Abelian subalgebras and the Jordan structure of a von Neumann algebra
aa r X i v : . [ m a t h - ph ] D ec ABELIAN SUBALGEBRAS AND THE JORDANSTRUCTURE OF A VON NEUMANN ALGEBRA
ANDREAS D ¨ORING AND JOHN HARDING
Abstract.
For von Neumann algebras M , N not isomorphic to C ⊕ C and without type I summands, we show that for an order-isomorphism f : AbSub
M →
AbSub N between the posets ofabelian von Neumann subalgebras of M and N , there is a uniqueJordan ∗ -isomorphism g : M → N with the image g [ S ] equalto f ( S ) for each abelian von Neumann subalgebra S of M . Theconverse also holds. This shows the Jordan structure of a vonNeumann algebra not isomorphic to C ⊕ C and without type I summands is determined by the poset of its abelian subalgebras,and has implications in recent approaches to foundational issues inquantum mechanics. Mathematics Subject Classifications (2010):
Key words: von Neumann algebra; Jordan structure; abelian sub-algebra; orthomodular lattice.1.
Introduction
We consider the question: given a von Neumann algebra M , howmuch information about M is encoded in the order structure of its col-lection of unital abelian von Neumann subalgebras? The set AbSub M of such subalgebras, partially ordered by set inclusion, becomes a com-plete meet semilattice in which every subset that is closed under finitejoins has a join. The task is to reconstruct algebraic information aboutthe algebra M from the order-theoretic structure of AbSub M . Moregenerally, we are interested in the interplay between these two levels ofalgebraic structure.When M is abelian, the projection lattice P roj M forms a com-plete Boolean algebra, and one can show that the poset AbSub M isisomorphic to the lattice of complete Boolean subalgebras of P roj M .Modifying a result of Sachs [26] that every Boolean algebra is deter-mined by its lattice of all subalgebras, to show each complete Booleanalgebra is determined by its lattice of complete subalgebras, one canthen obtain that P roj M is determined by AbSub M . That M isdetermined by P roj M is a consequence of the spectral theorem. For a non-abelian von Neumann algebra, the situation is more com-plicated. Reconstruction of the non-commutative product in M willnot generally be possible as there are non-isomorphic von Neumann al-gebras having the same Jordan product, hence exactly the same posetsof unital abelian subalgebras. However, we will show that the orderstructure of AbSub M does determine M as a Jordan algebra up to(Jordan) ∗ -isomorphism. This means that the poset AbSub M en-codes a substantial amount of algebraic information about M . Theproof goes along the same lines as the abelian case, using a result of[16] that an orthomodular lattice is determined by its poset of Booleansubalgebras. In fact, our result is somewhat stronger than we described. Theorem
Suppose M , N are von Neumann algebras without type I summands and f : AbSub
M →
AbSub N is an order-isomorphism.Then there is a unique Jordan ∗ -isomorphism F : M → N with f ( S ) equal to the image F [ S ] for each S . This result is particularly interesting with respect to the so-calledtopos approach to the formulation of physical theories [6, 7, 8, 9, 10, 19],where a mathematical reformulation of algebraic quantum theory issuggested. For a von Neumann algebra M , one considers the poset AbSub M of its abelian subalgebras and the topos of presheaves overthis poset. The idea is that each abelian subalgebra represents a ‘classi-cal perspective’ on the quantum system. By taking all classical perspec-tives together, one obtains a complete picture of the quantum system.Mathematically, this corresponds to considering the poset AbSub M and presheaves over it. These presheaves form the topos associatedwith the quantum system. The so-called spectral presheaf Σ M , whosecomponents are the Gelfand spectra of the abelian von Neumann sub-algebras of M , plays a key role in the topos approach. Physically,the spectral presheaf is interpreted as a generalized state space for thequantum system described by the algebra M . Mathematically, Σ M isa kind of spectrum of the non-abelian von Neumann algebra M . Itbecomes clear that, from the perspective of the topos approach, it isvery relevant to see how much information about the algebra M canbe extracted from the poset AbSub M .Since the appearance of the draft of this manuscript on ArXiv [5],several related manuscripts and papers have arisen. In [14] a relatedtask is undertaken for the poset of abelian subalgebras of a C ∗ -algebra,and in [15] the matter is considered from the viewpoint of associative BELIAN SUBALGEBRAS AND JORDAN STRUCTURE 3 subalgebras of a Jordan algebra. In [4] applications to the topos ap-proach to physical theories are considered further. In particular, it isshown that if
M, N are von Neumann algebras with no direct sum-mands of type I , then there is a Jordan ∗ -isomorphism F : M → N ifand only there is an isomorphism Φ : Σ N → Σ M between their spectralpresheaves in the opposite direction.2. Preliminaries
For a complex Hilbert space H , let B ( H ) be the C ∗ -algebra of allbounded operators on H . For a subset S ⊆ B ( H ), the commutant S ′ isthe set of all elements of B ( H ) that commute with each member of S .A von Neumann algebra is a subset M ⊆ B ( H ) with M = M ′′ . For avon Neumann algebra M , we use P roj M for the set of projections in M . The following well-known result [22, pg. 69] will be used repeatedly. Proposition 2.1.
For M a von Neumann algebra, M = ( P roj M ) ′′ . For any von Neumann algebra M the projections P roj M form acomplete orthomodular lattice (abbreviated: oml ). Our primary inter-est lies in subalgebras of von Neumann algebras, subalgebras of theirprojection lattices, and relationships between these and the originalvon Neumann algebra. We require several definitions. Definition 2.2.
A von Neumann subalgebra of a von Neumann algebra M is a subset S ⊆ M that is itself a von Neumann algebra.
We will only consider von Neumann subalgebras
S ⊆ M such thatthe unit elements in S and N coincide. (In particular, we will notconsider subalgebras of the form ˆ P M ˆ P for a non-trivial projectionˆ P ∈ M .) We remark that being a von Neumann subalgebra is equiva-lent to being a unital C ∗ -subalgebra that is closed in the σ -weak topol-ogy, equivalent to being a unital C ∗ -subalgebra that is closed undermonotone joins [1, pg. 101–110]. Definition 2.3.
For a von Neumann algebra M , we let Sub M bethe set of all von Neumann subalgebras of M ordered by set inclusion; AbSub M be the set of abelian von Neumann subalgebras of M orderedby set inclusion; and F AbSub M be the set of all abelian subalgebras of M that contain only finitely many projections, ordered by set inclusion. We note that
Sub M is a complete lattice, with meets given byintersections. The join of a family ( S i ) i ∈ I of subalgebras is the weakclosure of the algebra generated by the algebras S i , i ∈ I . Analogously, AbSub M is a complete meet semilattice where every subset that isclosed under finite joins has a join, and F AbSub M is a complete ANDREAS D ¨ORING AND JOHN HARDING meet semilattice where every meet is essentially finite. Yet, neither
AbSub M nor F AbSub M have a top element if M is non-abelian, soempty meets do not exist in these posets. Definition 2.4.
For an oml L , we let Sub L be the set of all subalge-bras of L ; BSub L be the set of Boolean subalgebras of L , and F BSub L be the set of finite Boolean subalgebras of L , all partially ordered by setinclusion. If L is complete we let CSub L be the set of complete subal-gebras of L , meaning subalgebras that are closed under arbitrary joinsand meets from L , and CBSub L be the set of complete Boolean sub-algebras of L . Again, these are considered as posets, partially orderedby set inclusion. For a von Neumann algebra M we can use the associative, but notnecessarily commutative, product on M to define a commutative, butnot necessarily associative product ◦ on M , called the Jordan product,by setting a ◦ b = 12 ( ab + ba ) . Suppose ϕ is a map between von Neumann algebras that is linear,bijective, and preserves the involution (adjoint) ∗ . We say ϕ is a ∗ -isomorphism if it satisfies ϕ ( ab ) = ϕ ( a ) ϕ ( b ); a ∗ -antiisomorphism ifit satisfies ϕ ( ab ) = ϕ ( b ) ϕ ( a ); and a Jordan isomorphism if it satisfies ϕ ( a ◦ b ) = ϕ ( a ) ◦ ϕ ( b ). The following is well known [21, 27]. Proposition 2.5.
Every Jordan isomorphism η : M → N betweenvon Neumann algebras M , N can be decomposed as the sum of a ∗ -isomorphism and a ∗ -anti-isomorphism. More concretely, there are central projections ˆ P , ˆ P ∈ M and ˆ Q , ˆ Q ∈N such that M and N are unitarily equivalent to M ˆ P ⊕ M ˆ P and N ˆ Q ⊕N ˆ Q , respectively, and η | M ˆ P : M ˆ P → N ˆ Q is a ∗ -isomorphism,while η | M ˆ P : M ˆ P → N ˆ Q is a ∗ -antiisomorphism.It follows from [3] that there is a von Neumann algebra that is not ∗ -isomorphic to its opposite, hence these two von Neumann algebrasare Jordan isomorphic, but not ∗ -isomorphic. So there can be two dif-ferent associative noncommutative products on a weakly closed set ofoperators, giving different von Neumann algebras, but the same Jor-dan structure. So the associative noncommutative product on a vonNeumann algebra cannot be recovered from the lattice of its subalge-bras as a von Neumann algebra and its opposite will have precisely thesame subalgebras. However, we will see that in the absence of type I summands (and excluding the case M = C ⊕ C ), the Jordan structure BELIAN SUBALGEBRAS AND JORDAN STRUCTURE 5 can be recovered. The following result by Dye [11], see also [13, Theo-rem 8.1.1], will be of key importance. We note that the uniqueness inthe version of this result given below follows from the spectral theorem.
Theorem 2.6.
Suppose M , N are von Neumann algebras without type I summands. Then for any oml -isomorphism ψ : P roj
M →
P roj N there is a unique Jordan ∗ -isomorphism Ψ :
M → N with Ψ( p ) = ψ ( p ) for each projection p of M . The reader should consult [1, 12, 17, 21, 27] for basics on von Neu-mann algebras, [2] for lattice theory, and [22] for oml s.3.
Main result
Lemma 3.1.
Let M be a von Neumann algebra. Then there is anorder-isomorphism Ψ :
F AbSub
M →
F BSub ( P roj M ) defined bysetting Ψ S = S ∩
P roj M .Proof. It follows from [1, Theorem 2.104] that the projections of anyabelian subalgebra of M form a Boolean subalgebra of P roj M . So Ψis indeed a map from F AbSub M to F BSub ( P roj M ). Clearly Ψ isorder-preserving. Suppose Ψ S ⊆ Ψ T . As S is a von Neumann algebra S = ( P roj S ) ′′ , and similarly for T . Therefore S = (Ψ S ) ′′ ⊆ (Ψ T ) ′′ = T , showing Ψ is an order-embedding.Suppose B is a finite Boolean algebra of projections in M with atoms p , . . . , p n , and consider the map Λ : C n → M defined by settingΛ( λ , . . . , λ n ) = P n λ i p i . One easily sees Λ is a normal, unital ∗ -isomorphism, so by [1, Lemma 2.100] its image S is a von Neumannsubalgebra of M . Clearly S is an abelian, has finitely many projections,and Ψ S = B . So Ψ is onto. (cid:3) Remark 3.2.
While not needed for our results, it is natural to considerseveral questions related to the above result. It is easy to see that asabove there is an order-embedding Ψ :
Sub
M →
CSub ( P roj M )that preserves all meets. A simple example with M being the boundedoperators on C shows this map need not preserve joins or be onto.A more difficult argument, using the notion of Bade subalgebras andresults from [24], shows there is an order-isomorphism Ψ : AbSub
M →
CBSub ( P roj M ). The result above follows from this more generalone, but is not needed here. Lemma 3.3.
For oml s L , M , each order-isomorphism µ : F BSub L → F BSub M extends uniquely to an isomorphism ¯ µ : BSub L → BSub M . ANDREAS D ¨ORING AND JOHN HARDING
Proof.
We define an ideal of
F BSub L to be a downset I of F BSub L where any two elements of I have a join, and this join belongs to I . Forany element x of BSub L , we have x ↓ ∩ F BSub L = { z ∈ F BSub L : z ⊆ x } is an ideal of F BSub L and the join of this ideal in
BSub L isequal to x . Further, each ideal of F BSub L is of this form as can beeasily seen from the compactness of finitely generated subalgebras in asubalgebra lattice.Define ¯ µ by setting ¯ µ ( x ) = W µ [ x ↓ ∩ F BSub L ]. This join is welldefined as the image under the isomorphism µ of an ideal is an ideal.Clearly ¯ µ is order preserving. Suppose ¯ µ ( x ) ≤ ¯ µ ( y ). Then for each z ∈ x ↓ ∩ F BSub L we have µ ( z ) ≤ W µ [ y ↓ ∩ F BSub L ]. Compactnessthen yields z ≤ y for each such z , giving x ≤ y . Thus ¯ µ is an order-embedding. To see ¯ µ is onto, note each element w of BSub M is thejoin of an ideal J of F BSub M . The preimage µ − [ J ] is an ideal of F BSub L , so has a join x in BSub L . Then ¯ µ ( x ) = w , showing ¯ µ isonto.Clearly ¯ µ extends µ . If ˜ µ is another isomorphism from BSub L to BSub M extending µ , then ˜ µ preserves joins, so ˜ µ ( x ) = W µ [ x ↓∩ F BSub L ] = ¯ µ ( x ). (cid:3) We are ready to provide our main result.
Theorem 3.4.
Suppose M , N are von Neumann algebras not iso-morphic to C ⊕ C and without type I summands, and suppose that f : AbSub
M →
AbSub N is an order-isomorphism. Then there isa unique Jordan ∗ -isomorphism F : M → N with f ( S ) equal to theimage F [ S ] for each S . Proof.
Consider a series of mappings, starting with the given
AbSub M f −−−→ AbSub N . We then restrict this to
F AbSub M . Note that the members of F AbSub M are precisely those members of AbSub M that have only finitely manyelements beneath them, and similarly for F AbSub N . Thus this re-striction g is also an order-isomorphism. F AbSub M g −−−→ F AbSub N . Lemma 3.1 gives order-isomorphisms Ψ M : F AbSub
M →
F BSub ( P roj M ) and Ψ N : F AbSub
N →
F BSub ( P roj N ) givenby Ψ M ( S ) = S ∩
P roj M and Ψ N ( T ) = T ∩
P roj N . It followsthere is a unique order-isomorphism h as below with h ( S ∩
P roj M ) = BELIAN SUBALGEBRAS AND JORDAN STRUCTURE 7 g ( S ) ∩ P roj N for each S ∈
F AbSub M . F BSub ( P roj M ) h −−−→ F BSub ( P roj N ) . Then by Lemma 3.3 this extends uniquely to an order-isomorphism
BSub ( P roj M ) j −−−→ BSub ( P roj N ) . The main result of [16] says that if
L, M are oml s without any 4-element blocks (a block is a maximal Boolean subalgebra), then forany order-isomorphism α : BSub L → BSub M there is a unique oml -isomorphism β : L → M with α ( D ) = β [ D ] for each Booleansubalgebra D of L . As M , N are neither isomorphic to C ⊕ C norto B ( C ⊕ C ) (the latter is a von Neumann algebra of type I ), thereare no 4-element blocks in P roj M or P roj N . So the map j definedabove gives a unique map k as shown below with j ( D ) = k [ D ] for eachBoolean subalgebra D of P roj M . P roj M k −−−→ P roj N . Finally, Theorem 2.6 gives a unique Jordan ∗ -isomorphism F as belowextending k . M F −−−→ N . Claim 1 : If S ∈
F AbSub M then f ( S ) ∩ P roj N = F [ S ] ∩ P roj N . Proof :
To see this, note that for such S , f ( S ) ∩ P roj N = g ( S ) ∩ P roj N = h ( S ∩
P roj M )= j ( S ∩
P roj M )= k [ S ∩
P roj M ]= F [ S ∩
P roj M ]= F [ S ] ∩ P roj N The first equality follows as g is the restriction of f ; the second by thedefinition of h ; the third as j extends h ; the fourth by the definition of k ; the fifth as F extends k ; and the sixth as F restricts to a bijectionbetween P roj M and P roj N . (cid:3) Claim 2 : If S ∈
AbSub M , then F [ S ] ∈ AbSub N . Proof : As F is Jordan and S is abelian, by [27, pg. 187] the re-striction F |S preserves the associative product. By [1, pg. 189] F is a ANDREAS D ¨ORING AND JOHN HARDING unital order-isomorphism, so it preserves monotone joins, and as S is avon Neumann subalgebra of M , the identical embedding of S into M preserves monotone joins. So the composite F |S preserves monotonejoins, hence is a normal unital one-one ∗ -homomorphism of S into N .So by [1, Lemma 2.100] the image F [ S ] is a von Neumann subalgebraof N that is clearly abelian. (cid:3) Claim 3 : If S ∈
AbSub M , then f ( S ) = F [ S ]. Proof :
A projection p belongs to F [ S ] if, and only if, it belongs to F [ U ] for some U ⊆ S with
U ∈
F AbSub M . The proof is essentiallythat of Lemma 3.1. By Claim 1, this is equivalent to p belonging to f ( U ) for some U ⊆ S with
U ∈
F AbSub M . As the members of F AbSub M are exactly the members of AbSub M with finitely manyelements beneath them, it follows from f being an order-isomorphismthat T = F [ U ] for some U ⊆ S with
U ∈
F AbSub M if, and only if, T ⊆ f ( S ) and T ∈
F AbSub N . So p belonging to F [ S ] is equivalentto p belonging to T for some T ⊆ f ( S ) with T ∈
F AbSub N , soequivalent to p belonging to f ( S ). By Claim 2, f ( S ) and F [ S ] are vonNeumann subalgebras of N , and they contain the same projections, so f ( S ) = F [ S ]. (cid:3) .To conclude the proof of the theorem, it remains to show uniqueness.Suppose G : M → N is a Jordan ∗ -isomorphism with f ( S ) = G [ S ] foreach S ∈
AbSub M . Using the spectral theorem, it follows that twoJordan ∗ -isomorphisms from M to N agreeing on the projections mustbe equal. So it is enough to show that F and G agree on P roj M . Fromthe uniqueness of the result in [16] it is enough to show F [ D ] = G [ D ]for each Boolean subalgebra D of P roj M , and by the uniqueness inLemma 3.3 it is enough to show this for finite Boolean subalgebras D of P roj M . Using Lemma 3.1, it is then enough to show F [ S ∩
P roj M ] = G [ S ∩
P roj M ] for each S ∈
F AbSub M , and this is adirect consequence of the assumption that F [ S ] = G [ S ]. This shows F = G , and concludes the proof of the theorem. (cid:3) We finally observe that the converse of the above result also holds(in fact, for arbitrary von Neumann algebras):
Proposition 3.5.
Let M , N be von Neumann algebras, and let F : M → N be a Jordan ∗ -isomorphism. Then F induces a unique orderisomorphism f : AbSub
M →
AbSub N with f ( S ) equal to the image F [ S ] for each S . BELIAN SUBALGEBRAS AND JORDAN STRUCTURE 9
Proof :
It is well-known that a Jordan ∗ -homomorphism F : M →N between von Neumann algebras preserves commutativity (see e.g.[27, 18]), so F maps abelian subalgebras of M to abelian subalgebrasof N in a bijective and order-preserving way. Hence, we obtain anorder-isomorphism f : AbSub
M →
AbSub N . (cid:3) Conclusions
There remain several directions for further research. First, it wouldbe of interest to see if the Jordan structure of a C ∗ -algebra is de-termined by its poset of abelian C ∗ -subalgebras. In this direction weremark that it is known that the lattice of C ∗ -subalgebras of an abelian C ∗ -algebra determines the C ∗ -algebra [23, Theorem 11]. Perhaps [25]may also be related to this question. [28] is concerned with abeliansubalgebras of partial C ∗ -algebras and von Neumann algebras.For a different direction, one might consider the matter of addingadditional information to the poset AbSub M in hopes of recoveringthe full von Neumann structure of M , rather than just its Jordanstructure. This seems very closely related to the subject of orienta-tion theory, very nicely described in [1]. From the perspective of thetopos approach, the natural question becomes whether orientationscan be encoded by presheaves (contravariant, Set-valued functors) over AbSub M , or maybe by covariant functors.5. Acknowledgements
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Andreas D¨oring, Clarendon Laboratory, Department of Physics,University of Oxford, Parks Road, OX1 3PU, Oxford, UK
E-mail address : [email protected] John Harding, Department of Mathematical Sciences, New MexicoState University, Las Cruces, NM 88003, USA
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