid:20) aa ∗ ab ∗ ba ∗ bb ∗ (cid:21) ( with the contractions a, b in Definition 6.5 ) .Proof. Thanks to (6.11) P is indeed a projection and P (cid:20) aξbξ (cid:21) = (cid:20) aξbξ (cid:21) holds true. Onthe other hand, Theorem 6.7 shows P (cid:20) ξ ξ (cid:21) = (cid:20) a ( a ∗ ξ + b ∗ ξ ) b ( a ∗ ξ + b ∗ ξ ) (cid:21) ∈ Γ( T ϕ )for ξ, ξ i ∈ H . Remark 6.9. (i) Obstruction for closability of T ϕ isΓ( T ϕ ) ∩ (0 ⊕ H ) ∼ = { bξ ; aξ = 0 } = b ker a. The polar decompositions of a, b (in Definition 6.5) are of the forms a = v (1 − h ) / , b = uh with a positive contraction h (because of (6.11)). Thus, we observeker a = ker(1 − h ) / = ker(1 − h ) = { ξ ∈ H ; ξ = hξ } and b ker(1 − h ) = u ker(1 − h ). Since ker(1 − h ) ⊆ R ( h ), the initial space of thepartial isomery u , the projection ˜ q onto b ker a is given by˜ q = u ˜ q u ∗ with the projection ˜ q onto ker(1 − h ) = ker a .63ii) In particular, T ϕ is closable if and only if ker a = 0 (see Definition 6.1 andProposition 6.2). Here, a = ( Dϕ : Dχ ) − i/ p is one of the contractions in Definition 6.5.(iii) We set ˜ p = 1 − ˜ q. (6.12)Then ( T ϕ ) c = ˜ p T ϕ (the obstruction ˜ q for closability is removed and ( T ϕ ) c isclosable) is known as the closable part or the operator part of T ϕ (see [39]). Wenote ˜ qba ∗ = u ˜ q u ∗ uh (1 − h ) / v ∗ = u ˜ q h (1 − h ) / v ∗ = uh (1 − h ) / ˜ q v ∗ = 0 , ˜ qbb ∗ = u ˜ q u ∗ uh u ∗ = u ˜ q h u ∗ = uh ˜ q u ∗ = bb ∗ ˜ q, showing ˜ pba ∗ = ba ∗ and ˜ pbb ∗ = bb ∗ ˜ p. Hence the projection onto the graph of ( T ϕ ) c (i.e., the characteristic matrix of( T ϕ ) c ) is given by (cid:20) aa ∗ ab ∗ ˜ p ba ∗ ˜ p bb ∗ (cid:21) = (cid:20) aa ∗ ab ∗ ba ∗ ˜ p bb ∗ (cid:21) . This matrix obviously commutes with u ′ ⊗ M ( C ) (with a unitary u ′ ∈ M ′ ),showing that ( T ϕ ) c is always affiliated with M .So far we have presented direct self-contained arguments (based on Corollary 6.8) inour special situation. A general theory on the (maximal) closable part of an operatorwas actually developed in [39], and it goes as follows: It is well-known that a denselydefined operator T is closable if and only if the adjoint T ∗ has a dense domain. Let (cid:20) p p p p (cid:21) be the projection onto the closure Γ( T ). Then, we have D ( T ∗ ) ⊥ = ker(1 − p ) (see[39, Theorem 3.1,(d)]), which corresponds to obstruction for closablity. In our specialcase p is bb ∗ and hence the projection onto ker(1 − p ) is nothing but ˜ q (= u ˜ q u ∗ )(in Remark 6.9,(ii)), i.e., ˜ p = 1 − ˜ q is the projection onto D ( T ∗ ). In [39] T c = ˜ p T isdefined as the closable part of T . It was shown there that T c is indeed closable withthe characteristic matrix (cid:20) p p ˜ p p ˜ p p (cid:21) = (cid:20) p p p ˜ p p (cid:21) . .3 Technical lemmas The projection ˜ p defined by (6.12) (in Remark 6.9,(iii)) plays a special role in the nextresult (and also in the rest of the section). Lemma 6.10.
For ϕ ∈ M + ∗ we have an increasing sequence { A n } n ∈ N in L p ( M , ψ ′ ) + satisfying A n ր ∆ / pϕψ ˜ p ∆ / pϕψ and A n ≤ n ∆ /pϕ ψ , and the map xξ ϕ ∈ M ξ ϕ ∆ / pϕψ ˜ p ∆ / pϕψ , ∆ / qϕ ψ x ∗ x ∆ / qϕ ψ i ∈ R + (6.13) is always lower semi-continuous.Proof. Recall that ˜ p T ϕ (= ( T ϕ ) c ) is closable and its closure ˜ p T ϕ is affiliated with M (see Remark 6.9,(iii)). Then, with ˜ p T ϕ (instead of T ϕ ) we can repeat almost identicalarguments as in the proof of (i) = ⇒ (ii) in Proposition 6.2. We will just sketch ar-guments with explanations on some differences. Using the spectral decomposition of | ˜ p T ϕ | we construct h n ∈ M + and A n = ∆ / pϕ ψ h n ∆ / pϕ ψ ∈ L p ( M , ψ ′ ) + . Note that (6.4)is modified to k ˜ p T ϕ J xJ ξ ϕ k = k J xJ ˜ p T ϕ ξ ϕ k = k J xJ ˜ p T ϕ ξ ϕ k = k J xJ ˜ p ∆ / pϕϕ ξ ϕ k and hence (6.5) changes totr (cid:16)(cid:16) ˜ p ∆ / pϕψ ∆ / qϕ ψ x ∗ (cid:17) (cid:16) ˜ p ∆ / pϕψ ∆ / qϕ ψ x ∗ (cid:17) ∗ (cid:17) = tr (cid:16) ˜ p ∆ / pϕψ ∆ / qϕ ψ x ∗ x ∆ / qϕ ψ ∆ / pϕψ ˜ p (cid:17) = tr (cid:16) ∆ / pϕψ ˜ p ∆ / pϕψ ∆ / qϕ ψ x ∗ x ∆ / qϕ ψ (cid:17) = h ∆ / pϕψ ˜ p ∆ / pϕψ , ∆ / qϕ ψ x ∗ x ∆ / qϕ ψ i . (6.14)Thus, we have h ∆ / pϕψ ˜ p ∆ / pϕψ , ∆ / qϕ ψ x ∗ x ∆ / qϕ ψ i = k ˜ p T ϕ J xJ ξ ϕ k = sup n ( h n J xJ ξ ϕ , J xJ ξ ϕ ) = sup n h A n , ∆ / qϕ ψ x ∗ x ∆ / qϕ ψ i . Here, the above last equality holds true since the computation (6.3) remains in the sameform. This means A n ր ∆ / pϕψ ˜ p ∆ / pϕψ as before (i.e., thanks to Lemma 6.3 again).Since h ∆ / pϕψ ˜ p ∆ / pϕψ , ∆ / qϕ ψ x ∗ x ∆ / qϕ ψ i = sup n ( h n J xJ ξ ϕ , J xJ ξ ϕ ) as seen above, lowersemi-continuity of the map defined by (6.13) is obvious.Lower semi-continuity of (6.13) shown above and the obvious inequality∆ / pϕψ ˜ p ∆ / pϕψ ≤ ∆ /pϕψ yield h ∆ / pϕψ ˜ p ∆ / pϕψ , ∆ / qϕ ψ x ∗ x ∆ / qϕ ψ i = inf (cid:16) lim inf n →∞ h ∆ / pϕψ ˜ p ∆ / pϕψ , ∆ / qϕ ψ x ∗ n x n ∆ / qϕ ψ i (cid:17) ≤ inf (cid:16) lim inf n →∞ h ∆ /pϕψ , ∆ / qϕ ψ x ∗ n x n ∆ / qϕ ψ i (cid:17) (6.15)65or each x ∈ M . Here, the infimum is taken over all sequences { x n } n ∈ N in M satisfying x n ξ ϕ → xξ ϕ in H .Actually, the three quantities in (6.15) are all identical. Lemma 6.11.
For each x ∈ M we have h ∆ / pϕψ ˜ p ∆ / pϕψ , ∆ / qϕ ψ x ∗ x ∆ / qϕ ψ i = inf (cid:16) lim inf n →∞ h ∆ /pϕψ , ∆ / qϕ ψ x ∗ n x n ∆ / qϕ ψ i (cid:17) , where the infimum is taken over all sequences { x n } n ∈ N in M satisfying x n ξ ϕ → xξ ϕ in H .Proof. For x ∈ M we have T ϕ J xJ ξ ϕ = J xJ ∆ / pϕϕ ξ ϕ (Definition 6.1). On the otherhand, let us recall the projection ˜ q = 1 − ˜ p , which corresponds to the intersection ofΓ( T ϕ ) and the y -axis 0 ⊕ H (as was discussed in Remark 6.9). Since (0 , ˜ qJ xJ ∆ / pϕϕ ξ ϕ )belongs to Γ( T ϕ ), there is a sequence { y n } in M satisfying J y n J ξ ϕ → T ϕ J y n J ξ ϕ → ˜ qJ xJ ∆ / pϕϕ ξ ϕ . We set x n = x − y n ∈ M . We observe x n ξ ϕ → xξ ϕ and T ϕ J x n J ξ ϕ (cid:16) = J x n J ∆ / pϕϕ ξ ϕ (cid:17) tends to J xJ ∆ / pϕϕ ξ ϕ − ˜ qJ xJ ∆ / pϕϕ ξ ϕ = ˜ pJ xJ ∆ / pϕϕ ξ ϕ = J xJ ˜ p ∆ / pϕϕ ξ ϕ . Hence, we have lim n →∞ k J x n J ∆ / pϕϕ ξ ϕ k = k J xJ ˜ p ∆ / pϕϕ ξ ϕ k , but this meanslim n →∞ h ∆ /pϕψ , ∆ / qϕ ψ x ∗ n x n ∆ / qϕ ψ i = h ∆ / pϕψ ˜ p ∆ / pϕψ , ∆ / qϕ ψ x ∗ x ∆ / qϕ ψ i by usual computations that have been carried out repeatedly (see (6.5) and (6.14)).Thus, the far right side in (6.15) is smaller than h ∆ / pϕψ ˜ p ∆ / pϕψ , ∆ / qϕ ψ x ∗ x ∆ / qϕ ψ i (i.e., thefar left side in (6.15)), that is, the three quantities there are all identical. In this subsection we obtain results on Lebesgue-type decomposition in L p ( M , ψ ′ ) + among others by combining various technical results obtained so far. We begin withdefinitions of absolute continuity and singularity. Definition 6.12. ( Absolute continuity and singularity ) We assume 1 ≤ p < ∞ .A positive element A ∈ L p ( M , ψ ′ ) + is absolutely continuous with respect to ∆ /pϕ ψ , or∆ /pϕ ψ -absolutely continuous in short, if there is an increasing sequence { A n } n ∈ N in L p ( M , ψ ′ ) + satisfying A n ր A and A n ≤ ℓ n ∆ /pϕ ψ for some ℓ n >
0. On the other hand, A is defined to be ∆ /pϕ ψ -singular if B ∈ L p ( M , ψ ′ ) + must be 0 whenever B ≤ A and B ≤ ∆ /pϕ ψ . 66ote that Proposition 6.2 and Remark 6.9,(ii) characterize absolute continuity interms of the associated operator T ϕ (given in Definition 6.1). Our proof of the nexttheorem is motivated by discussions in [46, §
3] and [63].
Theorem 6.13. ( Maximal absolutely continuous part ) We assume that ϕ isa faithful positive linear functional in M + ∗ . For a given positive operator ∆ /pϕψ ∈ L p ( M , ψ ′ ) + ( with ϕ ∈ M + ∗ ) the operator ∆ / pϕψ ˜ p ∆ / pϕψ ∈ L p ( M , ψ ′ ) + with the projec-tion ˜ p ∈ M defined by (6.12) ( see Remark 6.9 ) is ∆ /pϕ ψ -absolutely continuous. More-over, it is the maximum among all ∆ /pϕ ψ -absolutely continuous elements in L p ( M , ψ ′ ) + majorized by ∆ /pϕψ .Proof. The first statement is just Lemma 6.10. Hence, it remains to show maximalitystated in the last part. Let us assume that ∆ /pϕ ψ ∈ L p ( M , ψ ′ ) + (with ϕ ∈ M + ∗ ) is∆ /pϕ ψ -absolutely continuous and ∆ /pϕ ψ ≤ ∆ /pϕψ . Proposition 6.2,(iii) (applied for ϕ )says that the map xξ ϕ ∈ M ξ ϕ ∆ /pϕ ψ , ∆ / qϕ ψ x ∗ x ∆ / qϕ ψ i ∈ R + is lower semi-continuous so that we have h ∆ /pϕ ψ , ∆ / qϕ ψ x ∗ x ∆ / qϕ ψ i = inf (cid:16) lim inf n →∞ h ∆ /pϕ ψ , ∆ / qϕ ψ x ∗ n x n ∆ / qϕ ψ i (cid:17) for each x ∈ M . Here (and below) the infimum is taken over all sequences { x n } n ∈ N in M satisfying x n ξ ϕ → xξ ϕ . Thanks to ∆ /pϕ ψ ≤ ∆ /pϕψ the above right side is smallerthan inf (cid:16) lim inf n →∞ h ∆ /pϕψ , ∆ / qϕ ψ x ∗ n x n ∆ / qϕ ψ i (cid:17) , which is equal to h ∆ / pϕψ ˜ p ∆ / pϕψ , ∆ / qϕ ψ x ∗ x ∆ / qϕ ψ i according to Lemma 6.11. Thus, wehave shown h ∆ /pϕ ψ , ∆ / qϕ ψ x ∗ x ∆ / qϕ ψ i ≤ h ∆ / pϕϕ ˜ p ∆ / pϕϕ , ∆ / qϕ ψ x ∗ x ∆ / qϕ ψ i for each x ∈ M . Then Lemma 6.3 enables us to change ∆ / qϕ ψ x ∗ x ∆ / qϕ ψ (in the aboveinequality) to arbitrary positive elements in L q ( M , ψ ′ ) + and consequently we have∆ /pϕ ψ ≤ ∆ / pϕψ ˜ p ∆ / pϕψ as desired. Remark 6.14.
As was mentioned at the beginning of the section. Ando [3] defined a [ b ] = sup n ( a : ( nb )) for positive bounded operators a, b (to study Lebesgue decom-position). Since { a : ( nb ) } n is an increasing sequence majorized by a , this sequenceconverges to a [ b ] in the strong operator topology.Let us mimic this procedure in our non-commutative L p -space setting. We have tobegin by understanding the meaning of sup. We assume that { A n } is an increasingsequence in L p ( M , ψ ′ ) + satisfying A n ≤ A for some A ∈ L p ( M , ψ ′ ) + . Then, { A − A n } is a decreasing sequence in L p ( M , ψ ′ ) + . We set X = Inf n ( A − A n ). Indeed, we know67 ∈ L p ( M , ψ ′ ) + and A − A n → X in the strong resolvent sense and also in the L p -norm. (It is possible to use Proposition 4.2 here since Haagerup and Hilsum L p -spacesare isometrically order-isomorphic.) Therefore, A n = A − ( A − A n ) → A − X in the L p -norm. Since A − A n ≥ X , we have A − X ≥ A n . Conversely, if Y ≥ A n (for each n ), then A − A n ≥ A − Y and hence X ≥ A − Y , i.e., Y ≥ A − X . Thus, A − X is theminimal element majorizing all A n , i.e., A − X = sup n A n . Therefore, we have seenthat an increasing sequence having an upper bound (in L p ( M , ψ ′ ) + ) convergesto its supremum in the L p -norm.We recall (6.1) (which is the strong limit) and start from ∆ /pϕψ , ∆ /pϕ ψ in L p ( M , ψ ′ ) + .Parallel sums in non-commutative L p -spaces were studied in §
4. The increasing se-quence { ∆ /pϕψ : ( n ∆ /pϕ ψ ) } n ∈ N bounded by ∆ /pϕψ from the above converges to the supre-mum in the L p -norm (as was remarked above). Let us denote this supremum by∆ /pϕψ [∆ /pϕ ψ ] = sup n (cid:16) ∆ /pϕψ : ( n ∆ /pϕ ψ ) (cid:17) . As is explained at the beginning of § /pϕψ [∆ /pϕ ψ ] is themaximum of all ∆ /pϕ ψ -absolutely continuous operators (in L p ( M , ψ ′ ) + ) majorized by∆ /pϕψ , and hence we conclude ∆ /pϕψ [∆ /pϕ ψ ] = ∆ / pϕψ ˜ p ∆ / pϕψ , i.e., the maximal ∆ /pϕ ψ -absolutely continuous part captured in Theorem 6.13. Theorem 6.15. ( Lebesgue decomposition ) With the same notations as in theprevious theorem the difference ∆ /pϕψ − ∆ / pϕψ ˜ p ∆ / pϕψ = ∆ / pϕψ (1 − ˜ p )∆ / pϕψ (cid:0) ∈ L p ( M , ψ ′ ) + (cid:1) is ∆ /pϕ ψ -singular and hence ∆ /pϕψ = ∆ / pϕψ ˜ p ∆ / pϕψ + ∆ / pϕψ (1 − ˜ p ) ∆ / pϕψ gives us a Lebesgue decomposition of ∆ /pϕψ into ( maximal ) ∆ /pϕ ψ -absolutely continuousand ∆ /pϕ ψ -singular parts.Proof. Let us assume that B ∈ L p ( M , ψ ′ ) + satisfies B ≤ ∆ / pϕψ (1 − ˜ p )∆ / pϕψ and B ≤ ∆ /pϕ ψ . At first we observe that the sum of two ∆ /pϕ ψ -absolutely continuouselements remains to be ∆ /pϕ ψ -absolutely continuous, which is clear from the defini-tion. Note that B (cid:16) ≤ ∆ /pϕ ψ (cid:17) is obviously ∆ /pϕ ψ -absolutely continuous and hence sois the sum B + ∆ / pϕψ ˜ p ∆ / pϕψ . However, we have B + ∆ / pϕψ ˜ p ∆ / pϕψ ≤ ∆ /pϕψ because of B ≤ ∆ / pϕψ (1 − ˜ p )∆ / pϕψ . Thus, the maximality stated in the preceding theorem implies B + ∆ / pϕψ ˜ p ∆ / pϕψ ≤ ∆ / pϕψ ˜ p ∆ / pϕψ , showing B = 0.68ecall that via ∆ /pϕψ ↔ h /pϕ one can identify L p ( M , ψ ′ ) + with Haagerup’s L p ( M ) + .The latter is independent of the choice of ψ so that absolute continuity, Lebesguedecomposition and so on between ∆ ϕψ , ∆ ϕ ψ ∈ L p ( M , ψ ′ ) + are the same as thoseof ∆ ϕψ , ∆ ϕ ψ ∈ L p ( M , ψ ) + with a different faithful ψ ∈ M + ∗ . More precisely,everything is determined by just ϕ and ϕ (and no role is played by ψ ). Observe that T ϕ in Definition 6.1 and the projection ˜ p in Remark 6.9 are determined just by ϕ, ϕ .We point out that for a finite von Neumann algebra the concept of absolute continu-ity is not so meaningful. In fact, (whenever ϕ is faithful) any ∆ /pϕψ is ∆ /pϕ ψ -absolutelycontinuous. Indeed, we take a finite trace as ψ . Then the Hilsum L p ( M , τ ′ ) is justthe classical L p -space L p ( M , τ ) + (in the setting of semi-finite von Neumann algebras).Thus, what we are claiming is the following:If H, K ∈ L p ( M , τ ) + with K non-singular, then H is always K -absolutelycontinuous.This comes from the classical fact (which goes back to Murray-von Neumann) thatall (unbounded) operators affiliated with M can be manipulated without worryingdomain questions, closablity and so on, that is, everything is τ -measurable. Thus, X = K − / HK − / always makes a perfect sense. Thus, with the spectral projections { e λ } of X we have H n = K / (cid:18)Z n λ de λ (cid:19) K / ր K / XK / = H (as n → ∞ ) with the obvious estimate H n ≤ nK .On the other hand, a lot of pathological phenomena are known even in B ( H ) (see[47, §
10] for instance) and hence so is the case for properly infinite von Neumannalgebras
M ∼ = M ⊗ B ( H ). A Haagerup’s L p -spaces In this appendix we give a brief survey on Haagerup’s L p -spaces as a preliminary for § M be a general von Neumann algebra on a Hilbert space H . We fix a faithfulsemi-finite normal weight ϕ on M and the associated modular automorphims σ t (= σ ϕ t ). Let R := M ⋊ σ R be the crossed product (acting on L ( R , H ) = H ⊗ L ( R , dt )), and the dual action on R is given by θ s = Ad µ ( s ) | R ( s ∈ R ) with( µ ( s ) ξ )( t ) = e − ist ξ ( t ) (for ξ ∈ L ( R , H )) . (A.1)(See [70] for basics of crossed products.) Let T be the operator valued weight from R to M defined by T ( x ) := Z ∞−∞ θ s ( x ) ds ∈ c M + (for x ∈ R + ) , c M + is the extended positive part of M (see [29, 71]). We note the invariance T ◦ θ s = T ( s ∈ R ). Let P ( M , C ) denote the set of all semi-finite normal weights on M (as in § ϕ ∈ P ( M , C ) the composition b ϕ = ϕ ◦ T ∈ P ( R , C )(where ϕ is extended to a normal weight on c M + ) is nothing but the dual weight of ϕ (see [28] for details). Then, the modular automorphism group σ b ϕ t on R = M ⋊ σ R isimplemented by the translation operators λ ( t ), i.e., σ b σ t = Ad λ ( t ) with( λ ( t ) ξ )( s ) = ξ ( s − t ) ( ξ ∈ L ( R , H ))(or more precisely 1 ⊗ λ ( t ) on L ( R , H ) ∼ = H ⊗ L ( R , ds )). This is of course oneof generators in the crossed product M ⋊ σ R , and hence σ b ϕ t is inner, that is, R issemi-finite. Stone’s theorem says λ ( t ) = e itH with some self-adjoint operator H , and τ = b ϕ ( H − · ) (with the non-singular positive self-adjoint operator H = e H ) is a traceon R . This is often referred to as the canonical trace and possesses the followingtrace-scaling property: τ ◦ θ s = e − s τ ( s ∈ R ) . It is an easy exercise to show that via Fourier transformation F the operator H istransformed to F H F ∗ = m e t , (A.2)the multiplication operator induced by the function t e t . For a weight ϕ ∈ P ( M , C )let h φ be the Radon-Nikodym derivative of the dual weight b ϕ with respect to τ , i.e., b ϕ = τ ( h ϕ · ). Since b ϕ ◦ θ s = b ϕ from the definition, we have θ s ( h ϕ ) = e − s h ϕ . This scalingproperty is known to characterize all (densely defined) positive self-adjoint operators h (on L ( R , H )) of the form h = h ϕ for some ϕ ∈ P ( M , C ). We set H := the set of all positive self-adjoint operators h affiliated with R satisfying θ s ( h ) = e − s h for each s ∈ R so that ϕ ∈ P ( M , C ) ←→ h ϕ ∈ H (A.3)is an order preserving one-to-one bijective correspondence.One of the most important ingredients in Haagerup’s theory on non-commutative L p -spaces is the fact that h ϕ is a τ -measurable operator exactly when ϕ ∈ M + ∗ , that is,restricting (A.3) to M + ∗ we have an order preserving bijection ϕ ∈ M + ∗ ↔ h ϕ ∈ H ∩ R ,where R is the space of τ -measurable operators affiliated with R . For 0 < p ≤ ∞ Haagerup’s L p -space is defined by L p ( M ) := { a ∈ R ; θ s ( a ) = e − s/p a, s ∈ R } , (A.4)and its positive part is L p ( M ) + := L p ( M ) ∩ R + . In particular, L ∞ ( M ) = R θ (the θ -fixed point algebra) = M . Note L ( M ) = H ∩ R and the above bijection extends to70 ∈ M ∗ ↔ h ρ ∈ L ( M ) by linearity. Thus, a positive linear functional tr is defined on L ( M ) as tr( h ρ ) = ρ (1) ( ρ ∈ M ∗ ) . For 0 < p < ∞ the L p -(quasi)norm is defined by k a k p := (cid:0) tr( | a | p ) (cid:1) /p ( a ∈ L p ( M )) . Also k · k ∞ denotes the operator norm on M . When 1 ≤ p < ∞ , L p ( M ) is a Banachspace with the norm k · k p , and its dual Banach space is L q ( M ), where 1 /p + 1 /q = 1,by the duality form h a, b i p,q := tr( ab ) (= tr( ba )) for a ∈ L p ( M ), b ∈ L q ( M ) . (A.5)In particular, L ( M ) is a Hilbert space with the inner product( a, b ) := tr( b ∗ a ) (= tr( ab ∗ )) . Then the quadruple h π ℓ ( M ) , L ( M ) , ∗ , L ( M ) + i (where π ℓ means the left multiplication) is a standard form of M .Note that the space L p ( M ) is independent (up to an isometric isomorphism) of thechoice of ϕ . In fact, set R := M ⋊ σ ϕ R for another faithful semi-finite normal weight ϕ on M . Then there exists an ismorphism κ : R → R such that κ ◦ θ s ◦ κ − is the dualaction on R and τ ◦ κ − is the canonical trace on R (see [71, Chap. II, Theorem 37]).The κ extends to a homeomorphic isomorphism R → R , and it induces an isometricisomorphism between L p ( M ) in R and that in R . When M is semi-finite with a trace τ , L p ( M ) is naturally identified with the classical L p -space L p ( M , τ ) ([16, 53, 73])with respect to τ (see the last part of [71, Chap. II]). B Connes’ spatial theory
Materials in § § B.1 Spatial derivatives
Throughout let M be a von Neumann algebra acting on a Hilbert space H and χ bea (fixed) faithful semi-finite normal weight on the commutant M ′ . We will use the71ollowing standard notations: n χ := { y ′ ∈ M ′ ; χ ( y ′∗ y ′ ) < ∞} , H χ := the Hilbert space completion of n χ relative to the inner product( y ′ , y ′ ) χ ( y ′ ∗ y ′ ) , Λ χ := the canonical injection n χ → H χ ,π χ := the GNS regular representation of M ′ on H χ . For each ξ ∈ H we set R χ ( ξ )Λ χ ( y ′ ) = y ′ ξ (for y ′ ∈ M ′ ) , which is a densely defined ( D ( R χ ( ξ )) = Λ χ ( n χ )) operator from H χ to H . Obviously ξ R χ ( ξ ) is linear, and it is also plain to see R χ ( xξ ) = xR χ ( ξ ) for x ∈ M , (B.1) y ′ R χ ( ξ ) ⊆ R χ ( ξ ) π χ ( y ′ ) for y ′ ∈ M ′ . (B.2)A vector ξ ∈ H is said to be χ -bounded when R χ ( ξ ) is a bounded operator, that is, k y ′ ξ k ≤ C k Λ χ ( y ′ ) k = Cχ ( y ′∗ y ′ ) ( y ′ ∈ n χ )for some constant C >
0. We set D ( H , χ ) = { ξ ∈ H ; ξ is a χ -bounded vector } , which is M -invariant due to (B.1). One can show that D ( H , χ ) is a dense subspace in H ([13, Lemma 2]).When ξ ∈ D ( H , χ ), R χ ( ξ ) is extended to a bounded operator from H χ to H (forwhich the same symbol R χ ( ξ ) will be used). For ξ ∈ D ( H , χ ) we set θ χ ( ξ, ξ ) = R χ ( ξ ) R χ ( ξ ) ∗ ∈ M + . Indeed, R χ ( ξ ) R χ ( ξ ) ∗ is a bounded operator from H to itself, which commutes witheach y ′ ∈ M ′ (due to (B.2)). Thanks to (B.1) we have θ χ ( xξ, xξ ) = xθ χ ( ξ, ξ ) x ∗ (B.3)for x ∈ M and ξ ∈ D ( H , χ ). Remark B.1.
For a generic vector ξ ∈ H (not necessarily ξ ∈ D ( H , χ )) R χ ( ξ ) ∗ is a(possibly non-densely defined) closed operator from H χ to H . Therefore, the relation h ω ζ , θ χ ( ξ, ξ ) i = ( k R χ ( ξ ) ∗ ζ k if ζ ∈ D ( R χ ( ξ ) ∗ ) , + ∞ otherwiseuniquely determines an element θ χ ( ξ, ξ ) in the extended positive part c M + (see [29, § ω ζ means a positive linear functional( · ζ , ζ ). Then, (B.3) remains valid with the understanding h ω ζ , xθ χ ( ξ, ξ ) x ∗ i = h x ∗ ω ζ x, θ χ ( ξ, ξ ) i = h ω ζ ( x · x ∗ ) , θ χ ( ξ, ξ ) i = h ω x ∗ ζ , θ χ ( ξ, ξ ) i . ψ be a semi-finite normal weight on M . We set q ψ ( ξ ) = ψ ( θ χ ( ξ, ξ )) ∈ [0 , ∞ ] for ξ ∈ H (B.4)(where ψ is extended to a weight on c M + ). It is easy to see that ξ ∈ H 7→ q ψ ( ξ ) ∈ [0 , ∞ ]is a quadratic form, i.e., q ψ ( ξ + ξ ) + q ψ ( ξ − ξ ) = 2 q ψ ( ξ ) + 2 q ψ ( ξ ) , q ψ ( λξ ) = | λ | q ψ ( ξ )for ξ , ξ , ξ ∈ H and λ ∈ C . Moreover, q ψ ( · ) is lower semi-continuous (and hence q ψ isa positive form in the sense explained in §
2) as shown below.To show lower semi-continuity, we begin with the special case ψ = ω ζ with a vector ζ ∈ H . Since D ( R χ ( ξ )) = Λ χ ( n χ ), for ζ ∈ D ( R χ ( ξ ) ∗ ) we have q ω ζ ( ξ ) = k R χ ( ξ ) ∗ ζ k = sup χ ( y ′∗ y ′ ) ≤ | ( R χ ( ξ ) ∗ ζ , Λ χ ( y ′ )) | = sup χ ( y ′∗ y ′ ) ≤ | ( ζ , R χ ( ξ )Λ χ ( y ′ )) | = sup χ ( y ′∗ y ′ ) ≤ | ( ζ , y ′ ξ ) | . (B.5)This variational expression (B.5) remains valid for ζ
6∈ D ( R χ ( ξ ) ∗ ) as well. In this casewe have q ω ζ ( ξ ) = + ∞ from the definition and sup χ ( y ′∗ y ′ ) ≤ | ( ζ , R χ ( ξ )Λ χ ( y ′ )) | = + ∞ also holds true. Indeed, if this sup were C < ∞ , then we would get | ( ζ , R χ ( ξ )Λ χ ( y ′ )) | ≤ C / k Λ χ ( y ′ ) k for each y ′ ∈ n χ and hence ( ζ , R χ ( ξ )Λ χ ( y ′ )) = ( η, Λ χ ( y ′ )) for some vector η ∈ H χ , contradicting ζ
6∈ D ( R χ ( ξ ) ∗ ). Since the map ξ ( ζ , y ′ ξ ) is continuous foreach fixed y ′ , q ω ζ ( · ) is lower semi-continuous thanks to (B.5). Finally a general weight ψ can be expressed as ψ = P ι ∈ I ω ζ ι with a suitable family { ζ ι } ι ∈ I of vectors (see [26])so that q ψ = P ι ∈ I q ω ζι is also lower semi-continuous.We set D ( q ψ ) = { ξ ∈ H ; q ψ ( ξ ) < ∞} . Since ψ is semi-finite, we can prove that D ( q ψ ) is dense in H . This follows from thefollowing observation: for x ∈ n ψ = { x ∈ M ; ψ ( x ∗ x ) < ∞} and ξ ∈ D ( H , χ ) we have q ψ ( x ∗ ξ ) = ψ ( θ χ ( x ∗ ξ, x ∗ ξ )) = ψ ( x ∗ θ χ ( ξ, ξ ) x ) ≤ k θ χ ( ξ, ξ ) k ψ ( x ∗ x ) < ∞ due to (B.3) (see [13, Lemma 6] for details). Therefore, the restriction q ψ | D ( q ψ ) : ξ ∈ D ( q ψ ) q ψ ( ξ ) ∈ [0 , ∞ )is a densely defined quadratic form. Lower semi-continuity explained above means thatthis quadratic form is closed (see [61, Proposition 10.1] for instance).With preparations so far we are now ready to use the standard representation the-orem for densely defined closed quadratic forms (see [60, Theorem VIII.15] or [61,Theorem 10.7] for instance). Definition B.2.
The positive self-adjoint operator associated with the densely definedclosed quadratic form q ψ | D ( q ψ ) (see (B.4)) is denoted by dψ/dχ , the spatial derivative of73 (semi-finite normal) weight ψ on M relative to a (faithful semi-finite normal) weight χ on M ′ . Since we have captured dψ/dχ via a closed quadratic form, we have q ψ ( ξ ) = ( k ( dψ/dχ ) / ξ k when ξ ∈ D (( dψ/dχ ) / ) , + ∞ otherwisewith D ( q ψ ) = D (( dψ/dχ ) / ).A few remarks are in order: Remark B.3. (i) The extended positive part c M + is not used in [13] and slightly different expla-nation is given there. Namely, the quadratic form q ψ (see (B.4)) is just definedon D ( H , χ ). Then, it is extended to a lower semi-continuous quadratic form onthe whole space H (based on the idea borrowed from [63]). Its restriction tothe “finite part” (where finite values are taken) is a closable quadratic form dueto lower semi-continuity. Then the spatial derivative dψ/dχ is defined as thepositive self-adjoint operator associated with the closure of this closable form.Equivalence of the two definitions is explained in [71, Chap. III].(ii) The support (as a positive self-adjoint operator) of dψ/dχ is the support (as aweight) of ψ (see [13, Corollary 12]).(iii) The intersection D ( H , χ ) ∩ D (cid:0) ( dψ/dχ ) / (cid:1) is known to be a core for the squareroot ( dψ/dχ ) / (see [71, Chap. III, Proposition 22, 2)]).We record the following fact ([13, Proposition 3]) (that was needed in § Lemma B.4.
There exists a family { ξ ι } ι ∈ I of vectors in D ( H , χ ) , the set of χ -boundedvectors, such that X ι ∈ I θ χ ( ξ ι , ξ ι ) = 1 . Let us start from a standard form hM , H , J, Pi and ϕ, ϕ ∈ M + ∗ with ϕ faithful.Let ξ ϕ , ξ ϕ be unique implementing vectors in P for ϕ, ϕ respectively so that ξ ϕ iscyclic and separating. Here we recall the notion of the relative modular operator dueto Araki [7]. A densely defined (conjugate linear) operator S ϕϕ : xξ ϕ ∈ M ξ ϕ x ∗ ξ ϕ ∈ M ξ ϕ is closable, and its closure admits the polar decomposition of the form S ϕϕ = J ∆ / ϕϕ . The positive self-adjoint operator ∆ ϕψ here is referred to as the relative modular oper-ator (of ϕ relative to ψ ) in the literature. We set ϕ ′ ( y ′ ) = ( y ′ ξ ϕ , ξ ϕ ) ( y ′ ∈ M ′ )74o that ϕ ′ ∈ M ′ + ∗ is the same as ϕ ′ ( y ′ ) = ϕ ( J y ′∗ J ). We consider the spatial derivative dϕ/dϕ ′ in this special setting. We have of course n ϕ ′ = M ′ and H ϕ ′ , Λ ϕ ′ , π ϕ ′ can beidentified with H ϕ ′ = H , Λ ϕ ′ ( y ′ ) = y ′ ξ ϕ , π ϕ ′ ( y ′ ) ξ = y ′ ξ. An operator R ϕ ′ ( ξ ) (with ξ ∈ H ) from H to H ϕ ′ becomes an operator on H and isgiven by R ϕ ′ ( ξ )( y ′ ξ ϕ ) = y ′ ξ ( y ′ ∈ M ′ ) . If R ϕ ′ ( ξ ) is a bounded operator on H , then R ϕ ′ ( ξ ) sits in M ′′ = M (due to (B.2)) andthis means ξ = R ϕ ′ ( ξ ) ξ ϕ ∈ M ξ ϕ . Therefore, we simply have D ( H , ϕ ′ ) = M ξ ϕ ,R ϕ ′ ( xξ ϕ ) = x for x ∈ M ,θ ϕ ′ ( xξ ϕ , xξ ϕ ) = R ϕ ′ ( xξ ϕ ) R ϕ ′ ( xξ ϕ ) ∗ = xx ∗ . Therefore, the quadratic form q ϕ (used to define the spatial derivative dϕ/dϕ ′ , see(B.4)) is simply xξ ϕ ∈ M ξ ϕ q ϕ ( xξ ϕ ) = ϕ ( xx ∗ ) ∈ [0 , ∞ ) . We note ϕ ( xx ∗ ) = k x ∗ ξ ϕ k (cid:0) = k S ϕϕ xξ ϕ k = k ∆ / ϕϕ xξ ϕ k (cid:1) . Therefore, the spatial derivative dϕ/dϕ ′ in this special case is nothing but the relativemodular operator ∆ ϕϕ , i.e., dϕ/dϕ ′ = ∆ ϕϕ ( ϕ ∈ M + ∗ ) . (B.6) B.2 Hilsum’s L p -spaces Let M be a von Neumann algebra acting on H , and we fix a faithful semi-finite normalweight χ on the commutant M ′ . For a semi-finite normal weight ϕ we denote Connes’spatial derivative of ϕ relative to χ by dϕ/dχ (see Appendix § B.1 for details). Then, for p ∈ [1 , ∞ ) Hilsum’s L p -space L p ( M , χ ) consists of all (densely defined) closed operators T on H whose polar decompositions T = u | T | satisfy u ∈ M and | T | p = dϕ/dχ with ϕ ∈ M + ∗ (see [38]). In particular, we have L p ( M , χ ) + = (cid:8) ( dϕ/dχ ) /p ; ϕ ∈ M + ∗ (cid:9) . For a functional ρ ∈ M ∗ with the polar decomposition ρ = uϕ (with ϕ = | ρ | ∈ M + ∗ )as a functional the notations dρ/dχ = udϕ/dχ and Z dρ/dχ dχ = ρ (1) (= ϕ ( u ))are used in [38]. Hilsum’s L p -space L p ( M , χ ) can be identified with the Haagerup L p -space L p ( M ) (see Appendix A) via T ←→ uh /pϕ , T has the polar decomposition T = u ( dϕ/dχ ) /p . In fact, through this identifica-tion all the operations (such as sums, products and so on) among elements in L p ( M , χ )(which are unbounded operators) are justified (see [38, §
1] and [71, Chap. IV] for de-tails). Thus, one can manipulate them like τ -measurable operators. Via this iden-tification between L ( M , χ ) and L ( M ) the above R · dχ corresponds exactly to thetrace-like linear functional tr( · ) (explained in Appendix A). By this reason, we will usethe the same symbol tr (instead of the integral notation R · dχ ). For T ∈ L p ( M , χ ) its L p -norm is defined by k T k p = (cid:0) tr( | T | p ) (cid:1) /p . Thus, for T with the polar decomposition u ( dϕ/dχ ) /p we have k T k p = ϕ (1) /p .In particular, when M is represented in a standard form hM , H , J, Pi and χ = ϕ ′ with a faithful ϕ ∈ M + ∗ (where ϕ ′ ( x ′ ) = ϕ ( J x ′∗ J )), in view of (B.6) we can write L p ( M , ϕ ′ ) + = { ∆ /pϕϕ ; ϕ ∈ M + ∗ } , (B.7)and ∆ /pϕϕ = ( dϕ/dϕ ′ ) /p corresponds to h /pϕ in Haagerup’s L p -space L p ( M ).The well-known duality between L p ( M , ψ ′ ) and L q ( M , ψ ′ ) with 1 /p + 1 /q = 1 isgiven by the bilinear form h· , ·i on L p ( M , ψ ′ ) × L q ( M , ψ ′ ) defined by h A, B i (= h A, B i p,q ) = tr( AB ) for A ∈ L p ( M , ψ ′ ), B ∈ L q ( M , ψ ′ ) (B.8)(which of course corresponds to (A.5)). Here, suffixes p, q may be omitted when noconfusion is possible. The inner product in the L -space L ( M , ψ ′ ) is given by( A, B ) = tr( AB ∗ ) . We note that the quadruple h π ℓ ( M ) , L ( M , ψ ′ ) , ∗ , L ( M , ψ ′ ) + i (where π ℓ means the left multiplication) is a standard form. B.3 Canonical correspondence of P ( M , C ) and P ( B ( H ) , M ′ ) For convenience we will use the following notations: P ( M , C ) := the set of all faithful semi-finite normal weights on M ,P ( B ( H ) , M ′ ) := the set of all faithful semi-finite normaloperator valued weights from B ( H ) to M ′ . A canonical order reversing correspondence between P ( M , C ) and P ( B ( H ) , M ′ ) wasconstructed in [13], and brief explanation on this correspondence is presented.We take faithful semi-finite normal weights φ, φ on M and a faithful semi-finite nor-mal weight χ on M ′ . We will repeatedly use modular properties of spatial derivatives(see [13]): 76a) ( dφ/dχ ) − = dχ/dφ ,(b) We have σ φt ( x ) = ( dφ/dχ ) it x ( dφ/dχ ) − it for x ∈ M ,σ χt ( y ′ ) = ( dφ/dχ ) − it y ′ ( dφ/dχ ) it (cid:0) = ( dχ/dφ ) it y ′ ( dχ/dφ ) − it (cid:1) for y ′ ∈ M ′ , (c) ( Dφ : Dφ ) t = ( dφ/dχ ) it ( dφ /dχ ) − it .Firstly we start from φ ∈ P ( M , C ) (by fixing χ for a moment). We set ω =Tr(( dχ/dφ ) · ) (with the standard trace on B ( H ) and the density operator dχ/dφ ) whichis a weight on B ( H ). We note σ ωt = Ad ( dχ/dφ ) it and σ ωt | M ′ = σ χt (as was mentioned in the above (b)). Thus, there is a unique operator valued weight E ∈ P ( B ( H ) , M ′ ) satisfying ω = χ ◦ E ([29, Theorem 5.1]). We note that E does notdepend on χ . Indeed, for another faithful semi-finite normal weight χ ′ on M ′ we have( D ( χ ′ ◦ E ) : D ( χ ◦ E )) t = ( Dχ ′ : Dχ ) t = ( dχ ′ /dφ ) it ( dχ/dφ ) − it . This means that the density operator for the weight χ ′ ◦ E on B ( H ) is dχ ′ /dφ , that is, χ ′ ◦ E = Tr(( dχ ′ /dφ ) · ) and χ ′ gives us the same E . Let us use the notation E = φ − .Discussions so far means that the defining property for φ − ∈ P ( B ( H ) , M ′ ) is χ ◦ φ − = Tr(( dχ/dφ ) · ) (B.9)for each faithful semi-finite normal weight χ on M ′ . Note that the value of (B.9)against a rank-one operator ξ ⊗ ξ c isTr (( dχ/dφ )( ξ ⊗ ξ c )) = k ( dχ/dφ ) / ξ k = χ (cid:0) θ φ ( ξ, ξ ) (cid:1) (see (B.4)) so that (B.9) means φ − ( ξ ⊗ ξ c ) = θ φ ( ξ, ξ ) . This time we start from F ∈ P ( B ( H ) , M ′ ). We set ω ′ = χ ◦ F . This is a weight on B ( H ) and of the form Tr( K · ) with some density operator K . We have σ ω ′ t = Ad K it and σ ω ′ t ( y ′ ) = σ χt ( y ′ ) = ( dφ/dχ ) − it y ′ ( dφ/dχ ) it for y ′ ∈ M ′ , and hence D t = K − it ( dφ/dχ ) − it falls into M ′′ = M . We also note D t + s = K − it K − is ( dφ/dχ ) − is ( dφ/dχ ) − it = K − it ( dφ/dχ ) − it ( dφ/dχ ) it K − is ( dφ/dχ ) − is ( dφ/dχ ) − it = D t σ φt ( D s ) . This means that D t is a σ φ -cocycle so that D t = ( Dψ : Dφ ) t (cid:0) = ( dψ/dχ ) it ( dφ/dχ ) − it (cid:1) ψ on M ([11, Th´eor`eme 1.2.4]), showing K = ( dψ/dχ ) − = dχ/dψ and χ ◦ F = ω ′ (cid:0) = Tr( K · ) (cid:1) = Tr(( dχ/dψ ) · ) . This expression shows that ψ is determined by F and χ . However, ψ does not dependon χ . Indeed, for another faithful semi-finite normal weight χ ′ on M ′ we have( D ( χ ′ ◦ F ) : D ( χ ◦ F )) t = ( Dχ ′ : Dχ ) t = ( dχ ′ /dψ ) it ( dχ/dψ ) − it , showing χ ′ ◦ F = Tr(( dχ ′ /dψ ) · ). Let us denote the above ψ ∈ P ( M , C ) by F − .Discussions so far mean that the defining property of F − is χ ◦ F = Tr(( dχ/dF − ) · ) (B.10)for each faithful semi-finite normal weight χ on M ′ .It is easy to see ( φ − ) − = φ for φ ∈ P ( M , C ) and ( F − ) − = F for F ∈ P ( B ( H ) , M ′ ) by repeated use of the defining properties (B.9) and (B.10). Indeed,we have Tr (cid:0)(cid:0) dχ/d (( φ − ) − ) (cid:1) · (cid:1) = χ ◦ φ − = Tr (( dχ/dφ ) · ) ,χ ◦ ( F − ) − = Tr (cid:0)(cid:0) dχ/dF − (cid:1) · (cid:1) = χ ◦ F. The properties (B.9) and (B.10) also clearly show that taking the inverse is order-reversing, which can be also seen from( Dφ − : Dφ − ) t = ( D ( χ ◦ φ − ) : D ( χ ◦ φ − )) t = ( dχ/dφ ) it ( dχ/dφ ) − it (by (B.9))= ( dφ /dχ ) − it ( dφ /dχ ) it = ( Dφ : Dφ ) − t = ( Dφ : Dφ ) ∗− t (B.11)for φ , φ ∈ P ( M , C ). Remark B.5.
Let us clarify the reason why order-reversing property can be seen from(B.11). We set I − / = { z ∈ C ; − / ≤ Im z ≤ } and I / = { z ∈ C ; 0 ≤ Im z ≤ / } for convenience. Then we observe:( Dφ − : Dφ − ) t extends to a bounded continuous function on I − / whichis analytic in the interior if and only if ( Dφ : Dφ ) t has the same extensionproperty. Furthermore, when this extension property is satisfied, we have( Dφ : Dφ ) z = ( Dφ − : Dφ − ) ∗− z (for z ∈ I − / ) (B.12)and in particular ( Dφ : Dφ ) − i/ = ( Dφ − : Dφ − ) ∗− i/ .Therefore, the order reversing property holds true thanks to [12, Th´eor`eme 3].78o see the above extension property, let us assume that ( Dφ − : Dφ − ) t has theextension ( Dφ − : Dφ − ) z ( z ∈ I − / ) stated above. The obvious computation(( Dφ − : Dφ − ) ∗ z ξ, ζ ) = ( ξ, ( Dφ − : Dφ − ) z ζ ) = (( Dφ − : Dφ − ) z ζ , ξ )(for vectors ξ, ζ ) shows that ( Dφ − : Dφ − ) ∗ z is a bounded continuous function on I − / = I / which is analytic in the interior. Therefore, ( Dφ − : Dφ − ) ∗− z is a boundedcontinuous function on − I / = I − / which is analytic in the interior. The value ofthis function for z = t ∈ R is ( Dφ − : Dφ − ) ∗− t = ( Dφ : Dφ ) t thanks to (B.11). Thismeans that ( Dφ : Dφ ) t extends to a bounded continuous function on I − / which isanalytic in the interior and we have (B.12). Conversely, when ( Dφ : Dφ ) t has theextension ( Dφ : Dφ ) z ( z ∈ I − / ) as stated above, by similar arguments as above( Dφ : Dφ ) ∗− z is a bounded continuous function on I − / which is analytic in theinterior. Also for z = t ∈ R the value of this function is ( Dφ : Dφ ) ∗− t = ( Dφ − : Dφ − ) t , that is, we have the extension ( Dφ − : Dφ − ) z and ( Dφ − : Dφ − ) z = ( Dφ : Dφ ) ∗− z ( z ∈ I − / ). This equation is of course equivalent to (B.12).In [29] an order-reversing bijection (called α ) from P ( M , C ) onto P ( B ( H ) , M ′ ) wasconstructed in a less canonical fashion, and the formula ( Dα ( T ) : Dα ( T )) t = ( DT : DT ) t appears in [29, p. 360]. This is an obvious misprint, and ( Dα ( T ) : Dα ( T )) t =( DT : DT ) ∗− t is the correct formula. C Infima of decreasing sequences of weights
In the main body of the article we have encountered decreasing sequences of variousoperators such as τ -measurable operators and positive elements in non-commutative L p -spaces (see Theorem 3.8 and Proposition 4.2 for instance). More generally decreas-ing sequences of unbounded positive self-adjoint operators (or rather positive formsin the sense of §
2) were considered in [50] (motivated by [63]). Here, for the sake ofcompleteness we will study decreasing sequences of weights on a von Neumann algebra,and probably no such study has been made in literature.Let M be a von Neumann algebra acting on a Hilbert space H and χ be a (fixed)faithful semi-finite normal weight on the commutant M ′ . The modular automorphismgroup on M ′ induced by χ will be denoted by { σ ′ t } t ∈ R . Definition C.1 ([13, 38, 71]) . Assume γ ∈ R . A densely defined closed operator T on H with the polar decomposition T = u | T | is said to be γ -homogeneous (relative to χ )when u ∈ M and σ ′ γt ( y ′ ) | T | it = | T | it y ′ for each t ∈ R , y ′ ∈ M ′ .The next characterization is proved based on Carlson’s theorem for entire functionsof exponential type (see [43, Lemma 2.1] and [72, p. 338]). For instance it is difficultto see that the sum of γ -homogeneous operators possesses the same homogeneity fromDefinition C.1 whereas this property can be immediately seen from the characterizationin the next lemma. 79 emma C.2. A densely defined closed operator T on H is γ -homogeneous if and onlyif y ′ T ⊆ T σ ′ iγ ( y ′ ) for each y ′ ∈ M ′ analytic with respect to the modular automorphism group σ ′ t . This characterization for γ -homogeneity will be used to prove the next result. Lemma C.3.
Let T n , n = 1 , , · · · , be a sequence of positive self-adjoint operatorsconverging to a positive self-adjoint operator T in the strong resolvent sense ( i.e., ( T n +1) − → ( T + 1) − in the strong operator topology ) . If each T n is γ -homogeneous, thenso is T .Proof. We choose and fix ξ ∈ D ( T ) and set ξ n = ( T n + 1) − ( T + 1) ξ ∈ D ( T n + 1) = D ( T n )for each n = 1 , , · · · . Since ( T n + 1) − → ( T + 1) − strongly, we havelim n →∞ k ξ n − ξ k = lim n →∞ k (cid:0) ( T n + 1) − − ( T + 1) − (cid:1) ( T + 1) ξ k = 0 . (C.1)Note T n ξ n = ( T n + 1) ξ n − ξ n = ( T + 1) ξ − ξ n and hence (C.1) also meanslim n →∞ k T n ξ n − T ξ k = 0 . (C.2)We take an analytic element y ′ ∈ M ′ with respect to the modular automorphismgroup σ ′ t . Since y ′ T n ⊆ T n σ ′ iγ ( y ′ ) by Lemma C.2, we have σ ′ iγ ( y ′ ) ξ n ∈ D ( T n ) and y ′ T n ξ n = T n σ ′ iγ ( y ′ ) ξ n . Since T n σ ′ iγ ( y ′ ) ξ n − y ′ T ξ = y ′ T n ξ n − y ′ T ξ = y ′ ( T n ξ n − T ξ ) , (C.2) implies lim n →∞ k T n σ ′ iγ ( y ′ ) ξ n − y ′ T ξ k = 0 . (C.3)Since σ ′ iγ ( y ′ ) ξ n ∈ D ( T n ), we can set η n = ( T + 1) − ( T n + 1) σ ′ iγ ( y ′ ) ξ n ∈ D ( T + 1) = D ( T ) . We observe η n − σ ′ iγ ( y ′ ) ξ n = (cid:0) ( T + 1) − − ( T n + 1) − (cid:1) ( T n + 1) σ ′ iγ ( y ′ ) ξ n . We set ζ n = ( T n + 1) σ ′ iγ ( y ′ ) ξ n − (cid:0) y ′ T ξ + σ ′ iγ ( y ′ ) ξ (cid:1) = (cid:0) T n σ ′ iγ ( y ′ ) ξ n − y ′ T ξ (cid:1) + σ ′ iγ ( y ′ ) ( ξ n − ξ ) . The second expression, (C.3) and (C.1) guaranteelim n →∞ k ζ n k = 0 . (C.4)80y using this ζ n we rewrite the above η n − σ ′ iγ ( y ′ ) ξ n as follows: η n − σ ′ iγ ( y ′ ) ξ n = (cid:0) ( T + 1) − − ( T n + 1) − (cid:1) (cid:0)(cid:0) y ′ T ξ + σ ′ iγ ( y ′ ) ξ (cid:1) + ζ n (cid:1) . We estimate k η n − σ ′ iγ ( y ′ ) ξ n k ≤ k (cid:0) ( T + 1) − − ( T n + 1) − (cid:1) (cid:0) y ′ T ξ + σ ′ iγ ( y ′ ) ξ (cid:1) k + k ( T + 1) − − ( T n + 1) − k k ζ n k≤ k (cid:0) ( T + 1) − − ( T n + 1) − (cid:1) (cid:0) y ′ T ξ + σ ′ iγ ( y ′ ) ξ (cid:1) k + 2 k ζ n k so that the strong resolvent convergence of T n to T and (C.4) showlim n →∞ k η n − σ ′ iγ ( y ′ ) ξ n k = 0 , (C.5)that is, lim n →∞ k η n − σ ′ iγ ( y ′ ) ξ k = 0 (C.6)(see (C.1)).On the other hand, we have T η n − y ′ T ξ = ( T + 1) η n − η n − y ′ T ξ = ( T n + 1) σ ′ iγ ( y ′ ) ξ n − η n − y ′ T ξ = (cid:0) T n σ ′ iγ ( y ′ ) ξ n − y ′ T ξ (cid:1) + (cid:0) σ ′ iγ ( y ′ ) ξ n − η n (cid:1) , which enables us to conclude lim n →∞ k T η n − y ′ T ξ k = 0 (C.7)due to (C.5) and (C.3).Each η n sits in D ( T ) so that the convergences (C.6) and (C.7) mean( σ ′ iγ ( y ′ ) ξ, y ′ T ξ ) ∈ Γ( T ) , the closure of the graph of T (in H ⊕ H ) . Therefore, from the closedness of T we conclude σ ′ iγ ( y ′ ) ξ ∈ D ( T ) with T σ ′ iγ ( y ′ ) ξ = y ′ T ξ and we are done.If all of T n ’s and the limiting operator T are assumed to be non-singular, then LemmaC.3 is much easier to prove. Indeed, in this situation one can just use the condition(involving unitary operators T itn , T it ) in Definition C.1. Then, since T itn = exp( it log T n )and T it = exp( it log T ), one can use Trotter’s theorem (see [60, Theorem VIII.21]) asin the proof of [13, Corollary 15].In [13] an increasing sequence { ω n } of faithful normal weights was studied. The keypoint here is that the point-wise supremum ω = sup n ω n (i.e., ω ( x ) = sup n ω n ( x ), x ∈M + ) is normal. Indeed, normality means lower semi-continuity in the σ -weak topology,and the supremum of lower semi-continuous functions is lower semi-continuous. When ω is semi-finite, it is known that spatial derivatives dω n /dχ (increasingly) tend to dω/dχ
81n the strong resolvent sense and hence corresponding modular automorphisms behavein the expected way (see [13, Corollary 15] for details). We will deal with a decreasingsequence { ω n } n of normal weights instead in the next theorem. (The theorem is usedin Remark 5.8,(iii) of § ω (and hence all ω n ’s), butfaithfulness is not assumed (and hence Lemma C.3 is needed). The point-wise infimuminf n ω n is certainly a weight (in the algebraic sense), but one cannot expect lower-semicontinuity in the σ -weak topology (i.e., normality), see Remark C.5,(ii). Theorem C.4.
Let { ω n } n ∈ N be a decreasing sequence of semi-finite normal weights on M . There exists a semi-finite normal weight ω such that spatial derivatives dω n /dχ , n = 1 , , · · · , converge ( decreasingly ) to dω/dχ in the strong resolvent sense. Further-more, ω is the maximum of all semi-finite normal weights majorized by the point-wiseinfimum inf n ω n . Our proof is based on the order preserving one-to-one correspondence. ϕ ∈ P ( M , C ) ←→ dϕ/dχ ∈ M + − explained in § Proof.
We have the following decreasing sequence of spatial derivatives: dω /dχ ≥ dω /dχ ≥ dω /dχ ≥ · · · . We set T = Inf n dω n /dχ so that the sequence dω n /dχ converges to T in the strongresolvent sense (see [50, 63] for details). Since each dω n /dχ is ( − T thanks to Lemma C.3. Thus, we have T = dω/dχ for some semi-finite normalweight ω . Since dω /dχ ≥ dω /dχ ≥ dω /dχ ≥ · · · ≥ T = dω/χ , we know ω n ≥ ω .Conversely, we assume that a semi-finite normal weight ω ′ satisfies inf n ω n ≥ ω ′ .Then, we have ω n ≥ ω ′ for each for each n . Hence we have dω n /dχ ≥ dω ′ /dχ for each n so that the point-wise infimum inf n satisfiesinf n dω n /dχ ≥ dω ′ /dχ. Thus, the maximality of Inf n dω n /dχ (= T ) among all positive forms majorized by inf n (see the paragraph before Theorem 2.3) impliesInf n dω n /dχ (= dω/dχ ) ≥ dω ′ /dχ. This means ω ≥ ω ′ , showing the desired maximality of ω .A few remarks on the theorem are in order: Remark C.5. ω mentioned in the theorem does not depend upon the choice of afathful semi-finite normal weight χ on the commutant M ′ (which can be seenfrom the latter statement of the theorem). Let us use the notation ω = Inf n ω n , and the proof of Theorem C.4 says d (cid:16) Inf n ω n (cid:17) /dχ = Inf n ( dω n /dχ ) . (ii) When { ω n } n ∈ N is a decreasing sequence in M + ∗ , ω n tends to ω ∈ M + ∗ in norm.We have ω n ( x ) ց ω ( x ) for x ∈ M + and ω ( x ) = inf n ω n ( x ). However, generallythe point-wise infimum inf n ω n and the weight ω in the theorem could be quitedifferent. This phenomenon can be seen by considering the special case M = B ( H ). In this case a semi-finite normal weight is of the form Tr( T · ) with a denselydefined positive self-adjoint operator T and the standard trace Tr of B ( H ). Thevalue of this weight against a rank-one operator ξ ⊗ ξ c isTr( T ( ξ ⊗ ξ c )) = k T / ξ k (cid:0) = q T ( ξ ) (cid:1) , i.e., the canonical quadratic form associated with T . There are many examplesof decreasing sequences T ≥ T ≥ T ≥ · · · of densely defined closed quadraticforms such that the point-wise infimum inf n q T n is not closable, i.e., inf n q T n (cid:9) Inf n q T n (see [50, 63]).Let us recall the canonical order reversing correspondence ϕ ∈ P ( M , C ) ←→ ϕ − ∈ P ( B ( H ) , M ′ )explained in § B.3. We will interpret the meaning of Inf n ω n in terms of this correspon-dence.For simplicity let us start from the following situation: ω ≥ ω ≥ ω ≥ · · · ≥ ω with some faithful normal weight ω . Then we obviously have inf n ω n ≥ ω and themaximality in the preceding theorem shows Inf n ω n ≥ ω . Therefore, faithfulness ofall ω n ’s and also Inf n ω n are guaranteed (which is the only role played by ω ). Then,by using the above-explained order reversing one-to-one correspondence, we get theincreasing sequence ω − ≤ ω − ≤ ω − ≤ · · · ≤ ω − . of faithful semi-finite normal operator valued weights from B ( H ) to M ′ . Lemma C.6.
Under the above-explained situation the point-wise supremum sup n ω − n of the increasing sequence { ω − n } n ∈ N in P ( B ( H ) , M ′ ) ( which is obviously an operatorvalued weight from B ( H ) to M ′ ) is normal. roof. Let us assume x ι ր x in B ( H ) + andsup n ω − n ( x ι ) ր y (cid:18) ≤ sup n ω − n ( x ) (cid:19) taken in the extended positive part c M ′ + . For a fixed ι we have sup n ω − n ( x ι ) ≤ y andconsequently sup n ϕ ◦ ω − n ( x ι ) = ϕ (cid:18) sup n ω − n ( x ι ) (cid:19) ≤ ϕ ( y )for each normal weight ϕ on M ′ (which extends to c M ′ + , see [29, Proposition 1.10]).Therefore, we have sup ι (cid:18) sup n ϕ ◦ ω − n ( x ι ) (cid:19) ≤ ϕ ( y ) . (C.8)On the other hand, { ϕ ◦ ω − n } n ∈ N is an increasing sequence of normal weights on B ( H )so that sup n ϕ ◦ ω − n is normal (see the paragraph right before Theorem C.4). Thus,the left hand side of (C.8) is actually sup n ϕ ◦ ω − n ( x ) and we havesup n ϕ ◦ ω − n ( x ) ≤ ϕ ( y ) . The left hand side here is equal to ϕ (sup n ω − n ( x )) by [29, Proposition 1.10] and theabove inequality is valid for each ϕ so that we concludesup n ω − n ( x ) ≤ y . The following fact is probably worth pointing out, that is considered as a weightversion of the formula (2.5) in § Proposition C.7.
Let { ω n } n ∈ N be a decreasing sequence of semi-finite normal weightson M , and we assume ω ≥ ω ≥ ω ≥ · · · ≥ ω with some faithful normal weight ω . Then, the semi-finite normal weight ω = Inf n ω n in Theorem C.4 ( i.e., the maximum of all semi-finite normal weights majorized by thepoint-wise infimum inf n ω n ) is given by Inf n ω n = (cid:0) sup n ω − n (cid:1) − . Proof.
It suffices to show (Inf n ω n ) − = sup n ω − n . For each n we have χ ◦ ω − n = Tr (( dχ/dω n ) · )thanks to (B.9). Therefore, we have χ ◦ (cid:18) sup n ω − n (cid:19) = sup n χ ◦ ω − n (since χ is normal)= sup n Tr (( dχ/dω n ) · ) = Tr (cid:18)(cid:18) sup n dχ/dω n (cid:19) · (cid:19) . (C.9)84e observe that the density in the above right side issup n dχ/dω n = sup n ( dω n /dχ ) − = (cid:16) Inf n dω n /dχ (cid:17) − = (cid:16) d (cid:16) Inf n ω n (cid:17) /dχ (cid:17) − = dχ/d (cid:16) Inf n ω n (cid:17) . Here, the second equality comes from (2.5) while the third is just the definition ofInf n ω n (see Remark C.5,(i)). 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