Von Neumann algebra conditional expectations with applications to generalized representing measures for noncommutative function algebras
aa r X i v : . [ m a t h . OA ] J a n VON NEUMANN ALGEBRA CONDITIONAL EXPECTATIONSWITH APPLICATIONS TO GENERALIZED REPRESENTINGMEASURES FOR NONCOMMUTATIVE FUNCTION ALGEBRAS
DAVID P. BLECHER AND LOUIS E. LABUSCHAGNE
Abstract.
We establish several deep existence criteria for conditional expec-tations on von Neumann algebras, and then apply this theory to develop anoncommutative theory of representing measures of characters of a functionalgebra. Our main cycle of results describes what may be understood as a‘noncommutative Hoffman-Rossi theorem’ giving the existence of weak* con-tinuous ‘noncommutative representing measures’ for so-called D -characters.These results may be viewed as ‘module’ Hahn-Banach extension theorems forweak* continuous ‘characters’ into possibly noninjective von Neumann alge-bras. In closing we introduce the notion of ‘noncommutative Jensen measures’and show that as in the classical case, representing measures of logmodular al-gebras are Jensen measures. The proofs of the two main cycles of results relyon the delicate interplay of Tomita-Takesaki theory, noncommutative Radon-Nikodym derivatives, Connes cocycles, Haagerup noncommutative L p -spaces,Haagerup’s reduction theorem, etc. Introduction
Recall that if
A ⊂ C ( K ) is a function algebra or uniform algebra on a compactspace K , then the primary associated object is the set M A of (scalar valued) char-acters on A . These are the nonzero scalar valued homomorphisms on A . The basictexts on function algebras spend considerable time on the theory and applicationsof these characters, which are fundamental in this branch of function theory (seee.g. [15, 33]). In [9], inspired by Arveson’s seminal paper [1], we proposed a newgeneralization of characters applicable to operator algebras. By this last expressionwe mean a subalgebra A of a C ∗ -algebra or von Neumann algebra M . In trans-lating the classical theory of characters χ : A → C we may identify the range ofthe character with D = C A = C M , in which case the homomorphism becomesan idempotent map on A , and is a D -bimodule map. Thus in the operator algebra Date : 12 January, 2021.2000
Mathematics Subject Classification.
Primary: 46L10, 46L51, 46L52, 46L53, 47L30; Sec-ondary: 46E30, 46A22, 46J10, 47L45, 47N30.
Key words and phrases.
Operator algebras, conditional expectation, representing measure,subdiagonal algebra, non-commutative integration, weight on von Neumann algebras, Haagerup L p -spaces, noncommutative Radon-Nikodym theorem, Jensen measures, character of a functionalgebra.DB is supported by a Simons Foundation Collaboration Grant (527078). LL was supported bythe National Research Foundation KIC Grant 171014265824. Any opinion, findings and conclu-sions or recommendations expressed in this material, are those of the author, and therefore theNRF do not accept any liability in regard thereto. LL was also the recipient of a UCDP grantfor the period 2019-2020 which indirectly helped to fund a visit by the first author where some ofthis work was done. case we suppose that we have a C ∗ - (indeed in this paper usually von Neumann)subalgebra D of A and of M , and we define a D - character to be a unital contractivehomomorphism Φ : A → D . We also assume that Φ is a D -bimodule map (or equiv-alently, is the identity map on D ). The basic theory of D -characters is developedin [9]. One motivation for our study of such maps comes from [1], where Arvesongives many important examples of D -characters in his approach to noncommutativeanalyticity/generalized analytic functions/noncommutative Hardy spaces. In this‘generalized analytic function theory’, and in certain other operator algebraic situa-tions, it is very important that the ‘noncommutative character’ Φ is D -valued ratherthan merely B ( H )-valued (and the same comment holds for the ‘noncommutativerepresenting measures’ discussed below).In the present paper we simultaneously generalize important aspects of the theoryof conditional expectations of von Neumann algebras, and the theory of representingmeasures of characters of function algebras. To describe the first topic we begin byrecalling some fundamental probability theory. Let ( K, A , µ ) be a probability mea-sure space. ‘Admissable’ sub- σ -algebras B of A correspond (up to null sets) to vonNeumann subalgebras D of M = L ∞ ( K, A , µ ). For any such subalgebra D thereexists a unique conditional expectation E : M → D such that R E ( f ) dµ = R f dµ for f ∈ M . Also E is weak* continuous (or equivalently, in von Neumann algebralanguage, normal ), and faithful (that is, E ( f ) = 0 implies f = 0 if f ∈ M + ). In thevon Neumann algebra context we will by a conditional expectation mean a (usuallynormal) unital positive (or equivalently, by p. 132–133 in [2], contractive) idem-potent D -module map E : M → D from a C ∗ -algebra onto its C ∗ -subalgebra D .The unique conditional expectation for µ above will be called the weight-preservingconditional expectation . A better name might be ‘probabilistic conditional expec-tation’, but much of our paper is in the setting of ‘weights’ as opposed to ‘states’.If we write it as E µ then any other normal conditional expectation E : M → D isgiven by E ( x ) = E µ ( hx ) for a density function h ∈ L ( K, A , µ ) + with E µ ( h ) = 1.Such h is sometimes called a weight function . See e.g. [3, Section 6] for references.We shall show that a similar structure pertains in the noncommutative context.Turning to von Neumann algebras and noncommutative integration theory, recallthat here von Neumann algebras are regarded as noncommutative L ∞ -spaces. Soby fixing a faithful normal semifinite (often abbreviated as fns ) weight ω on M ,we may view the pair ( M , ω ) as a structure encoding a noncommutative measurespace. (We will define the notions of faithful, normal, semifinite, and weight inSection 2 below. For now recall that every von Neumann algebra has a fns weight[38, Theorem VII.2.7].) For an inclusion D ⊂ M of von Neumann algebras, theexistence of a normal conditional expectation E : M → D is a rich subject, and isoften a very tricky matter. The best result is as follows [38, Theorem IX.4.2]: If ω above restricts to a semifinite weight on D then there exists a (necessarily unique) ω -preserving normal conditional expectation E : M → D if and only if the modulargroup ( σ ωt ) t ∈ R of ω (see [38, Chapter VIII]) leaves D invariant. By ω -preservingwe here mean that ω ◦ E = ω on M . We will refer to this unique conditionalexpectation as the weight-preserving conditional expectation , and often write it as E ω or as E D . This is of course the noncommutative analog of the object withthe same name mentioned in the last paragraph. As a special case, if ω restrictsto a semifinite trace on D then there exists an ω -preserving normal conditionalexpectation E : M → D if and only if the modular group of ω acts as the identity ONDITIONAL EXPECTATIONS AND GENERALIZED REPRESENTING MEASURES 3 on D , or equivalently if D is contained in the centralizer (defined below) of ω . If inaddition ω is a normal state on M , then this centralizer condition is saying that ω ( dx ) = ω ( xd ) , x ∈ M , d ∈ D . We make a few contributions to the general theory of conditional expectations ofvon Neumann algebras in the early sections of our paper. See e.g. Theorems 3.5,3.7, 3.10, 3.12, and their corollaries. Also in Section 4 we develop the noncommuta-tive version of the ‘weight function’ mentioned in the last paragraph for a classicalconditional expectation. See the last two sections of [3] for some more facts andbackground about the above view of classical and von Neumann algebraic condi-tional expectations (and for a brief survey of some aspects of the present paper).We now turn to representing measures for characters of a function algebra
A ⊂ C ( K ). A positive measure µ on K is called a representing measure for acharacter Φ of A if Φ( f ) = R K f dµ for all f ∈ A . The functional e Φ( g ) = R K g dµ on C ( K ) is a state on C ( K ), and indeed representing measures for Φ are in a bijec-tive correspondence with the extensions of Φ to positive functionals on C ( K ). Thatis, representing measures for a character are just the Hahn-Banach extensions to C ( K ) of the character. The classical Hoffman-Rossi theorem proves the existenceof ‘normal’ or ‘absolutely continuous’ representing measures’: weak* continuouscharacters of a function algebra
A ⊂ M = L ∞ ( X, µ ) have weak* continuous pos-itive Hahn-Banach extensions to M . Such extensions correspond to measures on X which are absolutely continuous with respect to µ . The original Hoffman-Rossitheorem assumes that µ is a probability measure, but actually any positive measurewill do (see [4] for a swift proof of this result). In several sections of the presentpaper we prove noncommutative versions of the Hoffman-Rossi theorem; that is, weestablish the existence of ‘noncommutative representing measures’ Ψ : M → D forweak* continuous D -characters on a weak* closed subalgebra A of M , where by anoncommutative representing measure we mean the extension of a D -character toa positive D -valued map on all of M satisfying certain regularity conditions. Thesemay be viewed as ‘module’ Hahn-Banach extension theorems for weak* continuous D -characters into possibly noninjective von Neumann algebras. It is not at all clearthat ‘Hahn-Banach extensions’ into D exist, let alone weak* continuous ones, infact this is quite surprising. In particular it is important that A is an algebra andΦ is a homomorphism for such positive weak* continuous extensions to exist. In[4] the first author and coauthors proved such a theorem which works for all vonNeumann algebras M , but requires that the subalgebra D be purely atomic. Atthe end of that paper it was mentioned that we were still pursuing the special caseof the noncommutative Hoffman-Rossi theorem in the case that M has a faithfulnormal tracial state and D is any von Neumann subalgebra of A . In the presentpaper we supply this noncommutative generalization of the Hoffman-Rossi theo-rem, indeed we generalize to subalgebras of much larger classes of von Neumannalgebras.To be more specific, in Sections 5–7 we consider inclusions D ⊂ A ⊂ M for aweak* closed subalgebra A , and a von Neumann subalgebra D of our von Neumannalgebra M , together with a weak* continuous D -character Φ : A → D . (As wesaid, it is usually crucial in our results here that A is an algebra, as is the case inHoffman and Rossi’s original theorem, but in a few results A can be allowed to bea D -submodule of M containing D .) We as before view M as a structure encodinga noncommutative measure space by fixing a faithful normal semifinite weight ν DAVID P. BLECHER AND LOUIS E. LABUSCHAGNE on M . We then seek to find a (usually normal) extension of Φ to a positive mapΨ : M → D . Such extensions Ψ will be our ‘noncommutative representing mea-sures’. Note that Ψ is necessarily a conditional expectation since it is the identitymap on D and hence is idempotent and is a D -module map (see p. 132–133 in [2]for the main facts about conditional expectations and their relation to bimodulemaps and projections maps of norm 1). Thus our setting in Sections 5–7 simul-taneously generalizes the classical setting above (the case that M is commutativeand D = C
1) and the setting for von Neumann algebraic conditional expectations(the case A = D ). We will describe our specific results in more detail below. Italso generalizes, as we said above, Arveson’s profound noncommutative abstractionfrom [1, 7] of Hardy space theory on the disk. In particular it is very importantin Arveson’s theory that the ‘noncommutative representing measures’ take valuesin D , and not merely in the ambient B ( H ). We now describe the structure of ourpaper. In Section 2 we state and prove some results in noncommutative integrationtheory, some of which may be folklore. We also state some other general facts aboutvon Neumann algebras, noncommutative L p spaces, and extensions of conditionalexpectations to the latter. In Section 3 we develop some aspects (which we were notable to find in the von Neumann algebra literature) of the relation between normalconditional expectations and centralizers of normal states and weights. For exam-ple, we show how from any given state on M , one may use an averaging techniqueto construct a related normal state for which D is in the centralizer of that state. InSection 4, which is also purely von Neuman algebraic, we develop the noncommu-tative theory of the ‘weight function’ mentioned early in our Introduction, and giveexistence criteria and characterizations of normal conditional expectations in termsof such ‘weight functions’ h . This uses the delicate interplay of Tomita-Takesakitheory, noncommutative Radon-Nikodym derivatives, Connes cocycles, Haagerup L p -spaces, etc.In Sections 5–7 we prove the noncommutative Hoffman-Rossi theorems describedabove, that is the existence of ‘noncommutative representing measures’. The proofof such theorems involve three stages. In the first stage we show that ω ◦ Φ extendsto a normal state on M . In the second stage we show that we may assume thatthis normal state has D in its centralizer, by averaging over the unitary group of D . Finally in the third stage we use the technology thus developed to prove themain result. As we hinted at above, such states with D in its centralizer possessassociated D -valued normal conditional expectations.More specifically in Section 5 we develop our main ideas for proving the non-commutative Hoffman-Rossi theorem, and as an example apply these ideas to provethis result for all inclusions D ⊂ A ⊂ M where M is commutative. This gener-alizes the classical Hoffman-Rossi theorem (in which D = C M is finite or σ -finite, and D is contained inthe centralizer. The σ -finite case becomes quite technical, involving the technologyof Haagerup L p -spaces. In Section 7 we provide several results giving the existenceof ‘representing measures’ for general von Neumann algebras. The most generalresult one can hope for is that for inclusions D ⊂ A ⊂ M for which we knowthat there does exist a normal conditional expectation E ω : M → D , every weak*continuous D -character Φ on A extends to some normal conditional expectationΨ : M → D . We obtain several special cases of this. In our most general settings
ONDITIONAL EXPECTATIONS AND GENERALIZED REPRESENTING MEASURES 5 in Section 7, using the famous Haagerup reduction theorem [22] we are able to ob-tain a conditional expectation Ψ :
M → D extending Φ, that is, a ‘noncommutativerepresenting measure’. However at present we are in this generality not always ableto show that Ψ may be chosen to be normal.Finally in Section 8, we consider a class of ‘noncommutative representing mea-sures’ generalizing the Jensen measures and Arens-Singer measures in classical func-tion theory [15, 33]. We prove a noncommutative variant of the classical result thatif A is logmodular then every representing measure is a Jensen measure. The theoryachieved therefore provides a well-rounded foundation for the development of a newnoncommutative theory of function spaces.Turning to notation: We write P ( M ) for the projection lattice of a von Neumannalgebra M , and M + for the positive cone of M , namely { x ∈ M : x ≥ } .In this paper an operator algebra is a unital algebra A of operators on a Hilbertspace, or more abstractly a Banach algebra isometrically isomorphic to such analgebra of Hilbert space operators. Many authors consider an operator space struc-ture on an operator algebra, and replace the word ‘isometrically’ by ‘completelyisometrically’ in the last sentence–see e.g. [10] for definitions–but this will not beimportant in the present paper. Indeed in the introduction to [9] we explain why D -characters are automatically completely contractive. By 4.2.9 in [10], which is anearlier result of the present authors, any unital completely contractive projection ofa unital operator algebra onto a subalgebra D is a D -bimodule map. Thus one mayequivalently define the D -characters above to be the unital completely contractiveprojections of A onto D which are also homomorphisms. In this paper A is usuallyweak* closed, and D -characters are usually weak* continuous.We will sometimes abusively call a von Neumann algebra finite if it has a faithfulnormal tracial state (this is abuse, since in nonseparable situations it differs fromthe common usage meaning that the identity element is a finite projection). Werecall that a von Neumann algebra is σ -finite if it has a faithful normal state, and semifinite if it has a faithful normal tracial weight.2. Preliminaries on weights on von Neumann algebras A weight on a von Neumann algebra M is a [0 , ∞ ]-valued map on M + which isadditive, and satisfies ω ( tx ) = tω ( x ) for t ∈ [0 , ∞ ] (interpreting 0 · ∞ = 0). We usethe standard notation p ω = { x ∈ M + : ω ( x ) < ∞ , n ω = { x ∈ M : ω ( x ∗ x ) < ∞} , and m ω = Span p ω = n ∗ ω n ω . If here we wish to emphasize that we are only considering elements of m ω taken froma specific subset E of M , we will then write m ( E ) ω . The latter may be regardedas ‘the definition domain’ of ω : this is the part of M on which ω is naturallydefined as a linear functional (see around VII.1.3 in [38]). We say that ω is normal if it preserves suprema of bounded increasing nets in M + , faithful if ω ( x ) = 0implies x = 0 for x ∈ M + , and tracial (or is a trace ) if ω ( x ∗ x ) = ω ( xx ∗ ) for x ∈ M . Any normal weight is the supremum of an increasing family of positivenormal functionals (see [38, Theorem VII.]). There are several equivalent definitionsof semifiniteness for weights: that n ω is weak* dense in M , that m ω is weak*dense in M , that there exists an increasing net in p with supremum 1, and that DAVID P. BLECHER AND LOUIS E. LABUSCHAGNE p = { x ∈ M + : ω ( x ) < ∞} generates M . A weight is strictly semifinite if it isthe sum of a family of positive functionals with mutually orthogonal supports, orequivalently that its restriction to the centralizer (defined below) is semifinite.For a normal trace τ on a von Neumann algebra the τ -finite projections are adirected set by [37, Proposition V.1.6], and this gives a net with weak* limit 1 if τ is also semifinite. Lemma 2.1.
Let ω be a normal semifinite weight on a von Neumann algebra M ,with ω ( d d ) = ω ( d d ) for d , d ∈ m ω . Then ω is a trace on M .Proof. Let d ∈ M be given with ω ( d ∗ d ) < ∞ . That is d ∈ n ω . By semifinitenessthere must exist ( d t ) ⊂ m ω such that d t ր D with ω ( d t ) < ∞ . Then since d t d ∈ m ω and ω ( d ∗ d ) = lim t ω ( d ∗ d t d ), we have by hypothesis that ω ( d ∗ d ) = lim t ω ( d t dd ∗ d t ) = lim t ω ( d t | d ∗ | d t ) = lim t ω ( | d ∗ | d t | d ∗ | ) = ω ( dd ∗ ) , which is finite. On replacing d with d ∗ the above also shows that if ω ( dd ∗ ) < ∞ ,then ω ( d ∗ d ) = ω ( dd ∗ ). So ω ( d ∗ d ) < ∞ if and only if ω ( dd ∗ ) < ∞ , in which casethey are equal. It follows that ω is a trace. (cid:3) The important data for a weight ω is obviously p ω and the values of the weight ω there. Any self-adjoint element x ∈ M of course generates a minimal abelian vonNeumann subalgebra of M , which we shall denote by W ∗ ( x ). Given x ∈ p ω , wemay identify W ∗ ( x ) with L ∞ ( X, A , µ ) for a strictly localizable measure µ , and x with some f ∈ L ∞ ( X, µ ) [31, Theorem 1.18.1]. Any f ∈ L ∞ ( X, µ ) is an increasinglimit of simple functions corresponding to projections in p ω . This carries over tothe noncommutative case. Hence the important data for a normal weight ω is theset of projections in p ω and the values of ω on such projections. To see this notethat for any f ∈ p ρ and any ǫ >
0, we will have that ω ( e ǫ ) ≤ ǫ − ω ( f ) < ∞ where e ǫ = χ ( ǫ, ∞ ) ( f ). The element f may of course be written as an increasing limitof positive Riemann sums made up of sub-projections of projections like e ǫ . Bynormality the value of ω on f is determined by its value on those Riemann sums,thereby proving the claim.The support projection e = s ( ω ) of a normal weight ω is the complement of thelargest projection q in M with ω ( q ) = 0. For any projection q in M with ω ( q ) = 0we have that q ∈ m ω . Also for any x ∈ p ω we have ω ( qx ) = ω ( xq ) = ω ( qxq ) = 0 bythe Cauchy-Schwarz inequality, hence ω ( x ) = ω ( exe ) < ∞ . For any normal weight ω on M there exist canonical projections f ≤ e = s ( ω ) in M such that ω is faithfulon e M e , semifinite on f ⊥ M f ⊥ , and semifinite and faithful on ( e − f ) M ( e − f ). Forthis reason authors often assume that normal weights are faithful and semifinite.We shall not usually do this though.For a faithful normal semifinite weight ω on a von Neumann algebra M , the centralizer M ω of ω is the von Neumann subalgebra consisting of a ∈ M with σ ωt ( a ) = a for all t ∈ R . By [38, Theorem VIII.2.6], if ω is a state then M ω = { a ∈ M : ω ( ax ) = ω ( xa ) , x ∈ M} . For a faithful normal semifinite weight the centralizer is known to equal the set of x ∈ M with x m ω ⊂ m ω and m ω x ⊂ m ω , and ν ( xy ) = ν ( yx ) for all y ∈ m ω . If ω issemifinite but not necessarily faithful we will say that an element x ∈ M is ω - central if x commutes with the support projection e of ω and satisfies the conditions x m ω e ⊂ m ω , e m ω x ⊂ m ω , ONDITIONAL EXPECTATIONS AND GENERALIZED REPRESENTING MEASURES 7 and ν ( xy ) = ν ( yx ) for all y ∈ m ω . Since this definition differs somewhat fromcompeting definitions of ω -centrality for non-faithful weights, we shall in this casewrite M ω for the ω -central elements in M . In the applications later in the paperthe elements x will also be in m ω , as is the case for example if ω is a state, sothat one may drop the e ’s from the last centered equation defining ‘ ω -central’. Seeparticularly Lemma 2.3 below.If ω ( q ) = 0 for a projection q then q is dominated by, and hence commutes with,the complement of the support projection e of ω . Since m ω is an algebra it followsthat m ω is left and right invariant under multiplication by q , and hence by e . Itfollows that q and e are ω -central, and are in M ω . Indeed e lies in the center of M ω by the definition of the latter. It is easy to see that M ω is a unital ∗ -subalgebraof M . In fact it is a von Neumann subalgebra (we remark that we were not ableto see this without including in the definition of M ω the commutation with thesupport projection of ω , which plays a key role in the proof below). Lemma 2.2.
Let ω be a normal semifinite weight on a von Neumann algebra M with support projection e . Then M ω is a von Neumann subalgebra of M . If N = e M e then N ω = N ∩ M ω = e M ω e .Proof. We first show that N ω ⊂ M ω . Given x ∈ N ω , y ∈ m ω , then xye = exeye ∈ m ( N ) ω ⊂ m ( M ) ω . Similarly eyx ∈ m ( M ) ω . Moreover, ω ( xy ) = ω ( exeye ) = ω ( eyexe ) = ω ( yx ) . Thus N ω ⊂ M ω .If d ∈ M ω then d m ( M ) ω e ⊂ m ( M ) ω and ω ( dx ) = ω ( xd ) for x ∈ m ( M ) ω . Then ω ( edxe ) = ω ( exde ). If x ∈ p ( N ) ω ⊂ m ( M ) ω , then edexe ∈ e m ( M ) ω e ⊂ m ( N ) ω and ω ( edex ) = ω ( dx ) = ω ( xd ) = ω ( xede ). So ede ∈ N ω .We said that M ω is a unital ∗ -subalgebra of M with e in its center. Suppose that d t ∈ M ω with d t → d weak* in M . Then ed t e → ede weak*, and ed t e is in the vonNeumann algebra N ω , so that ede ∈ N ω ⊂ M ω . With q = e ⊥ we have qd t e = 0 sothat qde = 0. Similarly edq = 0. Also qdq ∈ m ( M ) ω with ω ( qdqx ) = 0 = ω ( xqdq )for x ∈ m ( M ) ω , so that qdq ∈ M ω . Thus d ∈ M ω . Hence M ω is a von Neumannalgebra. The rest is clear. (cid:3) Lemma 2.3.
Let ω be a normal weight on a von Neumann algebra M which issemifinite on a von Neumann subalgebra D , with ω ( xd ) = ω ( dx ) for all d ∈ m ( D ) ω and x ∈ m ( M ) ω . Then D ⊂ M ω . If e is the support projection of ω then e ∈ D ′ and D e is a von Neumann subalgebra of e M e . Also, ω is a normal semifinite traceon M ω , and also on each of D and D e .Proof. Since 1 − e ∈ p ( M ) ω as we said above, if d ∈ m ( D ) we see that (1 − e ) d ∗ ∈ m ( M ) ω , and 0 = ω ((1 − e ) d ∗ d ) = ω ( d (1 − e ) d ∗ ) = ω ( ed (1 − e ) d ∗ e ) . Hence ed (1 − e ) d ∗ e = 0 since ω is faithful on e M e . We therefore have ed (1 − e ) = 0,so that ed = ede , and this must equal de by symmetry. Thus e commutes with theweak* closure of m ( D ) ω and its support projection. Since e ∈ M ω and m ( M ) ω isan algebra, our hypothesis implies that m ( D ) ω ⊂ M ω . Since ω is semifinite on D ,we see that D ⊂ M ω and e ∈ D ′ . The map d de is a normal ∗ -homomorphismon D . Hence its range D e is weak*-closed, a von Neumann subalgebra of e M e . DAVID P. BLECHER AND LOUIS E. LABUSCHAGNE
By Lemma 2.1, ω is a normal semifinite trace on D . Since ω = ω ( e · e ), it iseasy to see that it is also a normal semifinite trace on D e . Setting D = M ω in theargument above shows that ω is a normal semifinite trace on M ω . (cid:3) We recall the Radon-Nikodym theorem of Pedersen and Takesaki [30]. Fix afaithful normal weight ω on a von Neumann algebra M . We use notation from[30], except that given a positive operator h affiliated to M ω , we will as in [38]write ω h for the weight(1) lim ǫ ց ω ( h / ǫ xh / ǫ ) , where h ǫ = h ( + ǫh ) − . If ψ is a normal semifinite weight on M then we saythat ψ commutes with ω if ψ ◦ σ ωt = ψ for all t . The Radon-Nikodym theoremstates that ψ commutes with ω if and only if ψ = ω h for a (necessarily unique)positive selfadjoint h affiliated with M ω . We call h the Radon-Nikodym derivative and write it as dψdω . See [30] for some theory of the Radon-Nikodym derivative, andalso [38], e.g. Corollary VIII.3.6 there and the connection to the Connes cocyclederivative, although there all weights are faithful. We also sometimes call theRadon-Nikodym derivative a density . More loosely, by a density we mean a positiveaffiliated operator which can be used to derive a new weight from the given referenceweight.In a few places we will assume that the reader is familiar with the extendedpositive part c M + of a von Neumann algebra M and its connection to weights andoperator valued weights, as may be found in [38, Section IX.4] and [20].Since we will refer to the following facts frequently, we label them as remarks.These facts are scattered in various papers in the literature. However in view oftheir importance, we choose to summarize what we need locally. See e.g. [17] formore details and background. Remark 2.4.
Let M be a von Neumann algebra acting on a Hilbert space H ,and let ν be a fns weight on M . Then M may be represented as an algebraacting as a von Neumann algebra on L ( R , H ) by means of the *-isomorphism a → π ν ( a ) where ( π ν ( a ) ξ )( s ) = σ ν − s ( a ) ξ ( s ) for every s ∈ R . We will usuallywrite π ν as π . One may also define left shift operators by ( λ t ( ξ ))( s ) = ξ ( t − s ).The dual action θ s of R on B ( L ( R , H )) is implemented by the unitary group w ( t ), where ( w ( t ) ξ )( s ) = e − its ξ ( s ). We have θ s ( π ( a )) = π ( a ) for all a ∈ M , and θ s ( λ t ) = e − its λ t . The crossed product M = M ⋊ ν R is the von Neumann algebragenerated by π ( M ) and the λ t ’s. The operator valued weight T ν from the extendedpositive part c M + ≡ \ π ν ( M ) + to c M + is given by T ( a ) = R ∞−∞ θ s ( a ) ds . The dualweight on M of a normal semifinite weight ρ on M (computed using ν as a referenceweight) is given by e ρ = ρ ◦ T ν . It is well known that σ e νt is implemented by λ t , andthat σ e νt ( π ( a )) = π ( σ νt ( a )) for any a ∈ M . (See Theorem 4.7 and Lemma 5.2 andits proof in [20, 21] for these facts.) By Stone’s theorem there exists k ν η M + sothat k itν = λ t for each t . By [38, Theorem VIII.3.14], M is semifinite. It followsfrom the proof of [38, Theorem VIII.3.14], that τ M = ν ( k − ν · ) will then be a faithfulnormal semifinite trace on M for which we have that k ν = d e νdτ (writing τ M as τ ). Itis precisely this trace that plays such a crucial role in the construction of Haagerup L p -spaces, as one may begin to appreciate already in the proof of [21, Lemma 5.2].For any normal semifinite weight ρ on M , the dual weight e ρ = ρ ◦ T ν above ischaracterized by the fact that the densities k ρ = d e ρdτ satisfy θ s ( k ρ ) = e − s k ρ for all ONDITIONAL EXPECTATIONS AND GENERALIZED REPRESENTING MEASURES 9 s . In fact there is bijective correspondence between the classes of normal semifiniteweights on M , dual weights of such normal semifinite weights, and densities k η M which satisfy θ s ( k ) = e − s k for all s . (See [39, Proposition II.4] or [17, Proposition6.67].) In view of this correspondence, we shall denote this class of densities by η L ( M ). See [17] for more on these topics.Observe that if the same construction is done for any other von Neumann algebra N with fns weight ϕ acting on H , then the crossed product N = N ⋊ ϕ R willshare the same left shift operators, which will here too implement the modularautomorphism group for the dual weight ˜ ϕ on N . Hence remarkable as it mayseem, the same argument as above will then yield the conclusion that the operator k for which we have that λ t = k it for all t , here satisfies k = d e ϕdτ N and hencethat d e ϕdτ N = d e νdτ M as operators affiliated to B ( L ( R , H )). See [17, Theorem 6.62].However we need to sound a word of warning here. Given two faithful normalsemifinite weights ν and ρ on M , one should carefully differentiate between thedual weight of ρ computed using the crossed product M ⋊ ν R , and that using thecrossed product M ⋊ ρ R . Let us write e ρ ( ν ) and e ρ ( ρ ) for these two dual weights. Theweight e ρ ( ν ) is an altogether different object to e ρ ( ρ ) , and if ν does not agree with ρ ,one should most certainly not expect that d e νdτ M agrees with d e ρ ( ν ) dτ M . Remark 2.5.
Let M be a von Neumann algebra acting on a Hilbert space H ,and let ν be a fns weight on M . Let D be a von Neumann subalgebra of M and E D a faithful normal conditional expectation from M onto D for which wehave that ν ◦ E D = ν . Then of course σ νt ( D ) = D and E D ◦ σ νt = σ νt ◦ E D for all t . The first claim is proved in [38, IX.4.2]. For the second note that forany a ∈ m ν , using ν ◦ E D = ν and ν ◦ σ νt = ν , it is an exercise to see that ν ( | E D ( σ t ( a )) − σ t ( E D ( a )) | ) = 0 for all t , and hence that E D ◦ σ νt = σ νt ◦ E D on m ν .By normality and the fact that m ν is weak* dense in this case, E D ◦ σ νt = σ νt ◦ E D .See also [20, Proposition 4.9].If now we compute the crossed product of D with respect to the modular au-tomorphism group determined by ν |D , it is clear from the definitions given in thepreceding remark that in this case π ν |D ( d ) = π ν ( d ) for all d ∈ D . Since D ⋊ ν |D R isgenerated by π ν |D ( D ) and the left shift operators, it follows that D ⋊ ν |D R is in thenaive sense a linear subspace of M ⋊ ν R . So in this particular case, the discussionin the preceding remark does indeed show that d e νdτ M = d g ν |D dτ D .By [22, Theorem 4.1], E D extends to a faithful normal conditional expectation E D from M ⋊ ν R to D ⋊ ν R which satisfies e ν ◦ E D = e ν and θ s ◦ E D = E D ◦ θ s forall s . In particular e ν is semifinite on D ⋊ ν R . The prescription given by [22] forthis extension is exactly the one given by [16, Proposition 4.3]. In fact Section 4.1of [16] is a precursor to [22, Theorem 4.1]. So [22, Theorem 4.1] asserts that [16,Proposition 4.3, Lemma 4.4, Proposition 4.6, Theorem 4.7] holds in the non- σ -finitesetting. By exactly the same proof as the one used by Goldstein, [16, Lemma 4.5& Lemma 4.8] hold in the non- σ -finite setting. Now observe that the proof of [29,Proposition 4.5] shows that we also have that τ ◦ E D = τ where τ is the trace on M ⋊ ν R .By [22, Remark 5.6] and the reasoning in Example 5.8 there, there is an inducedexpectation E (1) D on L ( M ), given by the continuous extension to all L ( M ), ofthe map k / ak / → k / E D ( a ) k / where a ∈ m ν , and where k = d e νdτ = d g ν |D dτ D . For any a ∈ m ν it follows from [18, Proposition 2.13(a)] that tr D ( E (1) D ( k / ak / )) = tr D ( k / E D ( a ) k / ) = ν ( E D ( a )) = ν ( a ) = tr ( k / ak / ) . Now recall that k / m ν k / is dense in L ( M ) [18, Proposition 2.11]. So by conti-nuity tr D ◦ E (1) D = tr .3. Normal conditional expectations and centralizers of normalstates and weights
Lemma 3.1.
Let ω be a normal state on a finite von Neumann algebra M whichis tracial on a von Neumann subalgebra D . Let U be the unitary group of D , andlet K ω be the norm closed convex hull of { u ∗ ωu : u ∈ U} in M ∗ . Then K ω is weakly compact, and contains a normal state ψ extending ω |D such that ψ ( dx ) = ψ ( xd ) , d ∈ D , x ∈ M . Proof.
We follow the idea in the proof of [38, Theorem V.2.4]. By that proof K ω is weakly compact, and the group of isometries ϕ u ∗ ϕu , for u ∈ U , has a fixedpoint ψ in K ω . Since ψ = u ∗ ψu , and since U spans D , we have ψ ( dx ) = ψ ( xd ) for d ∈ D , x ∈ M . It is easy to see that any element of K ω is a normal state on M extending ω |D . (cid:3) Remark 3.2.
If we drop the requirement that ω is tracial on D one obtains thesame conclusion except that ψ need not extend ω | D .We thank Roger Smith and Mehrdad Kalantar for conversations around thislast result. Indeed Roger Smith pointed us to [32, Lemma 3.6.5], and the followingresult about masas, and its proof, which was an important initial step in the presentinvestigation: Let M be a von Neumann algebra with a faithful normal tracialstate τ , and let D be a masa in M . Let Φ : A → D be a unital weak* continuousmap on a weak* closed subalgebra A of M containing D , which is a D -bimodulemap. Then Φ is the restriction to A of the unique τ -preserving normal conditionalexpectation E τ : M → D . To see this, since D is a masa, by Lemma 3.6.5 and theremark after it in [32], for x ∈ A there is a net x t with terms which are convexcombinations of u ∗ xu for various unitaries u ∈ D such that x t → E τ ( x ) weak*.Since Φ( u ∗ xu ) = u ∗ Φ( x ) u = Φ( x ), we deduce that Φ( x ) = E τ ( x ) for x ∈ A .On the other hand, after we had proved the last lemma and used it to obtain alater result, Mehrdad Kalantar showed us an alternative proof that gives ψ aboveas ω ◦ E D ′ ∩M . A generalization of his argument plays a role in the next result andthe point in its proof where his contribution is mentioned in more detail. Example 3.3.
One may ask if there is a variant of the last lemma for σ -finitevon Neumann algebras. Consider the case that D is the copy of L ∞ ([0 , σ -finite semifinite von Neumann algebra M = B ( L ([0 , ω be thenormal state on M defined by ω ( x ) = P k ∈ Z − k h xe k , e k i , for the standardorthonormal basis e n = e πnit of L ([0 , M f is multiplication by f ∈ L ∞ ([0 , ω ( M f ) = P k − k R f dx = R f dx , so ω restricts to a faithful normaltracial state on D . However there is no normal state ψ extending ω |D such that ψ ( dx ) = ψ ( xd ) , for d ∈ D , x ∈ M . Indeed if there were then by e.g. Theorem 3.5there would be a normal conditional expectation onto D , which is well known to ONDITIONAL EXPECTATIONS AND GENERALIZED REPRESENTING MEASURES 11 be false. Note that M ω in this example is also an abelian von Neumann algebra,namely the operators that are diagonal with respect to this orthonormal basis.The situation is better if we fix a faithful normal state ω on a σ -finite vonNeumann algebra M with D a von Neumann subalgebra of the centralizer M ω : Lemma 3.4.
Let ω be a faithful normal state on a von Neumann algebra M , andlet D be a von Neumann subalgebra of M ω . Let U be the unitary group of D , ψ anormal state extending ω |D , and let K ψ be the norm closed convex hull of { u ∗ ψu : u ∈ U} in M ∗ . Then K ψ contains a normal state ρ extending ω |D such that ρ ( dx ) = ρ ( xd ) , d ∈ D , x ∈ M . Indeed we may take ρ to be ψ ◦ E D ′ ∩M where E D ′ ∩M is the unique ω -preservingconditional expectation onto D ′ ∩ M . Finally, if ψ commutes with ω then so does ρ .Proof. First we claim that there exists an ω -preserving normal conditional expec-tation E D ′ ∩M onto D ′ ∩ M . We follow the proof strategy of [32, Lemma 3.6.5].Set k = k ω , the density d e ωdτ of the dual weight e ω on M ⋊ ν R . It is well known that k strongly commutes with elements of the centralizer of ω (see e.g. Proposition 4.2below). The embedding x xk of M into L ( M ) is weak* continuous, since itis easy to see that for any η ∈ L ( M ) the functional h · k , η i is weak* continuous(since the representation of M on L ( M ) is normal, and k ∈ L ( M )). Thusthe image of any norm closed ball in M is weakly compact. Let K D ( x ) be theweak* closed convex hull in M of { uxu ∗ : u ∈ U} . Hence K D ( x ) k is a weaklyclosed, hence k · k -norm closed by Mazur’s theorem, convex set in L ( M ) (andin M k ). Let φ ( x ) be the unique element of smallest k · k -norm (in L ( M )) in K D ( x ) k ⊂ L ( M ). We have φ ( x ) ∈ M k , and φ ( x ) = E ( x ) k for an element E ( x ) ∈ K D ( x ) ⊂ M . Clearly k E ( x ) k ≤ k x k . We also have ω ( E ( x )) = ω ( x ), since D ⊂ M ω . For a fixed unitary v ∈ U the map xk → vxk v ∗ is a k · k -normisometry, and so as in [32, Lemma 3.6.5] we have that E ( x ) ∈ D ′ ∩ M , and for anyunitaries v, w ∈ D ′ ∩ M we have K D ( wxv ) = wK D ( x ) v and E ( wxv ) = wE ( x ) v . Itfollows that ω ( E ( x + y ) v ) = ω (( x + y ) v ) = ω ( E ( xv )+ E ( yv )) = ω (( E ( x )+ E ( y )) v ) , x, y ∈ M . Hence this is true with v replaced by any element in D ′ ∩ M . Hence E ( x + y ) = E ( x ) + E ( y ). For a scalar t > K D ( tx ) = tK D ( x ), and φ ( tx ) = tφ ( x ). Itfollows that E is linear, and also E is a D ′ ∩ M -module map. Of course E (1) = 1and so E is an ω -preserving conditional expectation onto D ′ ∩ M . The uniquenessof ω -preserving conditional expectations is standard (see e.g. the proof of Theorem3.5).We have σ ωt ( d ′ ) d = σ ωt ( d ′ d ) = σ ωt ( dd ′ ) = dσ ωt ( d ′ ) , d ′ ∈ D ′ ∩ M , d ∈ D . Thus σ ωt ( D ′ ∩ M ) ⊂ D ′ ∩ M , and so by [38, Theorem IX.4.2] there does exist anormal ω -preserving conditional expectation onto D ′ ∩ M . Thus E is normal, bythe uniqueness above. Note that K D ( vxv ∗ ) = K D ( x ) for v ∈ U , and so E ( vxv ∗ ) = E ( x ). Hence ρ ( u ∗ xu ) = ρ ( x ) for all u ∈ U and x ∈ M , so that D ⊂ M ρ . We learned the following argument from M. Kalantar. We claim the normal state ρ = ψ ◦ E is in K ψ . By way of contradiction, assume ρ / ∈ K ψ . By the geometricHahn-Banach theorem, there exists x ∈ M and real t such thatRe( ρ ( x )) < t < Re( α ( x )) , α ∈ K ψ . Since E ( x ) ∈ K D ( x ) we see that the real part of ρ ( x ) = ψ ( E ( x )) is in the closedconvex hull of { Re ψ ( uxu ∗ ) : u ∈ U} ⊂ { Re α ( x ) : α ∈ K ψ } ⊂ [ t, ∞ ) . This contradicts the inequality above. So ρ ∈ K ψ .Since ρ ∈ K ψ it extends ω |D , since if d ∈ D then ρ ( d ) is in the norm closedconvex hull of terms of form ψ ( u ∗ du ) = ω ( d ).Finally, if ψ commutes with ω then ψ ( u ∗ σ ωt ( x ) u ) = ψ ( σ ωt ( u ∗ xu )) = ψ ( u ∗ xu ) , x ∈ M , u ∈ D . This will persist for the convex hull of { u ∗ ψu : u ∈ U} , hence for its norm closure K ψ . So ρ commutes with ω . (cid:3) Theorem 3.5.
Let ω be a normal state on a von Neumann algebra M which isfaithful and tracial on a von Neumann subalgebra D . Then ω ( dx ) = ω ( xd ) , d ∈ D , x ∈ M , if and only if there is a normal conditional expectation E ω from M onto D whichpreserves ω . Moreover such an ω -preserving normal conditional expectation onto D is unique, and is faithful if ω is faithful on M .Proof. For the ‘only if’ part, the hypothesis about ω ( dx ) = ω ( xd ) ensures that D ⊂ M ω . By [38, Theorem VIII.2.6], if ω is faithful (so M is σ -finite) thenthere exists a unique ω -preserving normal conditional expectation E D from M onto D . (Alternatively, by [38, Theorem IX.4.2] there exists a normal conditionalexpectation E from M onto M ω which preserves ω . From the the latter it followsthat E is faithful. By [37, Proposition V.2.36], there exists a faithful normalconditional expectation E from M ω onto D . The conditional expectation we seekis E = E ◦ E , as the reader can easily check.)Now consider the case where ω is not faithful. Let z be the support projectionof the state ω . By Lemma 2.3 and its proof we may make the following assertions: z ∈ D ′ , D z is in fact a W ∗ -algebra, a von Neumann subalgebra of z M z , and themap d dz is a normal ∗ -homomorphism from D onto D z . These do not require ω to be faithful on D . Also, if ω is faithful on D then the ∗ -homomorphism d dz is also faithful, hence is a ∗ -isomorphism from D onto D z , and ω is a trace on D z .We next show that as a subalgebra of z M z , z D z satisfies the hypothesis of theproposition. For any b ∈ M and d ∈ D , we have that ω (( zbz )( dz )) = ω ( bzd ) = ω ( dbz ) = ω ( db ) . Similarly ω (( dz )( zbz )) = ω ( bd ). Since ω ( bd ) = ω ( db ), the restriction of ω to z M z is a faithful state which as claimed, satisfies the hypothesis with respect to thesubalgebra D z . Therefore by the first part of the proof, there exists a faithfulnormal conditional expectation E : z M z → D z which preserves the restriction of ω to z M z .Since the map D → D z : d dz is an injective normal ∗ -homomorphism, itsinverse ι D : D z → D : zdz d is an injective normal ∗ -homomorphism. With Q ω ONDITIONAL EXPECTATIONS AND GENERALIZED REPRESENTING MEASURES 13 denoting the compression
M → z M z , we will show that E ω = ι D ◦ E ◦ Q ω is anormal conditional expectation from M onto D .The map E ω is clearly a unital map onto D . The map E ◦ Q ω is clearly positive,and ι D is positive. So E ω is a unital positive map, and hence is contractive.Given d ∈ D , we have that E ω ( d ) = ι D ◦ E ◦ Q ω ( d ) = ι D ◦ E ( dz ) = ι D ( dz ) = d. Thus E ω ◦ E ω = E ω . The map E ω is therefore a conditional expectation from M onto D . It is an exercise to see that ω ◦ i D = ω , and hence that ω ◦ E ω = ω . That E ω is normal follows since the maps E , Q ω , and i D are all normal.Given another such ω -preserving normal conditional expectation ρ onto D wehave ω (( ρ ( x ) − E ω ( x )) d ) = ω ( xd ) − ω ( xd ) = 0 , d ∈ D . Since ω is faithful we deduce that ρ = E ω .For the “if” part, suppose that there exists a normal conditional expectation E ω from M onto D which preserves ω . Then ω ( ax ) = ω ( E ω ( ax )) = ω ( a E ω ( x )) = ω ( E ω ( x ) a ) = ω ( E ω ( xa )) = ω ( xa ) , for a ∈ D and x ∈ M . (cid:3) Remark 3.6.
Putting the last result together with Lemma 3.1, if ω is a normalstate on a finite von Neumann algebra M , which is faithful and tracial on a vonNeumann subalgebra D , then there is a normal conditional expectation from M onto D which preserves ω ◦ E D ′ ∩M . The latter state is ‘an averaging of ω over theunitary group’ of D .Let ω be a normal weight on a von Neumann algebra M . We say that a vonNeumann subalgebra D is locally ω - central if ω ( pxpdp ) = ω ( pdpxp ) for all p ∈ P ( D )with ω ( p ) < ∞ and x ∈ M , d ∈ D . Note that pxp and pdp above are in the‘definition domain’ m ( M ) ω of ω (see [38, Definition VII.1.3]). We say that a normalconditional expectation E from M onto D is locally ω - preserving if we have that ω ( E ( pxp )) = ω ( pxp ) for every x ∈ M + and every projection p ∈ D such that ω ( p ) < ∞ .We next show that ‘local = global’ in these two definitions, under a reasonableassumption that always holds for example for faithful weights. Knowing that thesetwo properties coincide then provides us with a very powerful technique for liftingresults known for states to results pertaining to normal semifinite weights. Theorem 3.7.
Let ω be a normal weight on a von Neumann algebra M whichis faithful, semifinite, and tracial on a von Neumann subalgebra D . Then D is ω -central if and only if D is locally ω -central and the support projection of ω commuteswith D . If these hold then we have sup p ω ( pxp ) = ω ( x ) , x ∈ M + , where the supremum is taken over all projections in D with ω ( p ) < ∞ ; also if E : M → D is a locally ω -preserving normal conditional expectation then E is ω -preserving.Proof. Clearly if D is ω -central then it is locally ω -central: indeed if ω ( xd ) = ω ( dx )for all d ∈ m ( D ) ω and x ∈ m ( M ) ω , then ω ( pxpdp ) = ω ( pdpxp ) for all p ∈ P ( D ) with ω ( p ) < ∞ and x ∈ M , d ∈ D . The support projection of ω commutes with D by Lemma 2.3.For the converse, suppose that D is locally ω -central. First let ω be a fns weightwhich is a semifinite trace on D . Let A ω be the full left Hilbert algebra associatedwith ω . This contains the image of p M p for ω -finite p ∈ P ( D ). Since ω is tracialon D , the set P ( D ) ω of ω -finite projections in D is upwardly directed, and may beviewed as a net with weak* limit 1; that is p ր P ( D ) ω .Since the left and right representations of M on L ( M ) are weak* continuous, wesee that p t → L ( M ). Hence(2) k x − pxp k ≤ k x − px k + k p ( x − xp ) k ≤ k x − px k + k x − xp k → p ր p M p ⊂ D ♯ , and S ( pxp ) = px ∗ p for x ∈ M . Indeed D ♯ contains the set K of those x ∈ M which equal pxp for some p ∈ P ( D ) ω . Notethat K is a linear subspace, since if r ∈ P ( D ) ω dominates p and q in P ( D ) ω then y = pxp + qzq equals ryr . Also, by equation (2) above K is 2-norm dense in D ♯ , andhence also in H = L ( M ). Let R be the restriction of S to K , and let R be thelinear operator R = CR into ¯ H , where as in the proof of [38, Lemma VI.1.5], C isthe canonical anti-linear isometry from H to ¯ H . We claim that the closure of R is the closed operator CS . To see this note that, as was shown in the proof of [38,Lemma VI.1.5], CS is the closure of S = CS . Moreover if ( x, S x ) is in the graphof S then x ∈ D ♯ , and p t xp t → x in 2-norm as above. Since also p t x ∗ p t → x ∗ in2-norm, it is now clear that CS ( p t xp t ) → CS ( x ). It follows that (the graph of) CS is the closure of (the graph of) R , or that K is a core for CS .Note that if q ≤ p in P ( D ) ω , then ω ( pxpq ) = ω ( pxpqp ) = ω ( pqpxp ) = ω ( qpxp ) = h pxp, q i , x ∈ M . It follows from the above that for any ξ ∈ K and any q ∈ P ( D ) ω , we have h S ξ, q i = h q, ξ i , or equivalently h q, S ξ i = h ξ, q i . To see this recall that ξ is of the form ξ = pxp for some p ∈ P ( D ) ω . Since P ( D ) ω is upwardly directed, we may assumethat p ≥ q . It is then an exercise to conclude from the previously centred equationthat h q, S ξ i = ω (( px ∗ p ) ∗ q ) = ω ( qpxp ) = h ξ, q i as claimed. This ensures that thefunctional ξ
7→ h q, S ξ i where ξ ∈ K , will for any q ∈ P ( D ) ω , be bounded andlinear on K ⊂ A ω . Equivalently, the functional ξ
7→ h R ( ξ ) , Cq i is bounded andlinear on K , and for ξ ∈ K we have h R ( ξ ) , Cq i = h ξ, q i . Thus Cq ∈ Dom( R ∗ ) and R ∗ ( Cq ) = q . Since K is a core for CS , ( R ) ∗ = ( CS ) ∗ by an elementary lemma inthe theory of unbounded operators. Thus ( CS ) ∗ ( Cq ) = ( CS ) ∗ ( CSq ) = q , whence∆( q ) = q . Since q ∈ A ω it follows by the criterion in [38, Theorem VI.2.2 (i)] that q is in the Tomita algebra A . Hence by [35, 10.21 Corollary], q is in the Tomitaalgebra as described on p. 28 in [34].By [34, Proposition 2.13], the functionals a → ω ( ap ) and a → ω ( pa ) on m ( M ) ω will for any p ∈ P ( D ) ω extend to weak* continuous functionals on M . Since P ( D ) ω is upwardly directed we may construct a net ( q t ) in P ( D ) ω increasing to 1, witheach q t majorising p . Given a ∈ m ( M ) ω we then have ω ( q t ap ) = ω (( q t aq t )( q t pq t )) = ω (( q t pq t )( q t aq t )) = ω ( paq t ) . Since q t → σ -strong* topology, the continuity noted earlier ensures that ω ( ap ) = lim t ω ( q t ap ) = lim t ω ( paq t ) = ω ( pa ). Thus we have p ∈ M ω . Since ω istracial on D , the span of P ( D ) ω is weak* dense in D . In addition M ω is a von ONDITIONAL EXPECTATIONS AND GENERALIZED REPRESENTING MEASURES 15
Neumann algebra. Combining these two facts with what we have just proven thenshows that
D ⊂ M ω . That is, D is ω -central.Now for the non-faithful case. Here ω is a normal weight on a von Neumannalgebra M which is faithful, semifinite, and tracial on D , and D is locally ω -central.Let e be the support projection of ω . As in the proof of Lemma 2.3 and Theorem3.5 D e is a von Neumann subalgebra of e M e , and the map d de is a faithfulnormal ∗ -homomorphism from D onto D e . It follows that any projection in D e isof the form pe for p ∈ P ( D ), and ω ( p ) = ω ( pe ). If the latter is finite then ω ( pedpexepe ) = ω ( pdpexep ) = ω ( pexepdp ) = ω ( pexepedpe ) , d ∈ D , exe ∈ e M e. Thus e D is locally ω -central in e M e . By the first part of the proof, ep is in thecentralizer of ω in e M e , and e D is ω -central in e M e . Finally, if d ∈ m ( D ) then d ∈ M ω since ed = de and ω ( dx ) = ω ( edxe ) = ω ( edexe ) = ω ( exede ) = ω ( xd ) , x ∈ m ( M ) ω . Since M ω is a von Neumann algebra we have D ⊂ M ω .For the final assertions we will apply [30, Proposition 4.2] to the restriction of ω to e M e . For any x ∈ M + we have thatsup p ω ( pxp ) = sup p ω ( epxpe ) = sup p ω ( epexepe ) = ω ( exe ) = ω ( x ) , where the supremum is taken over p ∈ P ( D ) ω . We have used that e ∈ D ′ , that pe ր e , and the fact from earlier in the proof that ep is in the centralizer of ω on e M e . The ω -preserving assertion clearly follows from the second last centeredequality above, since ω ( E ( x )) = sup p ω ( pE ( x ) p ) = sup p ω ( E ( pxp )) = sup p ω ( pxp ) = ω ( x ) , for x ∈ M + . (cid:3) Remark 3.8.
Let ω be a normal weight on a von Neumann algebra M with supportprojection e , which is faithful, semifinite, and tracial on a von Neumann subalgebra D , and suppose that D is locally ω -central. One can show that the followingare equivalent: (1) e ∈ D ′ , (2) D is ω -central, (3) sup p ∈ P ( D ) ω ω ( pxp ) = ω ( x ) for x ∈ M + , (4) there exists a ω -preserving normal conditional expectation, and (5)there exists a locally ω -preserving normal conditional expectation E with E ( e ) = 1(or equivalently, with supp( E ) ≤ e , where supp( E ) is as defined above Lemma3.11). Theorem 3.9.
Let ω be a normal weight on a von Neumann algebra M which isa faithful semifinite trace when restricted to a von Neumann subalgebra D . Thenthere is a locally ω -preserving normal conditional expectation E ω from M onto D if and only D is locally ω -central. Moreover such a locally ω -preserving normalconditional expectation onto D is unique, and is faithful if ω is faithful on M .Proof. Suppose that there exists a locally ω -preserving normal conditional expec-tation E onto D . Let p ∈ D be a projection with ω ( p ) < ∞ . The conditionalexpectation E then restricts to a normal conditional expectation from p M p to p D p . Also the restriction of ω to p M + p is finite. Thus p M p ⊂ m ω , and ω ex-tends to a scalar multiple of a normal state ω p on p M p , with ω p on p D p . Wehave ω p ◦ E = ω p on p M p . So by Theorem 3.5 we have ω p ( dx ) = ω p ( xd ) for x ∈ p M p, d ∈ p D p . Thus ω ( pdpxp ) = ω ( pxpdp ) for d ∈ D , x ∈ M . Conversely, suppose that that for every projection p ∈ D with ω ( p ) < ∞ , wehave that ω ( pdpxp ) = ω ( pxpdp ) , d ∈ D , x ∈ M . The collection of all projections in D with finite ω -trace, is of course a net increasingto when ordered with the natural ordering. By Theorem 3.5, there will for suchprojection p exist a normal conditional expectation E p : p M p → p D p , preservingthe restriction of ω to p M p . We claim that for p ≥ p , ( E ) | p M p = E . Thisis a fairly straightforward consequence of the uniqueness assertion in Theorem 3.5,and the fact that p ≥ p ensures that p xp ∈ p M p and p dp ∈ p D p , with E ( p xp ) = E ( p p xp p ) = p E ( p xp ) p .We know that { p ∈ P ( D ) : ω ( p ) < ∞} is σ -strong* convergent to . So for any x ∈ M , { pxp : p ∈ P ( D ) , ω ( p ) < ∞} is σ -strong* convergent to x . Now let x ∈ M be fixed. We show that similarly { E p ( pxp ) : p ∈ P ( D ) , ω ( p ) < ∞} is σ -strong*convergent to some d x ∈ D . Observe that the set { E p ( pxp ) : p ∈ P ( D ) , ω ( p ) < ∞} is norm-bounded, and hence relatively weak*-compact. Let d be any weak* clusterpoint of the net { E p ( pxp ) : p ∈ P ( D ) , ω ( p ) < ∞} . By the relative compactness,there must exist some subnet { E p α ( p α xp α ) } converging weak* to p . For any fixed q ∈ P ( D ) with ω ( q ) < ∞ , the fact that qE p ( pxp ) q = E q ( qxq ) for each p ≥ q , ensuresthat { qE p ( pxp ) q : p ∈ P ( D ) , ω ( p ) < ∞} is uniformly convergent to E q ( qxq ). Butthis net also has weak* limit point qd q . Thus clearly E q ( qxq ) = qd q . Since { qd q : q ∈ P ( D ) , ω ( q ) < ∞} is σ -strong* convergent to d , the claim follows.We now define the map E : M → D by setting E ( x ) = lim p E p ( pxp ). Clearly E ( d ) = lim p E p ( pdp ) = lim p pdp = d, d ∈ D . It easily follows that E is a contractive unital positive projection. Hence byTomiyama’s result E is a conditional expectation.We next show that E is normal. Let { x α } ⊂ M + be given with x = sup α x α ∈M + . By positivity E ( x ) ≥ sup α E ( x α ). Suppose that E ( x ) − sup α E ( x α ) = 0. Wenow equip D with the weight induced by the restriction of ω to D . The semifinitenessof ( D , ω ) then ensures that there must exist a non-zero subprojection p of thesupport projection of E ( x ) − sup α E ( x α ) ≥
0, for which ω ( p ) < ∞ , and E ( pxp ) − sup α E ( px α p ) = p ( E ( x ) − sup α E ( x α )) p = 0 . But by construction E | p M p = E p . Since E p is known to be normal, and sup α px α p = pxp , it follows that E ( pxp ) − sup α E ( px α p ) = E p ( pxp ) − sup α E p ( px α p ) = 0 . This is a contradiction. Hence E must be normal.The fact that E | p M p = E p , ensures that E is locally ω -preserving. The claimsregarding uniqueness and faithfulness similarly follow from this fact. For exampleif it was indeed possible to find two distinct conditional expectations E and E fulfilling the criteria of the hypothesis, we would be able to find some x ∈ M + forwhich E ( x ) − E ( x ) = 0. Thus there must then exist some non-zero subprojection p of the support projection of E ( x ) − E ( x ) for which we still have that p ( E ( x ) − E ( x )) p = 0. But we then also have that p ( E ( x ) − E ( x )) p = E ( pxp ) − E ( pxp ) = E p ( pxp ) − E p ( pxp ) = 0 , a contradiction. Thus E must be unique. ONDITIONAL EXPECTATIONS AND GENERALIZED REPRESENTING MEASURES 17 If ω is faithful then by Theorem 3.5 each E p is also faithful. Let 0 = x ∈ M + begiven. We know that { pxp : p ∈ P ( D ) , ω ( p ) < ∞} is σ -strong* convergent to x . Sofor some p ∈ P ( D ) with ω ( p ) < ∞ , we must have that pxp = 0. The faithfulness of E p then ensures that 0 = E p ( pxp ) = pE ( x ) p , which in turn ensures that E ( x ) = 0.Hence E is faithful. (cid:3) Theorem 3.10.
Suppose that D is a von Neumann subalgebra of a von Neumannalgebra M . The following are equivalent: (1) D is semifinite (or equivalently, has a faithful normal semifinite trace τ )and there exists a normal conditional expectation (resp. faithful normal con-ditional expectation) Ψ :
M → D . (2) M has a normal (resp. faithful normal) weight ω with m ( D ) ω ω -centralsuch that ω is semifinite and faithful on D . (3) M has a normal (resp. faithful normal) weight ω with D locally ω -central,such that ω is a semifinite faithful trace on D .Moreover if (2) holds then we can choose E in (1) to be ω -preserving. Such an ω -preserving normal conditional expectation onto D is unique. Also Ψ τ ◦ Ψ isa bijective correspondence between the items in (1) and (2) . This correspondence isalso a bijection with the ω in (3) whose support projections commute with D .Proof. (1) ⇒ (2) Indeed if there exists a (faithful) normal conditional expectationΨ : M → D , and if D has a fns trace τ , then the composition ω = τ ◦ Ψ is a(faithful) normal semifinite weight with m ( D ) ω ω -central: ω ( xd ) = τ (Ψ( x ) d )) = τ ( d Ψ( x ))) = ω ( dx ) , d ∈ m ( D ) ω , x ∈ p ( M ) ω . (2) ⇒ (1) If M has a faithful normal weight ω with D ⊂ M ω , such that ω is semifinite on D , then there exists a ω -preserving faithful normal conditionalexpectation by [38, Theorem IX.4.2]. By Lemma 2.1, ω is a trace on D .Now suppose that M has a normal weight ω such that ω is semifinite and faithfulon D , with ω ( xd ) = ω ( dx ) for d ∈ m ( D ) ω , x ∈ m ( M ) ω . Let e be the supportprojection of ω , then Lemma 2.3 shows that D ⊂ M ω , that e ∈ D ′ and D e is a vonNeumann subalgebra of e M e , and the restriction of ω to D (resp. D e ) is a normal(resp. faithful normal) semifinite trace.By the earlier case there exists a (faithful) normal conditional expectation Ψ : e M e → D e , so that Ψ( e · e ) is a normal conditional expectation from M onto D e as in the proof of Theorem 3.5. The canonical map D 7→ D e is faithful, since ω ( ed ∗ de ) = ω ( d ∗ d ), and ω is faithful on D . As in the proof of Theorem 3.5, theassociated map E : M → D is a normal conditional expectation. Moreover ω ( E ( x )) = ω ( L − e Ψ( exe )) = ω (Ψ( exe )) = ω ( exe ) = ω ( x ) , x ∈ M + . (2) ⇒ (3) We saw above that (2) implies that ω is a trace on D . Since p M p ⊂ m ω (and similarly for D ), (3) is now clear.(3) ⇒ (1) This follows from Theorem 3.9. (Note that the ‘faithful case’ followsfrom the ‘faithful case’ of Theorem 3.9.)The uniqueness of an ω -preserving conditional expectation is just as in e.g. theproof of Theorem 3.5. This easily gives the bijectivity of the map Ψ τ ◦ Ψbetween the items in (1) and (2). Finally if e ∈ D ′ then D is locally ω -central ifand only if D is ω -central by Theorem 3.7. (cid:3) As noted in an early paragraph in the introduction we may call E in (1) in thelast result the (generalized) weight-preserving conditional expectation from M to D associated with the normal weight ω in (2) (by the bijective correspondence in thetheorem). We wrote it as E ω before (in the normal state case).Putting the last two results and their proofs together shows the weight-preservingconditional expectation E associated with a weight ω in this way, may be approx-imated by a net ( E p ) of expectations on the ‘finite’ von Neumann algebras p M p ,for ω -finite p ∈ P ( D ). In Theorem 3.7 we also saw that ω can be approximatedby a matching net of normal functionals: ω ( x ) = sup p ω ( pxp ) = lim p ω ( pxp ) for x ∈ M + . Another upshot of the last theorem is that for semifinite von Neumann subalge-bras, the existence of conditional expectations necessitates the ω -centrality condi-tion. Thus in our later generalization of the Hoffman-Rossi theorem to D -charactersfor a semifinite D , it is natural to assume that D ⊂ M ω . Of course for a normalstate the latter condition is just saying that ω ( xd ) = ω ( dx ) for x ∈ M , d ∈ D . Inone of our ‘best Hoffman-Rossi theorems’ (Theorem 6.5), D is a subalgebra of a σ -finite von Neumann algebra M satisfying the latter condition.For a normal positive map E : M → N we define the support projection supp ( E )of E to be the complement of the largest projection p ∈ M with E ( p ) = 0. All ofthe following result is no doubt known, even in this generality. We furnish a proofamounting to a very slight generalization of the ideas in [13, Lemma 1.2]. Lemma 3.11.
For a normal positive map E : M → N between von Neumannalgebras the support projection z above of E exists, and E ( x ) = E ( zxz ) for all x ∈ M . We have z = 1 if and only if E is faithful. If x ∈ M + then E ( x ) = 0 ifand only if zxz = 0 .If in addition, N ⊂ M , E ◦ E = E , and E is contractive, then zE ( x ) = E ( x ) z for x ∈ M .Proof. Let L E = { x : E ( x ∗ x ) = 0 } , the left kernel of E . For x ∈ L E we have E ( y ∗ x ) = 0 for all y ∈ M , since for all normal states ϕ on N we have | ϕ ( E ( y ∗ x )) | =0 by the Cauchy-Schwarz inequality for ϕ ◦ E . Then L E is a weak* closed left idealin M . The support projection f of L E is the “largest projection” mentioned inthe statement preceding the Lemma. Indeed clearly E ( f ) = 0, and for a projection p ∈ M with E ( p ) = 0 we have p ∈ L E so that p ≤ f . It follows that if z = f ⊥ and x ∈ M then E ( x ) = E ( xz ). Similarly E ( x ) = E ( zx ) = E ( zxz ). Clearly z = 1 ifand only if f = 0 if and only if E is faithful. If x ∈ M + then E ( x ) = 0 if and onlyif x = xf , and if and only if x = f xf . Indeed E ( x ) = 0 if and only if x ∈ L E ,and if and only if x = x f , which implies x = xf and x = f xf . Conversely, if x = f xf then E ( x ) = 0 by a fact a few lines back. Thus E ( x ) = 0 if and only if zxz = 0 (since the latter implies that x z = 0 and x f = x .Suppose that N ⊂ M , E ◦ E = E , and E is contractive. Let M = M ⊕ C .Then it is an exercise (or alternatively a consequence of [11, Theorem 2.3 (2)]) that E ′ ( x + λ , λ ) = ( E ( x ) + λ , λ ) , x ∈ M , λ ∈ C , defines a unital normal idempotent positive projection on M extending E . Sincethe projections in M are the projections in M together with their orthocomple-ments in M , it is easy to see that the support projection of E ′ equals z above.Thus zE ( x ) = zE ′ ( x ) = E ′ ( x ) z = E ( x ) z for x ∈ M by [13, Lemma 1.2 (2)]. (cid:3) ONDITIONAL EXPECTATIONS AND GENERALIZED REPRESENTING MEASURES 19
Theorem 3.12.
Let
D ⊂ M be an inclusion of von Neumann algebras. Let ω be a normal state on M , and let z be the support projection of ω |D in D . If wehave that ω ( ax ) = ω ( xa ) for every a ∈ D , x ∈ M , then there is an ω -preservingnormal idempotent contractive completely positive D -module map E ω from M ontoan ideal in D , with supp( E ω ) ≤ z . We have E ω ( M ) = z D . Indeed there is aunique ω -preserving normal idempotent contractive D -module map from M onto a ∗ -subalgebra of D with supp( E ) ≤ z .Proof. Let z be the support projection in D for ω |D . The left kernel of ω |D in D isan ideal in this case, so z is central in D . Then z D ⊂ z M z is an inclusion of vonNeumann algebras as in Theorem 3.5, and by that result there is a unique normalconditional expectation F from z M z onto z D which preserves ω | z M z .Composing with the normal idempotent contractive D -module map x zxz weget a normal idempotent contractive map E ω from M onto z D . Clearly E ω is a D -module map since e.g. E ω ( dx ) = F ( zdxz ) = zdF ( zxz ) = dF ( zxz ) = d E ω ( x ) , d ∈ D , x ∈ M . We have ω ( E ω ( x )) = ω | z M z ( E ω ( zxz )) = ω | z M z ( zxz ) = ω ( x ) , x ∈ M . The last equality holds since z is in the multiplicative domain of ω (see e.g. [10,Proposition 1.3.11]). Of course E ω (1 − z ) = 0, so that supp( E ω ) ≤ z .Suppose that E is an ω -preserving normal idempotent contractive D -modulemap from M onto an ideal R of D with supp ( E ) ≤ z . Now E ( z ) = zE (1) z ≤ z ,and ω ( z − E ( z )) = 0, so that z = E ( z ). Also E (1 − z ) = 0 and so E (1) = E ( z ) = z .The restriction of E to z M z is a conditional expectation onto z D which preserves ω | z M z .By the uniqueness assertion in the first paragraph the restriction of E to z M z is uniquely determined, and indeed E ( x ) = E ( zxz ) = F ( zxz ) = E ω ( x ) for x ∈M + . (cid:3) Remark 3.13.
The condition in the last result that supp ( E ) ≤ z is necessary,as is clear from the following example. Let M = M with ω ( a ) = a , and let D = D , the diagonal matrices. Then E ω ( a ) = ω ( a ) z . However if F is anyidempotent contractive D -module map from M onto a ∗ -subalgebra of D then E ( a ) = F ( a ) ⊕ ω ( a ) is an ω -preserving normal idempotent contractive D -modulemap onto a ∗ -subalgebra of D . Since there are many such maps F in general, theremay be many such E . Hence the ‘uniqueness’ statement at the end of the theoremis violated.We may call E ω in the last result the weight-preserving conditional expectation from M to D associated with the normal state ω on M . Corollary 3.14.
Let
D ⊂ M be an inclusion of von Neumann algebras. Let ω bea normal state on M such that ω ( ax ) = ω ( xa ) for every a ∈ D , x ∈ M , and let z be the support in D of ω |D . There is a bijective correspondence between surjectivenormal idempotent contractive D -module maps E : M → D z with supp( E ) ≤ z , andextensions of ω |D to normal states ρ on all of M which have D in the centralizer.Under this correspondence, the support projections of E and ρ in M are the same,and ρ = ρ ◦ E = ω ◦ E . Also, such maps E are completely positive. Proof. If E : M → D z is a surjective normal idempotent contractive D -modulemap with supp( E ) ≤ z then ω ◦ E is a normal contractive functional with D in thecentralizer, and ω ( E ( d )) = ω ( E ( dz )) = ω ( dz ) = ω ( d ) , d ∈ D . In particular ω ( E (1)) = 1 so that ω ◦ E is a state.Conversely, given an extension of ω |D to a normal state ρ on M which has D in the centralizer, then by Theorem 3.12 there is a unique surjective ρ -preservingnormal idempotent contractive completely positive D -module map from M onto D z with supp( E ) ≤ z .Note that E ( x ) = 0 if and only if ω ( E ( x )) = 0, for x ∈ M + . Thus the supportprojections of E and ω ◦ E are the same. (cid:3) Remark 3.15.
1) The ‘conditional expectations’ E in Corollary 3.14 need not benicely related to the conditional expectation E ω in Theorem 3.12, unlike the casementioned early in the introduction where we discussed ‘weight functions’ h , ande.g. in Remark 4.5 (1). A good example to see some of the issues that can arisein this ‘nonfaithful’ case is as follows. Let M = M and D = C I + C E , andlet ω ( x ) = x . The ‘conditional expectations’ E in Corollary 3.14 correspond tothe density matrices in M (i.e. positive trace 1 matrices in M ). Only some ofthese density matrices are related to E ω or to ‘Radon-Nikodym derivatives’ of ω , incontrast to the faithful case (see e.g. Corollary 4.6 or Corollary 4.8, and the remarkbetween these results).If however the support projections of ω and ω |D agree, then there is much morethat one can say. Indeed we may cut down to z M z in this case, on which algebra ω and ω |D are faithful, and on z M z simply apply the theory of von Neumannalgebraic conditional expectations, and the later result Corollary 4.6.2) The proof above shows that every surjective normal idempotent contractive D -module map E : M → D z with supp( E ) ≤ z is the ‘weight-preserving conditionalexpectation’ on M associated with a state on M (namely the state ω ◦ E ).The following simple result is similar to [9, Corollary 3.1]. Corollary 3.16.
Suppose that D is a von Neumann subalgebra of a von Neumannalgebra M , and that M has a normal weight ω such that ω is faithful on D , andsuch that there exists an ω -preserving conditional expectation E ω : M → D . Let
Φ :
A → D be an ω -preserving unital linear map on a unital subalgebra A of M containing D , which is a D -bimodule map. Then there is a unique ω -preservingnormal conditional expectation M → D extending Φ .Proof. The desired expectation is the map E ω . To see this let d ∈ D and a ∈ A begiven. It follows from the hypothesis that ω ( d Φ( a )) = ω (Φ( da )) = ω ( da ) = ω ( E ω ( da )) = ω ( d E ω ( a )) . Since ω is faithful on D and d ∈ D arbitrary, this equality suffices to show thatΦ( a ) = E ω ( a ) as claimed. The uniqueness is similar, and follows as in the proof ofTheorem 3.5. (cid:3) ONDITIONAL EXPECTATIONS AND GENERALIZED REPRESENTING MEASURES 21 Weight functions and characterizations of normal conditionalexpectations
Let D be a von Neumann subalgebra of a von Neumann algebra M , and let ν be afaithful normal semifinite weight on M . We again recall that as in [38], given a pos-itive operator h affiliated to M ω , we write ν h for the weight lim ǫ ց ω ( h / ǫ xh / ǫ ).We say that a normal conditional expectation E : M → D commutes with ν if ν ◦ E commutes with ν in the sense of [30] (so e.g. ν ◦ E ◦ σ νt = ν ◦ E for t ∈ R ). Lemma 4.1.
Let D be a von Neumann subalgebra of von Neumann algebra M ,and let ν be a faithful normal semifinite weight on M . Then a normal conditionalexpectation E : M → D commutes with ν if and only if E ◦ σ νt = σ νt ◦ E for all t ∈ R .Proof. Since ν ◦ σ νt = ν the one direction is easy. Conversely, suppose that ν ◦ E ◦ σ νt = ν ◦ E for t ∈ R . If a ∈ n ν ◦ E then ν ◦ E ( | σ νt ( a ) | ) = ν ◦ E ( | a | ). So forany t , σ νt ( a ) will then again belong to n ν ◦ E . Similarly if a ∈ n ν ◦ E , we may use theKadison-Schwarz inequality to see that ν ◦ E ( | E ( a ) | ) ≤ ν ◦ E ( E ( | a | )) = ν ◦ E ( | a | ).Thus E ( a ) will then again belong to n ν ◦ E . For any a ∈ n ν ◦ E and any t ∈ R each ofthe terms in the expansion of ν ( | E ( σ νt ( a )) − σ νt ( E ( a )) | ) as ν ( | E ( σ νt ( a )) | ) − ν ( E ( σ νt ( a )) ∗ σ νt ( E ( a ))) − ν ( σ νt ( E ( a )) ∗ E ( σ νt ( a ))) + ν ( σ νt ( | E ( a ) | )) , is finite. Repeated use of the facts that ν ◦ σ νt = ν and ν ◦ E ◦ σ νt = ν ◦ E thenshows that(3) ν ( | E ( σ νt ( a )) − σ νt ( E ( a )) | ) = ν ( | E ( σ νt ( a )) | ) − ν ( | E ( a ) | )We therefore have ν ( | E ( σ νt ( a )) | ) − ν ( | E ( a ) | ) nonnegative for each t ∈ R . Butsince σ νt ( a ) ∈ n ν ◦ E for any t , we must by symmetry also have that0 ≤ ν ( | E ( σ ν − t ( σ νt ( a ))) | ) − ν ( | E ( σ νt ( a )) | ) = ν ( | E ( a ) | ) − ν ( | E ( σ νt ( a )) | )for each t . Thus ν ( | E ( σ νt ( a )) | ) − ν ( | E ( a ) | ) = 0 for each t ∈ R , in which casewe will have by equation (3) that ν ( | E ( σ νt ( a )) − σ νt ( E ( a )) | ) = 0 for each t . Thefaithfulness of ν then ensures that E ( σ νt ( a )) = σ νt ( E ( a )) for all a ∈ n ν ◦ E . Bynormality this equality holds on all of M . Thus E ◦ σ νt = σ νt ◦ E . (cid:3) Thus E commutes with ν if and only if E commutes with ( σ νt ). In particular, bythe last remark (or [20, Proposition 4.9]), E D commutes with ν , where E D is theweight-preserving conditional expectation of M onto D associated with the weight ν as in the definition at the start of Section 3.We recall that two selfadjoint unbounded operators S, T on H commute strongly if all of their Borel spectral projections commute. This is equivalent to saying thatthere is a commutative von Neumann algebra N on H which both S and T areaffiliated with (see e.g. [25, Theorem 5.6.15]). For the one direction of this equiva-lence take the von Neumann algebra N generated by these two commuting familiesof spectral projections (we have S η W ∗ ( S ) ⊂ N so that S η N , and similarly for T ). For the other direction recall that W ∗ ( S ) is the smallest von Neumann algebraon H with which S is affiliated, and similarly for T . So all the spectral projectionsof S and T are in N , and hence they commute.Before proving the main theorem of this section, we need some insight into howone may describe the weights of the form described in equation (1) in Haagerup L p -space terms. This is the topic of the next result. Proposition 4.2.
Let ν be a faithful normal semifinite weight on M . A possiblyunbounded positive operator h is affiliated with M + ν if and only if h η M and h strongly commutes with k ν , where k ν denotes the density d e νdτ of the dual weight e ν on M ⋊ ν R . In particular, if D is a von Neumann subalgebra of M . Then D is contained in the centralizer of ν if and only if k ν commutes strongly with D .Moreover for any positive operator h affiliated with M + ν , the dual weight of ν h is given by the formula lim ǫ ց e ν ( h / ǫ · h / ǫ ) , with the strong product k = k ν · h corresponding to the density d f ν h dτ of the dual weight f ν h on M ⋊ ν R .Proof. Indeed if h η M then affiliation of h with M ν is equivalent to the conditionthat hχ [0 ,n ] ( h ) ∈ M ν for each n ∈ N , which in turn is equivalent to requiring that σ νt ( hχ [0 ,n ] ( h )) = hχ [0 ,n ] ( h ) for all t ∈ R and all n . On regarding M as a subalgebraof the crossed product M ⋊ ν R , the modular automorphism group will then beimplemented by k itν . The above equality may be reformulated as the claim that hχ [0 ,n ] ( h ) = k itν hχ [0 ,n ] ( h ) k − itν for all t ∈ R and all n ∈ N . This clearly shows that hχ [0 ,n ] ( h ) and k itν will commute for any n ∈ N and any t ∈ R , which in turn isequivalent to the fact that the spectral projections of h and k ν all commute.In particular, k ν commutes strongly with D if and only if for each d ∈ D andeach t ∈ R we have that k itν dk − itν = d . This holds if and only if σ νt ( d ) = d , that is,if and only if D ⊂ M ν .We next show that d f ν h d e ν exists and equals h . Let T denote the operator valuedweight from the extended positive part of M ⋊ ν R to that of M . If we combine[20, Proposition 4.9] with the fact that ν h commutes with ν , we have that f ν h ◦ σ e νt = ν h ◦ T ◦ σ e νt = ν h ◦ σ νt ◦ T = ν h ◦ T = f ν h for all t ∈ R . Thus f ν h must then commutewith e ν , which ensures that e h = d f ν h d e ν exists as an operator affiliated to ( M ⋊ ν R ) e ν .We go on to prove that e h = h . In the case where ν h is faithful this easily followsfrom the fact that h it = ( Dν h : Dν ) t = ( D f ν h : D e ν ) t = e h it for all t ∈ R . (The first and last equality follows from [38, Theorem VIII.2.11] and the definitionof the cocycle derivative; see the proof of [38, Corollary VIII.3.6]. The secondequality follows from [20, Theorem 4.7].) In the case where ν h is not faithful (i.e. e = supp( ν h ) is strictly smaller than ), we may pass to the weight ( ν h ) = ν h + ( − e ) .ν. ( − e ). It is an exercise to see that this is a faithful normalsemifinite weight on M with d ( ν h ) dν = h + ( − e ). The dual weight will then be ] ( ν h ) = f ν h + ( − e ) . e ν. ( − e ) using one of [39, Lemma II.1] or [17, Theorem6.55]. We also have h + ( − e ) = d ^ ( ν h ) d e ν , by the ‘faithful case’ above. But clearly d ( − e ) . e ν. ( − e ) d e ν = ( − e ). We also know from either [39, Lemmas II.1 and II.2,and Proposition II.4] or [17, Theorem 3.24, Theorem 6.55 and Proposition 6.67],that supp( f ν h ) = supp( ν h ) = supp( h ) = e . Therefore e . ] ( ν h ) .e = f ν h . Using all these facts it is now possible to verify that d f ν h d e ν = e ( h + ( − e )) e = h . This fact then ensures that f ν h = e ν h , that is that thedual weight of ν h is given by the formula lim ǫ ց e ν ( h / ǫ · h / ǫ ).Since the centralizer ( M ⋊ ν R ) τ is all of M ⋊ ν R , we may now apply the chainrule described in [30, Proposition 4.3] to the pair k ν and h , to see that hk ν = d f ν h d e ν d e νdτ = d f ν h dτ = k as required. (cid:3) ONDITIONAL EXPECTATIONS AND GENERALIZED REPRESENTING MEASURES 23
Remark 4.3.
Let ν be a faithful normal semifinite weight on M and let h be apositive operator affiliated to M ν . For x ∈ M + it is tempting to assign a meaningto the product h xh , perhaps as an element b of the extended positive part c M of M , and to then interpret ν h ( x ) as the canonical extension of ν to c M applied to b .In this regard the conclusions regarding h , k ν and k = k ν · h noted in thepreceding Proposition, may be interpreted as a realisation of exactly this objective.To see why this is so, we shall for the sake of simplicity pass to the case where ν is a normal state, and ν ( h ) < ∞ . (Note that the weight ν has a well-definedaction on h since it canonically extends to c M .) Since ν h ( ) = lim ǫ ց ν ( h ǫ ) = ν ( h ),this assumption ensures that ν h is a normal functional. The theory of Haagerup L p -spaces then ensures that for any x ∈ M + , we will have that ν ( x ) = tr ( k ν xk ν )and that ν h ( x ) = tr ( k xk ). When these equalities are considered alongside thefact that k = k ν · h , it is then clear that the equality ν h ( x ) = tr ( k xk ) maybe interpreted as a ‘rigorization’ of the formal equality ν h ( x ) = ν ( h xh ).In the case where ν is a trace, we can achieve this objective more directly, byusing basic facts about noncommutative integration with respect to a trace frome.g. Section IX.2 in [38] (see also the fact in [17, Remark 4.15], which informs usthe element b above will be in L ( M , ν )). Note that if we view h as an element of c M then the product c = x · h · x , defined as in [38, Definition IX.4.6], is in c M .The trace ν allows for an extension to c M , and for this extension normality of thetrace ensures that(4) ν h ( x ) = lim ǫ ց ν ( h ǫ xh ǫ ) = lim ǫ ց ν ( x h ǫ x ) = ν ( c ) . If this quantity is finite, that is if x ∈ p ν h , then we know from either [27, Remark1.2] or [17, Remark 4.15] that the product c = x · h · x above is in L ( M , ν ).But if x · h · x is a densely defined operator, then using the ideas in the proofof either [17, Lemma 7.39] or [18, Lemma 2.1], shows that h x is densely definedwith c = | h x | . Thus if ν h ( x ) is finite, then h x ∈ L ( M , ν ). We may thenuse the tracial property of ν to conclude that ν h ( x ) = ν ( c ) = ν (( h x ) ∗ ( h x )) = ν (( h x )( h x ) ∗ ) , giving what was claimed at the start of this remark with b = ( h x )( h x ) ∗ . Ifconversely h x is densely defined and in L ( M , ν ), then one may then use thetracial property alongside equation (4) to see that ν h ( x ) < ∞ .Finally we point out that the argument in this remark can be considerably sim-plified, and slightly strengthened, if h is also ν -measurable: in this case for any x ∈ M + , the operators x h ǫ x = | h ǫ x | are known to increase to the opera-tor x hx = | h x | inside the algebra of ν -measurable operators as ǫ ց ν to thisalgebra [17, Proposition 4.17] then ensure that ν h ( x ) = lim ǫ ց ν ( h ǫ xh ǫ ) = ν ( x hx ) = ν ( h xh ) , for all x ∈ M + . Theorem 4.4.
Let D be a von Neumann subalgebra of M , and ν a faithful normalweight on M , which is semifinite on D , and satisfies σ νt ( D ) ⊂ D for all t ∈ R .There is a bijective correspondence between the following objects: (a) Normal conditional expectations E onto D which commute with ( σ νt ) in thesense above, (b) densities h η M + ν which commute with all elements of D , and which satisfy ν h = ν on D + , (b ′ ) densities h η M + ν which commute with all elements of D , and which satisfy E D ( h ) = 1 (where the action on h is by the normal extension of E D to theextended positive part of M in e.g. [16, Proposition 3.1] ), (b ′′ ) densities k ∈ η L ( M ) which commute strongly with the element k ν in thelast result, for which h = k.k − ν commutes with all elements of D , and whichsatisfy E D ( k ) = E D ( k ν ) (where the action on k ν and k is by the normalextension of E D to the extended positive part of M ⋊ ν R ), (c) extensions of ν |D to normal semifinite weights ρ on all of M which commutewith ν in the sense of the Introduction, whose support projection z commuteswith D and satisfies σ ρt ( D z ) ⊂ D z for all t ∈ R .With respect to the correspondences above we also have the formulae ρ = ν ◦ E = ν h , h = dρdν = k.k − ν , and E ( x ) = lim ǫ ց E D ( h ǫ x h ǫ ) , x ∈ M + , where h ǫ = h ( + ǫh ) − and with the last limit being a ( σ -strong*) limit of anincreasing net in M .Proof. By [38, Theorem VIII.2.6] there is a unique ν -preserving normal conditionalexpectation E D : M → D . We noted immediately after Lemma 4.1 that E D commutes with ( σ νt ).(a) ⇒ (c) Given a normal conditional expectation E onto D which commuteswith ( σ νt ), let ρ = ν ◦ E . Then ρ ◦ E = ρ . This is a normal weight on M extending ν |D , for which we have that ρ ◦ σ νt = ρ since E commutes with σ νt . Thus ρ commuteswith ν in the sense of [30] (see also [38, Corollary VIII.3.6], although here all weightsare faithful). We recall the support projection z = supp( E ) from Lemma 3.11. Let z be the support projection of E . It is also the support projection of ρ , since ν isfaithful (so that for x ∈ M + we have E ( x ) = 0 if and only if ν ( E ( x )) = 0).The fact that ρ is semifinite follows from the facts that ν |D = ρ |D is semifiniteand ρ ◦ E = ρ . We sketch the proof of this claim: We firstly note that the left ideal n ( D ) ν admits a right approximate identity ( f λ ) of positive contractive elements (seefor example [12, Proposition 2.2.18]) which by [12, Lemma 2.4.19] must converge σ -strongly to some contractive element p of M . Since ν |D is faithful on D , thiselement in fact turns out to be the support projection of ν |D (in the sense of [38],see p. 57 there), which in this case is since ν |D is semifinite on D . Although weshall not need this fact, we pause to note that this approximate identity may in factbe chosen to consist entirely of entire analytic elements of n ( D ) + ν by [40, Lemma9]. One may now use this approximate identity to show that m + ρ is weak* dense in M + , which in turn ensures that ρ is semifinite. In particular given any a ∈ M + ,the σ -strong convergence of { f λ } to , ensures that ( f λ af λ ) converges σ -weakly(weak*) to a . For any λ , we moreover have that ρ ( f λ af λ ) = ν ( E ( f λ af λ )) = ν ( f λ E ( a ) f λ ) ≤ k a k ν ( f λ ) < ∞ . ONDITIONAL EXPECTATIONS AND GENERALIZED REPRESENTING MEASURES 25
By Lemma 3.11, z commutes with D . The map D → D z of multiplication by z is a surjective normal ∗ -homomorphism with right inverse E . Hence D ∼ = D z . Then ρ restricts to a faithful normal weight on z M z , which is semifinite on z M z and D z since if ( e t ) is an increasing ν -finite net in D with limit 1 then ( ze t ) is an increasing ρ -finite and ν -finite net in D z with limit z . Define E ′ ( zxz ) = E ( x ) z = E ( zxz ) z .Then E ′ : z M z → D z is a ρ -preserving normal conditional expectation. By [38,Theorem VIII.2.6] we have σ ρt ( D z ) ⊂ D z for all t ∈ R .(c) ⇒ (b) Now suppose that ρ is a normal semifinite weight on M which extends ν |D , and commutes with ν . The commutation with ν , ensures that h = dρdν exists,and that h η M ν [30]. We have ν h ( d ) = ρ ( d ) = ν ( d ) for d ∈ D + , so that ν h = ν on D + .Let z ρ = supp( ρ ) = supp( h ), which we sometimes write as z . Note that as in aproof two paragraphs above, z M z and D z are von Neumann subalgebras of z M z on which ρ and ν are semifinite faithful normal weights. Since h = dρdν is affiliatedto M ν , by the Borel functional calculus we have z ρ = χ (0 , ∞ ) ( h ) ∈ M ν . It remainsto show that h commutes with D . Let h n = hχ [0 ,n ] ( h ), f = z ⊥ ρ , ρ ′ = ρ + ν f and h = h + f . It is clear that ( h n ) increases to h . So by [30, Propositions 4.1 & 4.2],we have ρ ′ = ν h + ν f = sup n ≥ ν h n + ν f = sup n ≥ [ ν h n + ν f ] = sup n ≥ ν h n + f = ν h . Since ρ is faithful on z ρ M z ρ and ν f is faithful on f M f , the sum ρ ′ is faithfulon all of M . By [30, Theorem 4.6] the modular automorphism group of ρ ′ is h it σ νt ( · ) h − it . Now h it = h it + f it since hf = 0. So the modular group of ρ ′ is( h it + f it ) σ νt ( · ) ( h − it + f − it ) for all t .Theorem 4.6 in [30] ensures that the modular automorphism group of ν f is f it σ νt ( · ) f − it . Since σ νt preserves D and z = z ρ is a fixed point of σ νt , this auto-morphism group clearly preserves D f . So there exists a faithful normal conditionalexpectation E f from f M f onto D f such that ν f ◦ E f = ν f . Again by [30, Theo-rem 4.6], the modular group of ρ is h it σ νt ( · ) h − it , which by assumption preserves D z . Since ρ is faithful on z M z , this then ensures that there exists a unique faith-ful normal conditional expectation E z from z M z onto D z for which we have that ρ | zM z = ρ | z M z ◦ E z . It is now an exercise to see that the map E : M → D defined by E ( a ) = E z ( zaz ) + E f ( f af ) is a faithful normal conditional expecta-tion onto D for which we have that ρ ′ ◦ E = ρ ′ . By [38, Corollary IX.4.22], themodular automorphism group of the restriction of ρ ′ to D is just the restriction of( h it + f it ) σ νt ( · ) ( h − it + f − it ) to D . Since z ρ commutes with D and σ νt preserves D ,this group simplifies to h it σ νt ( · ) h − it + f it σ νt ( · ) f − it on D . But on D , ρ ′ = ρ + ν f agrees with ν g where g = 2 − z ρ . So this group must agree with the modular groupof ( ν g ) |D . We proceed to describe that group.We again use [30, Theorem 4.6] to see that the modular automorphism group of ν g is g it σ νt ( · ) g − it . Recall that σ νt preserves D and that z ρ commutes with D andthat z ρ is a fixed point of σ νt . So for any d ∈ D we have that g it σ νt ( d ) g − it = σ νt ( d ).So this modular group preserves D . Thus by [38, Theorem IX.4.2 & CorollaryIX.4.22], the modular group of ν g | D , is just the restriction of g it σ νt ( · ) g − it to D ,which as we have just seen is just σ νt . When considering this fact alongside what weshowed in the previous paragraph, it is clear that h it σ νt ( · ) h − it + f it σ νt ( · ) f − it agreeswith σ νt on D . So on multiplying with z ρ , we have that h it σ νt ( · ) h − it = σ νt ( · ) z ρ on D . Taking into account that σ νt preserves D , we have that h it dh − it = h it σ νt ( σ ν − t ( d )) h − it = σ νt ( σ ν − t ( d )) z ρ = dz ρ , d ∈ D , t ∈ R , or equivalently that h it d = dh it for all t . Thus h commutes strongly with each d ∈ D .(b) ⇔ (b ′ ) Since E D commutes with the modular group σ νt and each x ∈ M ν isa fixed point of this modular group, it now clearly follows that σ νt ( E D ( x )) = E D ( x )for each x ∈ M ν . Thus E D ( M ν ) ⊂ D ∩ M ν ⊂ D ν . Let E ′ be E D viewed asa conditional expectation from M ν onto D ν . The restriction of ν to M ν is a afaithful normal weight. It is also a trace (see III.4.6.2 in [2]).Most of the following part of the proof consists of a review of some aspects ofHaagerup’s theory of the extended positive part of a von Neumann algebra [20],and establishing a general consequence of [38, Proposition IX.4.11] (which maybe known to some experts). Write ˆ E D for the canonical extension of E D to theextended positive part ˆ M + of M . We may regard the extended positive part d M ν + as a subspace of ˆ M + , and may regard the extended positive part c D ν + as a subspaceof ˆ D + . It is easy to see that the restriction of ˆ E D to the extended positive partof M ν equals the canonical extension c E ′ of E ′ to d M ν + . Let S ν denote the setof selfadjoint positive operators a affiliated with M ν . These correspond to certainweights ν a on M by the Pedersen-Takesaki Radon-Nikodym correspondence [30].By [38, Proposition IX.4.11] we may view an element m in d M ν + (resp. in c D ν + )as a normal weight ν m on M (resp. normal weight ( ν |D ) m on D ). Then c E ′ inducesa ‘normal’ order preserving map j from S ν into the set of normal weights on D .Indeed j ( a ) = ( ν |D ) m with m = c E ′ ( a ). Let i be the map from S ν into the weightson D defined by i ( a ) = ( ν a ) |D + . We claim that j = i . If x and x are in ( M ν ) + then we have i ( x )( d ) = ( ν x )( d ) = ν ( x dx ) = ν ( d xd ) = ν ( d E D ( x ) d ) , d ∈ ( m ( D ) ν |D ) + . This equals ν E D ( x ) ( d ) since E D ( x ) ∈ c D ν + . Since normal weights on a semifinitealgebra are determined by their action on m + , we have j ( x ) = ( ν |D ) E D ( x ) = i ( x ).So i = j on ( M ν ) + . If a ≤ a (in the sense of unbounded operators as on p. 62 in[30]) in S ν then ν a ≤ ν a by [30]. Hence i ( a t ) ≤ i ( a ). So i is order preserving. If a t ր a (in the sense of unbounded operators as on p. 62 [30]) in S ν then ν a t ր ν a by [30]. Hence i ( a t ) ր i ( a ). So i is normal. Since every element of S ν is anincreasing limit of a sequence in ( M ν ) + , we see that i = j .Next suppose we are given h η M + ν with ν h = ν on D . Then by hypothesis i ( h )( d ) = ν h ( d ) = ν ( d ) for d ∈ D + . So in the notation of the last paragraph, i ( h ) = ( ν |D ) m where m = 1. On the other hand j ( h ) = ( ν |D ) n where n = c E ′ ( h ).Taking the Radon-Nikodym derivative with respect to ν |D , we obtain c E ′ ( h ) = 1.Thus ˆ E D ( h ) = 1.At this point it is easy to see that (b) is equivalent to (b ′ ). Indeed if E D ( h ) = 1and h commutes with elements of D + , then ν h ( d ) = lim ǫ ց ν ( h ǫ dh ǫ ) = lim ǫ ց ν ( d / h ǫ d ) = ν ( d / hd / )= ν ( E D ( d / hd / )) = ν ( d / E D ( h ) d / ) = ν ( d ) , d ∈ D + . ONDITIONAL EXPECTATIONS AND GENERALIZED REPRESENTING MEASURES 27 (b ′ ) ⇔ (b ′′ ) Given h η M + ν it is clear from Proposition 4.2 that h commutesstrongly with k ν . For any n ∈ N we may then use the Borel functional calculus tosee that θ s ( hχ [0 ,n ] ( h ) .k ν χ [0 ,n ] ( k ν )) = θ s ( hχ [0 ,n ] ( h )) .θ s ( k ν χ [0 ,n ] ( k ν )) , which equals hχ [0 ,n ] ( h ) .e − s k ν χ [0 ,n ] ( e − s k ν ) = e − s hχ [0 ,n ] ( h ) .k ν χ [0 ,e s n ] ( k ν ) . Letting n → ∞ now yields the conclusion that θ s ( hk ν ) = e − s for all s . That is hk ν ∈ η L ( M ).Conversely given some k ∈ η L ( M ) which commutes with k ν , a similar argumentshows that for h = k.k − ν we then have that θ s ( h ) = h for every s . For each n ∈ N we will then have that θ s ( χ [0 ,n ] ( h )) = χ [0 ,n ] ( θ s ( h )) = χ [0 ,n ] ( h ) for all s . Thus allthe spectral projections of h are in M , which ensures that h η M . Since h = k.k − ν clearly commutes with k ν , we in fact have that h η M ν .There is therefore a bijection between densities k ∈ η L ( M ) which commutestrongly with k ν and for which k.k − ν commutes with D , and densities h η M + ν which commute with D . This bijection is given by h → hk ν .If we are able to show that for densities h η M + ν which strongly commute with D and k ν we have that E D ( h ) = and only if E D ( hk ν ) = E D ( k ν ), then the classof densities described in (b ′′ ) will clearly be in bijective correspondence with theclass described in (b ′ ).Suppose first that E D ( k ν h ) = k ν . Note that the assumptions on h ensure that inthe crossed product it commutes strongly with D and with each λ t = k itν . Thus h commutes strongly with the von Neumann subalgebra generated by these operators,namely D ⋊ ν R .Recall that we noted in Remark 2.5 that τ ◦ E D = τ . Since by Proposition 4.2the strong product k ν h is the density k nu h of the weight e ν h , we may now use thisfact to conclude that e ν h ( d ) = τ (( hk ν ) d ( hk ν ) ) = τ ( d hk ν d ) = τ ( E D ( d hk ν d ))= τ ( d E D ( hk ν ) d ) = τ ( d k ν d )) = e ν ( d )for all d ∈ ( D ⋊ ν R ) + . Again by Remark 2.5, we have that e ν ◦ E D = e ν .This fact together with the commutation of h with D ⋊ ν R now ensures that e ν ( d ) = e ν h ( d ) equalslim ǫ ց e ν ( d / h ǫ d / ) = e ν ( d / hd / ) = e ν ( E D ( d / hd / )) = e ν ( d / E D ( h ) d / )for all d ∈ ( D ⋊ ν R ) + . Since ν is faithful and semifinite on D with e ν | ( D ⋊R ) = ν |D ◦ ( T ν ) | ( D ⋊R ) , e ν | ( D ⋊R ) the dual weight of ν |D , and therefore faithful and semifiniteon D ⋊ ν R . This then ensures that E D ( h ) = .Now suppose that E D ( h ) = . It then follows from the semifinite variant of [16,Lemma 4.8] mentioned in Remark 2.5, that E D ( k ν h ) = E D ( k ν h ) = k ν h ◦ E D . (Here k ν h ◦ E D is considered in the context of D ⋊ ν R .) Now notice that when combinedwith the fact that ν ◦ E D = ν , the commutation of h with D ensures that ν h ◦ E D ( a ) = sup ǫ ν ( h / ǫ E D ( a ) h / ǫ ) = sup ǫ ν ( E D ( a ) / h ǫ E D ( a ) / ) . But this equals ν ( E D ( a ) / hE D ( a ) / ) = ν ( c E D ( E D ( a ) / hE D ( a ) / )) = ν ( E D ( a )) . Thus we have that E D ( k ν h ) = k ν h ◦ E D = k ν as required.(b ′ ) ⇒ (a) Suppose that E D ( h ) = 1. It follows that for any x ∈ M + and any ǫ > E D ( h ǫ xh ǫ ) ≤ k x k E D ( h ǫ ) ≤ k x k E D ( h ) = k x k · h ǫ = h ( + ǫh ) − . Thus for any x ∈ M + , the net ( E D ( h ǫ xh ǫ )) isa norm bounded net in D .For any d ∈ D + we know from the equivalence of (b) with (b ′ ) that ν ( d ) = ν h ( d ).So for any d ∈ n ( D ) ν and any x ∈ M , we will have that d ∗ xd ∈ m ν h . For any d ∈ n ( D ) ν , it is now clear that ν ( d ∗ E D ( h ǫ xh ǫ ) d ) = ν ( d ∗ ( h ǫ xh ǫ ) d )= ν ( h ǫ d ∗ xdh ǫ ) ր ν h ( d ∗ xd ) < ∞ as ǫ ց
0. To see this recall that by [38, Lemma VIII.2.7] the net ( ν ( h ǫ d ∗ xdh ǫ ))increases to ν h ( d ∗ xd ). The above fact in particular also ensures that if ǫ > ǫ > d ∈ n ( D ) ν have that ν ( d ∗ [ E D ( h ǫ xh ǫ ) − E D ( h ǫ xh ǫ )] d ) ≥ . Now recall that in the GNS representation for ( ν |D , D ) the space η ( n ν ( D )) is adense subspace of H ν |D . On passing to this GNS representation if need be, theabove inequality corresponds to the statement that h [ E D ( h ǫ xh ǫ ) − E D ( h ǫ xh ǫ )] ξ, ξ i ≥ ξ in a dense subspace of H ν |D . By continuity we will then in facthave that h [ E D ( h ǫ xh ǫ ) − E D ( h ǫ xh ǫ )] ξ, ξ i ≥ ξ ∈ H ν |D . But this ensuresthat E D ( h ǫ xh ǫ ) ≥ E D ( h ǫ xh ǫ ). Thus the net ( E D ( h ǫ xh ǫ )) is a norm boundedincreasing net. Hence for each x ∈ M + , the limit lim ǫ ց E D ( h ǫ xh ǫ ) exists withconvergence taking place in the σ -strong* topology.For any x ∈ M + , we now define the element E ( x ) ∈ D + by E ( x ) = lim ǫ ց E D ( h ǫ xh ǫ ).We pause to note that for this element, it is clear from the above computations thatfor any d ∈ n ( D ) ν and any x ∈ M + , we have that(5) ν ( d ∗ E ( x ) d ) = lim ǫ ց ν ( d ∗ E D ( h ǫ xh ǫ ) d ) = lim ǫ ց ν ( h ǫ d ∗ xdh ǫ ) = ν h ( d ∗ xd ) . Notice that for x ∈ M + and any d ∈ D , the commutation of h with D clearlyensures that E ( d ∗ xd ) = lim ǫ ց E D ( h ǫ d ∗ xdh ǫ ) = d ∗ lim ǫ ց E D ( h ǫ xh ǫ ) d = d ∗ E ( x ) d, d ∈ D . The prescription E ( x ) = lim ǫ ց E D ( h ǫ xh ǫ ) ( x ∈ M + ), therefore yields a well de-fined D -valued operator valued weight on M + . Since also E ( ) = lim ǫ ց E D ( h ǫ ) = E D ( h ) = , E gives rise to a conditional expectation of M onto D that we continueto write as E (see [38, Lemma IX.4.13 (iii)]).Now suppose that x t ր x in M + . Since E is order preserving, we clearly havethat sup t E ( x t ) = lim t E ( x t ) ≤ E ( x ). But for any d ∈ n ( D ) ν it follows from ONDITIONAL EXPECTATIONS AND GENERALIZED REPRESENTING MEASURES 29 equation (5) and the fact that ν h ( d ∗ · d ) is a positive normal functional, that ν ( d ∗ ( E ( x ) − lim t E ( x t )) d ) = lim t ν h ( d ∗ ( x − x t ) d ) = 0 . On again passing to the GNS representation for ( ν |D , D ) if need be, this ensures that h sup t E ( x t ) ξ, ξ i = h E ( x ) ξ, ξ i for all ξ in a dense subspace of H ν |D . By continuitywe then in fact have that h sup t E ( x t ) ξ, ξ i = h E ( x ) ξ, ξ i for all ξ ∈ H ν |D . That canof course only be if sup t E ( x t ) = E ( x ), which proves the normality of E .It remains to show that ν ( E ( x )) = ν h ( x ) for all x ∈ M + . To see this recall thatfor x ∈ M + the net ( E D ( h ǫ xh ǫ )) increases to E ( x ). The normality of ν thereforeensures that ν ( E ( x )) = sup ǫ> ν ( E D ( h ǫ xh ǫ )) = sup ǫ> ν ( h ǫ xh ǫ ) = ν h ( x ) . Thus by the statement of the main correspondence in [30], ν ◦ E commutes with ν .We now prove the bijectivity of the correspondences. The map defined above‘from (b) to (c)’ is one-to-one by the bijectivity in the Pedersen-Takesaki Radon-Nikodym theorem [30].Let ρ = ν ◦ E , and z = supp ( E ) = supp ( ρ ). Define E z and E as in the proofof (b) above. Then as we saw in the proof of (a) above, E ( zxz ) z defines a normalconditional expectation from z M z onto D z with ρ ( E ( zxz ) z ) = ρ ( zxz ), for x ∈ M .By the uniqueness assertion where we defined E z above we have E z ( zxz ) = E z ( zxz ) z = E ( zxz ) z = E ( zxz ) z = E ( x ) z, x ∈ M . Since
D ∼ = D z via the ∗ -isomorphism of right multiplication by z we see from thelast formula that E may be retrieved from E z and hence from ρ . (See also [20,Lemma 4.8].)Thus to complete the proof that the correspondences defined between (a), (b),(c) are bijective it suffices to show that given h as in (b), if E h is the expectationobtained in the proof of (b ′ ) ⇒ (a), we then have d ( ν ◦ E h ) dν = h . But we saw in theproof of (b ′ ) ⇒ (a) that ν h = ν ◦ E h . Hence d ( ν ◦ E h ) dν = h as required. (cid:3) Remark 4.5.
1) Let D be a von Neumann subalgebra of M , and let ν be a faithfulnormal weight on M which is semifinite on D and satisfies σ νt ( D ) = D for real t . Let E D be the ν -preserving conditional expectation of M onto D (see the definition atthe start of Section 3). Let E be a faithful normal conditional expectation onto D which commutes with ( σ νt ). Then E is the weight-preserving conditional expectation associated with the faithful normal semifinite weight ρ = ν ◦ E . This weight satisfies σ ρt ( D ) = D for real t since by [20, Theorem 4.7 (1)] we have σ ν ◦ Et = σ νt . So we arein the setting of the theorem but with ω = ν ◦ E playing the role of ν , and then theassociated ω -preserving conditional expectation E D (guaranteed by [38, TheoremIX.4.2] as at the start of Section 3), is exactly E .2) An explicit example showing the necessity of ν being semifinite on D is thecase that D is the copy of L ∞ ([0 , B ( L ([0 , ν thetrace on B ( L ([0 , + . Here ν is not semifinite on D ∼ = L ∞ ([0 , B ( L ([0 , + and σ νt ( D ) = D .3) Let ν be a faithful normal semifinite trace on M and let h be a positiveoperator affiliated to M ν . Following on from the remark after Proposition 4.2 weshow that we may in the specialized setting of Theorem 4.4, similarly rewrite the limit E ( x ) = lim ǫ ց E D ( h ǫ xh ǫ ) as the claim that E D ( h xh ) = E ( x ), where by h xh we mean the operator b = ( h x )( h x ) ∗ ∈ L ( M ).Recall that the theorem ensures that ν ( d ) = ν h ( d ) for any d ∈ D + . Thus n ( D ) ν = n ( D ) ν h . So for any d ∈ n ( D ) ν and with x as before, we will have that d ∗ xd ∈ p ν h . The results of the remark following Proposition 4.2, therefore ensuresthat h d ∗ x ∈ L ( M , ν ). When combined with the fact that E D is ν -preserving,the strong commutation of h with D further ensures that ν ( d ∗ E D ( b ) d ) = ν ( d ∗ bd ) = ν (( h d ∗ x )( h d ∗ x ) ∗ ) = ν (( h d ∗ x ) ∗ ( h d ∗ x )) , while by similar considerations ν ( d ∗ E ( x ) d ) = lim ǫ ց ν ( E D ( d ∗ h ǫ xh ǫ d )) = lim ǫ ց ν ( h ǫ d ∗ xdh ǫ ) . In equation (4) we may by a tiny variant of that argument replace x with d ∗ x ,upon which it is then clear that the quantity in the last equation equals ν (( x d ) h ( x d ) ∗ ) = ν ( | h d ∗ x | ) . Hence ν ( d ∗ E ( x ) d ) = ν ( d ∗ E D ( b ) d ). We know from Remark 2.5 that E D ( b ) ∈ L ( M , ν ) + . Since ν ( E ( x )) = ν h ( x ) < ∞ , we also have that E ( x ) ∈ L ( M , ν ).It follows that E D ( b ) = E ( x ) = lim ǫ ց E D ( h ǫ xh ǫ ) as desired.We now turn to the special case that semifinite D is contained in the centralizer M ν . This is equivalent by [38, Theorem VIII.2.6] to the modular automorphismgroup ( σ νt ) restricting to the identity map on D for each real t . In this case thereagain exists the ν -preserving conditional expectation E D : M → D by [38, TheoremIX.4.2].
Corollary 4.6.
Let D be a von Neumann subalgebra of a von Neumann algebra M , and let ν be a faithful normal weight on M , which is semifinite on D and isalso D -central (that is, D ⊂ M ω ; or equivalently, by e.g. Lemma , ν is tracialon D with in addition ν ( xd ) = ν ( dx ) for any x ∈ m ( M ) ν and any d ∈ m ( D ) ν ).There is a bijective correspondence between the following objects: (a) Normal conditional expectations E onto D which commute with ( σ νt ) ( t ∈ R ), (b) densities hη M + ν which commute with all elements of D , and which satisfy ν h = ν on D + , (b ′ ) densities hη M + ν which commute with all elements of D , and which satisfy E D ( h ) = 1 (where the action on h is by the normal extension of E D to theextended positive part of M in e.g. [16, Proposition 3.1] ), (b ′′ ) densities k ∈ η L ( M ) which commute strongly with both k ν and D , andwhich satisfy E D ( k ) = E D ( k ν ) (where the action on k ν and k is by thenormal extension of E D to the extended positive part of M ⋊ ν R ), (c) extensions of ν |D to normal semifinite weights ρ on all of M which commutewith ν , and are D -central in the sense that ρ ( xd ) = ρ ( dx ) for any x ∈ m ( M ) ρ and any d ∈ m ( D ) ν .With respect to the correspondences above we also have the formulae ρ = ν ◦ E = ν h , h = dρdν = k.k − ν , ONDITIONAL EXPECTATIONS AND GENERALIZED REPRESENTING MEASURES 31 and E ( x ) = lim ǫ ց E D ( h ǫ xh ǫ ) , x ∈ M + , where h ǫ = h ( + ǫh ) − and with the last limit being a ( σ -strong*) limit of anincreasing net in M . Indeed the relation between E and h is uniquely determinedby the equation ν ( E ( exe )) = ν h ( exe ) for x ∈ M + and projections e ∈ D with ν ( e ) < ∞ .Proof. We saw earlier that
D ⊂ M ω is equivalent to the requirement that k ν commutes strongly with D . Under this a priori assumption the equivalence of (a),(b), (b ′ ) and (b ′′ ) now follow directly from Theorem 4.4. The only equivalence thatbears some investigation is (c).(a) ⇒ (c) As in the proof in Theorem 4.4 if E : M → D is a normal conditionalexpectation onto D which commutes with ( σ νt ) then ρ = ν ◦ E is a normal semifiniteweight on M extending ν |D . Again ρ ◦ σ νt = ρ and ρ commutes with ν in the senseof [30] (see also [38, Corollary VIII.3.6], although here all weights are faithful).Since ρ and ν agree on D , it is clear that m ( D ) + ν = m ( D ) + ρ , and hence by linearitythat m ( D ) ν = m ( D ) ρ . For d ∈ m ( D ) ρ , we then clearly have that d m ρ ⊂ m ρ and m ρ d ⊂ m ρ . Now let a ∈ m + ρ be given. Then E ( a ) ∈ m + ν since by the Kadison-Schwarz inequality ν ( E ( a ) ∗ E ( a )) ≤ ν ( E ( a ∗ a ) = ρ ( a ∗ a ) < ∞ . It is therefore clear that ρ ( ad ) = ν ( E ( ad )) = ν ( E ( a ) d ) = ν ( dE ( a )) = ν ( E ( da )) = ρ ( da )and hence that m ( D ) ρ is in the centralizer of ρ .(c) ⇒ (a) Take ρ commuting with ν as in (c). Let E be the ρ -preserving condi-tional expectation in Theorem 3.10 (1). We have ρ ( E ( σ νt ( x ))) = ρ ( σ νt ( x )) = ρ ( x ) if x ∈ M + . So E ◦ σ νt is a ρ -preserving conditional expectation, hence E ◦ σ νt = E bythe uniqueness in Theorem 3.10. The fact that E ◦ σ νt is the identity on D followsbecause it is normal and is the identity on m ( D ).For the final claim, it obviously follows from equation (5) that ν ( E ( exe )) = ν h ( exe ), for x ∈ M + and projections e ∈ D with ν ( e ) < ∞ .Conversely, if this equation holds then ν h and ν ◦ E are two normal semifiniteweights on M + that agree on e M e for each projection e ∈ D with ν ( e ) < ∞ . Notethat such e are in the centralizers of these weights (for ρ = ν h it follows from e.g.[30, Proposition 4.6]that σ ρt ( e ) = h it eh − it = e ). Since such e form an increasing net with limit 1, wemay deduce from [30, Proposition 4.2] (or [38, Lemma VIII.2.7]) that ν h = ν ◦ E .However we have seen above that ν h = ν ◦ E completely determines h in terms of E and vice versa. (cid:3) Remark 4.7.
1) The following parallels Remark 1) after Theorem 4.4. Let D bea von Neumann subalgebra of M , and ν a faithful normal semifinite weight on M ,which is semifinite on D and D -central in the sense of the corollary. Let E be afaithful normal conditional expectation onto D which commutes with ( σ νt ). As inRemark 1) after Theorem 4.4, E is the weight-preserving conditional expectation associated with the faithful normal semifinite weight ρ = ν ◦ E . We check that thisweight satisfies the appropriate centralizer condition. By [20, Theorem 4.7 (1)] we have σ ν ◦ Et = σ νt on D for all t , so that σ ν ◦ Et ( d ) = d for all d ∈ m ( D ) ν , and hence ν ( E ( xd )) = ν ( E ( x ) d ) = ν ( dE ( x )) = ν ( E ( dx )) , x ∈ m ( M ) ν , d ∈ m ( D ) ν . So we are in the setting of the corollary but with ρ = ν ◦ E playing the role of ν .Then the associated ρ -preserving conditional expectation E D (guaranteed by [38,Theorem IX.4.2] as at the start of Section 3), is exactly E .2) For a nonzero projection p ∈ D with ν ( p ) < ∞ , a multiple of ν is a faithfulnormal state ν p on p M p . As in Theorem 3.9 and its proof we may view the con-ditional expectation E as being approximated by an increasing net of conditionalexpectations E p ( x ) = E ( pxp ) = pE ( x ) p on the von Neumann algebras p M p . Pur-suing this line suggests that there is no doubt a variant of (b) (or (b ′ )) in the lastresult phrased in terms of the σ -finite von Neumann algebras p M p , for each such p , and ‘local densities’ ehe .3) Let D be a von Neumann subalgebra of M , and ν a faithful normal tracialweight on M , which is semifinite on D . It follows that we then have a bijectivecorrespondence between (a) normal conditional expectations E onto D , (c) exten-sions of ν |D to normal semifinite weights ρ on all of M which commute with ν inthe sense in the Introduction, and items (b) and (b ′ ) in the corollary as stated. Wemay also use the remark after Proposition 4.2 and Remark 3 after Theorem 4.4 towrite ν h and E in terms of h without using limits, as described there.The following is a special case of Corollary 4.6 and 3) of the last remark. Theexplicit formulation as a result characterizing D -valued normal conditional expec-tations will however prove to be extremely useful. Corollary 4.8.
Let D be a von Neumann subalgebra of M , and τ a faithful nor-mal tracial state on M . There is a bijective correspondence between the followingobjects: (a) Normal conditional expectations onto D , (b) densities h ∈ L ( M ) + which commute with D and satisfy E D ( h ) = , (c) extensions of τ |D to normal states ρ on all of M which have D in theircentralizer.The normal conditional expectation E in (a) and h in (b) are related by the formula E ( x ) = E D ( h xh ) , x ∈ M + . It also may be described by the relation τ ( E ( x )) = τ ( hx ) for all x ∈ M e . Also h in (b) is the Radon-Nikodym derivative dρdτ for ρ as in (c) .Proof. Since τ is a tracial state the commuting condition in Corollary 4.6 (a) and(c) is automatic, and m ( M ) ν = M and n ( D ) ν = D . With these in mind, the restis an exercise. (cid:3) The noncommutative Hoffman-Rossi theorem–preliminaryconsiderations
Proposition 5.1.
Consider the inclusions
D ⊂ A ⊂ M , where M is a von Neu-mann algebra with normal state ν which is faithful on von Neumann subalgebra D ,with D in the centralizer of ν , and A is a D -submodule of M . Let Φ :
A → D be amap with
Φ(1) = 1 , which is a D -bimodule map (if Φ is a homomorphism then thisis equivalent to Φ being the identity map on D ). ONDITIONAL EXPECTATIONS AND GENERALIZED REPRESENTING MEASURES 33 (1)
Assume that ν ◦ Φ extends to a normal state ρ on M with D in its centralizer(that is, ρ ( xd ) = ρ ( dx ) for every d ∈ D , x ∈ M ). Then there is a unique ρ -preserving normal conditional expectation M → D , and this expectationextends Φ . (2) Assume that ν is faithful and ν ◦ Φ extends to a normal state on M . Then ν ◦ Φ extends to a normal state ρ on M with D in its centralizer, andthere is a normal ρ -preserving conditional expectation M → D extending Φ . Moreover such a normal ρ -preserving conditional expectation is unique.Proof. (1) By Corollary 3.14 there is a ρ -preserving normal conditional expectation E : M → D . As usual the fact that it is ρ -preserving makes it unique, and by thesame argument E extends Φ.(2) Suppose that ν ◦ Φ extends to a normal state ω on M . Let K ω and ρ ∈ K ω be as in Lemma 3.4, so that ρ ( xd ) = ρ ( dx ) for every d ∈ D , x ∈ M . We have asequence ψ n = P k t k u ∗ k ωu k of convex combinations approximating ρ . For a ∈ A we have ψ n ( a ) = X k t k ω ( u k au ∗ k ) = X k t k ν (Φ( u k au ∗ k )) = X k t k ν (Φ( a )) = ν (Φ( a )) . Thus ρ ( a ) = ν (Φ( a )), so that ρ is a normal state extending ν ◦ Φ. Now apply (1)to obtain a normal ρ -preserving conditional expectation M → D extending Φ. (cid:3)
Remark 5.2.
Indeed we may take ρ in (2) to be ω ◦ E D ′ ∩M where E D ′ ∩M is theunique ω -preserving conditional expectation onto D ′ ∩ M . Corollary 5.3.
Consider the inclusions
D ⊂ A ⊂ M , where M is a von Neumannalgebra with faithful normal tracial state τ , A is a subalgebra of M , and D is a vonNeumann subalgebra containing the unit of M . Let Φ :
A → D be a unital map,which is a D -bimodule map (or equivalently is the identity map on D ). (1) Assume that τ ◦ Φ extends to a normal state ρ on M whose density withrespect to τ commutes with D . Then there is a unique ρ -preserving normalconditional expectation M → D extending Φ . (2) Assume that τ ◦ Φ extends to a normal state on M . Then there is a normalconditional expectation M → D extending Φ . To illustrate the efficacy of the technology developed above, we show how itmay be used to recover a commutative generalization of the classical Hoffman-Rossitheorem conditioned to the present context. The proof strategy of this corollary willin the next section serve as the template for the proof of the general noncommutativeHoffman-Rossi theorem.
Corollary 5.4.
Consider the inclusions
D ⊂ A ⊂ M , where M is a commuta-tive von Neumann algebra, A is a weak* closed subalgebra of M , and D is a vonNeumann subalgebra containing the unit of M . Let σ be a faithful normal semifi-nite weight on D . Let Φ :
A → D be a weak* continuous D -character. Then σ ◦ Φ extends to a normal semifinite weight ρ on M , such that there is a unique ρ -preserving normal conditional expectation M → D extending Φ . Indeed for anynormal semifinite trace ρ extending σ ◦ Φ , there is a unique ρ -preserving normalconditional expectation M → D extending Φ .Proof. First assume that D possesses a faithful normal state σ . Let τ be a faithfulnormal semifinite trace on M (integration with respect to µ if M = L ∞ ( µ )). By Banach space duality σ ◦ Φ extends to a weak* continuous functional on M , sothere exists r = ab ∗ ∈ L ( M ) , with a, b ∈ L ( M ), such that τ ( rx ) = σ (Φ( x )) forall x ∈ A . In particular τ ( rd ) = τ ( dab ∗ ) = σ ( d ) for d ∈ D , and τ ( rj ) = τ ( jab ∗ ) = 0for j ∈ J = Ker(Φ), so that b ⊥ F where F = [ Ja ] . We apply an idea that appearsto go back to Sarason and others, as discussed in [4]. Let c be the projection P E ( b )of b onto E = [ Aa ] . Then c ⊥ F since F ⊂ E . Note that τ ( jcc ∗ ) = τ ( jcb ∗ ) = 0 , j ∈ J, since jc ∈ J [ Aa ] ⊂ F ⊂ E . For d ∈ D we have σ ( d ∗ d ) = |h d ∗ da, b i| = |h d ∗ da, c i| = |h a, d ∗ dc i| ≤ k a k k d ∗ dc k . Thus σ ( d ∗ d ) ≤ k a k τ (( d ∗ d ) cc ∗ ) = k a k τ (( d ∗ d ) E D ( cc ∗ )) , d ∈ D . Thus ρ ( x ) = k c k τ ( xcc ∗ ) is a normal state on M restricting to a faithful normaltracial state ω on D , and ρ annihilates J .By e.g. Theorem 3.5 (or by Corollary 3.2 in [9]), there is a unique ρ -preservingnormal conditional expectation Ψ : M → D . Fix d ∈ D . Now ω (Ψ( a ) d ) = ρ ( ad ).This equals ω (Φ( a ) d ) for all a in J and in D , since both are zero on J and equal ω ( ad ) for a ∈ D . Since A = J ⊕ D we deduce that Ψ |A = Φ.Now that we know that there exists a normal conditional expectation Ψ : M → D extending Φ, redefining ρ = σ ◦ Ψ gives a normal state on M extending σ ◦ Φ. Also Ψis ρ -preserving. The last assertion follows from Theorem 3.5: for any normal state ρ on M extending σ ◦ Φ, there exists a ρ -preserving normal conditional expectationΨ : M → D . By the argument at the end of the last paragraph Ψ |A = Φ.In the general case suppose that ( p i ) are projections in D adding to 1 with σ ( p i ) < ∞ , as in the proof of Theorem 3.9. Let D i = p i D , which has a faithfulnormal state σ i = σ ( p i ) σ ( p i · ). We have Φ( p i ) = p i and Φ( p i A ) = D i . So by thefirst part there exists a unique σ i -preserving normal conditional expectation E i : M i = M p i → D i extending Φ( p i · ). Define Ψ : M → D by Ψ(( p i x )) = ( E i ( p i x )).As in the proof of Theorem 3.9 this is a normal conditional expectation M → D .Let ρ = σ ◦ Ψ on M + , a normal weight extending σ . Note that ρ is a semifiniteweight since xe t → x for all x ∈ M , and e t and hence xe t are in n ρ . We have σ (Ψ(( p i x ))) = σ ( X i E i ( p i x )) = X i σ ( E i ( p i x )) = X i ρ ( p i x ) = ρ ( x ) , x ∈ M + . If ρ is any normal weight on M extending σ ◦ Ψ, then e.g. Theorem 3.10 givesa unique normal conditional expectation Ψ from M onto D which is ρ -preserving.Fix d ∈ D , and p a projection in D of finite trace. Now ω (Ψ( a ) dp ) = ρ ( adp ). Thisequals ω (Φ( a ) dp ) for all a in J and in D , since both are zero on J and equal ω ( adp )for a ∈ D . Since A = J ⊕ D we deduce that Ψ( a ) p = Φ( a ) p . Taking a weak* limitalong net of such projections p increasing to 1, we obtain Ψ( a ) = Φ( a ).The uniqueness is as in the proof of Theorem 3.9: suppose that x ∈ M + and σ ( d ∗ Ψ( x ) d ) = σ ( d ∗ E ( x ) d ) for all d ∈ D . Choosing d a positive element in D i shows that σ ((Ψ( x ) − E ( x )) p i d ) = 0. Since such squares span D i , we see thatΨ( x ) p i = E ( x ) p i for all i so that Ψ( x ) = E ( x ). (cid:3) The last proof shows that for any normal state ω on D , ω ◦ Φ extends to a normalstate ρ on M , and there is a ρ -preserving normal conditional expectation M → D extending Φ. The latter is unique if ω is faithful on D . ONDITIONAL EXPECTATIONS AND GENERALIZED REPRESENTING MEASURES 35
One problem with the proof of the last result in the noncommutative case is thatnot every faithful normal tracial state on D extends to a faithful normal tracial stateon M (consider the diagonal algebra in M ). Indeed it need not even extend to afaithful normal state on M with D in the centralizer (see the example above Lemma3.4). We will overcome this difficulty in the form of Theorem 6.3 and Theorem 6.5.In the case that M is semifinite and noncommutative but D is finite the resultcannot be true without extra conditions (consider M = B ( L ([0 , A = D ,the latter being the copy of L ∞ ([0 , M as in in Example 3.3). Corollary 5.5.
Consider the inclusions
D ⊂ A ⊂ M , where M is a von Neumannalgebra, A is a subalgebra of M , and D is a von Neumann subalgebra containingthe unit of M . Let ω be a faithful normal tracial state on D . Let Φ :
A → D bea unital D -bimodule map (or equivalently is the identity map on D ). Assume alsothat A + A ∗ is weak* dense in M . Then there is a normal conditional expectation Ψ :
M → D extending Φ if and only if ω ◦ Φ extends to a normal state ψ on M .In this case Ψ is ψ -preserving.Proof. One direction is obvious. For the other, if ω ◦ Φ extends to a normal state ψ on M , then we have ω (Φ( ad )) = ω (Φ( a ) d ) = ω ( d Φ( a )) = ω (Φ( da )) for d ∈ D , a ∈A . Thus ψ ( xd ) = ψ ( dx ) for x ∈ A , hence for x ∈ M . Now appeal to Corollary3.16: there is a normal ψ -preserving conditional expectation Ψ : M → D . For anynormal conditional expectation Ψ :
M → D extending Φ we have ψ (Ψ( a )) = ψ (Φ( a )) = ω (Φ( a )) = ψ ( a ) , a ∈ A , so that Ψ is ψ -preserving. (cid:3) Remark 5.6.
The condition that A + A ∗ is weak* dense may not simply be droppedin the last result. The example mentioned before the corollary shows this (with D = A = L ∞ ( T )).In several results in the present paper, a condition on M (and perhaps also on D ) may be replaced by a condition on an intermediate subalgebra A between M and D , if A + A ∗ is weak* dense in M (a condition which is crucial for example forArveson’s subdiagonal algebras [1], or for much of the abstract analytic functiontheory from the 1960’s and 70’s [15, 7]). This follows e.g. by tricks seen in the lastproof.6. The noncommutative Hoffman-Rossi theorem for sigma-finitealgebras
Lemma 6.1.
Let ( K, µ ) be a measure space and g ∈ L ( K, µ ) + . Then ( Z K f dµ ) ≤ Z K f g dµ, f ∈ L ∞ ( K, µ ) + , if and only if g is µ -a.e. nonzero and R K g − dµ ≤ .Proof. Suppose that R K g − dµ ≤
1. Then by Cauchy-Schwarz,( Z K f g g − dµ ) ≤ ( Z K f g dµ ) ( Z K g − dµ ) ≤ Z K f g dµ. Conversely, suppose that the inequality holds. Letting E = { x ∈ K : g ( x ) = 0 } ,and f = χ E we see that µ ( E ) = 0. So we may assume that g is never 0, so that g −
16 DAVID P. BLECHER AND LOUIS E. LABUSCHAGNE is well defined and finite. Letting E n = { x ∈ K : g ( x ) − ≤ n } , and f n = g − χ E n ,we have ( Z E n g − dµ ) ≤ Z E n ( g − ) g dµ = Z E n g − dµ. Thus R E n g − dµ ≤
1. Letting n → ∞ we have R K g − dµ ≤ (cid:3) Remark 6.2.
We thank Vaughn Climenhaga for discussions around the last result.The following is the generalized Hoffman-Rossi theorem for subalgebras of ‘finite’von Neumann algebras. See [4] for the case when D is atomic. Theorem 6.3.
Consider the inclusions
D ⊂ A ⊂ M , where M is a von Neumannalgebra with faithful normal tracial state τ , A is a weak* closed subalgebra of M ,and D is a von Neumann subalgebra containing the unit of M . Let Φ :
A → D be aweak* continuous D -character. Then there exists a normal conditional expectation Ψ :
M → D extending Φ . Indeed τ ◦ Φ extends to a normal state ρ on M suchthat there exists a unique ρ -preserving normal conditional expectation Ψ :
M → D extending Φ .Proof. Proceeding as in the proof of Corollary 5.4 we obtain r = ab ∗ ∈ L ( M ) and c ∈ [ A a ] , such that x τ ( xcc ∗ ) annihilates J , and τ ( d ∗ d ) ≤ k a k τ (( d ∗ d ) cc ∗ ) = k a k τ (( d ∗ d ) E D ( cc ∗ )) , d ∈ D . Letting g = E D ( cc ∗ ) ∈ L ( D ) + and f = d ∗ d we have τ ( f ) ≤ k a k τ ( f g ) , f ∈ D + . Let M be the von Neumann algebra generated by g (see e.g. [25, p. 349]), acommutative subalgebra of D . For f ∈ ( M ) + we have τ ( f ) ≤ k a k τ ( f g ). Thisimplies that g − ∈ L ( M ) + by Lemma 6.1. Hence as a closed densely definedpositive operator affiliated with D , we have g − ∈ L ( D ) + .Let c = g − c ∈ L ( M ), and let h = g − cc ∗ g − (the so called ‘strong product’[38, p. 174]). Let ˆ E be Haagerup’s extension of E D to the extended positive partˆ M + . Since ˆ E ( cc ∗ ) = E ( cc ∗ ) = g ∈ L ( M ), by [16, Proposition 3.3] we haveˆ E ( h ) = g − · g · g − , where the latter is the strong product. But this is 1. Thusˆ τ ( h ) = τ ( g • g − ) = ˆ τ ( g − · g · g − ) = 1 , by Propositions 2.6 (used twice) and 3.2 in [16]. Thus h ∈ L ( M ) with k h k = 1 (seeRemark 1.2 in [27]), and c ∈ L ( M ) with k c k = 1. Then ω = τ ( · h ) is a normalstate on M which extends τ |D . Indeed τ ( dh ) = τ ( E D ( dh )) = τ ( dE D ( h )) = τ ( d ) for d ∈ D .We now check that ω annihilates J . Let d n be g − times the spectral projectionof [0 , n ] for g − . So d n ր g − , and this convergence is also in L -norm (as onecan see by spectral theory). Then, similarly to the above, τ (( g − − d n ) cc ∗ ( g − − d n )) = τ ( cc ∗ • ( g − − d n )) = τ ( g • ( g − − d n )) = τ ((1 − g d n ) ) . This equals τ ( p n ) , where p n is the spectral projection of [ n, ∞ ) for g − . Clearly1 − p n ր c ∗ d n → c ∗ g − and d n c → g − c in 2-norm.It follows that g − c ∈ E = [ Aa ] . We have jg − c ∈ F = [ Ja ] ⊂ E . Thus τ ( jg − cc ∗ ) = h jg − c, c i = 0 , ONDITIONAL EXPECTATIONS AND GENERALIZED REPRESENTING MEASURES 37 since b ⊥ F and c ⊥ F as before. Hence0 = lim n τ ( jg − cc ∗ d n ) = τ ( jg − cc ∗ g − ) = τ ( jh ) . So ω annihilates J .Since ω extends τ | D , we see that ω extends τ ◦ Φ. Next appeal to Corollary 5.3 (2)to obtain a normal conditional expectation Ψ :
M → D extending Φ. If ρ = τ ◦ Ψthen ρ extends τ ◦ Φ, and Ψ is ρ -preserving. The uniqueness is as before. (cid:3) Remark 6.4.
If Φ :
A → D is a homomorphism and projection onto a C ∗ -subalgebra D of A , and if Φ is faithful on A ∩ A ∗ , then it follows by an argumentof Arveson [1] that D = A ∩ A ∗ : Certainly D ⊂ A ∩ A ∗ . If x ∈ ( A ∩ A ∗ ) sa thenΦ(( x − Φ( x )) ) = (Φ( x − Φ( x ))) = 0. So ( x − Φ( x )) = x − Φ( x ) = 0, hence x = Φ( x ) ∈ D . Thus D = A ∩ A ∗ . This holds in particular if Φ is preserved by afaithful state.The following is the generalized Hoffman-Rossi theorem for certain subalgebrasof σ -finite von Neumann algebras–in this case D is still finite (that is, has a faithfulnormal tracial state τ D ), but M need not be finite. Theorem 6.5.
Consider inclusions
D ⊂ A ⊂ M , where M is a von Neumannalgebra with faithful normal state ω with D in its centralizer, A is a weak* closedsubalgebra of M , and D is a von Neumann subalgebra containing the unit of M .Let Φ :
A → D be a weak* continuous D -character. Then there exists a normalconditional expectation Ψ :
M → D extending Φ . Indeed ω ◦ Φ extends to a nor-mal state ρ on M such that there exists a unique ρ -preserving normal conditionalexpectation Ψ :
M → D extending Φ .Proof. Note that the fact that
D ⊂ M ω , ensures that D is the image of an ω -preserving faithful normal conditional expectation E D : M → D . We then knowfrom Remark 2.5 that D ⋊ ω R ⊆ M ⋊ ω R , whence ^ D ⋊ ω R ⊆ ^ M ⋊ ω R . Thus foreach p > L p ( D ) = { a ∈ ^ D ⋊ ω R : θ s ( a ) = e − s/p a for all s }⊆ { a ∈ ^ M ⋊ ω R : θ s ( a ) = e − s/p a for all s } = L p ( M ) . It is clear from Remark 2.4 (and perhaps the fact in Remark 2.5 that E D ◦ σ ωt = σ ωt ◦ E D ) that the dual action θ s corresponding to M agrees with the one correspondingto D . However the fact that D ⊂ M ω , also ensures that ω |D is a trace on D .So by for example pages 62-63 of [39] or the introductory discussion in Section2 of [27] (or Theorem 6.74 in [17] and Section 1 in [22]), D ⋊ ω R , the density k ω = d g ω |D dτ (which we observed in Remark 2.5 equals the density of the dual weight˜ ω on M ⋊ ω R ), and L p ( D ) will respectively up to Fourier transform canonicallycorrespond to D⊗ L ∞ ( R ), ⊗ exp and { a ⊗ (exp) /p : a ∈ L p ( D , ω ) } , where L p ( D , ω )denotes the tracial L p -space constructed using the trace ω |D . The density k ω ∈ L ( D ) then clearly commutes with D (see also Proposition 4.2 for another way tosee this).We momentarily follow the argument for Corollary 5.4: By Banach space duality ω ◦ Φ extends to a weak* continuous functional on M , so that there exists r = ab ∗ ∈ L ( M ) , with a, b ∈ L ( M ), such that tr ( rx ) = ω (Φ( x )) for all x ∈ A . In particular tr ( rd ) = tr ( dab ∗ ) = ω ( d ) , tr ( rj ) = tr ( jab ∗ ) = 0 , d ∈ D , j ∈ J = Ker(Φ) , so that b ⊥ F where F = [ Ja ] . We apply an idea that appears to go back toSarason and others, as discussed in [4]. Let c ∈ L ( M ) be the projection P E ( b ) of b onto E = [ A a ] . Then c ⊥ F since F ⊂ E . Note that tr ( jcc ∗ ) = tr ( jcb ∗ ) = 0 , j ∈ J, since jc ∈ J [ A a ] ⊂ F ⊂ E . For d ∈ D we then have ω ( d ∗ d ) = tr ( rd ∗ d ) = tr ( b ∗ d ∗ da ) = |h d ∗ da, b i| = |h d ∗ da, c i| = |h a, d ∗ dc i| , which is dominated by k a k k d ∗ dc k . Thus for d ∈ D we have ω ( d ∗ d ) ≤ k a k k d ∗ dc k = k a k tr (( d ∗ d ) cc ∗ ) = k a k tr (( d ∗ d ) E D ( cc ∗ )) . Letting g = E D ( cc ∗ ) ∈ L ( D ) + and f = d ∗ d we have ω ( f ) ≤ k a k tr ( f g ) , f ∈ D + . For the sake of clarity we will in the rest of this proof dispense with convention,and distinguish between D and the copy thereof, namely π ( D ), inside the crossedproduct. Similarly ˜ D may be identified with a subset of ^ D ⋊ ω R , which we will alsowrite as π ( ˜ D ). From the above discussion it is clear that up to Fourier transform,the action of π is to map each d ∈ D onto d ⊗
1. The above inequality should thenproperly be written as ω ( f ) ≤ k a k tr ( π ( f ) g ) for all f ∈ D + . We now reformulatethis inequality in terms of the L p ( D , ω ) spaces. By the discussion at the start ofthis proof, g is up to Fourier transform of the form g ⊗ exp = ( g ⊗ . ( ⊗ exp) fora unique g ∈ L ( D , ω ). So on taking the inverse transform, it then follows that g = π ( g ) k ω ≡ ( g ⊗ ⊗ exp) for a unique g ∈ L ( D , ω ). Thus the precedinginequality may be written in the form ω ( f ) ≤ k a k tr ( π ( f ) π ( g ) k ω ) = k a k ω ( f g ) , f ∈ D + . The last ω here is the natural extension of this trace on D to ˜ D + , hence to L ( D , ω ) + .Let D be the von Neumann subalgebra of D generated by g (see e.g. [25,p. 349]), a commutative subalgebra of D . For f ∈ ( D ) + we have ω ( f ) ≤k a k ω ( f g ). By Lemma 6.1 this implies that g − ∈ L ( D , ω ) + ⊂ L ( D , ω ) + .Thus as a closed densely defined positive operator affiliated with D , we have g − ∈ L ( D , ω ) + . Writing e g for k ω π ( g − ), this then ensures that e g / = k / ω π ( g − ) ∈ L ( D ). (Here we used the definition of Haagerup’s L ( D ), and the fact that k ω commutes strongly with π ( D ).)Let d n be g − e n where e n is the spectral projection of g − for [0 , n ]. So e n ր d n ր g − as n → ∞ . Notice that for m ≥ n we may use the fact that e n e m = e n to see that tr ( | c ∗ π ( d n ) − c ∗ π ( d m ) | )= tr ( π ( d n ) cc ∗ π ( d n ) − π ( d m ) cc ∗ π ( d n ) − π ( d n ) cc ∗ π ( d m ) + π ( d m ) cc ∗ π ( d m ))= tr ( π ( d m ) cc ∗ π ( d m ) − π ( d n ) cc ∗ π ( d n )) = tr ( E D ( π ( d m ) cc ∗ π ( d m ) − π ( d n ) cc ∗ π ( d n )))= tr ( π ( d m ) gπ ( d m ) − π ( d n ) gπ ( d n )) = tr ( π ( e m ) k ω − π ( e n ) k ω ) = ω ( e m − e n ) . (In the second equality we used the fact that since e n e m = e n , we for example havethat tr ( π ( d m ) cc ∗ π ( d n )) = tr ( π ( d n ) π ( d m ) cc ∗ ) = tr ( π ( d n ) cc ∗ ) = tr ( π ( d n ) cc ∗ π ( d n )) . ONDITIONAL EXPECTATIONS AND GENERALIZED REPRESENTING MEASURES 39
In the second last equality we used the fact that k ω commutes with π ( D ), and that g = π ( g ) k ω .) The projections e n increase to and so converge σ -strongly to . Itnow clearly follows from the above sequence of equalities thatlim n,m →∞ tr ( | c ∗ π ( d n ) − c ∗ π ( d m ) | ) = lim n,m →∞ | ω ( e m − e n ) | = ω ( − ) = 0 . So ( c ∗ π ( d n )) must converge in L -norm to some h ∗ ∈ L ( M ), with ( π ( d n ) c ) obvi-ously converging to h . Now let h = h ∗ h . This element of L ( M ) in some sensecorresponds to π ( g − / ) cc ∗ π ( g − / ) ∈ L ( M ), and must by construction be thelimit in L -norm of ( π ( d n ) cc ∗ π ( d n )). To see this observe that k h ∗ h − d n cc ∗ d n k = k h ∗ h − h ∗ c ∗ d n + h ∗ c ∗ d n − d n cc ∗ d n k ≤ k h k k h − c ∗ d n k + k c ∗ d n k k h ∗ − d n cd n k . For each x ∈ D , ( e n xe n ) is σ -weakly convergent to x . So for such an x , we may use L p -duality to see that tr ( E D ( h ) π ( x )) = lim n →∞ tr ( E D ( π ( d n ) cc ∗ π ( d n )) π ( x )) = lim n →∞ tr ( π ( d n ) gπ ( d n ) π ( x ))= lim n →∞ tr ( π ( e n ) k ω π ( e n x )) = lim n →∞ tr ( k ω π ( e n xe n )) = tr ( k ω π ( x )) . But then clearly E D ( h ) = lim n →∞ E D ( π ( d n ) cc ∗ π ( d n )) = k ω . Thus tr ( h ) = tr ( E D ( h )) = tr ( k ω ) = ω ( ) = 1 , thereby ensuring that h ∈ L ( M ) with k h k = 1, and h ∈ L ( M ) with k h k = 1.Then ϑ = tr ( · h ) is a normal state on M . Since ϑ ( d ) = tr ( π ( d ) h ) = tr ( E D ( π ( d ) h )) = tr ( π ( d ) k ω ) = ω ( d ) , d ∈ D , we have that ϑ agrees with ω on D .We now check that ϑ annihilates J . With ( d n ) as before we clearly have that( π ( d n ) c ) ⊂ E = [ π ( A ) a ] , and hence that h ∈ E = [ π ( A ) a ] . But then π ( j ) h ∈ F = [ π ( J ) a ] ⊂ E . Thus tr ( π ( j ) h c ∗ ) = h π ( j ) h , c i = 0, since as before b ⊥ F and c ⊥ F . Hence0 = lim n tr ( π ( j ) h ( c ∗ π ( d n ))) = tr ( π ( j ) h h ∗ ) = tr ( π ( j ) h ) . So ϑ annihilates J as required.Since ϑ extends ω |D and annihilates J , it is a normal state on M agreeing with ω ◦ Φ on A = D + J . Next appeal to Proposition 5.1 (2) to obtain a normalconditional expectation Ψ : M → D extending Φ. If ρ = ω ◦ Ψ then ρ extends ω ◦ Φ, and Ψ is ρ -preserving. The uniqueness is as before. (cid:3) Again it is easy to see the converse. Namely, if Φ is the restriction to A ofa normal conditional expectation of M onto D , then for any normal state ω on D , ω ◦ Φ extends to a normal state ρ on M , and there is a ρ -preserving normalconditional expectation M → D extending Φ. The latter is unique if ω is faithfulon D . Representing measures and the Hoffman-Rossi theorem forgeneral von Neumann algebras
Our first result gives a sufficient condition under which the main result of theprevious section holds for a general von Neumann algebra M . This result alsogeneralizes Corollary 5.4 to a large class of noncommutative situations. Theorem 7.1.
Consider the inclusions
D ⊂ A ⊂ M , where M is a von Neumannalgebra with faithful normal semifinite weight ω , A is a weak* closed subalgebraof M , and D is a von Neumann subalgebra of M ω . Let Φ :
A → D be a weak*continuous D -character. Suppose that is the sum of a collection { e t } of mutuallyorthogonal ω -finite projections in D , which are central in D . Then there exists anormal conditional expectation Ψ :
M → D extending Φ . Indeed τ ◦ Φ extends toa normal semifinite weight ρ on M for which there exists a unique ρ -preservingnormal conditional expectation Ψ :
M → D extending Φ . Note that the existence of the collection of projections { e t } above, ensures thatthe restriction of ω to D is still semifinite and hence by [38, Theorem IX.4.2] thereexists a normal conditional expectation E D onto D preserving τ , since D ⊂ M ω . Proof.
By Lemma 2.1 ω is tracial on D . We may apply Theorem 6.5 to each ofthe compressions e t M e t . Since ω ( e t de t xe t ) = ω ( e t de t xe t ) for all x ∈ M , d ∈ D , weobtain a normal conditional expectation Ψ e : e M e → e D e extending Φ t = Φ | e t A e t .(Note that since Φ( e t ae t ) = e t Φ( a ) e t for all a ∈ A , Φ | e M e maps e t A e t onto e t D e t .)We define the map Ψ : M → D by Ψ( x ) = ⊕ t Ψ t ( e t xe t ). Each Ψ t is normal,and so the sum Ψ will also be. We first show that this map extends Ψ. To seethis note that Ψ clearly extends ⊕ t Φ t . It is an easy exercise to see that for any a ∈ A we have that ⊕ t Φ t ( a ) = ⊕ t e t Φ( a ) e t = Φ( ⊕ t e t ae t ). The D -centrality ofthe e t ’s ensure that if t = s , then Φ( e t ae s ) = e t Φ( a ) e s = e s e t Φ( a ) = 0. SoΦ( a ) = Φ(( ⊕ t e t ) a ( ⊕ s e s )) = ⊕ t Φ t ( a ) for any a ∈ A . Thus Ψ extends Φ.It remains to show that Ψ is a conditional expectation. To see this, note thatfor any x ∈ M we will have thatΨ(Ψ( x )) = ⊕ t Ψ t ( e t ( ⊕ r Ψ r ( e r xe r )) e t ) = ⊕ t Ψ t (Ψ t ( e t xe t )) = ⊕ t Ψ t ( e t xe t ) = Ψ( x ) . The proof of the final claim now proceeds as in Theorem 6.3. (cid:3)
We pass to proving a noncommutative Hoffman-Rossi theorem for general vonNeumann algebras. We shall use the Haagerup reduction theorem to extract thisresult from the one for the finite algebras. In so doing we shall first prove a resultfor σ -finite algebras (because of the relative simplicity of the reduction theorem inthis setting) before indicating how the proof technique may be adaped to yield atheorem for general von Neumann algebras. Though we eventually do get a theoremfor a more general class of von Neumann algebras, it is at the cost of normality ofthe conditional expectation extending the given D -character. Theorem 7.2.
Consider the inclusions
D ⊂ A ⊂ M , where M is a σ -finite vonNeumann algebra with faithful normal state ϕ , A is a weak* closed subalgebra of M , and D is a von Neumann subalgebra containing the unit of M . Suppose that σ ϕt ( A ) = A and σ ϕt ( D ) = D for each t ∈ R . Let Φ :
A → D be a weak* continuouscompletely bounded D -character which commutes with ( σ ϕt ) . Then there exists apossibly non-normal expectation Ψ from M onto D extending Φ . ONDITIONAL EXPECTATIONS AND GENERALIZED REPRESENTING MEASURES 41
Proof.
We will closely follow the notation of [22]. (The proof was originally writtendown in May 1978 by Haagerup and circulated as a hand-written copy. An outlineof the proof appeared in print in 2005 (see [42]). However it was not until theappearance of [22] that a full proof appeared in print.) Write R for M ⋊ Q D where Q D are the diadic rationals. We shall identify M with the *-isomorphic copy thereofinside M ⋊ Q D . It is known that R is the weak* closure of span { λ t x : x ∈ M , t ∈ Q D } . We shall write b A and b D for the weak* closures of span { λ t a : a ∈ A , t ∈ Q D } and span { λ t d : d ∈ D , t ∈ Q D } respectively. Note that span { λ t a : a ∈ M , t ∈ Q D } is in fact a dense *-algebra, since λ t a λ s a = λ t + s λ − s a λ s a = λ t − s σ ϕ − s ( a ) a . The prescription b Φ( λ t a ) = λ t Φ( a ) defines a map on a dense subalgebra of b A , whichby the technique of [22, Theorem 4.1] extends to a bounded normal map from b A to b D . Another way to see these facts is to note that the complete boundedness of Φallows us to extend it to a map Φ ⊗ Id on A⊗ B ( L ( Q D )) with preservation of norm.We may now without loss of generality assume that M is in standard form, andhence that the modular group is implemented. Consulting [41, Part I, Proposition2.8] now gives a very precise description of how M ⋊ Q D may be realised as asubspace of M ⊗ B ( L ( Q D )). What remains now is to check that b A similarly livesinside A ⊗ B ( L ( Q D )) ⊂ M ⊗ B ( L ( Q D )), that Φ ⊗ Id is still normal and that thedemand that Φ commutes with σ ϕt , ensures that b A is an invariant subspace of theaction of Φ ⊗ Id on
A ⊗ B ( L ( Q D )). The restriction of Φ ⊗ Id to this invariantsubspace is the map b Φ we seek. Note that by definition (and normality) b Φ will map b A onto b D . The map b Φ is moreover a homomorphism. The easiest way to see this isto note the earlier computations may be modified to show that for any a , a ∈ A and t, s ∈ Q D we have that b Φ( λ t a λ s a ) = b Φ( λ t − s σ ϕ − s ( a ) a ) = λ t − s Φ( σ ϕ − s ( a ) a )= λ t − s σ ϕ − s (Φ( a ))Φ( a ) = λ t Φ( a ) λ s Φ( a ) = b Φ( λ t a ) b Φ( λ s a ) . Thus it is a b D -character.There exists an increasing sequence ( R n ) of finite von Neumann subalgebras of R for which ∪R n is weak*-dense in R . Moreover with b ϕ denoting the dual weight(it is a state) on R , there exist normal conditional expectations E M : R → M and E n : R → R n commuting with the automorphism group σ b ϕt [22, Lemma 2.4]. We will write A n and D n for E n ( b A ) and E n ( b D ) respectively. We claim that A n ⊂ b A and D n ⊂ b D for each n . To see this note that a n as defined prior to [22, Lemma 2.4] belongs tothe von Neumann subalgebra generated by the λ t ’s and hence to b A . The fact that σ b ϕt is implemented by the unitary group t → λ t , ensures that σ b ϕt ( b A ) = b A for every t . Hence by [22, Equation(2.4)] σ ϕ n t ( a ) ∈ b A for every t . So the definition of E n onpage 2133 of [22] ensures that A n ⊂ b A . Similarly D n ⊂ b D . In fact on modifyingthe ideas in [22, Lemmata 2.6 & 2.7], one can show that the A n ’s and D n ’s areincreasing with ∪A n and ∪D n are respectively dense in b A and b D . To see the claimabout being increasing, note that the last 5 lines of page 2133 of [22] shows thatif x ∈ R n , then σ ϕ n t ( x ) = x for any t . This suffices to show that if say a ∈ A n it follows from the definition of E n +1 (again on p 2133 of [22]) that E n +1 ( a ) = a . Soclearly A n ⊂ A n +1 for every n . Similarly D n ⊂ D n +1 .The fact that Φ commutes with σ ϕt , ensures that b Φ commutes with σ b ϕt (thiscan also be verified with the technique of [22, Theorem 4.1]). A version of [22,Theorem 4.1(i)] similarly holds for b Φ. When these facts are considered alongside[22, Equation(2.4)], it follows that σ ϕ n t ◦ b Φ = b Φ ◦ σ ϕ n t for all t . Thus again by thedefinition of E n , it follows that b Φ ◦ E n = E n ◦ b Φ, and hence that b Φ maps A n into D n .In fact for the same reason it also follows that E n ◦ b Φ = b Φ ◦ E n . We will write Φ n for the induced map from A n to D n . The inclusions A n ⊂ b A and D n ⊂ b D ensurethat each Φ n is a D n -character. So by the result for finite von Neumann algebras,each Φ n extends to a normal conditional expectation Ψ n : A n → D n .Now consider the maps Θ n = Ψ n ◦ E n . Each Θ n is a completely boundednormal conditional expectation which agrees with b Φ on A n . Thus they all belongto CB ( R , b D ). Take a weak* limit point b Ψ of the Θ n ’s. The definition of Φ n combined with the fact that E n ◦ b Φ = b Φ ◦ E n , ensures that in fact each Θ n isa normal conditional expectation from R to D n which agrees with E n ◦ b Φ on allof b A , and hence with E n ◦ Φ on A . But for each a ∈ A , E n ◦ Φ( a ) → Φ( a ) σ -strongly as n → ∞ . (This follows from for example [22, Lemma 2.7].) It followsthat b Ψ is a possibly non-normal conditional expectation from R to a subspace of b D , which agrees with Φ on A . So b Ψ( M ) must include D = Φ( A ). By construction b Ψ also maps ∪ n R n onto ∪ n D n . Thus if indeed b Ψ was normal, we would be ableto conclude that b Ψ maps R onto b D . However for now the most we can say is that D ⊂ b Ψ( M ) ⊂ b D . The conditional expectation E M maps b D (and hence also b Ψ( M ))onto D . So setting Ψ = E M ◦ b Ψ |M will yield a possibly non-normal conditionalconditional expectation from M onto D which agrees with Φ on A . (cid:3) We now present a version of the above theorem for general von Neumann al-gebras. Here we require the canonical weight on M to be strictly semifinite on D . We point out that this restriction is a consequence of the criteria required forthe application of the reduction theorem and not of the proof technique (see [22,Remark 2.8]). If this restriction can be lifted from the reduction theorem, it cantherefore also be lifted here. We also note that for tracial weights the concepts ofsemifinite and strictly semifinite agree. Hence in the tracial setting this theorem isprobably as general as it can be. Theorem 7.3.
Consider the inclusions
D ⊂ A ⊂ M , where M is an arbitrary vonNeumann algebra, A is a weak* closed subalgebra of M , and D is a von Neumannsubalgebra containing the unit of M . Suppose that M is equipped with a fns weight ϕ for which the restriction to D is strictly semifinite, and that σ ϕt ( A ) = A and σ ϕt ( D ) = D for each t ∈ R . Let Φ :
A → D be a weak* continuous completelybounded D -character which commutes with ( σ ϕt ) . Then there exists a possibly non-normal expectation Ψ from M onto D extending Φ .Proof. By [38, Exercise VIII.2(1)], the restriction of ϕ to D is semifinite on thecentralizer D ϕ . The restriction of the weight ϕ to D ϕ is again a trace, and hencethere exists a net of projections ( e α ) ⊂ D ϕ all with finite weight, increasing to . Amongst other facts, the inclusion ( e α ) ⊂ D ϕ ensures that each e α is a fixedpoint of σ ϕt . When in standard form, the fact of e α being a fixed point of themodular group, translates to the claim that e α commutes with the modular operator ONDITIONAL EXPECTATIONS AND GENERALIZED REPRESENTING MEASURES 43 ∆. Each compression e α M e α is a σ -finite von Neumann algebra with respect tothe restriction of ϕ . The commutation of e α with ∆ further ensures that themodular operator for the pair ( e α M e α , ϕ | e α M e α ) may be identified with ∆ | e α H ,and that the modular group of this pair may be identified with the restriction of σ ϕ α t = e α σ ϕt ( · ) e α to e α M e α . (The reader may find these facts in the proof of[38, Theorem VIII.2.11].) All of this ensures that σ ϕ α t ( e α A e α ) = e α A e α and that σ ϕ α t ( e α D e α ) = e α D e α . Moreover Φ | e α A e α is easily seen to be a e α D e α characteron e α A e α . We may therefore apply the preceding theorem to the compression tosee that Φ | e α A e α extends to a possibly non-normal expectation Ψ α from e α M e α to e α D e α which agrees with Φ | e α A e α on e α A e α . Now let Θ α be given by Φ α ◦ C α where C α is the compression M → e α M e α . Each Θ α belongs to CB ( M , D ). LetΨ be a weak* limit. First notice that for any a ∈ A we will have that Θ α ( e α ae α ) =Φ( e α ae α ) = e α Φ( a ) e α → Φ( a ) (since { e α } converges σ -strong* to ). So Ψ agreeswith Φ on A . Since each Θ α is a compressed expectation, and since (as we have justseen) Θ α ( e α de α ) → d for every d ∈ D , it follows that Ψ is a possibly non-normalexpectation from M to D . (cid:3) Jensen measures
Consider the inclusions
D ⊂ A ⊂ C , where C is a unital C ∗ -algebra with sub-algebra A and C ∗ -subalgebra D , containing 1 C . Let Ψ be a “noncommutativerepresenting measure” of a D -character Φ, that is Ψ is a D -valued normal con-ditional expectation extending Φ. Let ω be a state of C preserved by Ψ (that is ω ◦ Ψ = ω ). We say that ω , or the pair (Ψ , ω ), is a noncommutive Jensen measure for Φ, if it satisfies the Jensen-like inequality ω (log( | Φ( a ) | )) ≤ ω (log | a | ) , a ∈ A . We say that ω is an noncommutive Arens-Singer measure for Φ, if it satisfies thisinequality for all invertible a ∈ A . These inequalities may be rewritten in termsof the ω - geometric mean ∆ ω ( a ) = exp( ω (log | a | )) (see the introduction to [9]). orexample noncommutive Arens-Singer measures satisfy∆ ω (Φ( a )) ≤ ∆ ω ( a ) , a ∈ A − . As we will see in the proof below this inequality is actually an equality.In the classical setting we have that D = C and Ψ( b ) = ω ( b ) for all b ∈ M ,where ω ( b ) = R b dµ ω and µ ω is the probability measure associated with Ψ by theRiesz representation theorem. It is a straightforward exercise to see that underthese identifications the above condition yields exactly the classical definition ofJensen measures and Arens-Singer measures (see e.g. [15, 33], e.g. the argument onp. 108 in the latter text).In this section we will prove a theorem concerning the inequalities above whichis very closely related to the main result of [26]. We will need two facts noted aslemmata in [26]: Let a, b be positive invertible elements of a unital C ∗ -algebra C ,with [ a, b ] = 0. Then b + b − a ≥ a / . If for such a positive invertible element a ∈ C , we inductively define x = a, x n +1 = 12 ( x n + ax − n ) , then ( x n +1 ) decreases monotonically to a / with convergence taking place in thenorm topology. Lemma 8.1.
Let ω be a state on a C ∗ -algebra C . For any a ∈ C , the sequence ( ω ( | a | − n ) n ) is decreasing. If ω is in fact tracial, then (1) ω ( | a | p ) = ω ( | a ∗ | p ) for any < p < ∞ and any a ∈ C ; (2) given < p, q, r ≤ ∞ with p = q + r , we will for any a, b ∈ C have that ω ( | ab | p ) /p ≤ ω ( | a | q ) /q ω ( | b | r ) /r .Proof. Regarding the first claim note that for any a ∈ C , we may use the Cauchy-Schwarz inequality to see that ω ( | a | − n ) ≤ ω ( | a | − n +1 ) ω ( ) = ω ( | a | − n +1 ) .Now suppose that ω is tracial. Then the normal extension ˜ ω to M = C ∗∗ is stilla trace. As in the lines above Lemma 1.1 in [9], there is a central projection z ∈ M such that ˜ ω ( z · ) is a faithful normal tracial state on zM , and ˜ ω ( zx ) = ˜ ω ( x ) for all x ∈ M . So z M is a finite von Neumann algebra. Since the result which we areinterested in is true for faithful normal tracial states, we have ω ( | a | p ) = ˜ ω ( z | a | p ) = ˜ ω ( | za | p ) = ˜ ω ( | a ∗ z | p ) , and similarly ω ( | a ∗ | p ) = ˜ ω ( | a ∗ z | p ). We have also used the fact from Lemma 1.2 in[9] that | x | p z = ( | x | z ) p = | xz | p . In that lemma x was invertible, however this is notused in the proof. An alternative proof: By the Stone-Weierstrass theorem we mayselect a sequence of polynomials { p n } in one real variable which converge uniformlyto the function t → t p/ on [0 , k a k ]. It is an easy exercise to use the tracial propertyto see that ω (( a ∗ a ) m ) = ω (( aa ∗ ) m ) for all m ∈ { , , , . . . } . (In the case m = 0this boils down to the well known fact that ω ( s r ( a )) = ω ( s l ( a )), where s r ( a ) and s l ( a ) denote the left and tight supports of a . But then ω ( p n ( a ∗ a )) = ω ( p n ( aa ∗ ))for all n , whence ω ( | a | p ) = lim n →∞ ω ( p n ( a ∗ a )) = lim n →∞ ω ( p n ( aa ∗ )) = ω ( | a ∗ | p ).The final claim follows from [14, Theorem 4.9] and a similar argument to theargument starting the second paragraph of the proof (reduction to the finite vonNeumann algebra case). Thus for example, ω ( | ab | p ) /p = ˜ ω ( z | ab | p ) /p = ˜ ω ( | zazb | p ) /p = k ( za )( zb ) k p ≤ k za k q k zb k r , while k za k q = ˜ ω ( | za | q ) /q = ω ( | a | p ) /p , and so on. (cid:3) Remark 8.2.
Item (2) improves the H¨older inequality from [19] to include the caseof p < A of a unital C ∗ -algebra C is logmodular if every element b ∈ C such that b ≥ ǫ ǫ > , is a uniformlimit of terms of the form a ∗ a where a is an invertible element in A . If A islogmodular then C is the C ∗ -envelope of A by [5, Proposition 4.3]. In addition,by [5, Theorem 4.4] for any unital C ∗ -subalgebra D and D -character Φ on A , ifthere exists a positive map Ψ from C onto D extending Φ (which is necessarilya conditional expectation), then it is unique. Of course the earlier Hoffman-Rossitheorems can assist the existence of such an extension in some important cases.The latter is a generalization of the important unique normal state extensionproperty in the theory of noncommutative H ∞ (a la Arveson’s subdiagonal algebras)–see e.g. [7, 1]. Indeed if D ⊂ B ( H ) then there is a unique UCP map Ψ : C → B ( H )extending Φ, and of course Ψ( A + A ∗ ) ⊂ D . So if further C is a von Neumannalgebra with A + A ∗ weak* dense in M (which is the case in Arveson’s subdiagonalsetting), and if Ψ is weak* continuous, then Ψ maps into D , and by the above ONDITIONAL EXPECTATIONS AND GENERALIZED REPRESENTING MEASURES 45 it is the unique positive extension of Φ to M . Without the logmodular assump-tion certainly there may exist no conditional expectation from C onto D , i.e. no‘noncommutative representing measure’ in the sense of this paper. Theorem 8.3.
Let A be a unital norm closed logmodular subalgebra of a unital C ∗ -algebra C , and ω a state on C which is tracial on a C ∗ -subalgebra D of A . Let Φ be a ω -preserving D -character on A , which extends to an ω -preserving conditionalexpectation from C onto D . For any a ∈ A − , we have lim n →∞ ω ( | a | − n ) n ≥ lim n →∞ ω ( | Φ( a ) | − n ) n . These limits are decreasing, and equal exp( ω (log | a | )) and exp( ω (log | Φ( a ) | )) respec-tively. Thus we have Jensen’s equality ∆ ω ( a ) = ∆ ω (Φ( a )) , a ∈ A − , and ω is a noncommutative Arens-Singer measure for Φ .Proof. The proof uses induction. For any a ∈ A − ∪ ( A ∗ ) − , we clearly have bythe Kadison-Schwarz inequality that ω ( | a | ) = ω (Ψ( | a | )) ≥ ω ( | Φ( a ) | )) . Now suppose that for any a ∈ A − ∪ ( A ∗ ) − we have that(6) ω ( | a | − n ) n − ≥ ω ( | Φ( a ) | − n ) n − for all 0 ≤ n ≤ k. Here we interpret Φ( a ∗ ) = Φ( a ) ∗ if a ∈ A . We show that this inequality alsoholds for n = k + 1. So let a ∈ A − be given, and inductively define the sequence( x m ) ⊂ C + by x = | a | − k , x m +1 = 12 ( x m + | a | − k x − m ) . Then ( x m +1 ) decreases monotonically in norm to | a | − k by the facts noted priorto Lemma 8.1. Now let u a be the partial isometry in the polar decomposition a = u a | a | . Since a ∈ A − , we have that u a is a unitary in C ∗∗ , and u ∗ a a = | a | .Several times we will use silently the identity | x − | = | x ∗ | − , which is true since | x − | = ( xx ∗ ) − = | x ∗ | − . We will also use the operator theoretic fact (perhaps originally due to Macaev andPalant) that k x n − y k ≤ k x n − y k for positive operators x n , y . Hence if x n → x then x n → y . If z n → z in B ( H ) then z ∗ n z n → z ∗ z , and hence w ∗ z ∗ n z n w → w ∗ z ∗ zw for any operator w on H . Thus by the last fact, | z n w | → | zw | .Because A is logmodular, we may for each fixed m , select a sequence ( z ( m ) l ) ⊂A − with lim l →∞ | ( z ( m ) l ) − | = | x − k − m u ∗ a | . Note that x − k − m u ∗ a = ( u a x − k − m ) ∗ ∈ C , since x − k − m ∈ C ∗ ( | a | ). For this sequencewe have thatlim l →∞ | ( z ( m ) l ) − a | = lim l →∞ | | ( z ( m ) l ) − | a | = | | x − k − m u ∗ a | a | = | x − k − m u ∗ a a | = x − k − m | a | . Observe that we then also have (since all of our operators are bounded below) thatlim l →∞ | ( z ( m ) l ) ∗ | = lim l →∞ | ( z ( m ) l ) − | − = | x − k − m u ∗ a | − = | u a x k − m | = ( x k − m ) , and hence that lim l →∞ | z ( m ) l | = x k − m . It is clear from the induction hypothesisand Lemma 8.1, that12 ω ( | ( z ( m ) l ) ∗ | − k + | ( z ( m ) l ) − a | − k ) ≥ ω ( | Φ( z ( m ) l ) ∗ | − k + | Φ(( z ( m ) l ) − a ) | − k )= 12 ω ( | Φ( z ( m ) l ) | − k + | Φ( z ( m ) l ) − Φ( a ) | − k ) . (7)Recall that ω is a trace on D . Thus the p -seminorms on D defined in terms of thistrace, satisfy a H¨older inequality. Now since 2 k − + 2 k − = 2 k we may apply thisH¨older inequality with p = ( ) k and q = r = ( ) k − , to see that ω ( | Φ( a ) | − k ) k ≤ ω ( | Φ( z ( m ) l ) | − k ) k − ω ( | Φ( z ( m ) l ) − Φ( a ) | − k ) k − . This in turn translates to ω ( | Φ( z ( m ) l ) − Φ( a ) | − k ) ≥ ω ( | Φ( a ) | − k ) ω ( | Φ( z ( m ) l ) | − k ) . If we combine the above with inequality (7), it will then follow from the facts notedprior to Lemma 8.1 that12 ω ( | ( z ( m ) l ) ∗ | − k + | ( z ( m ) l ) − a | − k ) ≥ " ω ( | Φ( z ( m ) l ) | ) − k + ω ( | Φ( a ) | − k ) ω ( | Φ( z ( m ) l ) | − k ) ≥ ω ( | Φ( a ) | / k ) , the last inequality following from the AGM inequality. This fact alongside theabove limit formulas, now ensures that ω ( x m +1 ) = 12 ω ( x m + x − m | a | − k )= 12 lim l →∞ ω ( | ( z ( m ) l ) ∗ | − k + | ( z ( m ) l ) − a | − k ) ≥ ω ( | Φ( a ) | / k ) . Since ( x m +1 ) decreases uniformly and monotonically to | a | / k , it follows that ω ( | a | / k ) ≥ ω ( | Φ( a ) | / k ) for all a ∈ A − . It is an exercise to see that if A is logmodular, then so is A ∗ (see the remark before[5, Proposition 4.3]). Hence the same proof with the roles of A and A ∗ reversed,shows that we also have that ω ( | a | / k ) ≥ ω ( | Φ( a ) | / k ) for all a ∈ ( A ∗ ) − , as desired.Lemma 8.1 shows that the limits are decreasing. In fact ω ( | a | p ) is decreasing as p ց
0, as one may see by noting that ω restricts to a (tracial) state on C ∗ (1 , | a | ).We may thus reduce to the case of a positive invertible function f in C ( K ), with ω becoming a probability measure µ on K . In this case it is well known that k f k p ց exp( R K ln | f | dµ ) (see e.g. [15]). By the proof of [1, Lemma 4.3.6] we havethat ω ( | a | / k ) k → exp( ω (log | a | )). The matching assertions for Φ( a ) are betterknown, or are similar. Thus we have shown that∆ ω ( a ) ≥ ∆ ω (Φ( a )) , a ∈ A − , as desired. ONDITIONAL EXPECTATIONS AND GENERALIZED REPRESENTING MEASURES 47
Replacing a by a ∗ we have ω ( | a ∗ | / k ) ≥ ω ( | Φ( a ) ∗ | / k ), and letting k → ∞ yieldsexp( ω (log | a ∗ | )) ≥ exp( ω (log | Φ( a ) ∗ | )). Then replacing a by a − gives exp( ω (log | a | − )) ≥ exp( ω (log | Φ( a ) | − ))). By the functional calculus log | a | − = − log | a | since | a | isbounded below. We deduce that ∆ ω ( a ) ≤ ∆ ω (Φ( a )), giving Jensen’s equality. (cid:3) Remark 8.4. If τ is a trace on D , and Ψ : C → D is a positive extension (‘noncom-mutative representing measure’) of a D -character Φ on A then ω = τ ◦ Ψ is a stateon C which is tracial on D , and of course ω ◦ Ψ = ω ). We can also say that if ρ is astate on A which is tracial on D , and which is preserved by Φ, then setting τ = ρ D and ω = τ ◦ Ψ, then Ψ is ω -preserving, and ω ( a ) = ρ (Ψ( a )) = ρ (Φ( a )) = ρ ( a )for a ∈ A , so ω extends ρ . In these cases ω (and Ψ) satisfy the conditions of thetheorem. Acknowledgements.
We thank Roger Smith and Mehrdad Kalantar for assistancewith the von Neumann algebraic facts described in the Remark above Lemma 3.4.
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Email address , David P. Blecher: [email protected]
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