Order isomorphisms on order intervals of atomic JBW-algebras
aa r X i v : . [ m a t h . F A ] N ov Order isomorphisms on orderintervals of atomic JBW-algebras
Mark Roelands ∗ School of Mathematics, Statistics & Actuarial Science, University of Kent, Canterbury, CT27NX, United Kingdom
Marten Wortel † Department of Mathematics and Applied Mathematics, University of Pretoria, Private Bag X20Hatfield, 0028 Pretoria, South Africa
November 12, 2019
Abstract
In this paper a full description of order isomorphisms between effect algebras of atomicJBW-algebras is given. We will derive a closed formula for the order isomorphisms on theeffect algebra of type I factors by proving that the invertible part of the effect algebra of atype I factor is left invariant. This yields an order isomorphism on the whole cone, for whicha characterisation exists. Furthermore, we will show that the obtained formula for the orderisomorphism on the invertible part can be extended to the whole effect algebra again. Asatomic JBW-algebras are direct sums of type I factors and order isomorphisms factor throughthe direct sum decomposition, this yields the desired description.
Keywords:
Order isomorphisms, atomic JBW-algebras, effect algebra.
Subject Classification:
Primary 47B49; Secondary 46L70.
Let H be a complex Hilbert space. In [ˇS17], inspired by Ludwig ([Lud83, Section V.5]) and Moln´ar([Mol03, Corollary 4]), ˇSemrl investigated the order isomorphisms of the effect algebra[0 , I ] := { T ∈ B ( H ) sa : 0 ≤ T ≤ I } , which is an important object in quantum mechanics. For more details and an extensive historicalperspective we direct the reader to the excellent introduction of ˇSemrl’s paper [ˇS17]. The previousresults of Ludwig and Moln´ar characterised the order isomorphisms of the effect algebra that ∗ Email: [email protected] † Email: [email protected] B ( H ) is an atomic von Neumann algebra, in fact a factor, and its self-adjoint part istherefore an atomic JBW-algebra (the reader more familiar with the theory of C *-algebras shouldthink of the self-adjoint part of a von Neumann algebra whenever JBW-algebras are mentioned).The goal of this paper is to extend the above results on B ( H ) to arbitrary atomic JBW-algebras.Our approach is based on Drnovˇsek’s idea of using the order anti-isomorphism of the invertiblepart of the effect algebra with the translated positive cone. For this we need that the invertiblepart is invariant, and we show that this is true if the atomic JBW-algebra is a factor. Generalatomic JBW-algebras are direct sums (of arbitrary size) of factors, and if there are infinitelymany summands then the invertible part need no longer be invariant, but we also show thatorder decompositions correspond to JBW-algebra direct sum decompositions, a result that allowsus to treat each factor separately. We then apply a recent result by Roelands and van Imhoff[vIR19, Theorem 3.8] characterising order isomorphisms between the positive cones in atomicJBW-algebras to obtain a full description of order isomorphisms between their effect algebras.Another major contribution in this research area was made by Mori in [Mor19], who mostlyconsidered order isomorphisms between the effect algebra of (not necessarily atomic) von Neu-mann algebras without type I direct summand. He showed that the image of I is always locallymeasurably invertible, a concept from the theory of non-commutative integration. Furthermore,he characterised the order isomorphisms in the case where the image of I is invertible.We will now briefly outline the structure of our paper. Section 2 is our preliminary section wherewe cover the basics of JB(W)-algebras and order products. In Section 3 we investigate the orderstructure of the effect algebra. The main result of this section is that the JBW-algebra decomposi-tions are in bijection with the order decompositions of the space. This is used in Section 4, wherewe first show that the invertible part of the effect algebra is always invariant if and only if thecentre is finite-dimensional, and use this to obtain a full characterisation of order isomorphisms ofthe effect algebra in atomic JBW-algebras, Theorem 4.8. A Jordan algebra ( A, ◦ ) is a commutative, not necessarily associative algebra such that x ◦ ( y ◦ x ) = ( x ◦ y ) ◦ x for all x, y ∈ A. A JB-algebra A is a normed, complete real Jordan algebra satisfying, k x ◦ y k ≤ k x k k y k , (cid:13)(cid:13) x (cid:13)(cid:13) = k x k , (cid:13)(cid:13) x (cid:13)(cid:13) ≤ (cid:13)(cid:13) x + y (cid:13)(cid:13) x, y ∈ A . The canonical example of a JB-algebra is the set of self-adjoint elements ofa C ∗ -algebra equipped with the Jordan product x ◦ y := ( xy + yx ). By the Gelfand-Naimarktheorem, this JB-algebra is a norm closed Jordan subalgebra of the self-adjoint bounded operatorson a complex Hilbert space. Such a JB-algebra is called a JC-algebra .The elements x, y ∈ A are said to operator commute if x ◦ ( y ◦ z ) = y ◦ ( x ◦ z ) for all z ∈ A . In aJC-algebra, two elements operator commute if and only if they commute for the C ∗ -multiplicationby [AS03, Proposition 1.49]. In the sequel, the Jordan product of two operator commuting elements x, y ∈ A will be written as xy instead of x ◦ y . An element x ∈ A is said to be central if it operatorcommutes with all elements of A . The centre of A , denoted by Z ( A ), consists of all elementsthat operator commute with all elements of A , and it is an associative subalgebra of A . Everyassociative unital JB-algebra is isometrically isomorphic to C ( K ) for some compact Hausdorffspace K , see [HOS84, Theorem 3.2.2].The set of invertible elements of a unital JB-algebra A is denoted by Inv( A ). The spectrum of x ∈ A , which is denoted by σ ( x ), is defined to be the set of λ ∈ R such that λe − x is notinvertible in JB( x, e ), the JB-algebra generated by x and e , see [HOS84, 3.2.3]. Furthermore, thereis a continuous functional calculus, that is, JB( x, e ) is isometrically isomorphic to C ( σ ( x )) as aJB-algebra. The cone of elements with non-negative spectrum is denoted by A + , and equals theset of squares by the functional calculus, and its interior A ◦ + consists of all elements with strictlypositive spectrum. The cone A + induces a partial ordering ≤ on A by writing x ≤ y if y − x ∈ A + .An order interval in A is of the form [ x, y ] for x, y ∈ A with x ≤ y . For order intervals[ x, y ] and [ u, v ] an order isomorphism f : [ x, y ] → [ u, v ] is an order preserving bijection with orderpreserving inverse, that is, x ≤ y if and only if f ( x ) ≤ f ( y ). Since we have [ x, y ] = x + [0 , y − x ]and [ x, y ] is order isomorphic to [0 , y − x ] via translation by x , it follows that f : [ x, y ] → [ u, v ]is an order isomorphism if and only if ˆ f : [0 , y − x ] → [0 , v − u ] is an order isomorphism. Hencethe order isomorphisms between order intervals in JB-algebras are completely determined by orderisomorphisms between order intervals the form [0 , x ].The Jordan triple product {· , · , ·} is defined as { x, y, z } := ( x ◦ y ) ◦ z + ( z ◦ y ) ◦ x − ( x ◦ z ) ◦ y, for x, y, z ∈ A . The linear map U x : A → A defined by U x y := { x, y, x } will play an important roleand is called the quadratic representation of x .By the Shirshov-Cohn theorem for JB-algebras [HOS84, Theorem 7.2.5], the unital JB-algebragenerated by two elements is a JC-algebra, which shows all but the fourth of the following identitiesfor JB-algebras, since U x y = xyx in JC-algebras (for the rest of the paper, the operator-algebraicreader is encouraged to think of this equality whenever the quadratic representation appears). U y x ∈ A + ( ∀ x ∈ A + , y ∈ A ) U − x = U x − ( ∀ x ∈ Inv( A ))( U y x ) − = U y − x − ( ∀ x, y ∈ Inv( A )) (2.1) U y U x U y = U U y x ( ∀ x, y ∈ A ) U y e = y ( ∀ y ∈ A )A proof of the fourth identity can be found in [HOS84, 2.4.18], as well as proofs of the otheridentities. The following lemma also follows from the same idea combined with the continuousfunctional calculus. 3 emma 2.1. Let A be a unital JB-algebra and let y ∈ A . If f, g : σ ( y ) → R are continuousfunctions, then ( i ) U f ( y ) U g ( y ) = U [ fg ]( y ) ; ( ii ) U f ( y ) g ( y ) = [ f g ]( y ) .Proof. ( i ): For any x ∈ A the JB-algebra B generated by x , y , and e is a JC-algebra by theShirshov-Cohn theorem for JB-algebras and it contains JB( y, e ) ∼ = C ( σ ( y )). Hence in B we have U f ( y ) U g ( y ) x = f ( y ) g ( y ) xg ( y ) f ( y ) = [ f g ]( y ) x [ f g ]( y ) = U [ fg ]( y ) x which shows that the required identity holds in A as x was chosen to be arbitrary.( ii ): Since f ( y ) , g ( y ) ∈ JB( y, e ) ∼ = C ( σ ( y )), it follows that U f ( y ) g ( y ) = f ( y ) g ( y ) f ( y ) = [ f g ]( y ).A JBW-algebra is the Jordan analogue of a von Neumann algebra: it is a JB-algebra withunit e which is monotone complete and has a separating set of normal states, or equivalently,a JB-algebra that is a dual space. A positive functional ϕ ∈ M ∗ is called a state if ϕ ( e ) = 1,and it is said to be normal if for any bounded increasing net ( x i ) i ∈ I with supremum x we have ϕ ( x i ) → ϕ ( x ). The linear space of normal states on M is called the normal state space of M . Thetopology on M defined by the duality of M and the normal state space of M is called the σ -weaktopology . That is, we say a net ( x i ) i ∈ I converges σ -weakly to x if ϕ ( x i ) → ϕ ( x ) for all normalstates ϕ on M . The Jordan multiplication on a JBW-algebra is separately σ -weakly continuousin each variable and jointly σ -weakly continuous on bounded sets by [AS03, Proposition 2.4] and[AS03, Proposition 2.5]. Furthermore, for any x the corresponding quadratic representation U x is σ -weakly continuous by [AS03, Proposition 2.4].We will now collect some results on the functional calculus that are standard in von Neumannalgebras but for which there does not seem to be a good reference in the theory of JBW-algebras.Let M be a JBW-algebra and let x ∈ M . Then W ( x, e ), the JBW-algebra generated by x and e ,is an associative JBW-algebra and hence isomorphic to a monotone complete C ( K )-space and soits complexification M is a von Neumann algebra also generated by x and e . A standard resultin von Neumann algebras, which can for example easily be derived from [Con90, Theorem IX.2.3],now yields the bounded functional calculus f f ( x ) from the bounded Borel functions on σ ( x ) to M . It follows from this theorem and the dominated convergence theorem that if f n is a uniformlybounded sequence converging pointwise to f , then f n ( x ) converges to f ( x ) in the weak operatortopology and hence σ -weakly by [Tak02, Lemma II.2.5(i)]. Restricting this functional calculusto real-valued functions now yields the bounded functional calculus into W ( x, e ) with a similarproperty: if f n is a uniformly bounded sequence converging pointwise to f , then f n ( x ) converges σ -weakly to f ( x ).An element p in a JBW-algebra M is a projection if p = p . For a projection p ∈ M the orthogonal complement , e − p , will be denoted by p ⊥ and a projection q is orthogonal to p preciselywhen q ≤ p ⊥ , see [AS03, Proposition 2.18]. The collection of projections P ( M ) forms a completeorthomodular lattice by [AS03, Proposition 2.25] which means in particular that every set ofprojections has a supremum. We remark that these sets of projections need not have a supremumin M . The collection of projections P ( M ) is referred to as the projection lattice of M . For apositive element x ∈ M , the smallest projection p such that U p x = x is called the range projection of x and is denoted by r ( x ). See [AS03, Proposition 2.13]. By the proof of [HOS84, Lemma 4.2.6]and the bounded functional calculus, r ( x ) = (0 , ∞ ) ( x ). Furthermore, if p is a projection and4 ≤ x ≤ p , then r ( x ) ≤ r ( p ) = p by [AS03, Proposition 2.15 (2.7)], and U p x = x by [AS03,Proposition 2.15 (2.8)], or equivalently, x ∈ U p M .Any central projection p decomposes the JBW-algebra M as a direct sum of JBW-subalgebrassuch that M = U p M ⊕ U p ⊥ M , see [AS03, Proposition 2.41]. A minimal non-zero projection is calledan atom , and it follows from [AS03, Proposition 2.32, Lemma 3.29] that a non-zero projection p is an atom if and only if the order interval [0 , p ] is totally ordered. A JBW-algebra in which everynon-zero projection dominates an atom is called atomic . Every JBW-algebra decomposes as adirect sum of type I, type II, and type III JBW-algebras, and a JBW-algebra with trivial centreis called a factor . In this paper we will predominantly be working with atomic JBW-algebras,which by [AS03, Proposition 3.45] are a direct sum of type I factors. The type I JBW-factorshave been classified and are up to isomorphism either a spin factor, or Mat ( O ) sa , the self-adjoint3 × B ( H ) sa , the bounded self-adjoint operatorson H where H is a real, complex, or quaternionic Hilbert space of dimension d ≥
3. See [AS03,Theorem 3.39].A
Jordan isomorphism between JB-algebras is a bijection that preserves the Jordan structure.In [IRP95, Section 2], the Jordan isomorphisms of the type I JBW-factors (except Mat ( O ) sa ) werecharacterised: on a spin factor they are induced by unitaries on the Hilbert space, and on B ( H ) sa with a Hilbert space H of dimension d ≥ F = R , C , or H , every Jordan isomorphism J isof the form J x = uxu − , where u is a surjective real-linear isometry satisfying u ( λh ) = τ ( λ ) u ( h )( λ ∈ F , h ∈ H ) for some automorphism τ of F . So u is linear or conjugate linear if H is complex.The group of Jordan isomorphisms of Mat ( O ) sa is the exceptional Lie group F ([CS50]).For an atomic JBW-algebra M , we define the rank of M to be the cardinality of a maximalorthogonal collection of atoms. For example, rank( ℓ ∞ ( S )) = | S | and rank( B ( H ) sa ) = dim( H ) forany set S and for any real, complex or quaternionic Hilbert space H . In the sequel we will consider the category
Poset of partially ordered sets with monotone mapsas morphisms, and the category
Poset0 of posets with a distinguished element 0 (not necessarilythe least element) and monotone maps that preserve 0 as morphisms. Given a collection of posets { S i } i ∈ I in Poset or Poset0 , we define the order product Q i ∈ I S i to be the cartesian productof { S i } i ∈ I equipped with the product ordering. It is an obvious exercise to show that the orderproduct equipped with the canonical projections is the categorical product in Poset and
Poset0 ,and that suprema and infima exist in Q i ∈ I S i if and only if they exist coordinatewise, in whichcase they are given by the coordinatewise operations.Given { S i } i ∈ I in Poset0 , if s i ∈ S i for some i , then we denote by s i the element in Q i ∈ I S i which equals s i in the i -th coordinate and 0 elsewhere. Definition 2.2.
Let S ∈ Poset0 and { S i } i ∈ I be subobjects, i.e., subposets with 0. Then wedefine S to be the internal order product of { S i } i ∈ I if there is an order isomorphism between S and Q i ∈ I S i , denoted by s ∼ ( s i ) i ∈ I , such that s i ∼ s i for each i ∈ I and s i ∈ S i . Remark . Suppose that S ∈ Poset0 is the internal order product of { S i } i ∈ I and 0 is the leastelement of S . If s i ∈ S i for each i ∈ I , then( s i ) i ∈ I = _ i ∈ I s i ∼ _ i ∈ I s i . Hence ( s i ) i ∈ I ∈ Q i ∈ I S i is identified with W i ∈ I s i ∈ S .5 The order structure of [0 , e ] In this section we study the relation between the order structure of the effect algebra [0 , e ] ofa JBW-algebra M and that of its projection lattice P ( M ), as well as the connection betweendecompositions of the interval and algebraic direct summands of M .If T is a positive injective operator on a Hilbert space, then [ˇS17, Theorem 2.8] shows that [0 , I ]is order isomorphic to [0 , T ]. For self-adjoint operators being injective is equivalent to having denserange, which in turn is equivalent to having range projection I . Our next proposition generalisesthis theorem to JBW-algebras. In case of von Neumann algebras Douglas’ lemma yields a simplerproof, but the absence of a Hilbert space makes our proof more elaborate. Proposition 3.1.
Let M be a JBW-algebra and let x ∈ M + be non-zero with range projection r ( x ) . Then the quadratic representation U x / : [0 , r ( x )] → [0 , x ] is an order isomorphism.Proof. All limits in this proof are σ -weak limits. For n ∈ N define continuous functions f n by f n ( t ) := ( t − / if t ∈ [ n , ∞ ) n / t if t ∈ [0 , n ) . We also define g n ( t ) := tf n ( t ) = ( t ∈ [ n , ∞ ) n t if t ∈ [0 , n ) , h n ( t ) := t / f n ( t ) = ( t ∈ [ n , ∞ ) n / t / if t ∈ [0 , n ) . Since g n and h n are uniformly bounded and converge pointwise to (0 , ∞ ) , the bounded functionalcalculus yields that g n ( x ) , h n ( x ) → (0 , ∞ ) ( x ) = r ( x ).We claim that the map V : [0 , x ] → [0 , r ( x )] defined by V y := lim n U f n ( x ) y is the inverse of U x / . To show that V is well defined we have to prove that U f n ( x ) y actually converges. So let0 ≤ y ≤ x , then by Lemma 2.1( ii ), U f n ( x ) y ≤ U f n ( x ) x = g n ( x ) ≤ r ( x ), and as [0 , r ( x )] is σ -weaklycompact, there is a convergent subnet such that lim i U f i ( x ) y = z ∈ [0 , r ( x )]. Note that we have notshown yet that U f n ( x ) y converges. It follows that, using Lemma 2.1( i ) in the third equality and[AS03, p. 40, (2.2)] in the fourth equality, U x / z = U x / lim i U f i ( x ) y = lim i U x / U f i ( x ) y = lim i U h i ( x ) y = U r ( x ) y = y. (3.1)But U f n ( x ) y = U f n ( x ) U x / z = U h n ( x ) z → U r ( x ) z = z, so the limit exists and therefore V is well defined, z = V y and so (3.1) implies that U x / V y = y .Similarly, if z ∈ [0 , r ( x )], then V U x / z = lim n U f n ( x ) U x / z = U h n ( x ) z → U r ( x ) z = z showing that V is the inverse of U x / . Since U x / is positive it is order preserving. To show that V is order preserving, let y, z ∈ [0 , x ] be such that y ≤ z , then z − y ∈ [0 , x ] and since limits ofpositive elements are positive, V z − V y = lim n U f n ( x ) z − lim n U f n ( x ) y = lim n U f n ( x ) ( z − y ) ≥ . Hence U x / is an order isomorphism. 6t follows that any order interval in a JBW-algebra is order isomorphic to the order interval[0 , e ] of some other JBW-algebra, so in the sequel we will focus on the order structure of the orderintervals [0 , e ] of JBW-algebras.In a JBW-algebra M , the partially ordered set of projections P ( M ) forms a lattice, even thoughprojections generally do not have a supremum in M ; for example, B ( H ) is an anti-lattice, meaningthat any two non comparable elements have no supremum. However, we proceed to show that inthe effect algebra [0 , e ] any set of projections also has a supremum. If x ∈ [0 , e ], then x ⊥ denotesthe element e − x ; clearly x x ⊥ is an order anti-isomorphism of [0 , e ] that coincides with theusual orthogonal complement on P ( M ). Lemma 3.2.
Let M be a JBW-algebra. In [0 , e ] , the supremum and infumum of every set ofprojections exists and coincides with the supremum and infimum in P ( M ) .Proof. Since x x ⊥ is an order anti-isomorphism that leaves P ( M ) invariant, it suffices to showthe existence of either the supremum or the infimum. We start by showing the existence of finiteinfima, so let p, q ∈ P ( M ). With p ∧ q denoting the infimum in P ( M ), clearly p ∧ q ≤ p, q . Towardsshowing that p ∧ q is the greatest lower bound, let 0 ≤ x ≤ p, q . Then x ∈ U p M and x ∈ U q M (herethe fact that x ≥ x ∈ U p M ∩ U q M which is a JBW-subalgebra and hence, since k x k ≤ x is dominated by the identity of this JBW-subalgebra which is a projection r ∈ P ( M ).Since r ∈ U p M and p is the largest projection in U p M it follows that r ≤ p and similarly r ≤ q .Hence r ≤ p ∧ q (actually r = p ∧ q by [AS03, Proposition 2.32]), and so x ≤ r ≤ p ∧ q , showingthat p ∧ q is the greatest lower bound of p and q in [0 , e ].To show existence of arbitrary suprema of projections we employ the standard machinery. If { p i : i ∈ I } is an arbitrary collection in P ( M ), then for every finite F ⊆ I we let p F := sup i ∈ F p i ,where the supremum is taken in P ( M ) and hence coincides with the supremum taken in [0 , e ].The net ( p F ) F ∈F indexed by the finite subsets F ⊆ I is an increasing bounded net so it converges σ -weakly to its supremum p in [0 , e ], since [0 , e ] is σ -weakly closed. By the joint weak continuityof the multiplication on bounded sets, it follows that p is a projection, and a direct verificationshows that p is also the supremum of { p i : i ∈ I } .Next, we show that in atomic JBW-algebras internal order decompositions are related to algebradirect sum decompositions. Proposition 3.3.
Let M be an atomic JBW-algebra. Then the following are equivalent: ( i ) M = M ⊕ M as a direct sum of JBW-subalgebras; ( ii ) M ∼ = S × T as an internal order product; ( iii ) M + ∼ = C × C as an internal order product; ( iv ) [0 , e ] ∼ = [0 , x ] × [0 , y ] as an internal order product for some x, y ∈ M ; ( v ) [0 , e ] ∼ = [0 , z ] × [0 , z ⊥ ] as an internal order product for some central projection z ∈ M .Moreover, all these types of decompositions are in bijection with each other.Proof. ( i ) ⇒ ( ii ): Follows from the order on an algebra direct sum being defined coordinatewise.( ii ) ⇒ ( iii ): Take C := { s ∈ S : s ≥ } and C := { t ∈ T : t ≥ } .( iii ) ⇒ ( iv ): Let e ∼ ( x, y ) for some x ∈ C and y ∈ C . Then[0 , e ] ∼ [(0 , , ( x, y )] = [0 , x ] × [0 , y ] . iv ) ⇒ ( v ): Suppose [0 , e ] = [0 , x ] × [0 , y ] for some x, y ∈ M . We first show that x is a projection.If u ∼ ( a, b ) ∈ [0 , x ] × [0 , y ] is an atom and a and b are non-zero, then ( a, , (0 , b ) ∈ [0 , u ] areincomparable, contradicting the fact that u is an atom, and so either u ≤ x or u ≤ y . Since0 ≤ x ≤ e , it follows that σ ( x ) ⊆ [0 , λ ∈ σ ( x ) such that 0 < λ <
1. Let ε > < λ − ε < λ + ε <
1. Let p := ( λ − ε,λ + ε ) ( x ), which is non-zero by the spectraltheorem ([Con90, Theorem IX.2.2(b)]), and let u ≤ p be an atom. Then 0 < ( λ − ε ) u ≤ ( λ − ε ) p < x ,so ( λ − ε ) u is a common lower bound of both u and x , therefore u y and so u ≤ x . But then u = u ∧ x ≤ u ∧ (( λ + ε ) p + p ⊥ ) = ( λ + ε ) u < u, which is a contradiction. Hence σ ( x ) ⊆ { , } and it follows that x must be a projection. Similarly,we find that y is a projection. Next, we will show that x and y are orthogonally complemented.In [0 , x ] × [0 , y ], the only element z such that x ∧ z = 0 and x ∨ z = e is y . By Lemma 3.2, x ⊥ satisfies these properties, so y = x ⊥ .It remains to show that x is a central projection, and for this it suffices to show that x operatorcommutes with any projection. If u is an atom, then [0 , u ] is totally ordered, hence u ≤ x or u ≤ y = x ⊥ . In both cases x operator commutes with u . A standard argument now shows that x operator commutes with any projection, which we will give for convenience of the reader. To thatend, let p ∈ M be a projection. A standard application of Zorn’s lemma shows that there is amaximal set of pairwise orthogonal atoms { p i : i ∈ I } that are dominated by p . The net ( p F ) F ∈F directed by the finite subsets F ⊆ I with p F := X p i ∈ F p i for F ∈ F is increasing and converges σ -weakly to its supremum in M which is a projection q ≤ p as Jordan multiplication is jointly σ -weakly continuous on bounded sets by [AS03, Proposition 2.4]and [AS03, Proposition 2.5]. Suppose that there is an atom r ≤ p − q . Then r and q are orthogonaland so r is orthogonal to all atoms p i , contradicting the maximality of the set { p i : i ∈ I } . Hence q = p . By joint σ -weakly continuity of the multiplication, it follows that x operator commuteswith p , proving that x is a central projection.( v ) ⇒ ( i ): Central projections correspond to algebra direct sum decompositions.Regarding the bijection between the different decompositions, we show that if we start with anyof the types and pass through the chain of implications, we return to the original decomposition.Direct verification shows that the chain ( i ) ⇒ ( ii ) ⇒ ( iii ) ⇒ ( iv ) ⇒ ( v ) ⇒ ( i ) leads back to theoriginal decomposition M ∼ = M ⊕ M , and similarly for the case where we start with ( v ). If westart with a decomposition as in ( iv ), so [0 , e ] ∼ = [0 , x ] × [0 , y ], then our above proof shows that x isa central projection with orthogonal complement y , so these decompositions are in direct bijectionwith the decompositions in ( v ).Suppose we start with a decomposition of type ( ii ), so M ∼ = S × T as an internal order product.Going through our entire proof of ( ii ) ⇒ ( v ), we show that in this case the order interval [0 , e ] canbe decomposed as [0 , z ] × [0 , z ⊥ ] with z a central projection in S and z ⊥ ∈ T . In a similar way, for n ≥
2, using elements with spectrum { , n } ( n times a projection) instead of projections, it canbe shown that the order interval [0 , ne ] can be decomposed as [0 , nz n ] × [0 , nz ⊥ n ] with z n a centralprojection in S and z ⊥ n ∈ T . Since e ∼ ( z, z ⊥ ) ∈ [0 , ne ], so (0 , ≤ ( z, z ⊥ ) ≤ ( nz n , nz ⊥ n ), it followsthat z is a projection in U z n M and therefore z ≤ z n . Similarly, z ⊥ ≤ z ⊥ n , and so z = z n . Hence[0 , ne ] ∼ = [0 , nz ] × [0 , nz ⊥ ] for all n ≥
1. 8or n ≥
1, we now consider [ − ne, ne ]. Adding and subtracting ne ∼ ( nz, nz ⊥ ) is an orderisomorphism, so it preserves order products, and therefore[ − ne, ne ] ∼ = [0 , ne ] ∼ = [0 , nz ] × [0 , nz ⊥ ] ∼ = [ − nz, nz ] × [ − nz ⊥ , nz ⊥ ] ⊆ S × T. Now if x ∈ M , then x ∈ [ − ne, ne ] for some n ≥
1, and so x ∼ ( a, b ) with a ∈ [ − nz, nz ] ⊆ S and b ∈ [ − nz ⊥ , nz ⊥ ] ⊆ T . This shows that S = ∪ n ≥ [ − nz, nz ] and T = ∪ n ≥ [ − nz ⊥ , nz ⊥ ], and so S = U z M , T = U z ⊥ M , and M ∼ = U z M × U z ⊥ M is an internal order product, which is preciselywhat we get if we follow the chain ( ii ) ⇒ ( iii ) ⇒ ( iv ) ⇒ ( v ) ⇒ ( i ) ⇒ ( ii ).If we start with a decomposition of type ( iii ), then we just use the above argument without theneed to consider negative elements. This shows that all types of decompositions are in bijectionwith each other.We immediately obtain an analogue of this proposition for arbitrary decompositions. Corollary 3.4.
Let M be an atomic JBW-algebra and I an index set. Then the following areequivalent: ( i ) M = L i ∈ I M i as a direct sum of JBW-subalgebras; ( ii ) M ∼ = Q i ∈ I S i as an internal order product; ( iii ) M + ∼ = Q i ∈ I C i as an internal order product; ( iv ) [0 , e ] ∼ = Q i ∈ I [0 , x i ] as an internal order product for some x i ∈ M ; ( v ) [0 , e ] ∼ = Q i ∈ I [0 , z i ] as an internal order product for some central projections z i ∈ M withsupremum e .Moreover, all these types of decompositions are in bijection with each other.Proof. Except for ( iv ) ⇒ ( v ), the proof of this corollary is the same as the proof of Proposition 3.3.( iv ) ⇒ ( v ): By Remark 2.3, [0 , e ] ∼ = [0 , x i ] × [0 , sup { x j : j = i } ], and so it follows from Proposi-tion 3.3 that x i is a central projection. As before, Remark 2.3 implies that e = sup i x i .Since order isomorphisms preserve order products, the following lemma now follows immediatelyfrom Corollary 3.4. Lemma 3.5.
Let M and N be atomic JBW-algebras, let { z i } i ∈ I ∈ M be a collection of orthogonalcentral projections with supremum e M and let f : [0 , e M ] → [0 , e N ] be an order isomorphism. Then f can be decomposed as f = Q i ∈ I f i where f i : [0 , z i ] → [0 , f ( z i )] are the corresponding restrictionswhich are order isomorphisms for each i ∈ I , and { f ( z i ) } i ∈ I are orthogonal central projections withsupremum e N . Note that the central atoms of M yield totally ordered parts in the decomposition of the effectalgebra and in fact, these are the only ones. Lemma 3.6.
Let M be an atomic JBW-algebra and suppose that [0 , e ] = [0 , z ] × [0 , z ] . Then theorder interval [0 , z ] is totally ordered if and only if z is a central atom.Proof. If z is a central atom, then by definition the order interval [0 , z ] is totally ordered andorder isomorphic to the unit interval [0 , , z ] is totally ordered, then z is acentral projection by Proposition 3.3. But if z = p + q for some non-trivial orthogonal projections p and q , then p and q would be two incomparable elements of [0 , z ]. Hence z must be an atom.9 Order isomorphisms between effect algebras
The conditions for an order isomorphism f : [0 , e M ] → [0 , e N ] to map the invertible part (0 , e M ]onto the invertible part (0 , e N ], as well as a complete description for such f are given in thissection. It turns out that the invariance of the invertible parts of the effect algebras is related tothe dimension of the centre of the atomic JBW-algebra.In the following lemma we introduce an important class of order isomorphisms on effect algebrasof JB-algebras that are essential for the description of f . This result for operators on a real orcomplex Hilbert space can be found in [ˇS17, Lemma 3.10] and [Drn18, Lemma 2.1]. For the reader’sconvenience we include the analogous proof which holds for general unital JB-algebras. Lemma 4.1.
Let A be a unital JB-algebra. For any t < the function ϕ t : [0 , e ] → [0 , e ] definedby ϕ t ( x ) := x ( tx + (1 − t ) e ) − is an order isomorphism and the collection { ϕ t : t < } forms a group under composition whichsatisfies ϕ t ◦ ϕ s = ϕ t + s − ts . In particular we have ϕ − t = ϕ tt − .Proof. The statement trivially holds for t = 0. So, suppose t = 0. If 0 < t <
1, then we canrewrite ϕ t , by using the identity y ( y + λe ) − = e − λ ( y + λe ) − which holds for any y ∈ A + and λ >
0, to be ϕ t ( x ) = t e − − tt (cid:0) x + ( t − e (cid:1) − . Since we have that x ≤ y if and only if y − ≤ x − for all x, y ∈ A ◦ + by [AS03, Lemma 1.31], it followsimmediately from the alternative description of ϕ t that it is an order isomorphism. Similarly, if t <
0, then we can rewrite ϕ t to be ϕ t ( x ) = t e + − tt (cid:0) (1 − t ) e − x (cid:1) − and it is verified analogously that it is an order isomorphism in this case as well.If t, s <
1, then for x ∈ [0 , e ] it follows that ϕ t ( ϕ s ( x )) = ϕ t ( x ( sx + (1 − s ) e ) − ) = x ( sx + (1 − s ) e ) − (cid:0) t ( x ( sx + (1 − s ) e )) − ) + (1 − t ) e (cid:1) − = x (cid:0) tx + s (1 − t ) x + (1 − s )(1 − t ) e (cid:1) − = x (cid:0) ( t + s − ts ) x + (1 − ( t + s − ts ) e ) (cid:1) − = ϕ t + s − ts ( x ) , so ϕ t ◦ ϕ s = ϕ t + s − ts . Since ϕ is the identity function, we infer from the composition identity thatthe inverse of ϕ t satisfies ϕ − t = ϕ tt − . In unital JB-algebras the order isomorphisms on effect algebras [0 , e ] are related to order isomor-phisms on cones, where the invertible elements (0 , e ] of this effect algebra play an important role.If A and B are unital JB-algebras and f : [0 , e A ] → [0 , e B ] is an order isomorphism such that f leaves the invertible elements invariant, that is, the restriction f | (0 ,e A ] is an order isomorphism onto(0 , e B ], then this restriction yields an order isomorphism between the cones A + and B + . A key ob-servation is that (0 , e A ] and A + are order anti-isomorphic via the map (0 , e A ] ∋ x x − − e A ∈ A + .It then follows that x f | (0 ,e A ] (( x + e A ) − ) − − e B (4.1)10s an order isomorphism from A + onto B + . Here we use the fact that x ≤ y if and only if y − ≤ x − for all x, y ∈ A ◦ + by [AS03, Lemma 1.31]. The order isomorphisms between cones of atomic JBW-algebras are well understood and have been characterised in [vIR19, Theorem 3.8], which we canexploit to find a formula for f | (0 ,e A ] . Proposition 4.2.
Let M and N be atomic JBW-algebras and suppose that M does not containany central atoms. If f : (0 , e M ] → (0 , e N ] is an order isomorphism, then f ( x ) = (cid:0) U y J x − − y + e N (cid:1) − (cid:0) x ∈ (0 , e M ] (cid:1) for some y ∈ N ◦ + and a Jordan isomorphism J : M → N .Proof. Let f : (0 , e M ] → (0 , e N ] be an order isomorphism. Then the map ˆ f : M + → N + defined byˆ f ( x ) := f (( x + e M ) − ) − − e N is an order isomorphism, so by [vIR19, Theorem 3.8] there is a Jordan isomorphism J : M → N and a y ∈ N ◦ + such that ˆ f ( x ) = U y J x for all x ∈ M + . Rewriting this yields f (( x + e M ) − ) =( U y J x + e N ) − for all x ∈ M + , so we conclude that f ( x ) = ( U y J x − − y + e N ) − for all x ∈ (0 , e M ]as required. Corollary 4.3.
Let A be a unital JB-algebra. Suppose that for every y ∈ A ◦ + there exists an orderautomorphism f y of [0 , e ] such that f y ( x ) = (cid:0) U y x − − y + e (cid:1) − for all x ∈ (0 , e ] , then for all z ∈ (0 , e ) , the map f ( z − − e ) − / maps z to e and f ( z − − e ) / maps e to z . In particular, the order automorphism group of [0 , e ] acts transitively on (0 , e ) . The following lemma shows that (0 , e ] is order dense in [0 , e ]. Lemma 4.4.
Let A be a JB-algebra and x ∈ [0 , e ] . Then there exists a monotone decreasingsequence ( x n ) n ≥ in (0 , e ] with infimum x .Proof. For n ∈ N , define the continuous functions f n on [0 ,
1] by f n ( t ) := ( t if t ∈ [ n , , n if t ∈ [0 , n ) . By the continuous functional calculus x n := f n ( x ) defines a monotone decreasing sequence thatconverges to x in norm and has x as a lower bound. Suppose y is another lower bound for all x n .Then x − y = lim n x n − y ≥
0, showing that x is the greatest lower bound of ( x n ) n ≥ . Theorem 4.5.
Let M and N be atomic JBW-algebras and suppose that M does not contain anycentral atoms. Then f : [0 , e M ] → [0 , e N ] is an order isomorphism such that f ((0 , e M ]) = (0 , e N ] ifand only if f is of the form f ( x ) = ϕ t (cid:0) U ( z + e N ) / ( e N − ( e N + U z − J x ) − ) (cid:1) (cid:0) x ∈ [0 , e M ] (cid:1) for some t < , z ∈ N ◦ + , and a Jordan isomorphism J : M → N . roof. Suppose that f : [0 , e M ] → [0 , e N ] is an order isomorphism such that f ((0 , e M ]) = (0 , e N ].It follows from Proposition 4.2 that the restriction of f to (0 , e M ] is of the form f ( x ) = ( U y J x − − y + e N ) − for some y ∈ N ◦ + and a Jordan isomorphism J : M → N . Let λ > λe N − y ispositive and invertible. Then for x ∈ (0 , e M ], it follows from Lemma 2.1 and (2.1) that U ( λe N − y ) − / U y = U y / U ( λe N − y ) − / U y / = U U y / ( λe N − y ) − / , and so for z := U y / ( λe N − y ) − / we have f ( x ) − + ( λ − e N = U y J x − − y + λe N = U ( λe N − y ) / (cid:0) e N + U z J x − (cid:1) . Using the functional calculus on the JB-algebra generated by y and e N again, we find that z = y ( λe N − y ) − / = ( y ( λe N − y ) − ) / = ( λ ( λe N − y ) − − e N ) / hence z + e N = λ ( λe N − y ) − and so λ ( z + e N ) − = λe N − y . Therefore, U ( λe N − y ) / (cid:0) e N + U z J x − (cid:1) = U ( λ ( z + e N ) − ) / (cid:0) e N + U z J x − (cid:1) = U λ / ( z + e N ) − / (cid:0) e N + ( U z − J x ) − (cid:1) = λU ( z + e N ) − / ( e N + ( U z − J x ) − )and by using (2.1) together with the identity ( e N + ( U z − J x ) − ) − = e N − ( e N + U z − J x ) − toobtain the second equality, we find that λ (cid:0) f ( x ) − + ( λ − e N (cid:1) − = U ( z + e N ) / ( e N + ( U z − J x ) − ) − = U ( z + e N ) / ( e N − ( e N + U z − J x ) − ) . (4.2)Note that the identity ( e N + (( λ − f ( x )) − ) − = e N − ( e N + ( λ − f ( x )) − now yields λ (cid:0) f ( x ) − + ( λ − e N (cid:1) − = λλ − (cid:0) e N + (( λ − f ( x )) − (cid:1) − = λλ − (cid:0) e N − ( e N + ( λ − f ( x )) − (cid:1) = λλ − e N − λ ( λ − (cid:0) λ − e N + f ( x ) (cid:1) − = ϕ s ( f ( x )) (4.3)for s := λ − λ ∈ (0 , ϕ − s = ϕ t with t = ss − = 1 − λ ∈ ( −∞ , f ( x ) as f ( x ) = ϕ t ( ϕ s ( f ( x )) = ϕ t (cid:0) U ( z + e N ) / ( e N − ( e N + U z − J x ) − ) (cid:1) . (4.4)The expression for f in (4.4) is a well defined order isomorphism between the entire order intervals[0 , e M ] and [0 , e N ]. We will show that the only order isomorphism g : [0 , e M ] → [0 , e N ] for whichthe restriction g | (0 ,e ] coincides with f is of the form (4.4), proving the required description for f .Indeed, if g : [0 , e M ] → [0 , e N ] is an order isomorphism such that g | (0 ,e M ] = f , and x ∈ [0 , e M ], thenby Lemma 4.4 there is a monotone decreasing sequence ( x n ) n ≥ in (0 , e M ] with infimum x . Hence f ( x ) = f (cid:18) inf n ≥ x n (cid:19) = inf n ≥ f ( x n ) = inf n ≥ g ( x n ) = g (cid:18) inf n ≥ x n (cid:19) = g ( x )12s f and g are order isomorphisms. We conclude that f must be of the form (4.4).Conversely, suppose that f is of the form (4.4) for some t < z ∈ N ◦ + , and a Jordan isomor-phism J : M → N . Since ϕ s ( εe N ) = εs ( ε − e N ∈ (0 , e N ] for all s < ε >
0, it follows that ϕ s ((0 , e N ]) = (0 , e N ] for all s <
1. By (2.1) the quadratic representation U ( z + e N ) / maps invertibleelements to invertible elements. Lastly, for x ∈ [0 , e M ] it follows from the identity e N − ( e N + U z − J x ) − = ( U z − J x ) ◦ ( e N + U z − J x ) − in conjunction with the Shirsov-Cohn theorem that e N − ( e N + U z − J x ) − is invertible if and only if U z − J x is invertible. This in turn, is equivalent to saying that x is invertible. Hence f ( x ) ∈ (0 , e N ]if and only if x ∈ (0 , e M ], or equivalently, f ((0 , e M ]) = (0 , e N ]. (0 , e ] In this section we characterise for which atomic JBW-algebras all order isomorphisms satisfy f ((0 , e M ]) = (0 , e N ]. For B ( H ) it was shown by ˇSemrl [ˇS17] that every order isomorphism of[0 , e ] leaves the invertible part (0 , e ] invariant. For general atomic JBW-algebras this is no longerthe case. For example, on the effect algebra [0 , e ] ⊆ ℓ ∞ we construct an order isomorphism f : [0 , e ] → [0 , e ] as follows. For n ≥
1, let f n : [0 , → [0 ,
1] be an order isomorphism suchthat f n (1 /
2) = 2 − n . Then f ( λ n ) n ≥ := ( f n ( λ n )) n ≥ is an order isomorphism that maps e to(2 − n ) n ≥ / ∈ (0 , e ]. However, it turns out that all order isomorphisms between effect algebras ofJBW-factors of type I do leave the invertible parts invariant.Before proceeding, we will introduce some terminology and notation. For a JBW-algebra M ,we denote by P ( M ) the set of atoms of M . For an atom p in a JBW-algebra M and x ∈ M + , wesay that x dominates the atom p if there exists a λ > λp ≤ x . Proposition 4.6.
Let M and N be JBW-factors of type I and let f : [0 , e M ] → [0 , e N ] be an orderisomorphism. Then f (0 , e M ] = (0 , e N ] .Proof. Note that the rays generated by the atoms correspond exactly to extreme rays of the cone,and so the maximal totally ordered subsets S of [0 , e M ] such that [0 , x ] ⊆ S for all x ∈ S areprecisely of the form [0 , p ] for p ∈ P ( M ). Hence, if p ∈ P ( M ), then f ([0 , p ]) = [0 , q ] for some q ∈ P ( N ). In particular, f maps P ( M ) bijectively onto P ( N ).Now if x ∈ (0 , e M ], then x dominates every atom in M and hence by the above f ( x ) dominatesevery atom in N . Suppose N is of finite rank. Then the spectral decomposition yields that f ( x ) = n X i =1 λ i q i for some λ i ∈ [0 ,
1] and q i ∈ P ( N ). Since f ( x ) dominates every atom in N , all the λ i must bestrictly positive, and so 0 / ∈ σ ( f ( x )).It remains to consider the case where N is of infinite rank. Then by [HOS84, Theorem 7.5.11] N is of the form B ( H ) sa where H is a real, complex or quaternion Hilbert space. Assume H iscomplex. The atoms are precisely the rank one projections, and since f ( x ) dominates every atom,by Douglas’ lemma ([Dou66, Theorem 1]) it follows that the range of f ( x ) contains the range ofevery rank 1 projection. Hence f ( x ) is surjective, and f ( x ) has to be injective as well otherwise f ( x ) would not dominate a rank 1 projection in its kernel. Hence f ( x ) is invertible.As explained in [Dou66], Douglas’ lemma is also valid for real Hilbert spaces, so it remainsto verify Douglas’ lemma for quaternion Hilbert spaces. An inspection of Douglas’ proof shows13hat it only relies on the Closed Graph Theorem, which also holds in quaternion Hilbert spaces by[ACS15, Theorem 3.9] (an earlier but more archaic reference is [Bou81, Corollaire 5, p. I.19]).An alternative proof that avoids quaternion Hilbert spaces is as follows. As explained in[HOS84, 7.5.6, 7.5.10, 7.5.11] one can model the bounded self-adjoint operators on a quaternionHilbert spaces as follows: let H be a complex Hilbert space and let j : H → H be the conjugatelinear map defined by j ( ξ, η ) := ( η, ξ ), where ξ ξ is a conjugation on H . Then N , the boundedself-adjoint operators on a quaternion Hilbert space can be represented as the bounded self-adjointoperators on H that commute with j . For ξ ∈ H , let P ξ denote the projection onto the spanof ξ ; then one readily verifies that the self-adjoint operator T := P ξ ⊕ P ξ commutes with j andhence belongs to N . Since f ( x ) dominates every atom and T has at most rank 2 in N , it alsodominates T , and so it will also dominate P ξ ⊕ ⊕ P ξ for every ξ ∈ H ; by taking η := ξ , T also dominates 0 ⊕ P η for every η ∈ H . By Douglas’ lemma for complex Hilbert space operators,the range of f ( x ) contains the ranges of P ξ ⊕ ⊕ P η , and so the range of f ( x ) contains allvectors ( ξ, η ) ∈ H . Hence f ( x ) is surjective which as before shows that f ( x ) is invertible.Note that a consequence of Proposition 4.6 is that any order isomorphism between effect al-gebras of atomic JBW-algebras that are a finite direct sum of type I factors must preserve theinvertible parts as well. In fact, the following lemma shows that this characterises when it happens. Lemma 4.7.
Let M and N be atomic JBW-algebras such that [0 , e M ] is order isomorphic to [0 , e N ] . Then any order isomorphism f : [0 , e M ] → [0 , e N ] satisfies f (0 , e M ] = (0 , e N ] if and only ifthe centre Z ( M ) is finite dimensional.Proof. Suppose that all order isomorphisms f : [0 , e M ] → [0 , e N ] satisfy f (0 , e M ] = (0 , e N ] andthat Z ( M ) is infinite dimensional, then Z ( M ) must contain a countable set of pairwise orthogonalcentral projections ( p n ) n ≥ , so we can decompose M as a direct sum M = ∞ M n =1 U p n ( M ) ⊕ M ′ . The order interval [0 , e ] can now be written as the product [0 , e ] = Q ∞ n =1 [0 , p n ] × [0 , e M ′ ]. For n ∈ N let t n := (3 − n ), then t n < ϕ t n : [0 , p n ] → [0 , p n ]such that ϕ t n ( p n ) = 2 − n p n for n ∈ N . Together with the identity function Id [0 ,e M ′ ] on [0 , e M ′ ] byLemma 3.5 we obtain an order isomorphism g := ∞ Y n =1 ϕ t n × Id [0 ,e M ′ ] : [0 , e M ] → [0 , e M ] . For any order isomorphism f : [0 , e M ] → [0 , e N ] it follows that f ◦ g : [0 , e M ] → [0 , e N ] is an orderisomorphism as well, but ( f ◦ g )( e M ) / ∈ (0 , e N ] as 0 ∈ σ ( g ( e M )), so g ( e M ) / ∈ (0 , e M ] and f ([0 , e M ] \ (0 , e M ]) = [0 , e N ] \ (0 , e N ].Conversely, suppose that Z ( M ) is finite dimensional and let f : [0 , e M ] → [0 , e N ] be an orderisomorphism. Then M is a finite direct sum M = M ⊕ · · · ⊕ M n of type I factors which yields thedecompositions [0 , e M ] = Q nk =1 [0 , e M k ] and (0 , e M ] = Q nk =1 (0 , e M k ] as partially ordered sets. ByLemma 3.5 the image of the order isomorphism decomposes [0 , e N ] as the partially ordered set[0 , e N ] = n Y k =1 f [0 , e M k ] . e M k is a central projection that cannot be decomposed further as a non-trivial sum of centralprojections, it follows from Corollary 3.4 that f [0 , e M k ] = [0 , e N l ] where e N l is a central projectionwhich cannot be decomposed any further as a non-trivial sum of central projections either. Hence U e Nl ( N ) is a factor and f (0 , e M k ] = (0 , e N l ] by Proposition 4.6. We conclude that f (0 , e M ] = f n Y k =1 (0 , e M k ] ! = n Y k =1 f (0 , e M k ] = n Y l =1 (0 , e N l ] = (0 , e N ]as required. For a general atomic JBW-algebra M , consider the collection P ( M ) ∩ Z ( M ) of central atoms andwe define p := sup { q : q ∈ P ( M ) ∩ Z ( M ) } in the lattice of projections. Since Jordan multiplication is separately σ -weakly continuous, itfollows that p operator commutes with all projections, which in turn implies that p must becentral by [HOS84, Lemma 4.2.5]. Hence the central projection p decomposes M as an algebraicdirect sum M = U p ( M ) ⊕ U p ⊥ ( M ) where U p ⊥ ( M ) does not contain any central atoms. Followingthe terminology used in [vIR19], we call U p ( M ) the disengaged part of M and U p ⊥ ( M ) the engagedpart of M . Note that the disengaged part of M is precisely the associative part, i.e., the type I part. Furthermore, we will denote by M D the disengaged part of M and by M E the engagedpart of M . In order to abbreviate the notation and to incorporate the atoms in M D , we write D M := P ( M ) ∩ Z ( M ) and it follows from [vIR19, Proposition 3.5] that M D = M p ∈D M U p ( M ) = M p ∈D M R p, as U p ( M ) is one-dimensional by [AS03, Lemma 3.29]. Using the disengaged and engaged parts of M , we can decompose the effect algebra [0 , e M ] as[0 , e M ] = [0 , e M D ] × [0 , e M E ] = [0 , D M × [0 , e M E ] (4.5)and give the following description of the order isomorphisms between the effect algebras of atomicJBW-algebras. Theorem 4.8.
Let M and N be atomic JBW-algebras and let f : [0 , e M ] → [0 , e N ] be an orderisomorphism. For x ∈ M = M D ⊕ M E we write x = ( x p ) p ∈D M × x E as in (4.5) . ( i ) There exists an index set I such that M E decomposes as a direct sum of factors L i ∈ I M i and N E decomposes as a direct sum of factors L i ∈ I N i , and for x E = ( x i ) i ∈ I , f ( x ) = ( f p ( x p )) σ ( p ) ∈D N × Y i ∈ I ϕ t i (cid:16) U ( z i + e Ni ) / ( e N i − ( e N i + U z − i J i x i ) − ) (cid:17) for some t i < , z i ∈ ( N i ) ◦ + , a bijection σ : D M → D N , order isomorphisms f p : [0 , → [0 , for all p ∈ D M , and Jordan isomorphisms J i : M i → N i . ii ) If M E is a finite direct sum of factors, then f ( x ) = ( f p ( x p )) σ ( p ) ∈D N × ϕ t (cid:16) U ( z + e NE ) / ( e N E − ( e N E + U z − J x E ) − ) (cid:17) for some t < , z ∈ ( N E ) ◦ + , a bijection σ : D M → D N , order isomorphisms f p : [0 , → [0 , for all p ∈ D M , and some Jordan isomorphism J : M E → N E . ( iii ) If M E is a finite direct sum of factors and M D = { } , then f ( x ) = ϕ t (cid:0) U ( z + e N ) / ( e N − ( e N + U z − J x ) − ) (cid:1) for some t < , z ∈ N ◦ + , and some Jordan isomorphism J : M → N .Proof. ( i ): By (4.5) we can write[0 , e M ] = [0 , D M × [0 , e M E ] and [0 , e N ] = [0 , D N × [0 , e N E ] . Since the order isomorphism f must preserve the totally ordered parts of the decomposition, itfollows that f = f D × f E where f D : [0 , D M → [0 , D N and f E : [0 , e M E ] → [0 , e N E ] are thecorresponding order isomorphism restrictions of f by Lemma 3.5 and Lemma 3.6. Moreover, f D induces a bijection σ : D M → D N so that f D (( x p ) p ∈D M ) = ( f p ( x p )) σ ( p ) ∈D N where f p : [0 , → [0 , p ∈ D M .The atomic JBW-algebra M E is a direct sum L i ∈ I M i of type I factors by [AS03, Proposi-tion 3.45] and so the corresponding effect algebra is of the form [0 , e M E ] = Q i ∈ I [0 , e M i ]. Hence[0 , e N E ] is of the form Q i ∈ I [0 , f ( e M i )] and f E = Q i ∈ I f i where f i : [0 , e M i ] → [0 , f ( e M i )] are the re-striction order isomorphisms by Lemma 3.5. It follows from Proposition 3.3 that the effect algebra[0 , f ( e M i )] belongs to a factor N i in N E for each i ∈ I and N E = L i ∈ I N i .Since M i does not contain any central atoms and f i ((0 , e M i ]) = (0 , e N i ] by Lemma 4.7, itfollows from Theorem 4.5 that there is a t i <
1, an element z i ∈ ( N i ) ◦ + , and a Jordan isomorphism J i : M i → N i such that f i ( x i ) = ϕ t i (cid:16) U ( z i + e Ni ) / ( e N i − ( e N i + U z − i J i x i ) − ) (cid:17) which yields the required description of the order isomorphism f .( ii ): The proof is similar to ( i ), except now f E (0 , e M ] = (0 , e N ] by Lemma 4.7 and so we canuse the same arguments for f E that characterise f i .( iii ): Follows immediately from ( ii ).Note that the characterisation of order isomorphisms in Theorem 4.8 and the order isomor-phisms on the unit interval [0 ,
1] determine all order isomorphisms between effect algebras of atomicJBW-algebras.
Corollary 4.9.
Let M and N be atomic JBW-algebras. Then [0 , e M ] and [0 , e N ] are order iso-morphic if and only if M and N are Jordan isomorphic.Proof. If [0 , e M ] and [0 , e N ] are order isomorphic, then by decomposing [0 , e M ] = [0 , e M D ] × [0 , e M E ]and [0 , e N ] = [0 , e N D ] × [0 , e N E ], an order isomorphism f : [0 , e M ] → [0 , e N ] can be written as f = f D × f E where f D : [0 , e M D ] → [0 , e N D ] and f E : [0 , e M E ] → [0 , e N E ] are the corresponding order16somorphism restrictions of f by Lemma 3.5 and Lemma 3.6. Moreover, f D induces a bijection σ : D M → D N which can be used to define a Jordan isomorphism J D : M D → N D via J D (( x p ) p ∈D M ) := ( x p ) σ ( p ) ∈D N . By Theorem 4.8( i ) there is a Jordan isomorphism J i : M i → N i for each i ∈ I and we can definea Jordan isomorphism J E := L i ∈ I J i : M E → N E by J E (( x i ) i ∈ I ) := ( J i ( x i )) i ∈ I . We conclude that J := J D ⊕ J E : M → N is a Jordan isomorphism and so M and N are Jordan isomorphic.Conversely, every Jordan isomorphism J : M → N restricts to an order isomorphism between[0 , e M ] and [0 , e N ]. References [ACS15] Daniel Alpay, Fabrizio Colombo, and Irene Sabadini. Inner product spaces and Krein spaces in thequaternionic setting. In
Recent advances in inverse scattering, Schur analysis and stochastic processes , volume244 of
Oper. Theory Adv. Appl. , pages 33–65. Birkh¨auser/Springer, Cham, 2015.[AS03] Erik M. Alfsen and Frederic W. Shultz.
Geometry of state spaces of operator algebras . Mathematics: Theory& Applications. Birkh¨auser Boston, Inc., Boston, MA, 2003.[Bou81] Nicolas Bourbaki.
Espaces vectoriels topologiques. Chapitres 1 `a 5 . Masson, Paris, new edition, 1981.´El´ements de math´ematique. [Elements of mathematics].[Con90] John B. Conway.
A course in functional analysis , volume 96 of
Graduate Texts in Mathematics . Springer-Verlag, New York, second edition, 1990.[CS50] Claude Chevalley and R. D. Schafer. The exceptional simple Lie algebras F and E . Proc. Nat. Acad. Sci.U.S.A. , 36:137–141, 1950.[Dou66] R. G. Douglas. On majorization, factorization, and range inclusion of operators on Hilbert space.
Proc.Amer. Math. Soc. , 17:413–415, 1966.[Drn18] Roman Drnovˇsek. On order automorphisms of the effect algebra.
Acta Sci. Math. (Szeged) , 84(3-4):431–437,2018.[HOS84] Harald Hanche-Olsen and Erling Størmer.
Jordan operator algebras , volume 21 of
Monographs and Studiesin Mathematics . Pitman (Advanced Publishing Program), Boston, MA, 1984.[IRP95] Jos´e M. Isidro and ´Angel Rodr´ıguez-Palacios. Isometries of JB-algebras.
Manuscripta Math. , 86(3):337–348, 1995.[Lud83] G¨unther Ludwig.
Foundations of quantum mechanics. I . Texts and Monographs in Physics. Springer-Verlag, New York, 1983. Translated from the German by Carl A. Hein.[Mol01] Lajos Moln´ar. Order-automorphisms of the set of bounded observables.
J. Math. Phys. , 42(12):5904–5909,2001.[Mol03] Lajos Moln´ar. Preservers on Hilbert space effects.
Linear Algebra Appl. , 370:287–300, 2003.[Mor19] Michiya Mori. Order isomorphisms of operator intervals in von Neumann algebras.
Integral EquationsOperator Theory , 91(2):Art. 11, 26, 2019.[Tak02] M. Takesaki.
Theory of operator algebras. I , volume 124 of
Encyclopaedia of Mathematical Sciences .Springer-Verlag, Berlin, 2002. Reprint of the first (1979) edition, Operator Algebras and Non-commutativeGeometry, 5.[vIR19] Hendrik van Imhoff and Mark Roelands. Order isomorphisms between cones of JB-algebras.
Studia Math. ,2019. To appear. Preprint: https://arxiv.org/abs/1904.09278 .[ˇS17] Peter ˇSemrl. Order isomorphisms of operator intervals.
Integral Equations Operator Theory , 89(1):1–42, 2017., 89(1):1–42, 2017.