aa r X i v : . [ m a t h - ph ] A ug ObservablesIV : The Presheaf Perspective
Hans F. de Groote ∗ November 3, 2018
Contents R sa
63 Restrictions 74 The upper and lower observable presheaves 155 A unification of upper and lower observable presheaves 196 Global Sections 207 States 24
Abstract
In this fourth of our series of papers on observables we show thatone can associate to each von Neumann algebra R a pair of isomorphicpresheaves, the upper presheaf O + R and the lower presheaf O − R , on thecategory of abelian von Neumann subalgebras of R . Each A ∈ R sa induces a global section of O + R and of O − R respectively. We call them contextual observables . But we show that, in general, not every globalsection of these presheaves arises in this way. Moreover, we discussstates of a von Neumann algebra in the presheaf context. ∗ [email protected]; FB Mathematik, J.W.Goethe-Universit¨at Frankfurta. M. ¨ur Karin In this fourth paper of our series on “observables” ([4], [5], [6]) we generalizethe notion of a (bounded) quantum observable in the sense of contextuality.The main results where already announced in our overview article [3].The central idea in this article is the notion of restriction of operators from avon Neumann algebra R in L ( H ) (for an arbitrary Hilbert space H ) to a vonNeumann subalgebra M of R . This restriction of operators turns out to be afar reaching generalization of the notion of central support (or central carrier)of a projection. If P ∈ P ( R ), we call s M ( P ) := V { Q ∈ P ( M ) | Q ≥ P } the M -support of P . We can generalize this definition to selfadjoint operators A ∈ R sa by setting ̺ M A := ^ { B ∈ M sa | A ≤ s B } , where ≤ s denotes the spectral order on R sa ([2, 20]), called the restriction of A to M . One can prove directly from this definition that ̺ M A is a selfadjointelement of M : If B ∈ M such that A ≤ B , then (min sp ( A )) I ≤ s B , hencethe bounded completeness of the spectral order implies that ̺ M A ∈ M .However, in order to gain a deeper insight into the restriction process, weshow how the restriction of A is obtained in a natural way from the observablefunction of A . For the convenience of the reader we will revise the basicdefinitions and results on observable functions.Let R be a von Neumann algebra acting on a Hilbert space H and let A ∈R be selfadjoint. Moreover, let D ( R ) be the set of all dual ideals of theprojection lattice P ( R ) of R and let Q ( R ) ⊆ D ( R ) be the set of all maximaldual ideals (the “quasipoints ”). Q ( R ) is called the Stone spectrum of R .It is, equipped with the topology that is generated by the sets Q P ( R ) := { B ∈ Q ( R ) | P ∈ B } (0 = P ∈ P ( R )), a zero - dimensional Hausdorffspace, for which the clopen sets Q P ( R ) form a base. The space D ( R )bears the analogously defined topology. But note that D ( R ), equipped withthis topology, is (in general) not a Hausdorff space. If R is abelian, Q ( R )is homeomorphic to the Gelfand spectrum Ω( R ) of R ([4]). If A ∈ R sa and if E = ( E λ ) λ ∈ R is the spectral family corresponding to A , the function f A : D ( R ) → R , defined by f A ( J ) := inf { λ ∈ R | E λ ∈ J } , We use the shortcut “clopen ” for “closed and open ”.
2s called the observable function of A . Its restriction to the Stone spectrum Q ( R ) of R is continuous , whereas f A : D ( R ) → R is, in general, not. We notefurther that, if R is abelian, f A : Q ( R ) → R is (up to the homeomorphism Q ( R ) ∼ = Ω( R ) mentioned before) the Gelfand transform of A .We can characterise observable functions in an abstract way ([5]): Theorem 1.1. f : D ( R ) → R is an observable function if and only if thefollowing two properties hold for f :(i) ∀ J ∈ D ( R ) : f ( J ) = inf { f ( H P ) | P ∈ J } ,(ii) f ( T j ∈ J J j ) = sup j ∈ J f ( J j ) for all families ( J j ) j ∈ J in D ( R ) . Here H P denotes the principle dual ideal defined by P : H P := { Q ∈P ( R ) | P ≤ Q } . An observable function f A on D ( R ) induces a function r A : P ( R ) → R , given by r A ( P ) := f A ( H P ). This function is bounded andhas the property r A ( _ k ∈ K P k ) = sup k ∈ K r A ( P k )for all families ( P k ) k ∈ K in P ( R ). Therefore, it is called completely increasing .Conversely, every bounded completely increasing function r : P ( R ) → R comes from a unique selfadjoint operator A ∈ R sa : ∃ ! A ∈ R sa : r = r A . If M is a von Neumann subalgebra of the von Neumann algebra R , andif r : P ( R ) → R is a bounded completely increasing function, r can berestricted to a bounded function ̺ M r : P ( M ) → R . ̺ M r is obviously com-pletely increasing. It is shown that the selfadjoint operator correspondingto ̺ M r is ̺ M A , where A is the selfadjoint operator corresponding to r . It isin this sense that ̺ M A is a restriction of A .The foregoing definition of restricting selfadjoint operators from R to M has an equally natural counterpart: σ M A := _ { B ∈ M sa | A ≥ s B } . We show in the sequel that this type of restriction is induced by the mirroredobservable function g A of A . Mirrored observable functions were introducedby A. D¨oring in [9] and called antonymous functions. g A is defined on D ( R )by ∀ J ∈ D ( R ) : g A ( J ) := sup { λ ∈ R | I − E λ ∈ J } , E λ ) λ ∈ R denotes the spectral family corresponding to A .There is a simple relation between the observable function and the mirroredobservable function of A : g A = − f − A . Hence, if O ( R ) denotes the set of all observable functions, the set of allmirrored observable functions is just −O ( R ). Therefore, we prefer the name“mirrored observable function ” instead of “antonymous function ”. It followsimmediately from the foregoing relation between observable and mirroredobservable functions that a theorem analogous to theorem 1.1 is true: Theorem 1.2. g : D ( R ) → R is a mirrored observable function if and onlyif the following two properties hold for g :(i) ∀ J ∈ D ( R ) : g ( J ) = sup { g ( H P ) | P ∈ J } ,(ii) g ( T j ∈ J J j ) = inf j ∈ J g ( J j ) for all families ( J j ) j ∈ J in D ( R ) . The mirrored observable function g A induces a bounded function s A : P ( R ) → R , defined by s A ( P ) := g A ( H P ), which is completely decreasing : s A ( _ k ∈ K P k ) = inf k ∈ K s A ( P k )for all families ( P k ) k ∈ K in P ( R ). Using theorem 1.2, one can easily showthat every bounded completely decreasing function s : P ( R ) → R isinduced by a unique operator A ∈ R sa : s = s A .If M is a von Neumann subalgebra of R and A ∈ R sa is an operatorwith corresponding completely decreasing function s A : P ( R ) → R , weprove that σ M A is the selfadjoint operator corresponding to the restriction s A | P ( M ) of s A to P ( M ).In section 2 we present a new approach to the spectral order on R sa via observable functions. Section 3 contains a detailed discussion of therestriction processes defined above.In section 4 we define the upper and lower observable presheaf of a vonNeumann algebra R . These are presheaves on the category A ( R ) of abelianvon Neumann subalgebras of R (the context category ), hence presheaves inthe sense of Topos theory . The restrictions of the upper (lower) observablepresheaf are defined by restricting completely increasing (decreasing)functions corresponding to selfadjoint operators. It is shown that the upperobservable presheaf is isomorphic to the lower one. Of course A ( R ) can be regarded here simply as a semi-lattice.
4n section 5 we present a possible unification of upper and lower observablepresheaves to a presheaf of linear spaces and linear maps. This unificationis mathematical natural. However, we don’t have an operator - theoreticalinterpretation for it.In section 6 we discuss global sections of observable presheaves. Here, theissue is not their existence (every selfadjoint operator gives rise to a globalsection of the observable presheaf ) but the question whether every globalsection of the observable presheaf is induced by a single selfadjoint operator.Of course, this is not true when the von Neumann algebra contains a directsummand of type I , but we give an example that is different from thissituation. Thus the global sections of observable presheaves form a largerclass than the usual observables. We call the global sections of the upper(lower) observable presheaf upper (lower) contextual observables . Theseconstructions play an essential rˆole in the remarkable articles of A. D¨oringand C.J. Isham ([10, 11]) on a Topos - theoretical formulation of physicaltheories.In section 7 we consider states of a von Neumann algebra R from thepresheaf perspective. We show that each state of R induces a globalsection of the state presheaf S R . S R is a presheaf on the context category A ( R ) of abelian von Neumann subalgebras of R , which assigns to each A ∈ A ( R ) the space S ( A ) of all states of A ; the restrictions are the ordinaryrestrictions of functions. We show that, if R contains a direct summandof type I , not every global section of S R is induced by a state of R .But, contrary to the observable presheaves, this is the only exception. Itfollows from the generalization of Gleason’s theorem, due to Christensen,Yeadon et al. ([19]), that each global section of S R is induced by a stateof R , provided that R does not contain a summand of type I . Moreprecisely, we show that this generalization of Gleason’s theorem is equiva-lent to the property that every global section of S R is induced by a state of R .We emphasise that our results, except those of section 7, hold notonly for von Neumann algebras but also for arbitrary complete orthomodu-lar lattices: simply replace “operators” by “bounded spectral families” and“abelian von Neumann subalgebras” by “complete Boolean sublattices”.This has the interesting consequence that the whole theory is applicableto the lattice of causally closed subsets ([1]) of an arbitrary spacetime.Whether this has consequences for general relativity should be investigated. We discuss in this section only the upper observable presheaf. The same results holdfor the lower observable presheaf. A Canonical Lattice Structure on R sa We have seen that selfadjoint operators A ∈ R sa can be encoded in com-pletely increasing functions f : P ( R ) → R . In the sequel we will make nodistinction in the notation of completely increasing functions and observablefunctions. Let O ( R ) be the set of observable functions. Depending on thecontext these are either functions f : P ( R ) → R or f : D ( R ) → R or f : Q ( R ) → R .There is a canonical partial order on R sa : Definition 2.1.
Let
A, B be selfadjoint elements of the von Neumann algebra R and let f A , f B : P ( R ) → R be the observable functions corresponding to A and B respectively. Then we define A ≤ s B if and only if f A ≤ f B with respect to the pointwise defined ordering of real valued functions. It is easy to see that for f, g ∈ O ( R ) the relation f ≤ g does not dependwhether we view these functions as being defined on P ( R ) , D ( R ) or Q ( R ). Proposition 2.1.
Let
A, B ∈ R sa with spectral families E A and E B respec-tively. Then A ≤ s B ⇐⇒ ∀ λ ∈ R : E Bλ ≤ E Aλ . Proof:
Let A ≤ s B . By definition we have A ≤ s B ⇐⇒ ∀ P ∈ P ( R ) : inf { µ | P ≤ E Aµ } ≤ inf { µ | P ≤ E Bµ } . Let E Bλ = 0. Then inf { µ | E Bλ ≤ E Bµ } ≤ λ and therefore λ := inf { µ | E Bλ ≤ E Aµ } ≤ λ. But then E Bλ ≤ E Aλ ≤ E Aλ . Conversely, if E Bλ ≤ E Aλ for all λ ∈ R , thenobviously f A ( P ) = inf { λ | P ≤ E Aλ } ≤ inf { λ | P ≤ E Bλ } = f B ( P )for all P ∈ P ( R ). (cid:3) We call ≤ s the spectral order on R sa . It is obvious that ≤ s de-fines a partial order on R sa .The spectral order was defined and studied by M.P. Olson in [20] and6ndependently in [2]. Originally the spectral order has been defined directlyby means of the spectral families corresponding to the selfadjoint operators: A ≤ s B if and only if E Bλ ≤ E Aλ for all λ ∈ R . We think, however, that its most natural definition occurs here in connectionwith observable functions.
The lattice operations were defined as follows:Let ( A κ ) κ ∈ K be an arbitrary family in R sa and let E A κ be the spectral familycorresponding to A κ . Then λ ^ κ E A κ λ and λ ^ µ>λ _ κ E A κ µ are spectral families and the first of them defines the join W κ A κ , the secondthe meet V κ A κ of the family ( A κ ) κ ∈ K . With these operations of join andmeet R sa is a boundedly complete lattice. The abstract characterization of (quantum) observable functions leads to anatural definition of restricting selfadjoint elements of a von Neumann alge-bra R to a subalgebra M . Again we denote a completely increasing functionon P ( R ) and the corresponding observable function (on Q ( R ) or D ( R )) bythe same letter and speak simply of an observable function. Obviously wehave Remark 3.1.
Let M be a von Neumann subalgebra of a von Neumann alge-bra R and let f : P ( R ) → R be an observable function. Then the restriction ̺ M f := f |P ( M ) is an observable function for M . It is called the restriction of f to M . This definition is absolutely natural. However, if A is a selfadjoint oper-ator in R then the observable function f A : P ( R ) → R corresponding to A is a rather abstract encoding of A . So before we proceed, we will describethe restriction map ̺ M : O ( R ) → O ( M ) f A ̺ M f A in terms of spectral families.To this end we define 7 efinition 3.1. Let F be a filterbasis in P ( R ) . Then C R ( F ) := { Q ∈ P ( R ) | ∃ P ∈ F : P ≤ Q } is called the cone over F in R . Clearly C R ( F ) is a dual ideal and it is easy to see that it is the smallestdual ideal that contains F . A dual ideal
I ∈ D ( M ) is, in particular, afilterbasis in P ( R ), so C R ( I ) is well defined. Proposition 3.1.
Let f ∈ O ( R ) . Then ( ̺ M f )( I ) = f ( C R ( I )) for all I ∈ D ( M ) .Proof: From f ( J ) = inf P ∈J f ( P ), the definition of the cone and the factthat f is increasing on P ( R ) we obtain f ( C R ( I )) = inf { f ( Q ) | Q ∈ C R ( I ) } = inf { f ( P ) | P ∈ I} . (cid:3) Definition 3.2.
For a projection Q in R let c M ( Q ) := _ { P ∈ P ( M ) | P ≤ Q } and s M ( Q ) := ^ { P ∈ P ( M ) | P ≥ Q } .c M ( Q ) is called the M -core , s M ( Q ) the M -support of Q . The M -support is a natural generalization of the notion of central supportwhich is the M -support if M is the center of R . Note that if Q / ∈ M then c M ( Q ) < Q < s M ( Q ). The M -core and the M -support are related in asimple manner: Remark 3.2. c M ( Q ) + s M ( I − Q ) = I for all Q ∈ P ( R ) . Remark 3.3.
Core and support have the following properties: c M ( ^ k ∈ K P k ) = ^ k ∈ K c M ( P k ) , s M ( _ k ∈ K P k ) = _ k ∈ K s M ( P k ) and c M ( _ k ∈ K P k ) ≥ _ k ∈ K c M ( P k ) , s M ( ^ k ∈ K P k ) ≤ ^ k ∈ K s M ( P k ) . Lemma 3.1.
Let E = ( E λ ) λ ∈ R be a spectral family in R and for λ ∈ R define ( c M E ) λ := c M ( E λ ) , ( s M E ) λ := ^ µ>λ s M ( E µ ) . Then c M E := (( c M E ) λ ) λ ∈ R and s M E := (( s M E ) λ ) λ ∈ R are spectral familiesin M . roof: If λ < µ then c M ( E λ ) ≤ E λ ≤ E µ and therefore c M ( E λ ) ≤ c M ( E µ ). Moreover V µ>λ c M ( E µ ) ≤ V µ>λ E µ = E λ , hence ^ µ>λ c M ( E µ ) ≤ c M ( E λ ) ≤ ^ µ>λ c M ( E µ ) . The other assertions are obvious. Note, however, that λ s M ( E λ ) isn’t aspectral family in general! (cid:3) Proposition 3.2.
Let f ∈ O ( R ) and let E be the spectral family correspond-ing to f . Then c M E is the spectral family corresponding to ̺ M f .Proof: Let I be a dual ideal in P ( M ). Then ( ̺ M f )( I ) = f ( C R ( I )) and f ( C R ( I )) = inf { λ | E λ ∈ C R ( I ) } = inf { λ | ∃ P ∈ I : P ≤ E λ } = inf { λ | c M ( E λ ) ∈ I} . Thus the assertion follows from the theorem that an observable functiondefines a unique spectral family ([5]). (cid:3)
By theorem 2.6 in [5], the restriction map ̺ M : O ( R ) → O ( M ) in-duces a restriction map ̺ M : R sa → M sa A ̺ M A for selfadjoint operators. In particular, we obtain Corollary 3.1. ̺ M Q = s M ( Q ) for all projections Q in R . The corollary shows that the restriction map ̺ M : R sa → M sa hasthe important property that it maps projections to projections and acts asthe identity on P ( M ). It also shows that in general ̺ M is not linear: if P, Q ∈ P ( R ) such that P Q = 0 then it is possible that s M ( P ) s M ( Q ) = 0and therefore s M ( P + Q ) = s M ( P ) + s M ( Q ).We will now consider the special case M := Q R Q . This case willshow up the link to the restriction of ordinary continuous functions f : M → R on a topological space M to an open subspace U ⊆ M .9f course there is another natural way to restrict an operator to thesubalgebra Q R Q , namely the map R → Q R QA QAQ.
But this type of restriction does not have the property that it maps projec-tions to projections:
Remark 3.4. If P ∈ P ( R ) , then QP Q is a projection if and only if P commutes with Q .Proof: If QP Q is a projection, then P ∧ Q = lim n →∞ ( P Q ) n = lim n →∞ ( QP Q ) n = QP Q, where the limits are taken with respect to the strong topology. Hence( P − QP Q )( Q − QP Q ) =
P Q − QP Q = 0 , because Q − QP Q = Q ( I − P ) Q , so Q ( I − P ) Q equals ( I − P ) ∧ Q and, there-fore, is a subprojection of I − P . It is then obvious that P Q = QP holds. (cid:3) Proposition 3.3.
Let A ∈ R sa and let E A be the spectral family of A . Thenthe spectral family of the restriction ̺ Q R Q A is given by λ E Aλ ∧ Q .Proof: A projection P ∈ R is an element of Q R Q if and only if P ≤ Q .Hence if E ∈ P ( R ) and P ∈ P ( Q R Q ) such that P ≤ E then P ≤ E ∧ Q .This shows c Q R Q ( E ) = E ∧ Q . Therefore the proposition follows fromproposition 3.2. (cid:3) Note that for R = L ( H ) the subalgebra Q R Q is canonically isomor-phic to L ( Q H ), so ̺ Q L ( H ) Q A can be considered as the restriction of A ∈ L ( H ) to an operator in L ( Q H ).Now let f : M → R be a continuous function on a topological space M andlet U ⊆ M be an open nonvoid subset. The corresponding spectral family isgiven by σ f : λ int ( − f (] − ∞ , λ ])). Then λ int ( − f (] − ∞ , λ ])) ∩ U is aspectral family in T ( U ). Because of int ( − f (] − ∞ , λ ])) ∩ U = int ( − f (] − ∞ , λ ]) ∩ U )= int ( − ( f | U )(] − ∞ , λ ]))10his is the spectral family of the continuous function f | U : U R , therestriction of f to U . This also demonstrates that our definition of restrictionof operators is absolutely natural.Proposition 3.2 and lemma 3.1 suggest still another natural possibilityfor defining a restriction map σ M : R sa → M sa : if E A is the spectral familycorresponding to A ∈ R sa then σ M A is the selfadjoint operator defined bythe spectral family s M E A .Let us check what this means in the case M = Q R Q for some Q ∈ P ( R )different from I . First of all we have to determine the Q R Q -support ofa projection E ∈ R . Here a little bit of care is needed because Q R Q isnot a von Neumann subalgebra of R in the strict sense. It has a unity, Q , but this is different from I . This was irrelevant for the Q R Q -core but { P ∈ Q R Q | E ≤ P } = ∅ unless E ≤ Q . We can overcome this complicationby defining V Q R Q ∅ := Q . Then we obtain for all E ∈ P ( R ) s Q R Q ( E ) = ( E if E ∈ Q R QQ otherwiseand therefore the spectral family of σ Q R Q A is given by( s Q R Q E A ) λ = ( E Aλ if ∃ µ > λ : E Aµ ∈ Q R QQ otherwise . We will show that the restriction map σ M : R sa → M sa has a canonicalorigin too.Let g : P ( R ) → R be a mirrored observable function. There is aresult analogous to proposition 3.1 for the restriction σ M g of g to M : Proposition 3.4. ∀ I ∈ D ( M ) : ( σ M g )( I ) = g ( C R ( I )) . Proof: ( σ M g )( I ) = sup { s ( P ) | P ∈ I} = sup { s ( Q ) | Q ∈ C R ( I ) } . (cid:3) Let E be the spectral family of A ∈ R sa . It is now easy, to showthat σ M E := ( V µ>λ s M ( E µ )) λ ∈ R is the spectral family of the operator σ M A ∈ M sa corresponding to σ M g A . Corollary 3.2.
Let A ∈ R sa and let E be the spectral family of A . Then σ M E is the spectral family corresponding to σ M g A . roof: Indeed, we obtain for all dual ideals I in P ( M ): σ M g A ( I ) = g A ( C R I )= − f − A ( C R I )= − inf { λ ∈ R | ∃ P ∈ I : P ≤ c M ( I − E − λ − ) } = − inf { λ ∈ R | c M ( I − E − λ − ) ∈ I} = − inf {− λ ∈ R | c M ( I − E λ − ) ∈ I} = sup { λ ∈ R | c M ( I − E λ ) ∈ I} = sup { λ ∈ R | I − s M ( E λ ) ∈ I} = sup { λ ∈ R | I − ^ µ>λ s M ( E µ ) ∈ I} , where we have used some elementary properties of inf and sup. (cid:3) There is a simple relation between the two types of restrictions thatis quite analogous to that between observable and mirrored observablefunctions:
Proposition 3.5.
Let M be a von Neumann subalgebra of R . Then σ M A = − ̺ M ( − A ) holds for all A ∈ R sa .Proof: Because of − f σ M A ( P ) = − inf { λ ∈ R | ^ µ>λ s M ( E Aµ ) ≥ P } = − inf { λ ∈ R | s M ( E Aλ ) ≥ P } = − inf { λ ∈ R | I − c M ( I − E Aλ ) ≥ P } = sup { λ ∈ R | I − c M ( I − E A − λ ) ≥ P } = g ̺ M ( − A ) ( P )for all P ∈ P ( M ), we obtain f σ M A = − g ̺ M ( − A ) = − g − ( − ̺ M ( − A )) = f − ̺ M ( − A ) , and this implies σ M A = − ̺ M ( − A ) . (cid:3) M is an arbitrary von Neumann subalgebra of R and A ∈ R sa then ∀ λ ∈ R : ( ̺ M E A ) λ ≤ E Aλ ≤ ( σ M E A ) λ which means σ M A ≤ s A ≤ s ̺ M A and therefore σ M A ≤ A ≤ ̺ M A by [2]. Proposition 3.6.
Let M be a von Neumann subalgebra of the von Neumannalgebra R . Then, for all A ∈ R sa , we have σ M A = _ { B ∈ M sa | B ≤ s A } and ̺ M A = ^ { C ∈ M sa | A ≤ s C } , where σ M A, ̺ M A are considered as elements of R and W , V denote the great-est lower bound and the least upper bound with respect to the spectral order.Proof: Let
B, C ∈ M sa such that B ≤ s A ≤ s C and let E A , E B , E C bethe spectral families of A, B and C respectively. Then, by the definition ofthe spectral order, we have for all λ ∈ R E Cλ ≤ E Aλ ≤ E Bλ , and therefore, using E Bλ , E Cλ ∈ M , we obtain E Cλ = c M ( E Cλ ) ≤ c M ( E Aλ ) ≤ E Aλ ≤ s M ( E Aλ ) ≤ ^ µ>λ s M ( E Aµ ) ≤ ^ µ>λ s M ( E Bµ )= ^ µ>λ E Bµ = E Bλ . This shows B ≤ s σ M A ≤ s A ≤ s ̺ M A ≤ s C. (cid:3) This proposition shows that the two restriction mappings ̺ M and σ M from13 sa onto M sa are on an equal footing. Moreover, it shows that theserestrictions are generalisations of M - support and M - core to arbitraryselfadjoint operators. We call ̺ M A the upper M - aspect of A and σ M A the lower M - aspect of A .Let m A := inf { λ | E Aλ = 0 } and M A := min { λ | E Aλ = I } . Then m A I = σ C I A and M A I = ̺ C I A. Thus we recover via restrictions the well known simple inequality m A I ≤ A ≤ M A I. In general, σ M A, ̺ M A can be considered as lower and upper , respectively, coarse grainings of A . The following example makes this point of view ap-parent. Example 3.1.
Let A ∈ R sa and let λ , . . . , λ n ∈ sp ( A ) such that λ < · · · < λ n . We may assume that the corresponding spectral projections satisfy E Aλ < · · · < E Aλ n . Let A := A ( λ , . . . , λ n ) be the von Neumann subalgebra generated by { E Aλ , . . . , E Aλ n , I } , that is A = lin C { E Aλ , . . . , E Aλ n , I } . Setting E Aλ := 0 and E Aλ n +1 := I , we can representevery projection P ∈ A as a linear combination P = n +1 X k =1 a k ( E Aλ k − E Aλ k − ) with coefficients a k ∈ { , } . In order to avoid boring case distinctions, wefurther assume that m A < λ < λ n < M A . We can therefore set λ n +1 := M A . Then an easy, but somewhat tedious,discussion shows that for all λ ∈ R such that E Aλ = 0 we have c A ( E Aλ ) = if E Aλ < E Aλ E Aλ k if E Aλ k ≤ E Aλ < E Aλ k +1 ( k = 1 , . . . , n ) I if E Aλ = I and s A ( E A ) λ = E Aλ if E Aλ < E Aλ E Aλ k +1 if E Aλ k ≤ E Aλ < E Aλ k +1 ( k = 1 , . . . , n − I if E Aλ n ≤ E Aλ . herefore, the spectra of the restrictions ̺ A A and σ A A are sp ( ̺ A A ) = { λ , . . . , λ n , M A } and sp ( σ A A ) = { m A , λ , . . . , λ n } . It follows that the restrictions ̺ A A and σ A A have spectral representations ̺ A A = n +1 X k =1 λ k ( E Aλ k − E Aλ k − ) and σ A A = m A E Aλ + n X k =1 λ k ( E Aλ k +1 − E Aλ k ) respectively. These are finite approximations of the spectral representation A = R R λdE Aλ of A : ̺ A A is the upper and σ A A is the lower Riemann-Stieltjessum defined by the partition ( m A , λ , . . . , λ n , M A ) . Consider three von Neumann subalgebras A , B , C of R such that A ⊆ B ⊆ C .Then the corresponding restriction maps ̺ CB : C sa → B sa , ̺ BA : B sa → A sa and ̺ CA : C sa → A sa obviously satisfy ̺ CA = ̺ BA ◦ ̺ CB and ̺ AA = id A sa . (1)The set S ( R ) of all von Neumann subalgebras of R is a lattice with respectto the partial order given by inclusion. The meet of A , B ∈ S ( R ) is definedas the intersection, A ∧ B := A ∩ B , and the join as the subalgebra generated by A and B : A ∨ B := (
A ∪ B ) ′′ . The join is a rather intricate operation. This can already be seen in the mostsimple (non-trivial) example lin C { I, P } ∨ lin C { I, Q } for two non-commutingprojections P, Q ∈ R (see [16]). Fortunately we don’t need it really.The subset A ( R ) ⊆ S ( R ) of all abelian von Neumann subalgebras of R is also partially ordered by inclusion but it is only a semilattice : the meet15f two (in fact of an arbitrary family of) elements of A ( R ) always exists butthe join does not in general. Both S ( R ) and A ( R ) have a smallest element,namely O := C I . However, unless R is itself abelian, there is no greatestelement in A ( R ). Anyway, S ( R ) and A ( R ) can be considered as the sets ofobjects of (small) categories whose morphisms are the inclusion maps.In quantum physics the (maximal) abelian von Neumann subalgebras of L ( H )are called contexts . We generalize this notion in the following Definition 4.1.
The small category
CON ( R ) , whose objects are the abelianvon Neumann subalgebras of R and whose morphisms are the inclusion maps,is called the context category of the von Neumann algebra R . Since the morphisms of
CON ( R ) are so simple, we also speak of A ( R )as the context category or the category of abelian von Neumann subalgebrasof R .We define a presheaf O + R on the context category CON ( R ) of R bysending objects A ∈ A ( R ) to O + R ( A ) := A sa (or equivalently to O ( A )) andmorphisms A ֒ → B to restrictions ̺ BA : B → A . Due to 1 this gives acontravariant functor, i.e. a presheaf on
CON ( R ). Definition 4.2.
The presheaf O + R is called the upper observable presheaf of the von Neumann algebra R . Remark 4.1.
We can define a presheaf O −R by using the restrictions σ BA : B sa → A sa , A σ A A , for A ֒ → B in CON ( R ) . This presheaf is calledthe lower observable presheaf. Due to the next result, it has quite analogousproperties as O + R , so we will concentrate on O + R . Proposition 4.1.
The presheaves O + R and O −R are isomorphic.Proof: This is a direct consequence of the fact that −O ( R ) is the set ofmirrored observable functions. The most simple way to describe the isomor-phism is to regard observable functions as completely increasing functions.Then, for all A ∈ A ( R ), Φ A : O ( A ) → −O ( A ) f
7→ − f is obviously a bijection that commutes with restrictions: for all A , B ∈ A ( R )such that A ⊆ B we haveΦ B ( f ) |P ( A ) = Φ A ( f |P ( A ) )16or all f ∈ O ( B ). Hence Φ := (Φ A ) A∈ A ( R ) is an isomorphism from O + R onto O −R . (cid:3) Each observable function f ∈ O ( R ) induces a family ( f A ) A∈ A ( R ) , definedby f A := ̺ A f , which is compatible in the following sense: ∀ A , B ∈ A ( R ) : ̺ AA∩B f A = ̺ BA∩B f B . The problem whether each compatible family ( f A ) A∈ A ( R ) is induced by anobservable function in O ( R ) will be discussed in section 6.We will now show how the restriction maps ̺ BA and σ BA act on observablefunctions f : Q ( B ) → R or, in other words, how the Gelfand transformationbehaves with respect to the restrictions ̺ BA and σ BA . Lemma 4.1.
Let A , B be abelian von Neumann algebras such that A ⊆ B .Then π BA : β β ∩A maps Q ( B ) onto Q ( A ) . The mapping π BA is continuous,open and therefore also identifying. Moreover β ∩ A = { s A ( P ) | P ∈ β } . Proof:
This lemma is a special case of proposition 2.1 in [7]. (cid:3)
Proposition 4.2.
Let A be an abelian von Neumann algebra and J a dualideal in P ( A ) . Then J = \ { β ∈ Q ( A ) | J ⊆ β } . Proof:
Let Q J ( A ) := { β ∈ Q ( A ) | J ⊆ β } . Assume that there is E ∈ T Q J ( A ) such that E / ∈ J . Since J is a dual ideal, this implies P ( I − E ) = P − P E = 0 for all P ∈ J . The commutativity of A implies that( I − E ) J is a filter base. Since ( I − E ) P ≤ P, I − E , the cone C A (( I − E ) J )is a dual ideal in P ( A ) that contains J and I − E . But then any quasipointthat contains C A (( I − E ) J ), contains J , hence also E and, by construction, I − E , a contradiction. (cid:3) It is obvious that the proposition is true for all Boolean algebras B :every dual ideal in B is the intersection of quasipoints. Corollary 4.1.
Let A be an abelian von Neumann subalgebra of the abelianvon Neumann algebra B and let γ ∈ Q ( A ) . Then C B ( γ ) = \ { β ∈ Q ( B ) | γ ⊆ β } . orollary 4.2. Let A , B be as above and let f : Q ( B ) → R be an observablefunction. Then we have for all γ ∈ Q ( A ) :(i) ( ̺ BA f )( γ ) = sup { f ( β ) | γ ⊆ β } and(ii) ( σ BA f )( γ ) = inf { f ( β ) | γ ⊆ β } .Proof: ( i ) This follows immediately from propositions 3.1, 4.2 and generalproperties of observable functions:( ̺ BA f )( γ ) = f ( C B ( γ )) = f ( \ { β ∈ Q ( B ) | γ ⊆ β } ) = sup { f ( β ) | γ ⊆ β } . ( ii ) Since f is the Gelfand transform of some A ∈ B sa and f = f A = g A on Q ( B ), we obtain from proposition 3.6 and from theorem 1.2( σ BA f )( γ ) = g A ( C B ( γ )) = g A ( \ γ ⊆ β β ) = inf γ ⊆ β g A ( β ) = inf γ ⊆ β f ( β ) . (cid:3) Let π := π BA : Q ( B ) → Q ( A ) be the identifying mapping β β ∩ P ( A )from Q ( B ) onto Q ( A ). By proposition 2.1 in [7], π is continuous, open andsatisfies ∀ P ∈ P ( B ) : π ( Q P ( B )) = Q s A ( P ) ( A ) . The fibres − π ( γ ) , γ ∈ Q ( A ) , form a partition of Q ( B ) into closed subsets.Typically, they have empty interior. Since Q ( B ) is a Stonean space, there isa unique P ∈ P ( B ) such that int − π ( γ ) = Q P ( B ) . Hence, if int − π ( γ ) = ∅ , { γ } = π ( int − π ( γ )) = π ( Q P ( B )) = Q s A ( P ) ( A )and therefore { γ } is an open closed set. This means that γ is an atomicquasipoint of P ( A ). In this case, moreover, − π ( γ ) is open and closed, andthis implies P ∈ P ( A ). So we have proved: Remark 4.2.
For every γ ∈ Q ( A ) , the fibre − π ( γ ) of π := π BA is open andclosed if and only if γ is an atomic quasipoint. If γ is not atomic, then theinterior of − π ( γ ) is empty. If A has finite dimension, then every quasipoint in P ( A ) is atomic, so { − π ( γ ) | γ ∈ Q ( A ) } is a finite partition of Q ( B ) into open closed subsets.18 A unification of upper and lower observablepresheaves
We define a presheaf O R on A ( R ) such that O + R and O −R are subpresheavesof O R . We note first that the sum of two completely increasing functions f, h : P ( R ) → R is, in general, not completely increasing: Example 5.1.
Let
P, Q ∈ P ( R ) such that P Q = 0 and let f := r P , h := r Q , E := I − P, F := I − Q . Then f ( E ) = 0 , f ( F ) = 1 , h ( E ) = 1 , h ( F ) = 0 , so ( f + h )( E ) = ( f + h )( F ) = 1 , but f ( E ∨ F ) = h ( E ∨ F ) = 1 , hence ( f + h )( E ∨ F ) = 2 > f + h )( E ) , ( f + h )( F )) . We have seen that the presheaves O + R and O −R can be defined by the ordi-nary restriction of observable and mirrored observable functions, respectively.This restriction is a linear (and multiplicative) operation. The sets O ( R )and −O ( R ), however, have only poor algebraic structure: they are partiallyordered and closed with respect to multiplication by nonnegative real num-bers. Therefore, we are led to introduce the real vector space F R generatedby the set F + R of completely increasing functions. This space is generatedequally well by the set F −R of completely decreasing functions. Note that wecan construct F R also in the following way: Let M + R be the additive monoidgenerated by F + R . Since F + R is closed under multiplication by nonnegativereal numbers, M + R is a cone in the space of functions on P ( R ). Similarly,the additive monoid M −R generated by F −R is a cone and can be representedas M −R = − M + R . Eventually, we have F R = M + R + M −R . Moreover, it is easy to see that F R is isomorphic to the Grothendieck group([17]) of the monoid M + R . Definition 5.1.
We define a presheaf O R on A ( R ) by(i) O R ( A ) := F A for all A ∈ A ( R ) and ii) ̺ BA ( f ) := f |P ( A ) for A ⊆ B , f ∈ F B . O R is called the observable presheaf of R . Note that the presheaves O + R and O −R can be embedded in O R as sub-presheaves. This follows immediately from the fact that, by construction,the presheaf O + R is isomorphic to the presheaf of completely increasing func-tions on the context category CON ( R ) of R and, analogously, that thepresheaf O −R is isomorphic to the presheaf of completely decreasing functionson CON ( R ).In contrast with the upper and lower observable presheaves, O R is a presheafof real vector spaces and linear maps. But it is not at all obvious how tointerpret it at the level of operators or even physically. We leave this problemto future work. Let R be a von Neumann algebra acting on a Hilbert space H . We willnow consider the upper and lower observable presheaves O ±R on the contextcategory CON ( R ), defined in the previous section, more closely. We restrictour considerations to the upper presheaf because for the lower presheafresults and proofs are completely analogous.We have seen that every observable function f ∈ O ( R ) induces a family( f A ) A∈ A ( R ) of observable functions f A ∈ O ( A ), defined by f A := ̺ RA f . Thisfamily has the following compatibility property: ∀ A , B ∈ A ( R ) : ̺ AA∩B f A = ̺ BA∩B f B . (2)( f A ) A∈ A ( R ) is therefore a global section of the presheaf O + R in the followinggeneral sense. Definition 6.1.
Let C be a category and S : C → Set a presheaf, i.e. acontravariant functor from C to the category Set of sets. A global section of S assigns to every object a of C an element σ ( a ) of the set S ( a ) such thatfor every morphism ϕ : b → a of C σ ( b ) = S ( ϕ )( σ ( a )) holds. Not every presheaf admits global sections. An important example is the spectral presheaf of the von Neumann algebra R . This is the presheaf S : A ( R ) → CO from the category A ( R ) of abelian von Neumann subalgebrasof R to the category CO of compact Hausdorff spaces which is defined by20i) S ( A ) := Q ( A ) for all A ∈ A ( R ),(ii) S ( A ֒ → B ) := π BA , where the mapping π BA : Q ( B ) → Q ( A ) is defined inlemma 4.1.We know from theorem 3.2 in [4] that there is a canonical homeomor-phism ω A : Q ( A ) → Ω( A ). This homeomorphism intertwines the ordinaryrestriction r BA : Ω( B ) → Ω( A ) τ τ | A with π BA : r BA ◦ ω B = ω A ◦ π BA . This shows, according to a reformulation of the Kochen-Specker theoremby J. Hamilton, C.J. Isham and J. Butterfield ([12], [ ? ]) that the presheaf S : A ( R ) → CO admits no global sections.In the case of the observable presheaf O + R there are plenty of global sec-tions because each A ∈ R sa induces one. Here the natural question ariseswhether all global sections of O + R are induced by selfadjoint elements of R .This is certainly not true if the Hilbert space H has dimension two. For inthis case the constraints 2 are void and therefore any function on the complexprojective line defines a global section of O + L ( H ) . But Gleason’s (or Kochen-Specker’s) theorem teaches us that the dimension two is something peculiar.We will show, however, that the phenomenon, that there are global sectionsof O R that are not induced by selfadjoint elements of R , is not restricted todimension two. Definition 6.2.
We denote by Γ( O + R ) the set of global sections of the ob-servable presheaf O + R . The image of the canonical mapping Σ R : O ( R ) → Γ( O + R ) f ( ̺ RA f ) A∈ A ( R ) is denoted by Σ( O ( R )) . We will show in the sequel that Γ( O + R ) is strictly larger than Σ( O ( R )) for R = L ( C ). The example that we shall give for this case can be generalizedeasily to higher dimensions.We begin with a general Remark 6.1.
Let f ∈ O ( R ) such that Σ R ( f ) is a family of observable func-tions of projections. Then also f is the observable function of a projection. roof: Let A ∈ R sa such that f = f A . Then A ∈ A for some A ∈ A ( R )and therefore imf = im̺ RA f ⊆ { , } . Hence A is a projection. (cid:3) Now we consider the special case R := L ( H ) , H := C more closely. Remark 6.2.
Let E ∈ R be a projection of rank two and f := f E thecorresponding observable function. If P ∈ P ( L ( H )) , i.e. a projection ofrank one, then f ( P ) = ( if P = I − E otherwise . Hence f ( P ) = 0 for exactly one P ∈ P ( L ( H )) , namely P = I − E . Each twodimensional von Neumann subalgebra A of L ( C ) is abelianand generated by exactly one projection P ∈ P ( L ( C )). Moreover the max-imal abelian von Neumann subalgebras M of L ( C ) are threedimensionaland they are determined by orthogonal triples ( P , P , P ) ∈ P ( L ( C )) : M = lin C { P , P , P } . Two such triples determine the same algebra M ifand only if one is a permutation of the other.Now let P , P ∈ P ( L ( C )) be projections that do not commute (i.e. P P = 0) and for k = 1 , A k := {A ∈ A ( L ( C )) | P k ∈ A} . Define a family ( Q A ) A∈ A ( L ( C )) of projections Q A ∈ A by Q A := ( I − P k if A ∈ A k ( k = 1 , I if A / ∈ A ∪ A and let f A be the observable function of Q A . We show that ( f A ) A∈ A ( L ( C )) is aglobal section of O L ( C ) . In order to do that it is convenient to work directlywith the projections Q A instead with their observable functions because re-stricting a projection P to A ∈ A ( L ( C )) means passing to its A -support s A ( P ).We have to prove that the constraints 2 are satisfied for the family( Q A ) A∈ A ( L ( C )) . Since I remains unchanged by restriction we have to con-trol the behaviour of I − P k , ( k = 1 , I − P k ) | A = ( I − P k if P k ∈ A I otherwise , | A is a shortcut for the restriction to A . If A ∈ A k , B / ∈ A k and C ⊆ A ∩ B , then P k / ∈ C and therefore ( I − P k ) | C = I . If A ∈ A ( L ( C )) hasdimension two and is contained in A ∩ A , where A k ∈ A k ( k = 1 , A = lin C { Q, I − Q } with Q ∈ P ( L ( C )) orthogonal to P ∨ P . But then P , P / ∈ A and therefore ( I − P k ) | A = I for k = 1 ,
2. Altogether this showsthat ( f A ) A∈ A ( L ( C )) is a global section.This global section cannot be induced by an observable function f ∈O ( L ( C )): If this were the case then, by remark 6.1, f would be the ob-servable function of a projection E and, according to the definition of thefamily ( f A ) A∈ A ( L ( C )) , E must be of rank two. But then f ( P ) = f ( P ) = 0 , a contradiction to remark 6.2. Therefore we have proved: Proposition 6.1. Γ( O L ( C ) ) is strictly larger than Σ( O ( L ( C ))) . This leads us to the following
Definition 6.3.
Let R be a von Neumann algebra. The global sections ofthe observable presheaf O + R are called (upper) contextual observables . Clearly, Γ( O + R ) = Σ( O ( R )) if R is abelian.Contextual observables can be characterized as certain functions on P ( R ): Proposition 6.2.
Let R be a von Neumann algebra. There is a one-to-onecorrespondence between global sections of the observable presheaf O + R andfunctions f : P ( R ) → R that satisfy(i) f ( W k ∈ K P k ) = sup k ∈ K f ( P k ) for all commuting families ( P k ) k ∈ K in P ( R ) ,(ii) f | P R ) ∩A is bounded for all A ∈ A ( R ) .Proof: Let ( f A ) A∈ A ( R ) be a global section of O + R . Then the functions f A : P ( R ) ∩ A ( A ∈ A ( R )) can be glued to a function f : P ( R ) → R :Let P ∈ P ( R ) and let A be an abelian von Neumann subalgebra of R thatcontains P . Then f ( P ) := f A ( P )does not depend on the choice of A . Indeed, if P ∈ A ∩ B , then f A ( P ) = f B ( P ) by the compatibility property of global sections. It is23bvious that f satisfies properties ( i ) and ( ii ).If, conversely, a function f : P ( R ) → R with the properties ( i ) and( ii ) is given and if A is an abelian von Neumann subalgebra of R , then f A := f | P R ) ∩A is a completely increasing function. The family ( f A ) A∈ A ( R ) isthen, by construction, a global section of O + R . (cid:3) Note that the conditions in proposition 6.2 are much weaker than thecondition of being completely increasing, so it is not surprising that there areglobal sections of O R which are not induced by a single selfadjoint operator. Although “states” do not belong to our proper theme, we will include someremarks about quantum states emphasising again the presheaf perspective.A state of a von Neumann algebra R is a positive linear functional ϕ : R → C with ϕ ( I ) = 1. We denote the (convex and weak* compact) setof states of R by S ( R ).There is a natural restriction of states to von Neumann subalgebrasof R : Definition 7.1.
Let M be a von Neumann subalgebra of R . The usualrestriction of mappings defines a restriction map st RM : S ( R ) → S ( M ) ϕ ϕ | M .st RM is a surjective mapping (see [13], p.266) and for any three von Neu-mann subalgebras A , B , C of R such that A ⊆ B ⊆ C we have the obviousproperties st CA = st BA ◦ st CB and st AA = id S ( A ) . (3)If we consider in particular the abelian von Neumann subalgebras of R weobtain, due to 3, a presheaf S R that is in some sense dual to the observablepresheaf O R : Definition 7.2.
The contravariant functor S R : A ( R ) → Set , defined onobjects by S R ( A ) := S ( A ) and on morphisms by S R ( A ֒ → B ) := st BA , s called the state presheaf of the von Neumann algebra R . Each state ϕ ∈ S ( R ) gives rise to a global section ( ϕ A ) A∈ A ( R ) , defined by ϕ A := ϕ | A , of the presheaf S R . Also here arises the natural question whether all globalsections of S R are of this form.We will show in the sequel that the answer is affirmative, provided that R has no direct summand of type I . So here again the dimension twoforms the notorious exception. Nevertheless this contrasts to the situation ofthe observable presheaf where we can find counterexamples in all dimensions.Let ( ϕ A ) A∈ A ( R ) be a global section of S R . If A ∈ R sa then A ∈ A forsome A ∈ A ( R ). If A belongs also to B ∈ A ( R ) then A ∈ A∩B and therefore ϕ A ( A ) = ϕ A∩B ( A ) = ϕ B ( A ) . This shows that the global section ( ϕ A ) A∈ A ( R ) determines a function ϕ : R sa → R , defined by ∀ A ∈ R sa ∀ A ∈ A ( R ) : ( A ∈ A = ⇒ ϕ ( A ) = ϕ A ( A )) . Clearly ϕ ( I ) = 1 and ϕ ( A ∗ A ) ≥ A ∈ R . The salient point is the R -linearity of ϕ .Indeed, if R = L ( C ), then ϕ may fail to be linear. To see this, notethat the maximal abelian von Neumann subalgebras of L ( C ) are of the form A P := lin C { P, I − P } with a projection P = 0 , I . A state ϕ A P on A P is givenby prescribing an arbitrary value a P ∈ [0 ,
1] to P and the condition ϕ A P ( I − P ) = 1 − a P . Because the intersection of two different maximal abelianvon Neumann subalgebras of L ( C ) equals C I , there are no constraints for aglobal section of the state presheaf. Now assume that each global section ofthe state presheaf S L ( C ) is induced by a state of L ( C ). Let P, Q ∈ L ( C ) betwo noncommuting projections of rank one. Since L ( C ) is noncommutative, P + Q generates a maximal abelian von Neumann subalgebra A R of L ( C )and, because P does not commute with Q , the subalgebras A P , A Q and A R are pairwise different. Now P + Q = aR + b ( I − R ) (4)25ith uniquely determined real numbers a, b . To each choice of a P , a Q , a R ∈ [0 ,
1] there is a global section and therefore by assumption a state ϕ of L ( C )such that ϕ ( P ) = a P , ϕ ( Q ) = a Q , ϕ ( R ) = a R . Choosing a P = a Q = a R = 0, 4 implies b = 0 and the choice a P = a Q = 0 , a R = 1 leads to a = 0. This contradicts 4.We return to the discussion of the mapping ϕ : R sa → R defined by aglobal section ( ϕ A ) A∈ A ( R ) of S R . Let P , . . . , P n ∈ P ( R ) be pairwise orthog-onal. Then P , . . . , P n ∈ A for some A ∈ A ( R ) and therefore ϕ ( n X j =1 P j ) = ϕ A ( n X j =1 P j ) = n X j =1 ϕ A ( P j ) = n X j =1 ϕ ( P j ) . This implies that ϕ | P ( R ) : P ( R ) → [0 ,
1] is a (finitely additive) probabilitymeasure on the projection lattice P ( R ). Here we can apply a substantialgeneralization of Gleason’s theorem, due to Christensen, Yeadon et al.: Theorem 7.1. ([19], thm. 12.1) Let R be a von Neumann algebra withoutdirect summand of type I and let µ : P ( R ) → [0 , be a finitely additiveprobability measure on P ( R ) . Then µ can be extended to a unique state of R . It follows from the spectral theorem that a state ϕ of R is uniquelydetermined by its restriction to P ( R ). Hence we obtain from theorem 7.1and the previous discussion: Theorem 7.2.
The states of a von Neumann algebra R without direct sum-mand of type I are in one to one correspondence to the global sections of thestate presheaf S R . This correspondence is given by the bijective map Γ R : S ( R ) → Γ( S R ) ϕ ( ϕ | A ) A∈ A ( R ) . Clearly, the core of this theorem is the surjectivity of the mapping Γ R . Wewill show that this property implies that each probability measure on P ( R )can be extended to a state of R . Thus theorem 7.2 is indeed equivalent totheorem 7.1. Proposition 7.1.
The following properties of a von Neumann algebra R areequivalent:(i) Every finitely additive measure on P ( R ) extends to a state of R . ii) Every global section of S R is induced by a state of R .Proof: According to the foregoing discussion it remains to show that aprobability measure µ on P ( R ) defines a global section of S R . Let A ∈ A ( R ).Then µ A := µ | P ( A ) is a probability measure on P ( A ). We construct from µ A a state ϕ A of A that extends µ A . This construction is quite similar to thatused in the proof that the Gelfand spectrum of A is homeomorphic to itsStone spectrum ([4]). Let A := m X j =1 a j P j (5)where P , . . . , P m ∈ P ( A ) are pairwise orthogonal and a , . . . , a m are complexnumbers. Since we do not assume that the coefficients a j are all differentfrom zero, we can and do assume that P mj =1 P j = 1, i.e. that ( P , . . . , P m )is a partition of unity. We then call 5 a normalized representation of A ∈ lin C P ( A ). Each element of lin C P ( A ) has a normalized representation. Inorder to extend µ A linearly we are forced to define ϕ A ( A ) := m X j =1 a j µ A ( P j ) (6)for A ∈ lin C P ( A ) given by 5. To show that this is well defined, consideranother normalized representation P nk =1 b k Q k of A . If x ∈ im ( P j Q k ) then a j x = Ax = b k x , hence ∀ j, k : ( P j Q k = 0 = ⇒ a j = b k ) . (7)This implies X j a j µ A ( P j ) = X j,k a j µ A ( P j Q k ) = X j,k b k µ A ( P j Q k ) = X k b k µ A ( Q k ) . Clearly ϕ A ( aA ) = aϕ A ( A ) for all a ∈ C and A ∈ lin C P ( A ) in normal-ized representation. If A, B ∈ lin C P ( A ) have normalized representations P j a j P j , P k b k Q k respectively then A + B = X j a j P j + X k b k Q k = X j,k a j P j Q k + X j,k b k P j Q k = X j,k ( a j + b k ) P j Q k ϕ A ( A + B ) = X j,k ( a j + b k ) µ A ( P j Q k )= ϕ A ( A ) + ϕ A ( B ) . It is obvious that A ∈ lin C P ( A ) is positive if in a normalized representation P j a j P j of A all coefficients are nonnegative. Hence ϕ A : lin C P ( A ) → C isa positive linear functional with ϕ A ( I ) = µ A ( I ) = 1. Now | X j a j P j | = max j | a j | for every normalized representation P j a j P j of A ∈ lin C P ( A ) and | ϕ A ( X j a j P j ) | = | X j a j µ A ( P j ) |≤ X j | a j | µ A ( P j ) ≤ max j | a j | , so | ϕ A | = 1. This implies that ϕ A has a unique extension to A = lin C P ( A )and that this extension, which we also denote by ϕ A , is a state of A .If A , B ∈ A ( R ) such that A ⊆ B then trivially µ A = µ B | P ( A ) and therefore,according to the foregoing construction, ϕ A = ϕ B | A . So we have constructeda global section ( ϕ A ) A∈ A ( R ) from the probability measure µ on P ( R ). (cid:3) Each state ϕ A of A ∈ A ( R ) can be seen as a positive Radon measure on C ( Q ( A )): ϕ A : C ( Q ( A )) → C ψ ϕ A ( A ψ )where A ψ ∈ A is obtained from ψ ∈ C ( Q ( A )) by the inverse of the Gelfandtransformation. Therefore it induces a probability measure ν A on Q ( A ) suchthat ∀ ψ ∈ C ( Q ( A )) : ϕ A ( ψ ) = Z Q ( A ) ψdν A . (8)Identifying the Gelfand transform of A ∈ A with the (complexified) observ-able function f A A we can write this as ϕ A ( A ) = Z Q ( A ) f A A dν A . (9)28rom the definition of ν A we get immediately ∀ P ∈ P ( A ) : ϕ A ( P ) = ν A ( Q P ( A )) . (10)We denote by M ( Q ( A )) the set of probability measures on Q ( A ). Thesesets form a presheaf M R on the category A ( R ). For A , B ∈ A ( R ) such that A ⊆ B we define p BA : M ( Q ( B )) → M ( Q ( A )) ν π BA ν, where π BA ν denotes the image of the measure ν under the continuous mapping π BA : Q ( B ) → Q ( A ) , β β ∩ P ( A ). It is defined by ∀ ψ ∈ C ( Q ( A )) : Z Q ( A ) ψdπ BA ν := Z Q ( B ) ( ψ ◦ π BA ) dν (11)or, equivalently, by ( π BA ν )( M ) := ν ( − π BA ( M )) (12)for all Borel subsets M of Q ( A ). It is obvious that for A , B , C ∈ A ( R ) suchthat A ⊆ B ⊆ C we have p CA = p BA ◦ p CB and p AA = id M ( Q ( A )) . (13)So M R , together with the restriction mappings p BA , is a presheaf on A ( R ).We recall the following well known Definition 7.3.
Let R be a von Neumann algebra. A state ϕ of R is callednormal if ϕ (sup k ∈ K A k ) = sup k ∈ K ϕ ( A k ) for every increasing bounded net ( A k ) k ∈ K in R .If R is abelian, R = C ( Q ( R )) , then a probability measure ν on Q ( R ) iscalled normal if the corresponding state ϕ ν : ψ R Q ( R ) ψdν of R is normal. We call a global section of S R or M R normal if all of its members arenormal. Lemma 7.1.
Let ϕ be a state of a von Neumann algebra R . Then ϕ isnormal if and only if ϕ A := ϕ | A is normal for all A ∈ A ( R ) . roof: It is obvious that all ϕ A ( A ∈ A ( R )) are normal if ϕ is normal.For the converse we use the result that a state ϕ is normal if it is normalon P ( R ), i.e. if ϕ ( P k ∈ K P k ) = P k ∈ K ϕ ( P k ) for every orthogonal family( P k ) k ∈ K of projections in R ([14], Thm. 7.1.12). Because every such familyis contained in a suitable A ∈ A ( R ), the assertion follows. (cid:3) Proposition 7.2.
The presheaves M R and S R over A ( R ) are isomorphic.This isomorphism maps normal sections to normal sections.Proof: The isomorphism is simply given by the family (Φ A ) A∈ A ( R ) of(convex-linear) maps Φ A : M ( Q ( A )) → S ( A ) ν ϕ ν where ϕ ν is defined by ϕ ν ( A ) := R Q ( A ) F A ( A ) dν . The Riesz representationtheorem ensures that the mappings Φ A are convex-linear isomorphisms. Soit remains to show that the Φ A are compatible with restrictions: ∀ A , B ∈ A ( R ) : ( A ⊆ B = ⇒ (Φ B ( ν B )) | A = Φ A ( p BA ( ν B ))) . Due to linearity and continuity it suffices to check this for projections. If P ∈ P ( A ) then, using 10, we have(Φ B ( ν B ))( P ) = ν B ( Q P ( B ))= ν B ( − π BA ( Q P ( A )))= ( p BA ( ν B ))( Q P ( A ))= (Φ A ( p BA ( ν B ))( P ) . The normality assertion is obvious. (cid:3)
Let H be a separable Hilbert space of dimension greater than two and ϕ a normal state of L ( H ). The theorem of Gleason ([14]) assures that ϕ isinduced by a positive traceclass operator ̺ of trace one: ∀ A ∈ L ( H ) : ϕ ( A ) = tr ( ̺A ) . If ̺ = P C x with x ∈ S ( H ), then ϕ ( A ) = < Ax, x > , and in this case ϕ iscalled a vector state . We recall that a normal state of a maximal abelian vonNeumann subalgebra of a von Neumann algebra is always the restriction ofa vector state ([14]). The following result describes the situation when thisrestriction is a pure state. 30 heorem 7.3. Let H be a separable Hilbert space of dimension greater thantwo, M ⊆ L ( H ) a maximal abelian von Neumann subalgebra of L ( H ) and ϕ a normal state of L ( H ) . Then the Radon measure µ on Q ( M ) , induced by ϕ | M , is the point measure ε β for some β ∈ Q ( M ) , if and only if there is an x ∈ S ( H ) such that P C x ∈ M , β = β C x and ϕ is the vector state definedby x .Proof: Let x ∈ S ( H ) such that P C x ∈ M and let ρ := P C x . Let ψ bea real valued continuous function on Q ( M ) and let A ψ ∈ M be the corre-sponding hermitian operator. P C x commutes with A ψ , so x is an eigenvectorof A ψ . Let λ be the corresponding eigenvalue. Then tr ( ρA ψ ) = < A ψ x, x > = λ < x, x > = λ. On the other hand λ = f A ψ ( β C x ) = ε β C x ( f A ψ ) = ε β C x ( ψ ) . Therefore µ = ε β C x . Conversely, let ϕ be a normal state, ϕ = tr ( ̺ − ), and let M be a maximalabelian von Neumann subalgebra of L ( H ) such that the measure µ on Q ( M ),corresponding to ϕ | M is the point measure ε β for some β ∈ Q ( M ). Thenfor all P ∈ P ( M ) tr ( ρP ) = µ ( χ Q P ( M ) ) = χ Q P ( M ) ( β )and therefore ∀ P ∈ P ( M ) : ( P ∈ β ⇐⇒ tr ( ρP ) = 1) . Let P ∈ β and let ( e k ) k ∈ N be an P -adapted orthonormal basis of H , i.e. e k ∈ imP ∪ kerP for all k ∈ N . Then1 = tr ( ρP )= tr ( P ρ )= X k < P ρe k , e k > = X k < ρe k , P e k > = X e k ∈ imP < ρe k , e k > . < ρe k , e k > ≥ k ∈ N and trρ = 1 we conclude that ∀ e k ∈ kerP : < ρe k , e k > = 0 . Hence ρe k = 0 for all e k ∈ kerP and therefore ρ ( I − P ) = 0i.e. ρ = ρP. In particular ρP = ρ = ρ ∗ = P ρ.
This implies that ̺ commutes with M and therefore, since M is maximalabelian, ̺ ∈ M . Hence the range projection P im̺ belongs to P ( M ). Nowfor all y ∈ H ρy = ρP y = P ρy and therefore ∀ P ∈ β ∀ y ∈ H : ρy ∈ imP. This implies ∀ P ∈ β : imρ ⊆ imP, and from the maximality of β we conclude that P imρ ∈ β . P imρ is thereforethe minimal element of β . Let P ∈ L ( H ) be a projection of rank one suchthat P ≤ P imρ . Then P commutes with β and therefore with P ( M ), since Q ∈ β or I − Q ∈ β for all Q ∈ P ( M ). Hence imρ = imρ = C x for aunique line C x in H . Therefore P C x ∈ P ( M ) and β = β C x .There is a unique λ ∈ C such that ρx = λ x.ρ ≥ trρ = 1 imply λ = 1. Hence for all y ∈ H ρ y = ρ ( ρy ) = ρ ( λx ) = λρx = λx = ρy and therefore ρ = P C x . (cid:3) In classical physics, observables are continuous functions on some set of pure states . The connection with the definition of states of a von Neumannalgebra is simply established by considering a pure state of classical physics32s a point measure (evaluation functional) on a space of continuous func-tions. This is motivated by the definition of mixed states as probabilitymeasures (positive Radon measures of norm one) on the set of pure states.It should be discussed whether the linearity of states is due to this processof averaging . One advantage of the linearity of states is that the notion ofsuperposition of states makes no (mathematical) difficulties.We already mentioned in earlier parts of this work that, since observ-ables can be considered as continuous functions on the Stone spectrum, onemight think of the elements of the Stone spectrum as “pure quasistates”.This, however, is a bit naive. It is obvious that a vector state P C x canbe identified with the atomic quasipoint B x , but f A ( B x ) does give theexpectation value of A , if the physical system is in the state P C x , only when x is a normed eigenvector of A . We obtain the expectation value < Ax, x > as < Ax, x > = f P C x AP C x ( B x )and we can prove ([8]) that f P C x AP C x ( B x ) = min P ∈ B x f P AP ( B x ) . This leads to a possible generalization for arbitrary quasipoints of an arbi-trary von Neumann algebra:
Definition 7.4.
Let R be a von Neumann algebra, A ∈ R sa , B ∈ Q ( R ) and ˆ B ( A ) := inf P ∈ B f P AP ( B ) . The function ˆ B : R sa → R A ˆ B ( A ) is called the quasistate of R induced by B . Of course the same definition also applies to an arbitrary dual ideal in P ( R ). Remark 7.1.
This definition is in accordance with the description of vectorstates, but also with the notion of pure state for an abelian von Neumannalgebra R : If β ∈ Q ( R ) and A ∈ R sa , then ˆ β ( A ) = inf P ∈ β f P AP ( β ) = f A ( β ) since f P AP = f P f A in the abelian case, and f P ( β ) = 1 for P ∈ β . P, Q ∈ P ( R ) such that P ≤ Q does not implythat f P AP ≤ f QAQ . For this would mean that
P AP ≤ s QAQ , but even
P AP ≤ QAQ is not true in general. Moreover, according to theorem ofNeumark, it is hopeless to obtain the spectral family of A from the spectralfamily of P AP in a manageable way. Possibly, one can circumvent thesedifficulties by describing the observable function of A in terms of the operator A , without using its spectral family. References [1] H. Casini: The logic of causally closed space-time subsets,arXiv: gr-qc/02 05 013 v2[2] H.F. de Groote: On a Canonical Lattice Structure on the Effect Algebraof a von Neumann Algebra,arXiv: math-ph/04 10 018 v2 (2005)[3] H.F. de Groote: Observables,arXiv:math-ph/0507019 v1 7 Jul 2005[4] H.F. de Groote: Observables I : Stone Spectra,arXiv: math-ph/05 09 020[5] H.F. de Groote: Observables II : Quantum Observables,arXiv: math-ph/05 09 075[6] H.F. de Groote: Observables III : Classical Observables,arXiv: math-ph/06 01 011[7] H.F. de Groote: Stone spectra of finite von Neumann algebras of typeI n ,arXiv:math-ph/0605020v1w[8] H.F. de Groote: States (In preparation)[9] A. D¨oring: Observables as functions: antonymous functions,arXiv:quant-ph/0510102v1[10] A. D¨oring and C. J. Isham: A Topos Foundation for Theories of Physics:II. Daseinisation and the Liberation of Quantum Theory.arXiv:quant-ph/0703062v1 7 Mar 20073411] A. D¨oring and C. J. Isham: A Topos Foundation for Theories of Physics:III. The Representation of Physical Quantities With Arrows ˘ δ o ( A ) : Σ → R .arXiv:quant-ph/0703064v1 7 Mar 2007[12] J. Hamilton, C.J. Isham, J. Butterfield: A topos perspective on theKochen-Specker theorem: III. Von Neumann algebras as the base cate-gory,Int. J. Theor. Phys. (2000), 1413-1436[13] R.V. Kadison and J.R. Ringrose: Fundamentals of the Theory of Oper-ator Algebras, Vol. I (Elementary Theory)Am.Math.Soc. 1997[14] R.V. Kadison and J.R. Ringrose: Fundamentals of the Theory of Oper-ator Algebras, Vol. II (Advanced Theory)Am.Math.Soc. 1997[15] R.V. Kadison and J.R. Ringrose: Fundamentals of the Theory of Oper-ator Algebras, Vol. III Special Topics (Elementary Theory)Birkh¨auser Boston 1991[16] R.V. Kadison and J.R. Ringrose: Fundamentals of the Theory of Oper-ator Algebras, Vol. IV Special Topics (Advanced Theory)Birkh¨auser Boston 1992[17] S. Lang: AlgebraAddison-Wesley, 1966[18] S. MacLane and I. Moerdijk: Sheaves in Geometry and LogicSpringer Verlag, 1992[19] S. Maeda: Probability measures on projections in von Neumann algebrasRev. Math. Phys. (1990), 235-290[20] M.P. Olson: The Selfadjoint Operators of a von Neumann Algebra Forma Conditionally Complete LatticeProc. of the AMS28