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A Representation of Quantum Measurement in Order-Unit Spaces

Gerd Niestegge

Zillertalstrasse 39, 81373 Muenchen, Germany

gerd.niestegge@web.de

Abstract. A certain generalization of the mathematical formalism of quantum mechanics

beyond operator algebras is considered. The approach is based on the concept of conditional

probability and the interpretation of the Lüders - von Neumann quantum measurement as a

probability conditionalization rule. A major result shows that the operator algebras must be

replaced by order-unit spaces with some specific properties in the generalized approach, and it

is analyzed under which conditions these order-unit spaces become Jordan algebras. An

application of this result provides a characterization of the projection lattices in operator

algebras.

Key Words. Operator algebras, Jordan algebras, convex sets, quantum measurement,

quantum logic

1. Introduction

Despite of many efforts, quantum mechanics and relativity theory have been resisting their

unification for almost a century. Either both theories are universally valid and the right way

how to unify them has still not been found, or at least one of the two theories needs to be

extended to achieve their unification. Just as classical probability theory was not general

enough to cover the probabilities occurring in quantum mechanics, the current mathematical

formalism of quantum mechanics may still not be general enough for the unification, or this

may hold for relativity theory or for both theories. The present paper deals with the first case -

i.e., with a potential generalization of the mathematical formalism of quantum mechanics.

The starting point is a certain axiomatic approach to this formalism, developed by the

author in some recent papers [21-23], which is based on the concept of conditional

probabilities and the interpretation of the Lüders - von Neumann quantum measurement as a

probability conditionalization rule. This approach leads to the standard model of quantum

mechanics with the Hilbert space over the complex numbers. When the last one among the

axioms is dropped, it leads to Jordan algebras, which are only a little bit more general than the

standard Hilbert space model since most of them - but not all - have representations as

operator algebras on a complex Hilbert space. The approach provides further opportunities for

potential generalizations just by dropping some more axioms and keeping only those that

describe the basic properties of the conditional probabilities.

In the paper, it is shown how this results in a certain new mathematical structure - an

order-unit space with some specific additional properties. It still features some of the well-

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Gerd Niestegge A Representation of Quantum Measurement in Order-Unit Spaces

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known properties of Jordan operator algebras (e.g., existence and uniqueness of the

conditional probabilities), but not all (e.g., the spectral theorem does not hold anymore for all

elements of the order-unit space). Conditions under which the order-unit space becomes a

Jordan operator algebra are studied. An application of the main results of the paper provides a

characterization of the projection lattices in Jordan operator algebras, which modifies an earlier

result by Bunce and Wright [7].

Order-unit spaces were introduced by Kadison [17] and play an important role in the study

of convex sets [1,2,6] as well as of the order structure of operator algebras [14]. Alfsen and

Shultz [2] used them in their approach to a non-commutative spectral theory. Bunce, Wright

[7] and Pulmannová [25] studied the relation of Alfsen's and Shultz's results to quantum logic.

Edwards and Rüttimann [12] investigated the conditional probabilities assuming the Alfsen-

Shultz property (i.e., every exposed face of the state space is projective). All these results are

based on the notion of P-projections introduced by Alfsen and Shultz. A similar, but more

general type of projections will play a central role in the present paper; they become identical

with the P-projections only in specific situations.

The next section gives an overview of the basic notions from the author's recent papers

[21-23] as far as needed in the present paper. A first major result is the derivation of the order-

unit space and its specific properties from these basic notions in section 3. Observables and the

relationship with Alfsen and Shultz's spectral duality are studied in section 4. Further results in

the last section concern the conditions under which the order unit spaces become Jordan

algebras and the characterization of the projection lattices in operator algebras.

2. Events, states, and conditional probabilities

Our model of events (or quantum logic [7,25,28,29]) shall be a mathematical structure

which is as simple as possible, but has enough structure for the consideration of states. This

requires an orthogonality relation and a sum operation for orthogonal events. The precise

axioms for the system E of events were presented in [21] and look as follows.

E is a set with distinguished elements 0 and 1 I, an orthogonality relation ⊥ and a partial

binary operation + such that the following conditions hold for e,f,g∈E:

(OS1)

(OS2)

(OS3)

If e⊥f, then f⊥e; i.e., the relation ⊥ is symmetric.

e+f is defined for e⊥f, and then e+f=f+e; i.e., the sum operation is commutative.

If g⊥e, g⊥f, and e⊥f, then g⊥e+f, f⊥g+e and g+(e+f)=(g+e)+f; i.e., the sum operation

is associative.

0⊥e and e+0=e for all e∈E?

For every e∈E, there exists a unique e'∈E such that e⊥e' and e+e'=1 I.

There exists d∈E such that e⊥d and e+d=f if and only if e⊥

(OS4)

(OS5)

(OS6)

′ f .

Then 0'=1 I and e''=e for e∈E. A further relation ? is defined on E via e?f iff e⊥

relation will be needed for the definition of the conditional probabilities, but note that the

above axioms do not imply that it is an order relation. We call E orthogonally σ-complete if the

sum exists for any countable orthogonal subset of E, and we call E orthogonally complete if

the sum exists for any orthogonal subset of E.

A state is a map µ:E→[0,1] such that µ(1 I)=1 and µ(e+f) = µ(e) + µ(f) for orthogonal

pairs e and f in E. Then µ(0)=0 and µ(e1+...+ek) = µ(e1)+...+µ(ek) for orthogonal elements

e1,...,ek in E. When E is orthogonally σ-complete, the state µ is called σ-additive if

µ(Σnen)=Σnµ(en) for any orthogonal sequence en in E, and when E is orthogonally complete,

the state µ is called completely additive if µ(Σf∈F f) = Σf∈Fµ(f) for any orthogonal subset F of E.

′ f . This

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Denote by So the set of all states on E, by Sσ the set of σ-additive states on E and by Sc the set

of completely additive states on E, where it is assumed that E is orthogonally σ-complete and

orthogonally complete, respectively, in the latter two cases.

With a state µ and µ(e)>0 for some e∈E, another state ν is called a conditional probability

of µ under e if ν(f) = µ(f)/µ(e) holds for all f∈E with f?e. Now let S be either So or Sσ or Sc.

We shall consider the following axioms that were introduced in [21].

(UC1)

(UC2)

If e,f∈E and µ(e)=µ(f) for all µ∈S, then e=f.

If e∈E and µ∈S with µ(e)>0, there is one and only one conditional probability of µ

under e.

If these axioms are satisfied, E is called an S-UCP space (S = So, Sσ, or Sc) - named after

the major feature of this mathematical structure which is the existence of the unique

conditional probability - and the elements in E are called events. The unique conditional

probability of µ under e is denoted by µe and, in analogy with probability theory, we also write

µ(f|e) instead of µe(f) for f∈E. The above two axioms imply that there is a state µ∈S with

µ(e)=1 for each event e≠0, that the difference d in (OS6) becomes unique, that the relation ?

is anti-symmetric (but not necessarily transitive), and that e⊥e iff e⊥1 I iff e=0 (e∈E).

Note that the following identity which will be used later holds for convex combinations of

states µ,ν∈S (0<s<1):

(sµ+(1-s)ν)e = (sµ(e)µe+(1-s)ν(e)νe)/(sµ(e)+(1-s)ν(e)). (1)

Examples of the above structure can be obtained considering Jordan algebras. The

multiplication operation in a Jordan algebra A satisfies the condition a2?(a?b)= a?(a2?b) for

a,b∈A. A JB algebra is a complete normed real Jordan algebra A satisfying ||a?b||≤||a|| ||b||,

||a2||=||a||2 and ||a2||≤||a2+b2|| for a,b∈A. A partial order relation ≤ on A can then be derived by

defining its positive cone as {a2:a∈A}. If A is unital, we denote the identity by 1 I. A JB

Algebra A that owns a predual A* (i.e., A is the dual space of A*) is called a JBW algebra and is

always unital. A JBW algebra can also be characterized as a JB algebra where each bounded

monotone increasing net has a supremum in A and a normal positive linear functional not

vanishing in a exists for each a≠0 in A (i.e., the normal positive linear functionals are

separating). A map is normal if it commutes with the supremum. It then turns out that the

normal functionals coincide with the predual. The self-adjoint part of any W*-algebra (von

Neumann algebra) equipped with the Jordan product a?b:=(ab+ba)/2 is a JBW algebra, but

not each JBW algebra is the complete self-adjoint part of a W*-algebra. Moreover, there are

exceptional JBW algebras that cannot be represented as an algebra of self-adjoint operators at

all (e.g., the algebra of hermitean 3×3 matrices over the octonions equipped with the Jordan

product).

The monographs [14] and [26] are recommended as excellent references for the theory of

JB/JBW algebras and W*-algebras, respectively. Historically important references are Jordan,

von Neumann and Wigner's work [16] which initiated the theory of Jordan algebras and covers

the finite-dimensional case, as well as Alfsen, Shultz and Størmer's achievement for the infinite-

dimensional case [5].

The idempotent elements of a JB algebra are called projections, and the projections in a

JBW algebra A form a complete lattice. These projection lattices now become examples of Sc-

UCP spaces if A does not contain a type I2 part. The conditional probability has the shape

µ(f|e)=? µ({e,f,e})/µ(e), where {a,b,c} := a?(b?c) - b?(c?a) + c?(a?b) is the Jordan triple

product, and in the case of a W*-algebra we get µ(f|e)=? µ(efe)/µ(e). This was shown in [21]

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and reveals the link to the Lüders - von Neumann quantum measurement. Note that ? µ on A is

the unique linear extension of the state µ on the projection lattice; this extension exists by

Gleason's theorem [13] and its later enhancements to W*-algebras and JBW algebras

[8,9,11,20,30,31]. A complete characterization of the projection lattices in JBW algebras

without type I2 part as Sc-UCP spaces with some further properties will be a major result at the

end of this paper.

3. Order-unit spaces

A partially ordered real vector space A is an order-unit space if A contains an order-unit 1 I

and if A is Archimedean [1,6,14]. The order-unit 1 I is positive and, for all a∈A, there is t>0

such that -t1 I ≤ a ≤ t1 I. A is Archimedean if na ≤ 1 I for all n∈IN implies a≤0. An order-unit

space A has a norm given by a = inf {t>0: -t1 I ≤ a ≤ t1 I}. Each x∈A can be written as x=a-b

with positive a,b∈A (e.g., choose a = ||x||1 I and b = ||x||1 I - x). A positive linear functional

ρ:A→IR on an order-unit space A is norm continuous with ||ρ||=ρ(1 I) and, vice versa, a norm

continuous linear functional ρ with ||ρ||=ρ(1 I) is positive. Note that unital JB algebras are

order-unit spaces.

The order-unit spaces A considered in the following are dual spaces of base-norm spaces V

such that the unit ball of A is compact in the weak-*-topology σ(A,V). For ρ∈V and x∈A

define ? ρ(x):=x(ρ); the map ρ→? ρ is the canonical embedding of V in its second dual V**=A*.

Then ρ∈V is positive iff ? ρ is positive on A. Moreover, A is monotone complete and

? ρ(sup xα)=lim? ρ(xα) holds for ρ∈V and any bounded monotone increasing net xα in A; in the

JBW/W*-algebra setting one would say that ρ∈V is normal.

For any set K in A, denote by linK the σ(A,V)-closed linear hull of K and by convK the

σ(A,V)-closed convex hull of K. For a convex set K, denote by ext K the set of its extreme

points which may be empty unless K is compact. A projection is a linear map U:A→A with

U2=U and, for a≤b, we define [a,b] := {x∈A: a≤x≤b}. Suppose that E is a subset of [0,1 I] in A

such that

(a) 1 I∈E,

(b) 1 I-e∈E if e∈E, and

(c) d+e+f∈E if d,e,f,d+e,d+f,e+f∈E.

Define e':=1 I-e and call e,f∈E orthogonal if e+f∈E. Then E satisfies (OS1),...,(OS6) such that

we can consider So as in section 2. Since [0,1 I] is monotone complete, any sum of orthogonal

elements in E σ(A,V)-converges in [0,1 I]. If these sums converge in E, we call E orthogonally

complete, and we call E orthogonally σ-complete, if only the countable sums converge in E. In

these cases, we can again consider Sσ and Sc as in section 2.

Proposition 3.1: Suppose that A is an order-unit space with order unit 1 I and that A is

the dual of the base-norm space V. Moreover, suppose that E is a subset of [0,1 I] satisfying

the three above conditions (a), (b), (c). Consider the cases S=So, S=Sσ, or S=Sc, assuming that

E is orthogonally σ-complete or orthogonally complete, respectively, in the latter two cases,

and suppose that the following two conditions hold:

(i) A = linE, and for each µ∈S there is a σ(A,V)-continuous positive linear functional ? µon

A with ? µ(e) = µ(e) for e∈E.

(ii) For each e∈E there is a σ(A,V)-continuous positive projection Ue:A→A such that Ue1 I=e,

UeA = lin{f∈E: f≤e} and ? µ=? µUe for µ∈S with µ(e)=1.

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Then E is an S-UCP space. The conditional probabilities have the shape µ(f|e) = ? µ(Uef)/µ(e)

for e,f∈E and µ∈S with µ(e)>0.

Proof. For e,f∈E with e≠f there is ρ∈V+ with ρ(e-f)≠0. The restriction of ρ/ρ(1 I) to E then

yields a state µ∈S with µ(e)≠µ(f). Therefore (UC1) holds.

Suppose e∈E and µ∈S with µ(e)>0. It is rather obvious that the map g → ? µ(Ueg)/µ(e) on

E provides a conditional probability of µ under e. Now assume that ν is a further conditional

probability of µ under e. Then ν(e)=1 and thus ? ν=? νUe. From Ueg∈lin{f∈E: f≤e} we get that

ν(g) = ? ν(Ueg) = ? µ(Ueg)/µ(e) for g∈E. Therefore, (UC2) holds as well.q.e.d.

Note the similarities between the second part of condition (i) in the preceding proposition

and the Gleason theorem. We shall now see that the situation of Proposition 3.1 is universal for

the S-UCP spaces; i.e., each S-UCP space has such a shape as described there.

Theorem 3.2: Suppose that E is a S-UCP space with S=So, S=Sσ, or S=Sc. Then E is a

subset of [0,1 I] in some order-unit space A with predual V as described in Proposition 3.1.

Proof. Define V:={sµ-tν: µ,ν∈S, 0≤s,t∈IR }, which is a linear subspace of the

orthogonally additive real-valued functions on E, and consider for ρ∈V the norm ||ρ|| :=

inf{r∈IR : r≥0 and ρ ∈ r conv(S∪-S)}. Then |ρ(e)| ≤ ||ρ|| for every e∈E. Let A be the dual

space of the base-norm space V and let ? µ be the canonical embedding of µ∈V in V**=A*. If

? µ(x)≥0 for all µ∈S, the element x∈A is called positive and we write x≥0. Equipped with this

partial ordering, A becomes an order-unit space with the order-unit1 I:=π(1 I), and the order-

unit norm coincides with the dual space norm such that sup{|? µ(x)| : µ∈S} = ||x|| for x∈A. With

e∈E define π(e) in A via π(e)(ρ) := ρ(e) for ρ∈V. Then ||π(e)|| ≤ 1, and the finite additivity, σ-

additivity, or complete additivity of π in the different cases follow immediately from this

definition. Moreover, A is the σ(A,V)-closed linear hull of π(E).

We now define Ue for e∈E. Suppose x∈A and sµ-tν∈V with µ,ν∈S and 0≤s,t∈IR . Then

define (Uex)(sµ-tν) := sµ(e)? µe(x)-tν(e)? νe(x). Here, ? µe and ? νe are the canonical embeddings

of the conditional probabilities µe and νe in A*; they do not exist in the cases µ(e)=0 or ν(e)=0

and then define µ(e)? µe(x):=0 and ν(e)? νe(x):=0, respectively. We still have to show that Ue is

well defined for sµ-tν = s'µ'-t'ν' with µ,µ',ν,ν'∈S and 0≤s,s',t,t'. Then s-t = (sµ-tν)(1 I) =

(s'µ'-t'ν')(1 I) = s'-t' and s+t'=s'+t. If s+t'=0, s=s'=t=t'=0 and Uex is well-defined. If s+t'>0, then

either sµ(e) + t'ν'(e) = s'µ'(e) + tν(e) = 0 and sµ(e) = t'ν'(e) = s'µ'(e) = tν(e) = 0, or

(sµ+t'ν')/(s+t') = (s'µ'+tν)/(s'+t) ∈ S and, calculating the conditional probability under e for

both sides of this identity by using (1), we get sµ(e)µe+t'ν'(e) ′ νe = s'µ'(e) ′

cases, Ue is well defined.

If µ(e)=1 for µ∈S, then µ=µe and ? µ(Uex) = (Uex)(µ) = ? µ(x) such that ? µ=? µUe. Thus,

(UeUex)(µ) = µ(e)? µe(Uex) = µ(e)? µe(x) = (Uex)(µ) for all µ∈S and hence for all ρ∈V such that

UeUe=Ue, i.e., Ue is a projection. Its positivity, σ(A,V)-continuity as well as Ue1 I=π(e) and

Ueπ(f)=π(f) for f∈E with f≤e follow immediately from the definition.

Therefore lin{π(f): f∈E, f≤e} ⊆ UeA. Assume Uex∉lin{π(f): f∈E, f≤e} for some x∈A. By

the Hahn-Banach theorem, there is ρ∈V with ? ρ(Uex)≠0 and ρ(f)=0 for f∈E with f≤e. Suppose

ρ=sµ-tν with µ,ν∈S and 0≤s,t∈IR . Then sµ(f)=tν(f) for f∈E with f≤e and thus sµ(e)µe(f) =

tν(f)νe(f). The uniqueness of the conditional probability implies sµ(e)µe(f) = tν(f)νe(f), i.e.,

µe+tν(e)νe. In all

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? ρ(Uef)=0 for all f∈E such that ? ρUe=0 which contradicts ? ρ(Uex)≠0. This completes the proof

of Theorem 3.2 after identifying π(E) with E. q.e.d.

Lemma 3.3: If e≤f with e,f∈E, then Uef=e=Ufe and UeUf=UfUe=Ue. If e,f∈E are

orthogonal, then Uef=0=Ufe and UeUf =UfUe=0.

Proof. Suppose e≤f. Then ? µ UeUf=µ(e)? µeUf=µ(e)? µe=? µ Ue for µ∈S, where we have used

that 1=µe(e)≤µe(f)≤1 for the conditional probability µe implies µe(f)=1 and hence ? µeUf=? µe.

Thus UeUf=Ue. The identity UfUe=Ue immediately follows from (ii) in Proposition 3.1.

Moreover e=Ue1 I=UeUf1 I=Uef and e=Ue1 I=UfUe1 I=Ufe.

Now assume that e and f are orthogonal. Then e≤

that UeUf =0. In the same way it follows that UfUe=0. Therefore Uf vanishes on UeA such that

UfUe=0. The identity UeUf =0 follows in the same way.

′ f and e=Ue

′ f =Ue(1 I-f)=e-Uef such

q.e.d.

In the following two sections, it shall be investigated when we have spectral duality in the

meaning of Alfsen and Shultz [2] and when A becomes a JBW algebra. This requires the notion

of an observable.

4. Observables and spectral duality

A bounded real observable X is a bounded spectral measure allocating eB∈E to each Borel

set B in IR . Since a spectral measure is σ-additive, we must assume now that E is an an S-UCP

space with S=Sσ or S=Sc. This definition of an observable can be found e.g., in [22,25,28,29].

Some authors use other definitions; however, note that only bounded real-valued sharp

observables are considered in the present paper.

The spectral radius of X is r(X) := inf{t≥0:e[-t,t]=1 I} and is finite for bounded observables.

The probability measure µX with µX(B):=µ(eB) is the distribution of X in the state µ, and the

measure integral ∫ t dµX is the expectation value of the observable X. Then there is a unique

element x in A such that ∫ t dµX = ? µ (x) for all µ∈S because the map V∋ρ→∫ t dρX yields an

element x in V*=A with ||x||=r(X). Note that each e∈E and each sum Σtkek with orthogonal

events e1,...,en in E and real numbers t1,...,tn can be represented by an observable in this way.

Such a linear combination of orthogonal events is called a primitive element in A. Each x∈A

that represents an observable can be approximated by a norm convergent sequence of primitive

elements. In general, however, not each element in A represents an observable, which is an

important difference to the Jordan algebras of self-adjoint operators usually considered in

quantum mechanics.

We are now in the position to formulate some further potential postulates for a system of

quantum events. The numbering of the axioms shall remain consistent with the numbering in

earlier papers [22,23]; this is the reason why we now come to (A2), (A3) and (A4) while (A1)

will be considered in section 5.

(A2)

For e,f∈E there is a real observable X such that µ(f|e)µ(e) = ∫ t dµX for all µ∈S with

µ(e)>0.

If Y and Z are bounded real observables, there is another real observable X such that

∫ t dµY + ∫ t dµZ = ∫ t dµX for all µ∈S.

(A3)

The first axiom means that the element Uef in A represents an observable, but is formulated

in a more basic way using only the notions of conditional probabilities and observables. The

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second one postulates the existence of a reasonable sum within the class of real bounded

observables. These two axioms were already considered in [22,23], but in a stronger form;

there the uniqueness of the observable X was postulated, which we can dispense with in the

present paper. For two real observables,

≤ ∫ t dµY for all µ∈S. This is equivalent to x≤y for the elements in A that represent the

observables.

X≤Y shall mean that

∫

t dµX

(A4σ)

If Yn is a bounded (i.e., r(Yn)≤s for some s) monotone increasing sequence of

bounded real observables, there is a real observable X with

limn ∫ t d

µ

= ∫ t dµX

for all µ∈S.

the same as (A4σ), but with bounded monotone increasing nets instead of sequences.

Yn

(A4c)

We shall now study the relationship between the UCP spaces considered here and Alfsen

and Shultz's spectral duality [2]. They introduced the so-called P-projections and called P(1 I)

with a P-projections P a projective unit. The Ue considered here are similar to the P-

projections, but the two concepts are not identical. A P-projection P has a quasicomplement Q

such that Px=x iff Qx=0 (and Qx=x iff Px=0) for x≥0. If Uex=x, then Ue'x=0 by Lemma 3.3, but

Ue'x=0 does not imply Uex=x.

In the case of spectral duality of V and A, the system of projective units satisfies (UC1),

(UC2), (A2), (A3) and (A4σ) if and only if the states on the projective units have linear

extensions to A (as with the Gleason theorem, or in condition (i) of Proposition 3.1). Vice

versa, however, what is necessary to get spectral duality from a UCP space? The answer is

given in the following proposition.

Proposition 4.1: Suppose that E is a Sσ-UCP space satisfying (A2), (A3) and (A4σ). Let

L consist of those elements in A that represent observables. Note that L is a monotone σ-

complete linear space by (A3) and (A4σ). Then the following conditions are equivalent:

(i) Ue and Ue' are quasicomplementary P-projections on L for each e∈E (i.e., L and V are in

weak spectral duality).

(ii) If µ(f)=1 holds for each state µ∈Sσ with µ(e)=1, then e≤f (e,f∈E).

In both cases, E ⊆ ext{x∈L: 0≤x≤1 I}.

Proof. Assume (i) and that µ(f)=1 holds for each state µ∈Sσ with µ(e)=1 (e,f∈E). This

means µ(e)=µ(e)µe(f)=? µ (Uef) for µ∈Sσ; thus e=Uef and 0=Ue

quasicomplementary P-projections, we have for positive x in L that Uex=0 iff Ue'x=x.

Therefore, e' ≥Ue'

′ f =

′ f and e≤f.

Now suppose (ii). This means that 0=Ue

′ f implies e≤f for any e,f∈E. If now 0=Uex for

some positive x in L, then 0=Uep for any p in a spectral resolution of x such that e≤p' and p≤e';

therefore Ue'p=p for all p in the spectral resolution and finally Ue'x=x.

Assume e∈E and e=sx+ty with x,y∈L, 0≤x,y≤1 I and real number s,t>0, s+t=1. Then

0=Ue'e=sUe'x+tUe'y such that Ue'x=0=Ue'y. The quasicomplementarity yields x=Uex≤e and

y=Uey≤e such that the identity e=sx+ty is possible only if e=x=y. Therefore e is an extreme

point in {x∈L: 0≤x≤1 I}.

′ f . Since Ue and Ue' are

q.e.d.

Note that Alfsen and Shultz distinguish between spectral duality and weak spectral duality;

spectral duality means existence and uniqueness of the spectral resolution, while the weak form

does not require the uniqueness [2]. We would achieve spectral duality in condition (i) in the

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above proposition, when we would assume uniqueness for the observables in (A2) (A3) and

(A4σ).

For the results in the following section, it will neither be necessary to assume that

condition (ii) of Proposition 4.1 holds nor that the Ue are P-projections. This is why the

approach of the present paper differs from other approaches that assume spectral duality [2-

4,7,15] or at least the existence of sufficiently many P-projections [12].

5. Jordan operator algebras

We now arrive at the last missing ingredient to make A a JBW algebra. This is the

following axiom.

(A1)

µµµµµµµµ

( ) ( )( ) ( )( ) ( )( ) ()

f ee f e

′ ′

e e ff e f

′

f

+

′ =

+

′′ for e,f∈E and µ∈S.

(A1) is equivalent to U f

for the first time in [4] for the P-projections considered there. Note that (A1) holds in W*-

algebras and JBW algebras.

U f

e

U e U

f

e

ef

+ ′ =+′

′′

for e,f∈E. In this form, (A1) occurred

Theorem 5.1: Suppose that E is a S-UCP space with S=Sσ or S=Sc and that (A1), (A2),

(A3) hold. Then A can be equipped with a (generally non-associative) commutative

multiplication operation ? such that A becomes a JBW algebra and E is a σ(A,V)-dense

subset of the projection lattice in A; 1 I∈E becomes the unit element of the JBW algebra.

Proof. Denote by L the subset of A containing all elements which represent observables

and by M its norm closure. Because of (A3) L and thus also M are linear subspaces of A.

Define Tex := ½ (x + Uex - Ue'x) for e∈E and x∈A. Then Tex∈L for primitive elements x by

(A2) and (A3) and thus Tex∈M for x∈M due to the norm continuity of Ue and Ue'. Moreover,

Ue[-1 I,1 I] ⊆ [-e,e] and Ue'[-1 I,1 I] ⊆ [-e',e'], thus (Ue - Ue')[-1 I,1 I] ⊆ [-1 I,1 I] and therefore

||Ue-Ue'||≤1. Hence ||Te||≤1.

(A1) implies Tef = Tfe for e,f∈E. For x∈M and a primitive element y=Σtkek with orthogonal

events e1,...,en in E and real numbers t1,...,tn define Tyx:=Σt T x

primitive elements x and y, and this also implies that Ty is well-defined. Furthermore,

||Tye||=||Tey||≤||y|| for e∈E and thus ||Tyx|| ≤ ||x|| ||y|| for all primitive elements x,y since x with

||x||≤1 is a convex combination of elements from E and -E. Now define x?y for x,y∈M as the

norm-continuous extension of Txy to M.

Then e2:=e?e=e and e?f=0 for orthogonal events e,f∈E. For a primitive element y=Σtkek

we have yn=Σtk

sup{|? µ(x)| : µ∈S} we get ||x2||≤||x2+y2|| for primitive elements x and y. Due to norm continuity,

||y2||=||y||2 and ||x2||≤||x2+y2|| then hold for x,y∈M and, since the primitive elements are power-

associative, each element in M is power-associative. By a result in [16] or [27], this implies

e?(f?y)=f?(e?y) for orthogonal idempotent elements e and f and for any y. Therefore we get

for a primitive element x=Σtkek with orthogonal events e1,...,en in E and y∈M:

k ek

for x∈A. Then Tyx=Txy for

nek. This implies ||y2||=max{tk2}=(max{|tk|})2=||y||2. Moreover, from ||x|| =

x x y

?

t t e

l k l k

ey t t e

l k l l

eyxxy

klkk

2222

???????

()()()()

=== Σ ΣΣ Σ

.

Due to norm continuity, this identity then holds for all x,y∈M such that M become a JB

algebra.

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Gerd NiesteggeA Representation of Quantum Measurement in Order-Unit Spaces

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We now consider the seminorms x→? µ(x2)1/2, µ∈S, on M and the topology defined by them

on M which is called the s-topology. Norm convergence implies s-convergence. Due to the

Cauchy-Schwarz inequality, s-convergence implies σ(M,V)-convergence, the product x?y is s-

continuous separately in each factor and jointly s-continuous in both factors on bounded

subsets of M. Let N comprise all those elements of A that are the σ(A,V)-limit of a bounded net

in M which is a Cauchy net with respect to the seminorms defining the s-topology. Then

? µ(x2)≥0 for µ∈S, x∈N, and N becomes a JB algebra since it inherits all the properties from M.

Note that the σ(A,V)-closure of M is not automatically a JB algebra since the map x→x2 is not

σ(A,V)-continuous.

If now xα is a bounded monotone increasing net in N, then xα σ(A,V)-converges to sup xα

in A. Since (x-y)2≤||x-y||(x-y) for y≤x in a JB algebra, the net xα s-converges to sup xα such that

sup xα∈N. Therefore N is monotone complete and the restrictions of the positive elements of V

to N provide a separating family of normal functionals such that N becomes a JBW algebra. On

the other hand, if ρ is any positive normal functional on N with ρ(1 I)=1, its restriction to E

belongs to S such that V becomes identical with the normal functionals on N or with the

predual of N. Thus A = V* = N is a JBW algebra.

Since the unit ball of M is contained in the norm closure of conv(E-E), the σ(A,V)-closure

of the unit ball of M, which is the unit ball of A, must be included in the σ(A,V)-closure

conv(E-E). Therefore [-1 I,1 I]=conv(E-E). Applying the σ(A,V)-continuous affine function

x→(x+1)/2 to both sides and using the identity (e-f+1)/2=(e+

The Krein-Milman theorem then implies that ext[0,1 I]⊆ E . Since, in a JBW algebra, ext[0,1 I]

coincides with the projection lattice which contains E, we finally have the σ(A,V)-density of E

in ext[0,1 I].

′ f )/2 we get [0,1 I]=conv(E).

q.e.d.

Theorem 5.1 is an improvement of an earlier result by the author [22] in two ways. Only

the existence of the observable X must be assumed in (A2) and (A3), the uniqueness postulate

becomes redundant. Furthermore, in Theorem 5.1, the embedding of E in the JBW algebra is

σ-additive and completely additive in the different cases due to Theorem 3.2, while the one

considered in [22] is only finitely additive.

Corollary 5.2: Suppose that E is a Sc-UCP space and that (A1), (A2), (A3), (A4c) hold.

Then E is identical with the projection lattice of a JBW algebra.

Proof. Consider the subset L of A containing all elements which represent observables.

Due to (A4c) L is monotone complete and thus norm complete since sequential monotone

completeness already implies norm completeness (e.g., see [2] or [25]). In the proof of

Theorem 5.1 we have seen that L is a JB algebra then. The positive part of V provides a

separating family of normal functionals. Thus L is a JBW algebra (and L=N=A). Since each x in

L represents an observable, its spectral resolution belongs to E; applying this to a projection

x=p yields p∈E. q.e.d.

Corollary 5.3: Suppose that E is a Sσ-UCP space and that (A1), (A2), (A3), (A4σ) hold.

Then E is identical with the projection lattice of a monotone sequentially complete JB algebra

with a separating family of σ-normal states.

This follows in the same way as the preceding corollary. The monotone sequentially

complete JB algebras appearing in Corollary 5.3 are by far not as well-known as the JBW

algebras, but are nevertheless considered here because they appear a more natural analogue of

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Gerd NiesteggeA Representation of Quantum Measurement in Order-Unit Spaces

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the measurable function spaces of probability theory than the JBW algebras; indeed, the

measurable functions form a monotone sequentially complete JB algebra, but not a JBW

algebra in general. The monotone sequentially complete C*-algebras were studied by

Christensen [10], Kadison [18], Kehlet [19] and Pedersen [24].

Corollary 5.2 provides a complete characterization of the projection lattices in JBW

algebras without type I2 part similar to the one provided by Bunce and Wright [7]. However,

they use other characterizing properties; besides the countable chain condition and an "elliptic"

state space they assume spectral duality in the meaning of Alfsen and Shultz [2] which

presumes the existence of sufficiently many P-projections and the uniqueness for the spectral

measure. They do not use (A1) which is replaced by the "ellipticity". Their result is based on

other characterizations that do not concern the projection lattice, but the state space of an

operator algebra [2-4,15]. These characterizations involve very interesting geometric

properties of the state space and are extremely satisfying from a mathematical viewpoint, but

they become less satisfactory if one's concern is the axiomatic foundation of quantum

mechanics because plausible interpretations and physical meanings are hard to find for these

properties.

6. Conclusions

In classical probability theory, the observables or random variables are modeled by

measurable functions. When proceeding to quantum mechanics, self-adjoint operators must be

used as model instead. With the still more general framework of the present paper, the

observables become elements of an order-unit space A like in Proposition 3.1, but not each

a∈A represents an observable. This is a significant difference to standard quantum mechanics

(and even to the JBW algebra model). Moreover, the observables cannot anymore be

represented as self-adjoint linear operators on a Hilbert space such that discussions about the

so-called "collapse of the wave-function" and its interpretation would become needless.

A concrete example of a UCP space that does not have a Hilbert space representation is

given by the projection lattice in the algebra of hermitean 3×3 matrices over the octonions

which is an exceptional Jordan algebra. An example of a UCP space where A is not a JBW

algebra is not known at present.

Although the axioms used in the present approach and in the characterization of the

projection lattices in section 5 have plausible interpretations and physical meanings, it is still

not clear whether each single one is an absolute must in an axiomatic foundation of quantum

mechanics. The axiom (A1) turns out to be very useful, but does not appear to be a really

natural one. The postulate (A3) concerning the existence of a reasonable sum within the class

of real observables is usually considered a quite natural axiom, but wouldn't it then be as

natural for the vector-valued observables where it is already violated by standard quantum

mechanics and disproved by physical experiments?

In the present paper, two extremes have been considered - the UCP spaces with only the

basic properties and those with so much structure that they become projection lattices in JBW

algebras. There may be further interesting structures between the two extremes satisfying other

combinations of the axioms; the combination (UC1) and (UC2) together with (A1) appears

interesting in so far as these three postulates require only the basic notion of conditional

probability and do not need the notion of an observable.

Alfsen and Shultz's spectral duality of an order-unit space and a base-norm space [2] has

been identified as a candidate of a mathematical structure lying between the two extremes; in

this case, (A1) does not hold, but all other axioms would do, if the Gleason extension theorem

were satisfied for the states on the projective units. However, all the concrete examples

considered by Alfsen and Shultz either are related to JBW algebras, or they contain at most

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Gerd NiesteggeA Representation of Quantum Measurement in Order-Unit Spaces

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two orthogonal non-zero events (like the type I2 JBW factors) such that they cannot satisfy the

Gleason extension theorem unless being identical with the Boolean algebra {0,e,e',1 I}.

Another candidate structure are the GL spaces considered by Edwards and Rüttimann [12]; in

this case, only (UC1) and (UC2) would hold, if the Gleason extension theorem were available,

and the Ue become P-projections.

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