Dye's theorem for tripotents in von Neumann algebras and JBW* triples
aa r X i v : . [ m a t h . OA ] J a n Dye’s Theorem for tripotents in vonNeumann algebras and JBW ∗ -triples Jan Hamhalter
Czech Technical University in PragueDepartment of MathematicsFaculty of Electrical EngineeringTechnicka 2, 166 27 Prague 6Czech [email protected]
Abstract. We study morphisms of the generalized quantum logic of tripotents inJBW ∗ -triples and von Neumann algebras. Especially, we establish generalization ofcelebrated Dye’s theorem on orthoisomorphisms between von Neumann lattices to thisnew context. We show one-to-one correspondence between maps on tripotents preserv-ing orthogonality, orthogonal suprema, and reflection u → − u , on one side, and theirextensions to maps that are real linear on sets of elements with bounded range tripo-tents on the other side. In a more general description we show that quantum logicmorphisms on tripotent structures are given by a family of Jordan *-homomorphismson 2-Peirce subspaces. By examples we exhibit new phenomena for tripotent mor-phisms that have no analogy for projection lattices and demonstrated that the abovemention tripotent versions of Dye’s theorem cannot be improved. On the other hand,in a special case of JBW ∗ -algebras we can generalize Dye’s result directly. Besides weshow that structure of tripotents in C ∗ -algebras determines their projection poset andis a complete Jordan invariant for von Neumann algebras. key words: tripotent poset, quantum logic morphisms, JBW ∗ algebras, JBW ∗ -triples.MSC2010 classification: 17C65, 46L10, 81P10 The aim of the present paper is to explore the concept of orthogonality andorder in structures associated with von Neumann algebras and Jordan triples.Especially, we study morphisms of projection lattices of JBW ∗ -algebras andgeneralized quantum logics of tripotents in JBW ∗ -triples. The former struc-ture is going back to John von Neumann’s work on continuous geometry [20],foundations of operator algebras [15], and foundations of quantum theory [19].The latter structure is younger and stems from infinite dimensional holomorphy,1ordan theory and its applications to mathematical physics [5, 4, 25]. Main re-sults of the paper concern generalizations of celebrated Day’s theorem [9] for theabove indicated order structures showing their intimate connection with theirunderlying linear structure.Dye’s theorem is a generalization of famous Wigner’s theorem about sym-metries of quantum system [26]. Ulhorn’s logic theoretic version [24] of thisprinciple states that orthogonality relation in the set P ( H ) of projections act-ing on a Hilbert space, H with dim H ≥
3, determines dynamics of the system.More precisely, if ϕ : P ( H ) → P ( H ) is a bijection preserving orthogonality inboth directions, i.e. pq = 0 ⇐⇒ ϕ ( p ) ϕ ( q ) = 0 , then there is a unitary or antiunitary operator u acting on H such that ϕ ( p ) = u ∗ pu , for all p ∈ P ( H ) . In unifying reformulation avoiding distinguishing between unitary and antiu-nitary case, we can say that there is a Jordan *-isomorphism J : B ( H ) → B ( H ),where B ( H ) is the algebra of all bounded operators acting on H , such that ϕ ( p ) = J ( p ) , for all p ∈ P ( H ) . Following advances made by John von Neumann in his project of continuous ge-ometry [20], Day tackled the problem of describing orthoisomorphisms betweenprojection lattices of von Neumann algebras which involves Ulhron’s result as avery special case. His principal result, known today as Dye’s theorem, reads asfollows
Let M and N be von Neumann algebras,where M does not have Type I direct summand. Let ϕ : P ( M ) → P ( N ) beorthoisomorphism between projection lattices of M and N , respectively. Thenthere is a Jordan *-isomorphism J : M → N extending ϕ . The assumption on absence of Type I direct summand corresponds to as-sumption dim H ≥ x, y ) → ( xy + yx ) and the standard *-operation. This result was thengeneralized beyond von Neumann algebras by the author in [11].The main message of Dye’s theorem is the fact that any map preservingorthogonality relation can be ”linearised” by finding its linear extension. Oncewe have linearity, the fact that the extension has to be Jordan *-isomorphismcan be deduced relatively easily as a consequence of preserving projections andtheir orthogonality. L.J.Bunce and J.D.M.Wright fully realised this fact in theirwork [3] that was a turning point for further investigation. Namely, they showedthat Gleason’s theorem on extending probability measure on projections to alinear map can be applied to generalizing Dye’s theorem in a few directions.2t first it enabled to show a version of Dye’s theorem for non-bijective mapsbetween projections. Moreover, they established validity of Dye’s theorem forprojection structures in JW-algebras that are more general than von Neumannalgebras. (JW-algebra is a weakly closed subspace of the real space of self-adjoint operators acting on a Hilbert space that is closed under forming thesquares a → a .) We continue this research by showing Dye’s theorem for evenmore general JBW ∗ -algebras. Let A and B be JBW ∗ -algebras, where A doesnot have Type I part. We show that any map ϕ : P ( A ) → P ( B ) betweenprojection lattices of JBW ∗ -algebras such that ϕ preserves zero, orthogonalityand suprema of orthogonal projection (such a map will be called quantum logicmorphism) is a restriction of a Jordan *-homomorphism J : A → B .Principal contribution of our paper concerns Dye’s theorem in the contextof JBW ∗ -triples. JBW ∗ -triples are Banach spaces that generalize von Neumannalgebras. However, they involve also Hilbert spaces and structures of operatorsbetween different Hilbert spaces (rectangular matrices), which are seldom vonNeumann algebras. The theory of JBW ∗ -triples has been developing rapidlyrecently and is vital for infinite dimensional complex analysis, differential geom-etry and mathematical physics. Projection lattice is a prominent order structureassociated with classifications of von Neumann algebras. There is another poset,called tripotent poset, that underlies theory of JBW ∗ -triples in a similar way.An element u in a von Neumann algebra M is a tripotent if it is partial isometry,i.e. if u = uu ∗ u . This class of operators involves unitary operators as well asprojections. Order is defined as follows u ≤ v ⇔ u = uv ∗ u . The resulting poset ( U ( M ) , ≤ ) of all tripotents in M includes projection lattice P ( M ) as a principal ideal. Besides, there is a natural orthogonality relation on U ( M ). The tripotents u and v are orthogonal if uu ∗ v = 0. It it the same as tosay that kernels of u and v are orthogonal subspaces in the underlying Hilbertspace and the same holds for ranges of u and v . The importance of the structure( U ( M ) , ≤ , ⊥ ) was recognized by Edwards and R¨utimann who studied tripotentposets from the perspectives of orthomodular structures and obtained manydeep results between functional analysis and theory of quantum logics [7, 8].Especially, they showed that tripotent poset of a JBW ∗ -triple is Dedekind com-plete (i.e. each upper bounded set has supremum) and found connection of thisposet with facial geometry of the unit ball. Unlike von Neumann projectionlattice, triple poset is not upward directed and has many maximal elements.However, it can be organized into generalized quantum logic (nonunital versionof orthomodular poset). In further development a fruitful interplay of the theoryof orthomodular posets and JBW ∗ -triples initiated by Edwards and R¨utimannseemed to be neglected until recent paper [13]. We believe that it a pity inthe light of recent development in JBW ∗ -triple theory as well in foundationsof quantum theory [16]. For this reason we would like to revive this line ofthe research by studying morphisms between tripotent posets and appropriate3orms of Dye’s theorem for these structures.In the beginning of our research, we realised that one cannot generaliseDye’s theorem verbatim to Jordan triple structures. Indeed, easy counterex-amples show that there is an orthoisomorphism between tripotents that doesnot preserve order and cannot be so extended to a Jordan morphism. That iswhy we have to consider quantum logic morphisms (see [3])), i.e maps preserv-ing suprema of orthogonal elements, in addition to preserving orthogonality.Further, our counterexamples show that even if we exclude Type I part as inoriginal version of Dye’s theorem, the theorem is not valid. In other words, quan-tum logic morphisms are always more general than restrictions of linear Jordanmaps. However, we show that any quantum logic morphism of triple struc-tures extends to a map that is additive on relevant parts. Most importantly,we establish one-to-one correspondence between quantum logic morphisms pre-serving reflection u → − u and real homogeneous maps preserving tripotentsthat are additive with respect to elements whose range tripotents have upperbound. Such maps are called local Jordan morphisms. Another approach howto describe all quantum logic morphisms elaborated in this paper is based on afamily of linear Jordan maps that are defined on homotopes and are consistentin some sense. Beside generalization of Dye’s theorem we also bring some newresults on tripotent posets and their morphisms.Tripotent poset constitute invariant in the category of JB ∗ -triples. Even if itis not in the main focus of our paper, we also discuss briefly a natural questionof whether it is a complete invariant in case of JBW ∗ -triples. We show thatthe answer is in the positive in case of triple posets in von Neumann algebrasalgebras. More specifically, we prove that the following conditions are equiva-lent: (i) von Neumann algebras M and N are Jordan *-isomorphic (2) tripotentposets U ( M ) and U ( N ) are isomorphic as generalised quantum logics (iii) pro-jection lattices P ( M ) and P ( N ) are isomorphic as quantum logics. We alsoshow that the tripotent posets of two C ∗ -algebras are isomorphic if and onlyif their projection posets are isomorphic. As a consequence, tripotent posetscarry the same amount of information as projection lattices. This seems to beinteresting because tripotent poets include also non normal elements and aremuch larger than projection posetsOur paper is organised as follows. After introduction and recalling basicnotions we introduce orthomodular order structures in the second section andprove some easy statements needed later. In the third part we generalise Dye’stheorem to projection lattices of JBW ∗ algebras. In the forth section of thepresent note we focus on ordered tripotent structures and their morphisms. Weexhibit some examples showing that our later results are optimal. Sections 5 and6 contain main results describing quantum logic morphisms between tripotentstructures in terms of families of Jordan maps and local Jordan maps. Con-cluding section is an invitation to further research and contains discussion oncomplete order invariant of von Neumann algebras.4et us now recall a few concepts and fix the notation. For the theory of C ∗ -algebras and von Neumann algebras the reader is referred to monographs[15, 22]. For fundamentals of the theory of Jordan algebras and Jordan triplesystems we recommend monograph [5, 6, 21, 25, 23].Given a normed space X , B ( X ) shall denote its unit ball. By the symbol B ( H ) we shall denote the algebra of all bounded operators on a complex Hilbertspace H . Jordan algebra is a real or complex commutative algebra endowed witha Jordan product, ◦ , satisfying the identity x ◦ ( x ◦ y ) = x ◦ ( x ◦ y ). By aJB-algebra we mean a real Banach space that is simultaneously a (real) Jordanalgebra, for which we have k x ◦ y k ≤ k x k · k y k , k x k = k x k and k x − y k ≤ max {k x k , k y k} . By a JC-algebra we understand a real closed subalgebra ofthe self-adjoint part of B ( H ) that is closed under the squares and whose Jordanproduct is x ◦ y = ( xy + yx ). Let A be a JB-algebra. Positive elements in A are elements of the form a , a ∈ A . Elements x, y ∈ A are said to operatorcommute if x ◦ ( y ◦ z ) = y ◦ ( x ◦ z ) for all z ∈ A . The center Z ( A ) of A is theset of all elements operator commuting with each element of A . A JW-algebrais a JC-algebra that is moreover closed in the weak operator topology on B ( H ).JBW-algebra is a JB-algebra that has a (unique) predual. Given a ∈ A weshall define an operator U a acting on A by U a ( x ) = 2 a ◦ ( a ◦ x ) − a ◦ x. Let usremark that operator U a is positive in the sense that leaves the positive cone of A invariant.Let A be a complex Jordan algebra endowed with an involution ∗ . Then onecan define the triple product, ◦ , on A by { a, b, c } = a ◦ ( b ∗ ◦ c ) + c ◦ ( a ◦ b ∗ ) − b ∗ ◦ ( a ◦ b ) . (1.1)Let A be JC ∗ -algebra, that is a closed complex Jordan subalgebra of B ( H )which is invariant with respect to adjoints and endowed with the Jordan product a ◦ b = ( ab + ba ). Then we have that { a, b, c } = 12 ( ab ∗ c + cb ∗ a ) . JB ∗ -algebra is a Jordan *-algebra that is simultaneously a Banach spacewhose norm satisfies: k a ∗ k = k a k , k a ◦ b k ≤ k a kk b k , k{ a, a, a }k = k a k . The self-adjoint part of a JB ∗ -algebra is the set H ( A ) = { a ∈ A : a = a ∗ } . Itis a JB-algebra and each JB-algebra can be obtained in this manner [27]. If aJB ∗ -algebra admits a predual, then it is called a JBW ∗ -algebra. A projectionin a JB ∗ -algebra (resp. JB-algebra) is a self-adjoint idempotent (resp. idem-potent). The set of projections in a JB ∗ -algebra or in a JB-algebra A will bedenoted P ( A ). A linear functional f on a JB ∗ -algebra A is called positive if ittakes positive values on elements in the positive part of A , that is if f ( a ) ≥ a ∈ H ( A ). If f is moreover norm one, then it is called state.A Jordan triple is a complex space E endowed with triple product ( a, b, c ) →{ a, b, c } which is symmetric and linear in the first and the third variable andconjugate linear in the second variable and satisfies the identity[ L ( a, b ) , L ( c, d )] = L ( { a, b, c } , d ) − L ( c, { d, a, b } ) == L ( a, { b, c, d } ) − L ( { c, d, a } , b ) , where [ · , · ] denotes the commutator and L is the mapping from E × E into thespace of linear operators on E defined by L ( a, b ) c = { a, b, c } . A Jordan triple E is said to be a JB ∗ -triple if the following holds: • E is a Banach space and L is a continuous map from E × E into the space B ( E ) of bounded operators acting on E . • For each a ∈ E , L ( a, a ) is a hermitian operator with nonnegative spectrumand satisfies k L ( a, a ) k = k a k . (Let us recall that a bounded operator T acting on some complex Banach space is called hermitian if k e itT k = 1 forall t ∈ R . )The JBW ∗ -triple is a JB ∗ -triple that is a dual Banach space. Any JBW ∗ -algebra endowed with the triple product (1.1) is a JBW ∗ -triple.Tripotent u in a JB ∗ -triple E is an element satisfying { u, u, u } = u . The setof all tripotents of E will be denoted by U ( E ). Each tripotent u is responsiblefor decomposition of E into closed subspaces E = E ( u ) ⊕ E ( u ) ⊕ E ( u ) , where E i ( u ) is the eingenspace of L ( u, u ) corresponding to the eigenvalue i . Acomplete tripotent is a tripotent u ∈ E for which E ( u ) = { } . It is knownthat x is an extreme point of the unit ball of a JBW ∗ -triple if and only if it isa complete tripotent. The space E ( u ) can be made into JBW ∗ -algebra withrespect to the following involution ∗ u and Jordan product ◦ u : x ◦ u y = { x, u, y } , x ∗ u = { u, x, u } x, y ∈ E ( u ) . The tripotent u is the unit in the JB ∗ -algebra E ( u ). The triple product inducedby the Jordan product ◦ u via (1.1) coincides with the original triple productrestricted to E ( u ). We shall denote the JB ∗ -algebra defined in this way bythe symbol E ( u ). Sometimes E ( u ) is called homotope of E corresponding to u .Tripotent u in E is called unitary if E ( u ) = E . In that case the triple producton E is coming from the underlying JB ∗ -algebra E ( u ).Let H and K be Hilbert spaces and B ( H, K ) the space of all boundedoperators from H to K . A J ∗ -algebra is a closed subset of B ( H, K ) which isclosed under the product a → aa ∗ a . When endowed with the triple product6 a, b, c } = ( ab ∗ c + cb ∗ a ), J ∗ -algebra is a JB ∗ -triple. If E is a J ∗ -algebra, then u ∈ E is a tripotent if and only if it is a partial isometry, that is an element u for which uu ∗ and u ∗ u are projections (in the corresponding spaces). We shalldefine initial projection of a tripotent u as p i ( u ) = u ∗ u and final projection of u by p f ( u ) = uu ∗ . In this case we have E ( u ) = p f ( u ) Ep i ( u ) . If a J ∗ -algebra A is closed in the weak*-topology, then it is a JBW ∗ -triple. Anexample of a J ∗ -algebra is a JC ∗ -algebra. An example of a J ∗ -algebra that maynot be a JB ∗ -algebra is the JBW ∗ -triple B ( H ) a of all antisymmetric operators in B ( H ). Let us recall that an operator X ∈ B ( H ) is antisymmetric if X t = − X t ,where X → X t is the transpose operation with respect to a fixed orthonormalbasis of H . It is known that this JB ∗ -triple system is a JB ∗ -algebra if and onlyif H does not have odd finite dimension.Let x be a nonzero element in JBW-algebra M . Its range projection p is thesmallest projection in M such that p ◦ x = x . This projection is always con-tained in a JBW-subalgebra of M generated by x . For each norm one element x in a JBW ∗ -triple E , r ( x ) will denote its range tripotent. It is the smallesttripotent e in M for which x is a positive element in E ( e ). If x is a generalnonzero element then its range tripotent is the range tripotent of x k x k . We set r (0) = 0. The range tripotent is always contained in a the JBW ∗ -subtriple of E generated by x . Suppose that x is a positive element in E ( u ) for some tripo-tent u ∈ E . Then its range tripotent coincides with its range projection in E ( u ).Let ( A, ◦ ) and ( B, ◦ ) be JB ∗ -algebras. A linear map J : A → B is calleda Jordan *-homomorphism if J ( a ◦ b ) = J ( a ) ◦ J ( b ) and J ( a ∗ ) = J ( a ) ∗ forall a, b ∈ A . It is called a Jordan *-isomorphism if it is a bijective Jordan*-homomorphism. Jordan homomorphism J : ( A, ◦ ) → ( B, ◦ ) between JB-algebras A and B is a map preserving product, that is J ( a ◦ b ) = J ( a ) ◦ J ( b ).If J is is bijective it is called a Jordan isomorphism. A linear map J : E → F between JB ∗ -triples E and F is called a Jordan triple homomorphisms if itpreserves triple product, that is J { a, b, c } = { J ( a ) , J ( b ) , J ( c ) } . If a Jordantriple homomorphism is a bijection, then we are talking about Jordan tripleisomorphism. Celebrated Kaup’s theorem assures that a surjective linear op-erator between JB ∗ -triples is an isometry if and only if it is a triple isomorphism.It is well known that Dye’s theorem does not hold for algebra of two by twomatrices. In fact, it does not hold for JBW-algebras of Type I which is muchwider class. We do not give original definition of these algebras since we arenot going to use it. However, we describe what Type I means for readers notfamiliar with classification theory of Jordan algebras. Let H n , where n < ∞ ,be an n -dimensional real Hilbert space. We define JB-algebra V n = H n ⊕ R asa Banach space with norm k x ⊕ λ k = k x k + | λ | and with multiplication( a + λ ◦ ( b + µ
1) = ( µa + λb ) ⊕ ( h a, b i + λµ ) . C ( X, V n ) be theJBW-algebra of continuous functions from a hyperstonean space X into spinfactor V n with pointwise defined Jordan multiplications and maximum norm.Type I algebra is isomorphic to a nonzero direct sum ∞ X k =0 A n k , where ( n k ) is a strictly increasing sequence of integers and each A n k is eitherzero or the algebra C ( X k , V n k ), where X n k is a hyperstonean space. We shall saythat a JBW-algebra is regular if it does not contain any direct summand of Type I . A JBW ∗ -triple E is said to be regular if H ( E ( u )) is a regular JBW-algebrafor each complete tripotent u ∈ E . If M is a von Neumann algebra, then it isregular as a JBW ∗ -triple if and only if it does not contain any Type I directsummand. Indeed, having a complete tripotent u in a von Neumann algebra M , we shall show in the proof of Lemma 7.3 that there is a unital Jordan tripleisomorphisms between M = M (1) and M ( u ). This isomorphism is a Jordan *-isomorphism. Any Jordan *-isomorphism preserves Type I direct summands.Therefore M does not have any Type I direct summand if and only if the sameholds for M ( u ).Jordan triple isomorphism Φ : E → F . In this section we gather standard definitions and notations concerning orderedstructures. Let ( P, ≤ ) be a partially ordered set (poset in short). Given a, b ∈ P we shall denote [ a, b ] = { x : a ≤ x ≤ b } . This set will called the interval. By a ∧ b and a ∨ b we shall mean the join (infimum) and meet (supremum) ofthe set { a, b } , respectively. In case of general set S ⊂ P , we shall denote therespective meet and join by V S and W S . A subset of a poset is called boundedif it has lower and upper bound (that is, if it is a subset of some interval).Further, a subset S ⊂ P is called upper bounded if it has an upper bound. Bya conditionally complete poset we mean a poset for which every upper boundednonempty set has supremum. An upward directed poset is a poset in whichevery two point set has an upper bound. Now we shall recall basic concepts ofmorphisms between posets. Let P and Q be posets, and ϕ : P → Q a map. Then ϕ iscalled • an order morphism if a ≤ b implies ϕ ( a ) ≤ ϕ ( b ) for each a, b ∈ P . We alsosay that ϕ preserves order in one direction; • an embedding of posets if a ≤ b if and only if ϕ ( a ) ≤ ϕ ( b ) for each a, b ∈ P .We also say that ϕ preserves order in both directions. Order embeddingis always an injective map; 8 an order isomorphism if it is a surjective order embedding.The same terminology will be employed in case of order reversing maps.For example, a map ϕ : P → Q is an order antimorphism if a ≤ b implies ϕ ( a ) ≥ ϕ ( b ) in Q .We shall be mainly interested in poset endowed with some concept of or-thogonality or orthocomplementation. Let P be poset with a least element 0. Orthogonality relation, ⊥ , on P is a relation satisfying the following conditions:(i) ⊥ is a symmetric relation.(ii) 0 ⊥ a for each a ∈ P .(iii) a ⊥ a implies a = 0.(iv) If a ⊥ b , then c ⊥ b whenever c ≤ a . Let P be a poset with a least element 0 and a greatest element1. • P is called orthoposet if there is an operation a → a ⊥ on P , called ortho-complementation, fulfilling the following conditions for each a, b ∈ P :(i) a ≤ b implies b ⊥ ≤ a ⊥ (ii) a ⊥⊥ = a (iii) a ∧ a ⊥ = 0 and a ∨ a ⊥ = 1.We say that two elements a, b in an orthoposet are orthogonal, written a ⊥ b , if a ≤ b ⊥ . Let us remark that ⊥ is an orthogonality relation on P in the sense of Definition 2.2. • An orthoposet ( P, ≤ , ⊥ ) is called an orthomodular poset or quantum logicif a ∨ b exists whenever a ⊥ b and the following orthomodular law is satisfied: b = a ∨ ( b ∧ a ⊥ ) , whenever a ≤ b . An orthomodular lattice is an orthomodular poset thatis a lattice. • A generalized orthomodular poset P (or a generalized quantum logic) isa poset with a least element 0, such that each interval [0 , a ], a ∈ P is anorthoposet endowed with orthocomplemantation x → x ⊥ a such that thefollowing conditions hold:(i) ([0 , a ] , ≤ , ⊥ a ) is a quantum logic for each a ∈ P .9ii) If a ≤ b , then x ⊥ a = x ⊥ b ∧ a for all x ∈ [0 , a ].Let us remark that if P is a quantum logic, then it is canonically a generalizedquantum logic with local orthocomplementation on each interval [0 , a ] given by b ⊥ a = a ∧ b ⊥ b ≤ a . Therefore quantum logics can be viewed as unital generalised quantum logics.Further, having generalized quantum logic P , we can induce a canonical orthog-onality relation, ⊥ , on P by setting a ⊥ b if there is a v ∈ P with v ≥ a, b suchthat a and b are orthogonal in quantum logic [0 , v ], that is b ≤ a ⊥ v . It can beshown that this definition does not depend on v . A typical example of a gener-alized quantum logic is the poset P ( A ) of projection in a (possibly nonunital) C ∗ -algebra A . Indeed, for each projection p ∈ A we introduce orthocomplemen-tation ⊥ p on [0 , p ] by setting q ⊥ p = p − q , for q ≤ p . Let P and Q be posets endowed with relation of orthogonal-ity, and ϕ a map ϕ : P → Q . • ϕ is called orthomorphism if it preserves orthogonality relation in onedirection, that is if for each a, b ∈ Pϕ ( a ) ⊥ ϕ ( b ) whenever a ⊥ b . • ϕ is called orthoisomorphism if it is a bijection preserving orthogonalityrelation in both directions, that is if ϕ ( a ) ⊥ ϕ ( b ) if and only if a ⊥ b . Let P and Q be generalized orthomodular posets and ϕ : P → Q . Then ϕ is called a quantum logic morphism if for each orthogonal a, b ∈ P we have(i) ϕ (0)=0(ii) ϕ ( a ) ⊥ ϕ ( b )(iii) ϕ ( a ∨ b ) = ϕ ( a ) ∨ ϕ ( b ).If ϕ is a bijection such that both ϕ an ϕ − are quantum logic morphisms, then ϕ is called quantum logic isomorphism.We have the following simple observation. Let ϕ : P → Q be a quantum logic morphism between gen-eralized orthomodular posets P and Q . Then ϕ is an orthomorphism and ordermorphism. roof. By definition ϕ preserves orthogonality. Take a ≤ b in P . Hence b = a ∨ a ⊥ b . Then ϕ ( b ) = ϕ ( a ) ∨ ϕ ( a ⊥ b ). It says that ϕ ( a ) ≤ ϕ ( b ).On the other hand, in the unital case we see that orthoisomorphism is aquantum logic isomorphism. Let P and Q be quantum logics and ϕ : P → Q an orthoi-somorphism. Then ϕ is an order isomorphism preserving orthocomplements.Proof. As 0 is the only element that is orthogonal to every element of the poset,we see that ϕ (0) = 0. Further, as the largest element 1 can be characterized asthe only element that is orthogonal only to 0, we can conclude that ϕ (1) = 1.Let a ∈ P . Then ϕ ( a ) , ϕ ( a ⊥ ) are orthogonal. Put b = ϕ ( a ) ∨ ϕ ( a ⊥ ) . Then ϕ − ( b ⊥ ) is orthogonal to both a and a ⊥ . Therefore ϕ − ( b ⊥ ) ⊥ b = ϕ ( a ) ∨ ϕ ( a ⊥ ) = 1. Put c = ϕ ( a ⊥ ). Then c ⊥ ϕ ( a ) and ϕ ( a ) ∨ c = 1. As c ≤ ϕ ( a ) ⊥ there is, by the orthomodular law, d ∈ Q with c ⊥ d and c ∨ d = ϕ ( a ) ⊥ . Therefore d ⊥ ϕ ( a ) and so d ⊥ ( c ∨ ϕ ( a )) = 1. Hence d = 0and so ϕ ( a ⊥ ) = ϕ ( a ) ⊥ . We have shown that ϕ preserves orthocomplementation.As a ≤ b in P if and only if a ⊥ b ⊥ and ϕ preserves orthocomplements andorthogonality, we have that ϕ preserves the order. It is also clear from symmetrythat ϕ preserves the order in both directions. This completes the proof. In this part we shall generalize Dye’s theorem to projection lattices of JBW ∗ -algebras. For each JB ∗ -algebra ( A, ◦ ) we can associate its projection poset P ( A ), where order of projections is given by p ≤ q if p ◦ q = p . In this case p + q = p ∨ q . If A is unital, then P ( A ) becomes an orthomodular poset withorthocomplementation p → − p . (If A is nonunital, then P ( A ) can be en-dowed with the structure of generalized orthomodular poset, but we shall notneed this level of abstraction.) We can define orthogonality relation on P ( A ) by p ⊥ q if p ◦ q = 0. If A is unital, then orthogonality relation on P ( A ) is the oneinduced canonically by the orthocomplementation, i.e. p ⊥ q if p ≤ − q . Asa consequence of Proposition 2.7 we have that any orthoisomorphism betweenprojection structures of unital JB ∗ -algebras is order isomorphisms preservingorthocomplements. It is well known that if A is a JBW ∗ -algebra, then P ( A ) isa complete lattice.In Theorem 3.2 we present non-bijective version of Dye’s theorem for JBW ∗ -algebras. Our sharpest weapon will be deep Gleason’s theorem for Jordan alge-bras proved by Bunce and Wright in [1, 2]. Let us recall that positive finitelyadditive measure on the projection poset P ( A ) of a JB ∗ - algebra A is a map ̺ from P ( A ) into an interval [0 , ∞ ] such that ̺ ( p + q ) = ̺ ( p )+ ̺ ( q ) whenever p ⊥ q .11 .1. Theorem. (Jordan version of Gleason’s Theorem) Let W be a JBW-algebra such that W does not contain any Type I directsummand. Let ̺ be a finitely additive positive measure on P ( W ) . Then ̺ extends to a unique positive functional on W . The following theorem has been proved in [3] for JW-algebras.
Let A and B be JBW ∗ -algebras such that H ( A ) does not con-tain any Type I direct summand. Suppose that ϕ : P ( A ) → P ( B ) is a quan-tum logic morphism. Then ϕ extends uniquely to a Jordan *-homomorphism J : A → B .
Proof.
We shall follow ideas of the proof in [3]. First let us note that any Jordanhomomorphism between self-adjoint parts of JBW ∗ -algebras (viewed as JBW-algebras) can be canonically extended to a Jordan *-homomorphism betweenwhole algebras. Therefore we restrict ourselves to JBW-algebras M = H ( A ) and N = H ( B ) and show that ϕ extends to a Jordan homomorphism J : M → N .If p and q are orthogonal projections, then their supremum is their sum p + q .Hence, by the property of quantum logic morphism we have that ϕ ( p + q ) = ϕ ( p ) + ϕ ( q ) whenever p ⊥ q .Let us take a positive functional f on N . Composition f ◦ ϕ is a positivefinitely additive measure on P ( A ). According to Theorem 3.1, we have thatthere is a positive functional ˆ f on M that extends f ◦ ϕ . Let L ( M ) be thelinear span of P ( A ) in M . Pick up x ∈ L ( M ) and suppose that we have twoexpressions of x as linear combinations of projections, say x = n X i =1 λ i p i = m X j =1 µ j q j , where λ , . . . , λ n , µ , . . . , µ m ∈ R and p , . . . , p n , q , . . . , q m are projections.Then we have that ˆ f ( x ) = n X i =1 λ i ˆ f ( p i ) = m X j =1 µ j ˆ f ( q j ) . In other words, f ( m X i =1 λ i ϕ ( p i )) = f ( m X j =1 µ j µ ( q j )) . By the Hahn Banach Theorem and the fact that dual of N is spanned by positivefunctionals, we infer that m X i =1 λ i ϕ ( p i ) = m X j =1 µ j ϕ ( q j ) . T : L ( M ) → L ( N ) : n X i =1 λ i p i → m X i =1 λ i µ ( p i ) . We shall show that this map is positive. For a contradiction suppose that P ni =1 λ i p i is positive while P ni =1 λ i ϕ ( p i ) is not positive. In this case there isa positive functional f on N such that P ni =1 λ i f ( µ ( p i )) <
0. Therefore ˆ f is apositive functional on M with ˆ f ( P ni =1 λ i p i ) <
0, but this is a contradiction.Positivity of T implies its boundedness. Indeed, it follows from the inequality k T x k ≤ k T (1) k whenever x ∈ L ( M ) is a positive element with norm less thenone. Consequently, T can be extended to a bounded linear map (denoted againby T ) from M to N . Finally, as T preserves projections, it is a Jordan ho-momorphism. This fact was shown in Theorem A.4 in [18] for von Neumannalgebras, but the proof for JBW ∗ -algebras is the same.Now we shall consider bijective variant of the previous result, which is adirect generalization of Dye’s theorem for von Neumann algebras. Let A be a JBW ∗ -algebra such that H ( A ) does not containany Type I direct summand and B is another JBW ∗ -algebra. Suppose that ϕ : P ( A ) → P ( B ) is an orthoisomorphism. Then ϕ extends to a Jordan *-isomorphism J : A → B .Proof. By Theorem 3.2, ϕ extends to a Jordan ∗ -homomorphism J : A → B .The image J ( P ( A )) contains P ( B ). Since any image of a Jordan*-homomorphismof a JB ∗ -algebra is closed and the closed linear span of P ( B ) is dense in B , wehave that J ( A ) = B . Let us demonstrate that J is injective. We know thatKer J is a JB ∗ -algebra and so it is linearly generated by positive elements. Soif Ker J = { } , then there is a positive norm one element x ∈ J . Now we cancontinue in the same way as in the proof of [10, Theorem 8.1.2 p. 256]. By thespectral theory we can write x = X n n p n , where p n ’s are mutually commuting projections in H ( A ). As 0 = J ( x ) ≥ n J ( p n ), we have that J ( p n ) = 0 for all n . However, at leat one p n has to benonzero. This is a contradiction with injectivity of ϕ on the projection lattice. We shall recall a few standard definitions. Let E be a JB ∗ -triple. By U ( E ) weshall denote the set of all tripotents of E , that is the set of all elements u ∈ E for13hich u = { u, u, u } . This is always nonempty subset as zero is a tripotent. Incase when E is a JB ∗ -algebra, then tripotents are just partial isometries, that iselements u with u = uu ∗ u . Then p i ( u ) = u ∗ u and p f ( u ) = uu ∗ are projections,called initial and final projection, respectively. We shall be mainly interested inthe following orthogonality relation on U ( E ). Let E be a JB ∗ -triple. Two tripotents e, f ∈ E are orthogonalif L ( e, f ) = 0 . In the next proposition we shall gather basic simple characterizations oforthogonality of tripotents that we shall use in the sequel without further com-ments (see e.g. Lemma 2.1 in [13]).
Let E be a JB ∗ triple, and let e, f be tripotents in E . Thenthe following assertions are equivalent: (i) e ⊥ f (ii) f ⊥ e (iii) e ∈ E ( f )(iv) E ( e ) ⊂ E ( f )(v) { e, e, f } = 0(vi) Both e + f and e − f are tripotents. If p and q are projections in a JB ∗ -algebra A , then p and q are orthogonal asprojections ( p ◦ q = 0) if and only if they are orthogonal as tripotents. In case ofJ ∗ -algebras orthogonality is equivalent to pairwise orthogonality of initial andfinal projections. This relation is known under the name double orthogonality. Two tripotents u and v in a unital J ∗ -algebra A are or-thogonal if and only if they have orthogonal initial and final projections, thatis v ∗ u = uv ∗ = 0 . Proof.
Suppose that v ⊥ u . This is equivalent to { v, v, u } = 0 . Therefore vv ∗ u + uv ∗ v = 0 . (4.1)By multiplying from the left by v ∗ we obtain0 = v ∗ vv ∗ u + v ∗ uv ∗ v = v ∗ u + v ∗ uv ∗ v = v ∗ u [1 + v ∗ v ]14owever, as v ∗ v is a projection and therefore positive element, we have that1 + v ∗ v is invertible and so v ∗ u = 0 by the previous identity. Multiplying nowthe identity (4.1) by v ∗ from the right, we arrive similarly to[ vv ∗ + 1] uv ∗ = 0which gives uv ∗ = 0 in the same way as above. The reverse implication isobvious.Now we shall define key order considered in this paper. Let E be a JB ∗ -triple, and let e, f ∈ E be tripotents. We saythat e is less then f , written e ≤ f , if f − e is a tripotent orthogonal to e .We gather a few characterizations of the order relation that we shall usefrequently. Let E be a JB ∗ -triple, and let u, v be tripotents in E . Thefollowing assertions are equivalent (see e.g Proposition 2.4 in [13]): (i) u ≤ v (ii) u = { u, v, u } (iii) u = { u, u, v } (iv) u is a projection in E ( v )(v) E ( u ) is a JB ∗ -subalgebra of E ( v ) . Having a JB ∗ -triple E we shall always consider U ( E ) the set of all tripotentsin E as a poset with order defined above. It is a generalized orthomodular poset,where local orthocomplementation in interval [0 , e ] is given by f ⊥ e = e − f . It can be observed immediately that e ⊥ f for two tripotents e, f exactlywhen e ≤ u − f = f ⊥ u in each interval [0 , u ] containing e and f . (Such intervalexists because e + f is supremum e ∨ f .) In other words, orthogonality is inducedby order an local orthocomplementation.Let E be a JBW ∗ -triple. It is a consequence of Proposition 4.5 that any in-terval [0 , e ] in U ( E ) is order isomorphic to the projection poset P ( E ( e )), whichis an orthomodular poset. Moreover if e, f are orthogonal tripotents and w is atripotent w ≥ e, f , then e and f become orthogonal projections in E ( w ). (Espe-cially this holds for w = e + f .) Therefore the poset U ( E ) can be seen as pastingorthomodular posets that are isomorphic to projection posets of JB ∗ -algebras.If A is a JBW ∗ -algebra, then P ( A ) is a complete lattice. This is far frombeing true in case of tripotent order structures. The reason is that this poset is15ot upward directed in a typical situation. To see it, let us recall that maximalelements in U ( E ), where E is a nonzero JBW ∗ triple, are precisely completetripotents, that is, extreme points of the unit ball. If dim E ≥
2, there must betwo different maximal tripotents u and v . It is then straightforward to concludethat there is no upper bound of the set { u, v } .The relation of orthogonality for tripotents is an orthogonality relation inthe sense of our Definition 2.2. Further, tripotent poset U ( E ) is an exampleof generalized quantum logic. Even if it is not a lattice, a deep analysis givenby Edward and R¨uttimann in [7] showed that U ( E ) is a conditionally completelattice. More specifically, they showed that there is an order anti-isomorphismsbetween U ( E ) and the set of nonempty weak ∗ closed faces of the unit ball of E ordered by set inclusion. This antiisomorhism is given by the map e ∈ U ( E ) → e + B ( E ( e )) . In this light, let us look at the lattice operation. The supremum of two elements e, f ∈ U ( E ) exists if and only if the intersection of the faces e + B ( E ( e )) and f + B ( E ( e )) is nonempty. However, it may easily happen that this intersectionis empty. For example, one can take two distinct extreme points f, g of the unitball of E (they correspond to maximal tripotents) and consider singleton faces { e } , { f } . In contrast to this, given two weak ∗ closed nonempty faces E and F ,there is always their suremum, namely the smallest weak ∗ closed face contains F ∪ G . It means that infimum e ∧ f in E ( U ) always exists and corresponds toa weak ∗ -closed face generated by two faces.As an illustration of the triple order, let us consider a von Neumann algebra M . Then U ( M ) is the set of all partial isometries and the order relation is givenby u ≤ v if and only if u = uv ∗ u = uu ∗ v . All unitaries are maximal tripotents, however there might be non-unitarymaximal tripotents in case of infinite algebras (see [13] for deeper analysis ofthis phenomenon.)The following description of triple order was given in Proposition 4.6 in [13].
Let u, v be partial isometries in a von Neumann algebra M .The following assertions are equivalent: (i) u ≤ v (ii) There is a unique projection p ≤ p f ( v ) such that u = pv . (iii) There is a unique projection q ≤ p i ( v ) such that u = vq . , v ] in U ( E ) is order isomorphic to interval[0 , p f ( v )] (and [0 , p i ( v )]) in the projection lattice P ( M ).Let M be a von Neumann algebra acting on a Hilbert space H . Suppose u, w ∈ U ( M ). In order to understand better the way how intervals in U ( M )may overlap, we shall describe I = [0 , u ] ∩ [0 , w ] = [0 , u ∧ w ]. This gives a spacialdescription of infima that is not referring to facial structure of the unit ball asEdwards and R¨uttimann did in [7]. By Proposition 4.6 we see that a tripotent t is in this intersection if and only if t = pu = qw , where p is a projection under p f ( u ) and q is a projection under p f ( w ). Multi-plying the previous equation by u ∗ from the right, and using the fact p f ( u ) = uu ∗ ≥ p , we have p = puu ∗ = qwu ∗ . (4.2)However, this means that range of p is contained in the range of q and so p ≤ q .By symmetry argument p = q . Therefore we have, I = { pu : p ≤ p f ( u ) ∧ p f ( w ) , pu = pw. } In other words, infimum u ∧ w is of the form u ∧ w = hu = hw , where h = sup { p ∈ P ( M ) : p ≤ p f ( u ) ∧ p f ( w ) , pu = pw } .This is an expression for infima in terms of the projection lattice. Let usnow explore special geometric meaning of elements in I . Take p ≤ p f ( u ) with pu = pw . By (4.2) we have that p = pwu ∗ . (4.3)This implies that wu ∗ restricts to identity on p ( H ). Indeed, let us take ξ ∈ H with pξ = ξ . By (4.3) we have that ξ = pwu ∗ ξ . Suppose that wu ∗ ξ = pwu ∗ ξ . Then, obviously k pwu ∗ ξ k < k wu ∗ ξ k ≤ k ξ k . This is a contradiction. Therefore p ( H ) is an invariant subspace of wu ∗ andthis map is identity on it. By the same arguments, p ( H ) is invariant for uw ∗ =( wu ∗ ) ∗ and so p ( H ) ⊥ is also invariant for uw ∗ . So we obtain the followingorthogonal decomposition: uw ∗ = identity on p ( H ) ⊕ some contraction on (1 − p )( H ) . Now we turn to morphisms between tripotent posets.17 .7. Example.
Let
Φ : E → F be a Jordan triple homomorphism between JB ∗ -triples E and F . Then Φ preserves tripotents and restricts to a quantum logicmorphism ϕ : U ( E ) → U ( F ) . The proof of this statement is straightforward. In the opposite directionJordan triple homomorphisms between JBW ∗ -triples can be characterised aslinear maps preserving tripotents. This is the content of the following proposi-tion. We expect this fact to be known but we give the argument for the sake ofcompleteness. Let E and F be JBW ∗ - triples. Let Φ : E → F be a boundedlinear map preserving tripotents. Then Φ is a Jordan triple homomorphism.Proof. First we show that Φ restricts to a quantum logic morphism between U ( E ) and U ( F ). Let e and f be orthogonal tripotents in E . Then e + f and e − f are tripotents in E , implying that Φ( e ) + Φ( f ) and Φ( e ) − Φ( f )are tripotents in F . By Proposition 4.2 again we have that Φ( e ) and Φ( f ) areorthogonal. Therefore, Φ preserves orthogonality of tripotents. This is enoughfor showing that Φ is a triple homomorphism. Indeed, in view of polarizationidentities (see e.g [5]) it suffices to show that Φ preserves cubic powers. Let ustake take an element x and try to show thatΦ( x ) = Φ( x ) . By the spectral theorem and continuity of Φ we can suppose that x = n X i =1 λ i e i , where e , . . . , e n are orthogonal tripotents and λ , . . . , λ n ∈ C . As Φ preservesorthogonality of tripotents we have that { Φ( e i ) , Φ( e j ) , Φ( e k ) } = 0whenever { i, j, k } is not singleton. Based on it we can computeΦ( x ) = Φ (cid:18) n X i =1 λ i λ i λ i e i (cid:19) = n X i =1 λ i λ i λ i Φ( e i ) == n X i,j,k =1 λ i λ j λ k { Φ( e i ) , Φ( e j ) , Φ( e k ) } = n X i =1 λ i λ i λ i { Φ( e i ) , Φ( e i ) , Φ( e i ) } = Φ( x ) . Therefore, in case of linear maps the situation is clear. However, in thecontext of JBW ∗ -triplets duality between triple morphisms and Jordan triplemorphisms breaks down. Indeed, next series of examples shows that orderautomorhisms of tripotent poset may not be coming from restrictions of linearmaps. Also we demonstrate that the relationship between orthogonality andorder is more delicate for tripotents than for projections in Jordan algebras.18 .9. Example. (i) Orthoisomorphisms not extendable to a homogeneous map:
Let A be a JB ∗ -algebra. Then the star operation x → x ∗ is a bijectionthat preserves Jordan product, and so its triple product as well. There-fore, when restricted to U ( A ) and taking into account Proposition 4.5 andProposition 4.6, we obtain an orthoisomorphism and order isomorphismthat is not extendable to any linear map.(ii) Orthoisomorphism not extendable to an additive map:
Let M be a von Neumann algebrawith dim E ≥
3. Let T be the unit circlein C . Let us introduce equivalence relation on U ( M ) by putting u ∼ v if there is a complex unit λ such that u = λv . Choose an arbitrary map T : U ( E ) / ∼ → T . Let S : U ( E ) / ∼ → U ( E ) be a selection function.Finally, define a map ϕ : U ( M ) → U ( M ) by ϕ ( λS ([ u ])) = λT ([ u ]) S ([ u ]), u ∈ U ( E ), λ ∈ T . Then ϕ is a bijection for which ϕ ( S ([ u ]) = T ([ u ]) S ([ u ]).Moreover, ϕ preserves orthogonality and order in both directions. Indeed,let u ⊥ v in U ( E ), or equivalently { u, u, v } = 0. There are complex units λ and µ such that ϕ ( u ) = λu and ϕ ( v ) = µv . Then { ϕ ( u ) , ϕ ( u ) , ϕ ( v ) } = λλµ { u, u, v } = 0 . Similarly, it can be verified that ϕ preserves orthogonality in the oppo-site direction. We can now specify the map ϕ so that it has no extensionto any additive map acting on M . Indeed, take two nonzero orthogonaltripotents u and v in M . Modify parameters in definition of ϕ so that ϕ ( u ) = − u and ϕ ( v ) = v . There is a λ ∈ T such that ϕ ( u + v ) = λ ( u + v )Since ϕ ( u ) + ϕ ( v ) = v − u , we can see that ϕ ( u + v ) = ϕ ( u ) + ϕ ( v ). So ϕ cannot have an additive extension over M . Therefore there are orthoiso-morphisms of U ( M ) which cannot be extended to any additive map from M to M .(iii) Orthoisomorphism that is not preserving the order:
We shall show that there is a tripotent orthoisomorphism that is not pre-serving the order. This cannot happen in the projection poset of JB ∗ -algebras (see Proposition 2.7). We shall use the preceding example (ii).Let us take linearly independent tripotents u and v in M with u ≤ v . Wecan certainly choose the map ϕ above so that ϕ ( u ) = u and ϕ ( v ) = − v .Then u = { u, v, u } . But { ϕ ( u ) , ϕ ( v ) , ϕ ( u ) } = − u and so ϕ ( u ) is not un-derneath ϕ ( v ) as u = ϕ ( u ) = { ϕ ( u ) , ϕ ( v ) , ϕ ( u ) } .There is another example showing considerable nonlinearity of tripotentorthoisomorphisms. 19et M = B ( H ) a , where dim H = 5. By a rank, d ( u ), of a tripotent u in M we mean dimension of it initial (and so final) projection. First weobserve that if u ∈ M is a nonzero tripotent, then the following cases mayoccur (see e.g. [13]): either d ( u ) = 2 or d ( u ) = 4. In the former case u isminimal. Indeed, let e ≤ u be a nozero tripotent. Then p f ( e ) and p f ( u − e )are orthogonal projections underneath p f ( u ) and with even dimensions.This immediately implies that e = u . In the latter case we infer in asimilar way that u is a maximal tripotent. Let us now consider a bijection ϕ : U ( M ) → U ( M ) that is fixing tripotents with rank two and preserveszero and tripotents of rank 4. As nontrivial tripotents are orthogonal onlyif they have rank two, we can see that ϕ is an orthoisomorphism. Let usnow fix tripotents u ≤ v such that dim p i ( u ) = 2 and dim p i ( v ) = 4. Wecan further specify ϕ to send v to a tripotent whose initial projection isnot above p i ( u ). Then ϕ ( v ) is not above ϕ ( u ) and therefore ϕ is not orderpreserving.(iv) Orthoisomorphism on a regular JBW ∗ -triple having no linear extension. Let us now consider M = B ( H ) a , where dim H = 3. Let u be a nonzerotripotent of M . Then its rank must be two. Moreover, tripotents u and v are orthogonal if and only if at least one of them is zero. Therefore,any bijection Φ acting on U ( M ) fixing zero is an orthoisomorphism. Ofcourse, ϕ may not have any linear extension to M . On the other hand, M is regular. To see it we consider a homotope M ( u ) where u is a maximaltripotent (i.e. u has rank two). Simultaneously u is an atom and so E ( u )is isomorphic to C . For this reason E ( u ) cannot contain any Type I direct summand and so M is regular. Let E and F be JB ∗ -triples, and let ϕ : U ( E ) → U ( F ) be amap. The system of maps Φ ( ϕ ) = (Φ u ) u ∈ U ( E ) is a consistent system of Jordan*-homomorphisms corresponding to ϕ if the following conditions are satisfied:(i) Each Φ u : E ( u ) → F ( ϕ ( u )) is a unital Jordan *-homomorphism betweenalgebras E ( u ) and F ( ϕ ( u )).(ii) If u ≤ v in U ( E ) then Φ u and Φ v coincide on E ( u ).Remark that the map ϕ in the above definition is uniquely determined bythe system of maps (Φ u ) u ∈ ( U ( E )) as ϕ ( u ) = Φ u ( u ) . On the other hand, to a given ϕ there is only one possible consistent system Φ ( ϕ ). Indeed, suppose we have two such consistent systems Φ ( ϕ ) and Ψ ( ϕ ).20onsider a tripotent u ∈ E . We know that Φ u and Ψ u coincide with ϕ on[0 , u ] = P ( E ( u )). However projections in E ( u ) span a dense linear subspace.So by continuity Φ u = Ψ u . Let Φ ( ϕ ) be a consistent system of Jordan *-homomorphismsbetween JB ∗ -triples E and F . Then the map ϕ is a quantum logic morphism.Proof. Let u and v be orthogonal tripotents. Then w = u + v = u ∨ v is atripotent. By the assumption Φ w gives Φ u and Φ v on E ( u ) and E ( v ), respec-tively. Tripotents u and v becomes projections in E ( w ) and are mapped toorthogonal projections Φ w ( u ) = ϕ ( u ) and Φ w ( v ) = ϕ ( v ) as Φ w is a Jordan *-homomorphism. Moreover, we have ϕ ( u ∨ v ) = ϕ ( u + v ) = Φ w ( u + v ) = Φ w ( u ) + Φ w ( v ) = ϕ ( u ) + ϕ ( v ) = ϕ ( u ) ∨ ϕ ( v ) . Let E and F be JB ∗ -triples. The consistent system of Jordan*-homomorphisms Φ ( ϕ ) is called a consistent system of Jordan *-isomorphismsif each map Φ u is a Jordan *-isomorphism and ϕ is a bijection. Let Φ ( ϕ ) be a consistent system of Jordan *-isomorphisms.Then ϕ is a quantum logic isomorphism.Proof. By Proposition 5.2 the corresponding map ϕ is a bijection that is a quan-tum logic isomorphism. Let us now realize that the system of maps (Φ − w ) w ∈ U ( F ) it is a consistent system of Jordan *-homomorphism corresponding to ϕ − . Forthis reason ϕ − is a quantum logic morphism as well.Now we prove that any quantum logic morphism between structure of tripo-tents extends uniquely to a consistent system of Jordan maps. Let E and F be JBW ∗ -triples, where E is regular. Let ϕ : U ( E ) → U ( F ) be a quantum logic morphism. Then there is a unique consistentsystem of Jordan *-homomorphisms Φ ( ϕ ) = (Φ u ) u ∈ U ( E ) corresponding to ϕ .Proof. We know that ϕ preserves the order. Let us fix u ∈ U ( E ) and take acomplete tripotent w ∈ E with w ≥ u . Then ϕ maps [0 , u ] = P ( E ( u )) into[0 , ϕ ( u )] = P ( E ( ϕ ( u ))) and [0 , w ] = P ( E ( w )) into [0 , ϕ ( w )] = P ( E ( ϕ ( w ))) . Ac-cording to Theorem 3.2 there is a unital Jordan *-homomorphism Φ w : E ( w ) → F ( φ ( w )). As E ( u ) is a JB ∗ -subalgebra of E ( w ), Φ w restricts to a Jordan *-homomorphism Φ u with domain E ( u ) whose restriction to [0 , u ] coincides with ϕ . Such a map is unique and so does not depend on the choice of w . Therefore,we have that for each u ∈ U ( E ) the restricted map ϕ : [0 , u ] → [0 , ϕ ( u )]) extendsuniquely to a unital Jordan ∗ -homomorphism Φ u between E ( u ) and F ( ϕ ( u )). Itremains to verify that Φ ( ϕ ) is a consistent system of Jordan *-homomorphisms.But this follows immediately from the construction.21 .6. Theorem. Let E be a regular JBW ∗ -triple and F a JBW ∗ -triple. Let ϕ : U ( E ) → U ( F ) be a quantum logic isomorphism. Then there is a uniqueconsistent system Φ ( ϕ ) of Jordan *-isomorphisms.Proof. We know that there is a consistent system of Jordan *-homomorphisms Φ ( ϕ ) with corresponding function ϕ . We have to show that each Φ u is a Jordan*-isomorphism. Fix u ∈ U ( E ). Then the restriction ϕ : E ( u ) → E ( ϕ ( u )) is aquantum logic isomorphism. So by Theorem 3.3 we have that Φ u is a Jordan*-isomorphisms. In this part we describe morphisms of tripotent posets using one single maprather than a family of Jordan maps. It turns out that this global map ispartially linear in the sense of definitions below.
We say that a set S in a JBW ∗ -triple E is triple bounded ifthe set { r ( s ) : s ∈ S } has upper bound in U ( E ). Let E and F be JBW ∗ -triples. Let J : E → F be a map. Wesay that J is a local triple Jordan homomorphism if the following conditions aresatisfied.(i) J is real homogenous, i.e. J ( λx ) = λJ ( x ) for all x ∈ E and λ ∈ R .(ii) J is partially additive in the sense J ( x + y ) = J ( x ) + J ( y ), whenever theset { x, y } is triple bounded.(iii) J preserves tripotents.First we observe that in view of Proposition 4.8 any linear local Jordanmorphism is a Jordan triple homomorphism. Further we see that any localJordan isomorphim restricts to a quantum logic morphism preserving reflections. Suppose that E and F are JBW ∗ -triples. Let J : E → F be a local triple Jordan homomorphism between JBW ∗ -triples E and F . Then J restricts to quantum logic morphism ϕ : E → F such that ϕ ( − u ) = − ϕ ( u ) for all u ∈ U ( E ) .Proof. Suppose that e and f are orthogonal tripotents. Tripotent e + f and e − f is supremum of the set { e, f } and { e, − f } , respectively. Therefore the sets { e, f } and { e, − f } are triple bounded. By assumption J ( e + f ) = J ( e ) + J ( f )and J ( e − f ) = J ( e ) − J ( f ). As e ± f are tripotents, we conclude that J ( e ) ± J ( f )are tripotents as well. This shows that J ( e ) and J ( f ) are orthogonal tripotents.Moreover we see that ϕ ( e ∨ f ) = ϕ ( e ) ∨ ϕ ( f ). The proof is completed.22n order to prove the opposite statement we shall need the following auxiliarylemma. Let E be a JBW ∗ -triple. Let w ∈ U ( E ) . The following state-ments hold: (i) If x is a positive element in E ( w ) , then r ( x ) ≤ w . (ii) Let x , . . . , x n ∈ E be such that r ( x i ) ≤ w for all i = 1 , . . . , n , then r ( P ni =1 x i ) ≤ w .Proof. (i) Let x be a positive element in E ( w ). As E ( w ) is a JBW ∗ -subtripleof E , we have that the range tripotent r ( x ) belongs to E ( w ). However, r ( x ) isthe smallest tripotent such that x is positive in E ( r ( x )). Hence, r ( x ) ≤ w .(ii) We have that x i is a positive element in E ( r ( x i )) and so also in E ( w )because E ( r ( x i )) is a *-subalgebra of E ( w ). For this reason P ni =1 x i is positivein E ( w ) and (i) applies.Let us remark that the previous proposition does not hold without assumingpositivity of x . Indeed, any tripotent in E ( w ) that is not a projection providesa counterexample. Let E and F be JBW ∗ -triples. Suppose that E is regular. Let ϕ : U ( E ) → U ( F ) . Then the following conditions are equivalent: (i) ϕ is a quantum logic morphism such that ϕ ( − u ) = − ϕ ( u ) for all u ∈ U ( E ) . (ii) There is a local Jordan triple morphism
Φ : E → F extending ϕ . (iii) There is a local Jordan triple morphism
Φ : E → F extending ϕ such that moreover Φ { x, y, x } = { Φ( x ) , Φ( y ) , Φ( x ) } , whenever the set { x, y } is triple bounded.Further, if ϕ : E → F is a quantum logic morphism, then it extends to a partiallyadditive map between E and F . Proof. (i) ⇒ (iii). Let ϕ be a quantum logic morphism. By Theorem 5.5 wehave a consistent system of Jordan *-homomorphisms Φ ( ϕ ) corresponding to ϕ . Define now the map Φ in the following way: Let x ∈ E , setΦ( x ) = Φ r ( x ) ( x ) . Let us verify that Φ is a local Jordan triple morphism. Let us fix x ∈ E and λ ∈ R . If λ > r ( x ) = r ( λx ). Therefore,Φ( λx ) = Φ r ( λx ) ( λx ) = λ Φ r ( x ) ( x ) = λ Φ( x ) . Let us discuss the case when λ <
0. It holds that r ( − x ) = − r ( x ). Indeed, forany tripotent u ∈ E we have that E ( u ) = E ( − u ) and x ∗ u = x ∗ − u , x ◦ u y = − x ◦ − u y for all x, y ∈ E ( u ). This gives that an element z ∈ E ( u ) is positivein E ( u ) if and only if − z is positive in E ( − u ). Therefore − x is positive in E ( − r ( x )). On the other hand, suppose that − x is positive in E ( u ) for a tripotent u . By the above x is positive in E ( − u ) and so r ( x ) ≤ − u . Equivalently, r ( x ) = { r ( x ) , − u, r ( x ) } and so − r ( x ) = {− r ( x ) , u, − r ( x ) } . This means that − r ( x ) ≤ u . Hence, r ( − x ) = − r ( x ). Further, we shall need the fact thatΦ u = Φ − u . Let us remark that by the assumption both Φ u and Φ − u act between E ( − u ) and E ( ϕ ( − u )). Let us take a projection p ∈ E ( − u ). Then − p is aprojection in E ( u ). Indeed, the *-operation is the same and so p is self-adjointin E ( u ). The idempotency of p in E ( − u ) means that p = { p, − u, p } . Then( − p ) ◦ u ( − p ) = { p, u, p } = − p . Therefore, − p is a projection in E ( u ). In fact,by symmetry, q is a projection in E ( u ) if and only if − q is projection in E ( − u )Let us fix a projection p ∈ E ( − u ). ThenΦ u ( p ) = − Φ u ( − p ) = − ϕ ( − p ) = ϕ ( p ) . It means that Φ u and Φ − u coincide on projections in E ( − u ). Moreover, Φ u preserves Jordan product in E ( − u ). For this, let us take x, y ∈ E ( − u ) andcomputeΦ u ( x ◦ − u y ) = Φ u ( − x ◦ u y ) = − Φ u ( x ) ◦ ϕ ( u ) Φ u ( y ) == Φ u ( x ) ◦ − ϕ ( u ) Φ u ( y ) = Φ u ( x ) ◦ ϕ ( − u ) Φ u ( y ) . Since any Jordan *-homomorphism is uniquely deteremined by its value onprojections, we have that really Φ u = Φ − u . Finally, we can compute,Φ( λx ) = Φ r ( λx ) ( λx ) = Φ − r ( x ) ( λx ) = Φ r ( x ) ( λx ) = λ Φ r ( x ) ( x ) = λ Φ( x ) . We have shown that Φ is real homeogeneous.Let us investigate additivity of Φ. Take a triple bounded set { x, y } . Let w be a tripotent with w ≥ r ( x ) , r ( y ). By Lemma 6.4 we have r ( x + y ) ≤ w . Usingconsistency of the system of Jordan *- homomorphisms Φ ( ϕ ) we have24( x + y ) = Φ r ( x + y ) ( x + y ) = Φ w ( x + y ) == Φ w ( x ) + Φ w ( y ) = Φ r ( x ) ( x ) + Φ r ( y ) ( y ) = Φ( x ) + Φ( y ) . (6.1)This shows additivity of Φ on triple bounded sets.Suppose now that { x, y } is a triple bounded set. That is, there is a tripo-tent w with r ( x ) , r ( y ) ≤ w . As x and y are positive in E ( r ( x )) and E ( r ( y )),respectively, we can see that x and y are positive in E ( w ) as well. Observe that U x ( y ) = { x, y, x } is positive in E ( w ). Using Lemma 6.4 once again, we obtainthat r ( { x, y, x } ) ≤ w . Now we can compute, using the properties of a consistentsystem of Jordan maps, thatΦ { x, y, x } = Φ r ( { x,y,x } ) { x, y, x } = Φ w { x, y, x } == { Φ w ( x ) , Φ w ( y ) , Φ w ( y ) } = { Φ( x ) , Φ( y ) , Φ( x ) } . The proof of the present implication is completed.(ii) ⇒ (i) follows immediately from Proposition 6.3 and (iii) ⇒ (ii) is trivial.The fact that any quantum logic morphism ϕ : U ( E ) → U ( F ) extends toa partially additive map Φ : U ( E ) → U ( F ) is contained in the proof of theimplication (i) ⇒ (iii).The proof is completed. Let E and F be JBW ∗ -triples. A map J : E → F is a localJordan triple isomorphism if it is a bijection such that both J and J − are localJordan triple homomorphisms. Let J : E → F be a local Jordan triple isomorphism. Then itsrestriction ϕ to U ( E ) is a quantum logic isomorphism between U ( E ) and U ( F ) such that ϕ ( − u ) = − ϕ ( u ) for all u ∈ U ( E ) .Proof. It follows immediately from Proposition 6.3 that ϕ is a quantum logicisomorphism. Let E and F be JBW ∗ -triples. Suppose that E and F areregular. Let ϕ : U ( E ) → U ( F ) be a quantum logic isomorphism such that ϕ ( − u ) = − ϕ ( u ) for all u ∈ U ( E ) . Then ϕ extends to a localProof. Let us denote by Φ and Ψ local Jordan triple homomorphisms corre-sponding to ϕ and ϕ − respectively, as constructed in the proof of Theorem 6.5.25e shall prove that they are mutually inverse maps. For this let us take x ∈ E and consider y = Φ( x ) = Φ r ( x ) ( x ) . We know that Φ r ( x ) is a Jordan *-isomorphism from E ( r ( x )) onto E ( ϕ ( r ( x )))whose inverse is the map Ψ ϕ ( r ( x )) . As x is positive in E ( r ( x )), we can see that y = Φ( x ) is positive in E ( ϕ ( r ( x )). Hence, r ( y ) ≤ ϕ ( r ( x )). Now we can computeΨ( y ) = Ψ ϕ ( r ( x )) ( y ) = Φ − r ( x ) ( y ) = x . So we have established that Ψ ◦ Φ is identity on E . By the symmetry we havethat Φ and Ψ are really mutually inverse maps. Now by Theorem 6.5 Φ is alocal Jordan triple isomorphism. If we have two JB ∗ -triples E and F that are Jordan triple isomorphic, thenthe tripotent structures U ( E ) and U ( F ) are isomorphic as generalized quantumlogics. This holds because of the fact that Jordan triple isomorphism implementsquantum logic isomorphism. Therefore the tripotent poset is invariant in thetheory of Jordan triples. We know that morphism between tripotent structuresis not extendable to a Jordan triple morphism in all cases. However, it doesnot exclude that tripotent structure determines the triple structure itself. Ithappens in case of projection lattices. Indeed, even if Dye’s theorem does nothold for type I von Neumann algebras, we still have that such algebras areJordan *-isomorphic if and only if their projection lattices are orthoisomorphic.We have the following result proved in [17, Cor. 9.2.9, p. 193] and [12, Theorem2.3] Let M and N be von Neumann algebras. Then the followingconditions are equivalent: (i) P ( M ) and P ( N ) are orthoisomorphic. (ii) M and N are Jordan isomorphic (that is there is a Jordan *-isomorphismbetween M and N ). Therefore, projection lattice with orthogonality relation is a complete Jordaninvariant for von Neumann algebras. It is natural to ask whether the same holdsfor tripotent posets. We shall give an affirmative answer below. Let us first stateauxiliary facts. The following one is well known and we state it for the sake ofcompleteness.
Let A be a J ∗ -algebra in B ( H ) . Suppose that v ∈ U ( A ) . Then A ( v ) is triple isomorphic to A ( p i ( v )) . roof. We know that A ( v ) = p f ( v ) Ap i ( v ) and A ( p i ( v )) = p i ( v ) Ap i ( v ). It canbe easily verified by a direct computation that the map x ∈ A ( v ) → v ∗ x ∈ A ( p i ( v ))is a Jordan triple isomorphism.We show that for unital C ∗ -algebras the structure of tripotents determinesthe structure of projections. It is based on the following lemma. Let A be a unital C ∗ -algebra and u a complete tripotent in A .Then P ( A ) is quantum logic isomorphic to the interval [0 , u ] in U ( A ) .Proof. By Lemma 6.1 in [14] there is a Hilbert space H and an isometric unitalJordan *- homomorphism ψ : A → B ( H ) such that ψ ( u ) ∗ ψ ( u ) = 1. Thereforewe can replace A by JC ∗ -algebra ψ ( A ) that is Jordan *-isomorphic to A . Thisway we can suppose that p i ( u ) = 1. By Lemma 7.2 there si a unital tripleisomorphism between A ( u ) and A ( p i ( u )) = A (1) = A . Therefore A ( u ) and A (1)are isomorphic as JB ∗ -algebras. Consequently, P ( A ) = [0 ,
1] is quantum logicisomorphic to [0 , u ] in A ( u ). The proof is completed. Let A and B be unital C ∗ -algebras such that U ( A ) and U ( B ) are quantum logic isomorphic. Then P ( A ) and P ( B ) are quantum logicisomorphic.Proof. Let ϕ : U ( A ) → U ( B ) be an orthoisomorphism. Then u = ϕ (1) is a com-plete tripotent in B . Indeed, if it is not true, then there is a nonzero tripotent h in B orthogonal to ϕ (1). Then the preimage ϕ − ( h ) is a nonzero tripotentorthogonal to 1, which is not possible. Now ϕ restricts to an orthoisomorphismbetween [0 ,
1] and [0 , u ] . These posets are quantum logic isomorphic to P ( A )and P ( B ), respectively by Lemma 7.3.Since it is known that projection lattice as a quantum logic is a complete Jor-dan invariant for von Neumann algebras (see Proposition 7.1) we can concludethat the same holds for tripotent poset. Let M and N be von Neumann algebras. Suppose that U ( M ) and U ( N ) are quantum logic isomorphic. Then M and N are Jordan *-isomorphic. As a conclusion, even if the tripotent poset is larger than projection poset, itcontains the same amount of information about Jordan parts of von Neumannalgebras as their projection lattices.
Acknowledgement:
This work was supported by the project OPVVV CAASCZ.02.1.01/0.0/0.0/16 019/0000778 27 eferences
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