STONE SPECTRA OF VON NEUMANN ALGEBRAS OF TYPE I 5 n 0 0 2 ANDREASDO¨RING IAMPH,FACHBEREICHMATHEMATIK n J.W.GOETHE-UNIVERSITA¨TFRANKFURT,GERMANY a J 8 Abstract. TheStonespectrumofavonNeumannalgebraisageneralization 1 oftheGelfandspectrum,aswasshownbydeGroote. Inthisarticleweclarify thestructureoftheStonespectraofvonNeumannalgebrasoftypeIn. ] A O . 1. Introduction h t In a new approachtying together classical and quantum observables, de Groote a m has developed the theory of Stone spectra of lattices and observable functions. In thissection,wewillgivetherelevantdefinitionsandcitesomeofdeGroote’sresults. [ For details, the reader is referred to de Groote’s work [deG01, deG05]. 1 v Whenspeakingofalattice,wewillalwaysmeanaσ-completelattice atleast. A 8 latticeLalwayshasazeroelement0andaunitelement1. Thestartingpointisthe 7 2 new notion of a quasipoint of a lattice, which is nothing but a maximal filter base, 1 generalizing to arbitrary lattices what Stone did in the 1930s for Boolean algebras 0 [Sto36]: 5 0 Definition 1. A subset B of a lattice L is called a quasipoint of L if it has the / following properties: h t (i) 0∈/ B, a (ii) ∀a,b∈B ∃c∈B:c≤a∧b, m (iii) B is maximal with respect to (i) and (ii). : v It is easily seen that for a quasipoint B of the lattice L, we have i X ∀a∈B ∀b∈L:(a≤b=⇒b∈B). r a In particular, ∀a,b ∈ B : a∧b ∈ B, so quasipoints are maximal dual ideals also [Bir73]. The set of quasipoints of a lattice L is denoted by Q(L) and is equipped with a natural topology (also inspired by Stone): for a∈L, let Q (L):={B∈Q(L) | a∈B}. a Obviously, we have Q (L)=Q (L)∩Q (L), a∧b a b so {Q (L) | a∈L} is the base of a topology on L. a Definition2. ThesetQ(L)ofquasipoints ofalatticeL, equipped withthetopology givenbythesetsQ (L)definedabove, iscalled theStone spectrum of the lattice a L. Date:17. January2005. 1 2 ANDREASDO¨RING One can show that the Stone spectrum Q(L) is a zero-dimensional, completely regularHausdorffspace. FortheexampleL=L(H), thelatticeofclosedsubspaces of a complex separable Hilbert space H, Q(L) is not compact if dimH > 1. If H is infinite-dimensional, then Q(L) is not even locally compact [deG01]. The lattice L(H) plays an important role in the foundations of quantum theory, which was first recognized by Birkhoff and von Neumann [BirvNeu36]. L(H) is isomorphic the P(H), the lattice of projections onto closed subspaces of H. Let R be a unital von Neumann algebra, given as a subalgebra of the algebra L(H) of bounded operators on some Hilbert space H. The Stone spectrum Q(R) of R means the Stone spectrum Q(P(R)) of the projection lattice P(R) of R. In quantumtheory,includingquantummechanicsinthevonNeumannrepresentation, quantumfieldtheoryandquantuminformationtheory,observables arerepresented byself-adjointoperatorsAinsomevonNeumannalgebraR, the algebra of observ- ables. The set of observables R forms a real linear space in the algebra R. sa De Groote shows that if R is abelian, there is a homeomorphism between the StonespectrumQ(R)=Q(P(R))andtheGelfandspectrumΩ(R)ofR. Hence,for anarbitraryvonNeumannalgebraR, theStonespectrumQ(R)isageneralization of the Gelfand spectrum. Observable functions are introduced in the following way: Definition 3. Let A ∈ Rsa, and let EA = (EλA)λ∈R be the spectral family of A. The function f :Q(R)−→R A defined by f (B):=inf{λ∈R | EA ∈B} A λ is called the observable function corresponding to A. Ob(R) := {f | A ∈ A R } denotes the set of observable functions of R. sa One can show that the image of f is the spectrum of A. Observable functions A are continuous functions, so Ob(R)⊆C (Q(R),R), b where C (Q(R),R) denotes the set of continuous bounded real-valued functions b on the Stone spectrum Q(R) of R. De Groote shows that equality only holds for abelian von Neumann algebras R. Moreover, if R is abelian, then the observable functions turn out to be the Gelfand transforms of the self-adjoint elements of R, so for an arbitrary von Neumann algebra, the observable functions are generalized Gelfand transforms. There are many more results on Stone spectra and observable functions and their relationto physics [deG01, deG05]. For example,observable functions canbe characterizedintrinsically,withoutreferencetoself-adjointoperators. Stonespectra also play some role in the proof of the generalized Kochen-Speckertheorem, which is an important no-go theorem on hidden variables in quantum theory [KocSpe67, Doe04]. STONE SPECTRA OF VON NEUMANN ALGEBRAS OF TYPE In 3 The elements of the Stone spectrum Q(R), the quasipoints, are defined using Zorn’s lemma. As usual, it is not easy to get some intuition of such objects. Some extra structure of the von Neumann algebra is needed to clarify the properties of the quasipoints and the Stone spectrum. The quasipoints and the Stone spectrum of type-I -factors R = L(H), where dimH = n ∈ {0,1,...} is finite, are known n [deG01]. We need the following: an isolated point B of the Stone spectrum Q(R) iscalledanatomicquasipoint. IfR=L(H),thentheatomicquasipointsofP(R)(= P(H)) are of the form BCx ={P ∈P(H) | P ≥PCx}, wherex∈H,|x|=1andPCx istheprojectionontothe lineCx. While forinfinite- dimensionalH,P(H)alsohasnon-atomicquasipoints,forfinite-dimensionalHthe situation is simple: Proposition 4. If H is finite-dimensional, there are only atomic quasipoints in P(H), and we have Q(P(H))={BCx | x∈S1(H)}, where S1(H) denotes the unit sphere in Hilbert space. For finite-dimensional H, P(H)≃L(H)istheprojectionlatticeP(R)ofatypeI factorR,wheren=dimH. n This type I factor simply is represented as M (C), the n×n complex matrices n n acting on H. Ofcourse,theinnerproductofHplaysnorolehere,butonlythelinearstructure, so we have also characterized the quasipoints of the lattice of subspaces of a finite- dimensional vector space. There wereno results on the structure of quasipoints and Stone spectra ofmore general von Neumann algebras up to now. In this article, we will examine von NeumannalgebrasoftypeI forfiniten. TypeI algebrasincludeallvonNeumann n n algebrasonfinite-dimensionalHilbertspacesandallabelianvonNeumannalgebras. (ThelatterareofthoseoftypeI .) Wewillmakeuseofthefactthatsuchanalgebra 1 is of the form M (A), where A is the center of R. Note that while R ≃ M (A) n n is given by “finite” n×n-matrices, the center A of R may be represented on an infinite-dimensional Hilbert space. Drawingonaresultonabelianquasipoints (section2),i.e. quasipointscontaining an abelian projection, a fairly complete characterization of the Stone spectrum of atype I algebraisobtained. Inparticular,allquasipointsofatypeI algebraare n n abelian (Thm. 34) and the orbits of the action of the unitary group on the Stone spectrumQ(R) areparametrizedby the quasipoints ofthe center ofR (Thm. 36). 2. Abelian quasipoints of von Neumann algebras In this section, we will regard quasipoints containing an abelian projection. It will be shown that there is a close relationship between the abelian quasipoints of a von Neumann algebra R and the quasipoints of the center of R. This result will be central to the classification of Stone spectra of type I von Neumann algebras. n Definition 5. A quasipoint B ⊆ P(R) is called abelian if it contains an abelian projection E ∈ R. The set of abelian quasipoints of a von Neumann algebra R is denoted by Qab(R). 4 ANDREASDO¨RING Definition 6. The E-trunk B (E ∈B) of a quasipoint B is the set E B :={F ∈B | F ≤E}. E Obviously, B is a filter basis. E Lemma 7. The E-trunk B uniquely determines the quasipoint B. E Proof. LetB ,B be twoquasipointswhoseE-trunkisB . LetF be aprojection 1 2 E in B . Then we have E∧F ∈B ⊂ B . If a quasipoint contains a projection, it 1 E 2 contains all larger projections, so F ∈B and B =B follows. (cid:3) 2 1 2 This lemma holds analogously for any lattice L, since no features of the von Neumann algebra are used. Definition8. Let R⊆L(H) be a von Neumannalgebra, B⊂Q (R) a quasipoint E containing E and θ ∈R a partial isometry such that E =θ∗θ. We set θ(B ):={θFθ∗ | F ∈B }. E E Lemma9. IfR⊆L(H)is a vonNeumannalgebra and θ ∈Ris a partial isometry such that E :=θ∗θ, then for all projections P ∈R such that P ≤E it holds that U U θP θ∗ =P . U θU Proof. For x∈U, we have θP θ∗θx=θP Ex=θx=P θx. U U θU If y ∈(θU)⊥, then θ∗y ∈U⊥ and thus θP θ∗y =0=P y. U θU (cid:3) Obviously, θ(B ) is the θEθ∗-trunk of a quasipoint of R (given that θ∗θ =E). E We will denote the quasipoint induced by θEθ∗ by θ (B ). Q E Remark 10. Since in general θ∗θ ∈/ B for an arbitrary partial isometry θ and an arbitrary quasipoint B, we have no action of the set of partial isometries on the Stone spectrum Q(R). On the other hand, if θ is unitary, we can define an operation, see subsection 3. Let B ∈ Qab(R) be an abelian quasipoint, and let E ∈ B be an abelian pro- jection. Each F ∈B is a subprojection of the abelian projection E and hence of E the form F = QE, where Q ∈ R is a central projection. Then Q ∈ B holds, so Q∈B∩C. If, conversely,Q∈B∩C holds, then QE ∈B , therefore we have E B ={QE | Q∈B∩C}. E The mapping ζ :R′E −→R′C , E E T′E 7−→T′C E STONE SPECTRA OF VON NEUMANN ALGEBRAS OF TYPE In 5 is a ∗-isomorphism (see Prop. 5.5.5 in [KadRinI97]). Since PE∧QE =PQE =(P ∧Q)E, PE∨QE =(P +Q)E−PQE =(P ∨Q)E, ζE|BE is a lattice isomorphism from BE onto (B∩C)CE. B∩C is a quasipoint of P(C): obviously, B∩C is a filter basis P(C). Let β be a quasipoint in P(C) containing B∩C. Assume that P ∈ β\(B∩C). Then PC ∈ β\(B∩C) holds, E therefore PE ∈/ B . Then there is some QE ∈ B such that PQE = 0, so E E PC QC = PQC = 0, but that contradicts PC ,QC ∈ β. Thus B∩C is a E E E E E quasipoint in P(C), and hence ζ (B )=(B∩C) . E E CE In this manner, each abelian quasipoint B ∈ Qab(R) is assigned a quasipoint β(B):=B∩C ∈Q(C) of the center C of R. Moreover,the mapping ζ :Q(R)−→Q(C) B7−→B∩C is surjective, since each quasipoint β ∈ Q(C) (being a filter base in P(R)) is con- tained in some quasipoint B∈Q(R). Let B,B∈Qab(R) be abelian quasipoints such that e β :=B∩C =B∩C. LetE ∈B,E ∈Bbe abelianprojections. Sinece CE,CE ∈β, CECE ∈β holds and C C E ∈ B,C C E ∈ B are abelian projections with the same central carrier E E e Ee E e e C C . Hence,withoutlossofgenerality,onecanassumeC =C . Itfollowsthat E Ee e e e E E E and E are equivalent (see Prop. 6.4.6 in [KadRinII97]). Let θ ∈ R be a partial e e isometry such that θ∗θ =E,θθ∗ =E, therefore θEθ∗ =E. It follows that e θBEθ∗ ={θQEθ∗e| Q∈β}={QθEθ∗ |eQ∈β} ={QE | Q∈β}=B , E so e e θ (B)=B. Q Conversely,letB,Bbeabelianquasipoints,anedletθ ∈Rbeapartialisometrysuch that E := θ∗θ ∈ B,E := θθ∗ ∈ B. From this, θ (B) = B as shown. Let F ∈ B e Q be abelian, F ≤ E. Then θF is a partial isometry from (θF)∗θF = FEF = F to e e e θFθ∗ ∈ B. Since θFθ∗ is abelian, too, we can assume without loss of generality that E and E are abelian. From the definition of θ , it follows that e Q BeE =θBEθ∗ ={θQEθ∗ | Q∈B∩C}={QE | Q∈B∩C} holds, so e e e {PE | P ∈B∩C}={QE | Q∈B∩C}, and hence, since C =C , E eE e e {PCe | P ∈B∩C}={QC | Q∈B∩C} E E ⇐⇒(B∩C)CE =e(B∩C)CE, that is, B∩C =B∩C.eSumming up, it is proven that: e 6 ANDREASDO¨RING Theorem 11. Let R be a von Neumann algebra with center C. Then the mapping ζ :Q(R)−→Q(C) B7−→B∩C is surjective. If B,B ∈ Qab(R) are two abelian quasipoints, then ζ(B) = ζ(B) holds if and only if there is a partial isometry θ ∈R with θ (B)=B. e Q e 3. The action of the unitary group on the Stone specterum Q(R) A unitary operator transforms a quasipoint in the obvious way: Definition 12. Let T ∈ U(H) be a unitary operator. T acts on B ∈ Q(R) (R ⊆ L(H)) by T.B:={TET∗ | E ∈B}. Lemma13. T.Bis aquasipoint ofthevon Neumannalgebra TRT∗. IfT ∈U(R), then T.B∈Q(R). Proof. We have T(E∧F)T∗ ≤TET∗∧TFT∗. Moreover, T∗(TET∗∧TFT∗)T ≤E∧F, so T(E∧F)T∗ = TET∗∧TFT∗. Thus T.B is a filter basis and hence contained in some quasipoint B′ ∈TRT∗. T∗.B′ also is a filterbasis. We have T∗.(T.B)=B⊆T∗.B′. From the maximality of B, equality holds. (cid:3) 4. The Stone spectrum of a type I von Neumann algebra n Let R be a von Neumann algebra of type I , n finite. We will show that every n quasipoint B∈Q(R) of R is abelian, i.e. contains an abelian projection. In order to do so, we will use the fact that R is (isomorphic to) a n×n-matrix algebra, albeitwithentriesfromanothervonNeumannalgebra,the centerofR. We regard Rasacting onthe Hilbert moduleAn,whichgeneralizesthe vectorspaceCn. The abelian projections will be those projecting onto “lines” of the form aA, where A := C(R) is the center of R. Of course, A is not a field and An is not a vector space,sowecannotuseargumentsforsubspacelatticesoffinite-dimensionalvector spaces directly (in which case every quasipoint is abelian). But we will introduce equivalencerelationsonAandAn thatturnthemintoafieldandann-dimensional vector space, respectively, and show that after taking equivalence classes, enough of the structure remains intact to allow the conclusion that every quasipoint of R is abelian. The intuition from linear algebra carries through. From Thm. 11, we know that the abelian quasipoints can be mapped to quasipoints of the center of R via ξ :Qab(R)−→Q(C), B7−→B∩C, where two quasipoints B,B are mapped to the same quasipoint of the center if and only if there is a partial isometry θ ∈R such that θ (B)=B. Using the fact e Q e STONE SPECTRA OF VON NEUMANN ALGEBRAS OF TYPE In 7 that R is a finite algebra,we canreplace partialisometrieswith unitary operators. This will allow us to specify the orbits of the unitary group U(R) acting on Q(R) (Thm. 36). 4.1. Hilbert modules and the projections E . Itiswellknownthateachtype a I von Neumann algebra R is ∗-isomorphic to M (A), the matrix algebra with n n entries from A = C(R), the center of R (see Thm. 6.6.5 in [KadRinII97]). Let An be the free right module over A consisting of n copies of A. Another common notation for M (A) is End (An), the algebra of A-linear endomorphisms of An. n A M (A)actsontheHilbertspaceH:= nH ,the n-folddirectsumofH ,which n A A is the Hilbert space A acts on. We wilLl not make use of H and the representation e of M (A) on it, because we will regard M (A) as an algebra that acts on the A- n n e module An from the left. Elements a = (a ⊕...⊕a )t of An are regarded as 1 n column vectors. The operation of M (A) on An is a “matrix×vector” operation. n (Since A is commutative, An can be regardedas a left module as well. The chosen convention fits the natural structure of An as an M (A)-A-bimodule.) n An has a canonical basis with basis elements j ↓ e :=(0⊕...⊕0⊕1⊕0⊕...⊕0)t, j where 1 is the unit of A. With respect to this basis, a ∈ An is denoted as a=(a ⊕...⊕a )t. The sign of transposition will be omitted from now on. 1 n There is an A-valued product defined on An such that An becomes a Hilbert- A-module. Since An is a rightmodule, the inner product isA-linear with respect to the second variable: n (a|b)=(a ⊕...⊕a |b ⊕...⊕b ):= a∗b , 1 n 1 n k k Xk=1 (a|bα)=(a|b)α=α(a|b)=(aα∗|b) for a,b ∈ An,α ∈ A. In the second line the commutativity of A was employed. The inner product induces a norm on An by 1 |a|:=|(a|a)|2, where the norm on the right hand side is the norm on A. Let Ω := Q(A) be the Stone spectrum of A. Without loss of generality, we can assume A = C(Ω). Let (Ω ,...,Ω ) be a partition of Ω into closed-open sets 1 n Ω 6=∅, and let a :=χ . Then for all β ∈Ω, k k Ωk |a (β)|2 =1 k Xk and hence |a ⊕...⊕a |=1. Moreover,for k 6=j, 1 n (a |a )=a∗a =0, k j k j 8 ANDREASDO¨RING thatis,thea arepairwiseorthogonal. But(theanalogueof)Pythagoras’theorem k does not hold, since for our example one obtains n |a ⊕...⊕a |2 =1<n= |a |2. 1 n k Xk=1 In general, operators on Hilbert modules are −different from those on Hilbert spaces− not (all) adjointable, which is due to the lack of self-duality of Hilbert modules, see for example [WeO93, p 240]. A mapping T : E → E from a Hilbert module E to itself is called adjointable if there is a mapping T∗ such that (Ta|b)=(a|T∗b) for all a,b∈E. The mapping T∗ is called the adjoint of T. One can show that if T is adjointable, then T∗ is unique, T∗∗ =T and both T and T∗ are module maps which are bounded with respect to the operator norm ([WeO93, Lemma 15.2.3]). For our purpose, we have to characterize the projections in R≃M (A). n Lemma14. TheelementsofM (A)areadjointable, T =T ∈M (A)hasadjoint n jk n (T∗) =T∗ . jk kj Proof. Let a,b∈An,T =T ∈M (A). We have jk n (Ta|b)= (T a )∗b = a∗T∗ b kj j k j kj k Xk,j Xk,j = a∗(T )∗b =(a|T∗b). j jk k Xj,k (cid:3) The adjointofT ∈M (A) inthe Hilbertmodule sensethusisthe usual,Hilbert n space adjoint of T. It follows that the projections of the von Neumann algebra M (A) are the projections of the algebra B(An) of adjointable operators of the n Hilbert module An. (A projection P is A-linear, so it is contained in M (A).) n WewillnowintroduceprojectionsE thatmapfromAn onto“lines”oftheform a aA: let a∈An be such that p:=(a|a)∈A is a projection. Then ∀k ≤n:pa =a , k k sinceforβ ∈Q(A)suchthata (β)6=0,oneobtainsp(β)= a∗a 6=0andhence k j j j p(β)=1,sincepisaprojection. Hereandinthefollowing,tPhecomponentsak ∈A of a are identified with their Gelfand transforms. We get ∀β ∈Q(A) ∀k ≤n:(a p)(β)=a (β)p(β)=a (β), k k k so ap=a. Now define E :An −→An, a b7−→a(a|b). STONE SPECTRA OF VON NEUMANN ALGEBRAS OF TYPE In 9 For all b,c∈An, (E b|c)=(a(a|b)|c)=(a|b)∗(a|c) a =(b|a)(a|c)=(b|a(a|c)) =(b|E c), a so we have E∗ =E . Moreover,for all b∈An, a a E2b=a(a|E b)=a(a|a(a|b)) a a =a(a|a)(a|b)=ap(a|b) =a(a|b)=E b, a so E2 = E , that is, E is a projection with imE ⊆ aA. In fact, equality holds: a a a a let b=aα∈aA. Then a(a|b)=a(a|a)α=aα, therefore aA⊆imE . The central carrier C of E is I (a|a): obviously, I (a|a) a Ea a n n is a central projection, since I A≃A is the center of M (A). It holds that n n E b(a|a)=a(a|b)(a|a)=a(a|a)(a|b) a =a(a|b)=E b, a so C ≤ I (a|a). Conversely, let I q be a central projection such that qE = Ea n n a E q =E . Then for all b∈An, a a E b=a(a|b)=qE b=qa(a|b) a a =a(a|qb)=E (qb). a In particular, one obtains E (qa)=E a a a ⇐⇒a(a|qa)=a(a|a)=ap=a ⇐⇒a(a|a)q =a ⇐⇒aq =qa=a =⇒q(a|a)=qp=(a|a)=p =⇒p≤q, therefore, the central carrier of E is C =I p=I (a|a). a Ea n n The E are of interest, because they are abelian projections: a Lemma 15. E is an abelian projection from An onto aA with central carrier a I (a|a). n Proof. It only remains to show that E is abelian. Let A,B ∈ M (A). Then it a n holds for all b∈An that E AE BE b=E AE (Ba)(a|b)=E (Aa)(a|Ba)(a|b) a a a a a a =a(a|Aa)(a|Ba)(a|b)=a(a|Ba)(a|Aa)(a|b) =E (Ba)(a|Aa)(a|b)=E BE (Aa)(a|b) a a a =E BE AE b, a a a so E AE BE =E BE AE . (cid:3) a a a a a a 10 ANDREASDO¨RING E is a projection in M (A) if (a|a) is a projection in A. The converse is also a n true: Remark 16. Let a ∈ An be such that E is a projection. Then (a|a) ∈ A is a a projection. Proof. From E2 =E , a(a|b)=a(a|E b)=a(a|a)(a|b) for all b∈An. For b=a, a a a a(a|a)=a(a|a)2. This means that (a|a)∈{0,1} holds on the support supp a:= supp a =supp(a|a). k k[≤n If β ∈ Ω = Q(A) is such that (a|a)(β) 6= 0, then a (β) 6= 0 holds for at least k one k ≤ n and thus (a|a)(β) = 1. So (a|a) = 1 holds on supp(a|a) and (a|a) is a projection. (cid:3) If a ,...,a ∈ A are projections and a := a e , then E is a projection if 1 n k k k a and only if the ak are pairwise orthogonal, bPecause, according to its definition and the above remark, E is a projection if and only if (a|a) is a projection. For a a:= a e ,wehave(a|a)= a ,andthisisaprojectionifandonlyifthea are k k k k k k pairwPise orthogonal. FurthermPore, kakEek = kEakek (the ak are projections again): for all b∈An, it holds thatP P ( a E )b= a e (e |b)= a2e (e |b) k ek k k k k k k Xk Xk Xk = a e (e |ba )= a e (a e |b) k k k k k k k k Xk Xk = E b. akek Xk Lemma 17. A := n a E is a projection if and only if all the a are projec- k=1 k ek k tions. In this case, Pthe central carrier of A is CA =In( kak). W Proof. From (e |e )=δ e we get j k jk k ∀c∈An :E E c=e (e |E c)=e (e |e )(e |c)=δ E c, ej ek j j ek j j k k jk j so E E =δ E . A := n a E is a projection if and only if a∗ =a holds ej ek jk ej k=1 k ek k k for all k ≤n and if A2 =AP. Since we have n n A2 =( a E )2 = a a E E k ek j k ej ek Xk=1 jX,k=1 n n = a a δ E = a2E , j k jk ej j ej jX,k=1 Xj=1 this holds if and only if all the a ∈A are projections. C =I ( a ) holds then, j A n k k obviously. W (cid:3) Remark 18. With respect to the basis (e ,...,e ) of An, E has the matrix 1 n a (E ) =(a a∗) . a jk j k j,k≤n