Operator valued probability theory

Per K. Jakobsen, V.V. LychaginUniversity of Tromso,9020 Tromso, Norway

September 27, 2004

Abstract

This is where the abstract will come

Contents

1 Introduction 1

2 Extended probability spaces 3

3 Random vectors 43.1 The space of random vectors . . . . . . . . . . . . . . . . . . . . 43.2 The expectation of random vectors . . . . . . . . . . . . . . . . . 63.3 Conditional expectation . . . . . . . . . . . . . . . . . . . . . . . 7

4 Densities and random operators 84.1 The Hilbert module of half densities . . . . . . . . . . . . . . . . 94.2 Random operators . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5 The category of extended probability spaces 165.1 Morphisms of extended probability spaces . . . . . . . . . . . . . 165.2 The Naimark functor . . . . . . . . . . . . . . . . . . . . . . . . . 23

6 Monoidal structure on the category of extended probabilityspaces 276.1 Product of extended probability spaces and morphisms . . . . . . 296.2 The monoidal structure . . . . . . . . . . . . . . . . . . . . . . . 34

1 Introduction

In this paper we present a extension of standard probability theory. A extendedprobability space is de�ned to be a normalized positive operator valued mea-sure de�ned on a measurable space of events. This notion of extended probabil-ity space include probability spaces and spectral measures as important special

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cases. The use of the word probability in this context is justi�ed by showing thatextended probability spaces enjoy properties analog to all the basic propertiesof classical probability spaces. Random vectors are de�ned as a generalizationof the usual Hilberts space of square integrable functions. This generalizationis well known in the literature and was �rst described by Naimark. Expecta-tion and conditional expectation is de�ned for extended probability spaces byorthogonal projections in complete analogy with probability spaces.The introduction of probability densities presents special problems in the

context of extended probability spaces. For the case of probability spaces aprobability density is any normalized positive integrable function, whereas forthe case of extended probability spaces it turns out that the right notion isnot a density but a half density. These half densities are elements in a Hilbertmodule of length one. Special cases of such half densities are well known inquantum mechanics where they are called wave functions. We de�ne a randomoperator to be a linear operator on the space of half densities. The expectationof random operators are operators acting on the Hilbert space underlying theextended probability space. For the case of probability spaces the notion ofrandom vectors and random operators coincide.We introduce mappings or morphisms of extended probability spaces throught

a generalization of the notion of absolute continuity in probability theory. Halfdensities plays a pivotal role in this generalization. We show that the morphismscan be composed and that extended probability spaces and morphisms forms acategory just as for probability spaces. The Naimark construction extends tomorphisms and in fact de�nes a functor on the category of extended probabilityspaces.Extended probability spaces can be multiplied and we furthermor show that

this multiplication can be extended to morphisms in such a way that it de�nesa monoidal structure on the category of extended probability spaces. This is incomplete analogy with the case of probability spaces and testify strongly to thenaturalness of our constructions.We do not in this paper attempt to give any interpretation of extended prob-

abilities beyond the one implied by the strong structural analogies that we haveshown to exists between the categories of probability spaces and extended proba-bility spaces. It is well known that the interpretation of the classical Kolmogorovformalism for standard probability theory is not without controversy as the olddebate between frequentists and Bayesians, among others, clearly demonstrate.Our theory of extended probability spaces is evidently a generalization of theKolmogorov framework and it might be hoped that this enlarged frameworkwill put some of the controversy in a di¤erent light. As a case in point notethat extended probabilities are in general only partially ordered. The notionof partially ordered probabilities has been discussed and argued over for a verylong time. In our theory of extended probability spaces, ordered and partiallyordered probabilities lives side by side and enjoy the same formal categoricalproperties.

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2 Extended probability spaces

In this section we will make some technical assumptions that will assumed tohold throughout this paper. These asumptions are not neccesarily the mostgeneral ones possible.A measurable space [5] is a pair X = hX ;BXi where X is a set and

BX is a �-algebra on X . A measurable map f : X ! Y is a map of setsX ! Y such that f�1(A) 2 BX for all A 2 BY . Let be a set and let� be a topology on . In this paper the term topology is taken to mean asecond countable,locally compact Hausdorf topology [3]. Note that any suchspace is metrizable,Polish and �-compact.The Borel structure corresponding toa topology � is the smallest �-algebra containing the topology � and is denotedby B(�). A Borel space is a measurable space where the �-algebra is a Borelstructure. Any continuous map f : hX ; �Xi ! hY ; �Y i is measurable withrespect to the Borel structures B(�X) and B(�Y ). Borel sets are the observableevents to which we must assign probabilities.Let now hX ;B(�X)i be a borel space and let O(HX) be the real C� algebra

[4] of bounded operators on the real Hilbert space HX . A positive operatorvalued measure (POV) [1] de�ned on hX ;B(�X)i is a map FX from B(�X)to O(HX) such that FX(;) = 0,FX(X) = 1. The map FX is assumed to be�nitely additive on disjoint union of sets and for any increasing sequence of setsfVig satisfy the following continuity condition

FX( limi�!1

Vi) = supfFX(Vi) j i = 1; 2; 3; ::::g

where the supremum is taken with respect to the usual partial ordering ofself adjoint operators. The supremum always exists since the sequence fFX(Vi)gis increasing and bounded above by FX(limi�!1 Vi). The continuity conditionimplies that FX is additive on countable disjoint unions.

FX(U1i=1Vi) =

1Xi=1

FX(Vi)

where the sum converges in the strong operator topology, that is, pointwiseconvergence in norm.A positive operator valued measure is a spectral measure if FX(V ) is a

projector for all V 2 B. A neccesary and su¢ cient condition for a POV, FX ,tobe a spectral measure is that it is multiplicative

FX(V1 \ V2) = FX(V1)FX(V2)

We are now ready to de�ne our �rst main object

De�nition 1 A extended probability space X is a triple X = hX ;B(�X); FXiwhere FX : B(�X) �! O(HX) is a positive operator valued measure.

Note that a probability space X = hX ;B(�X); �Xi can be identi�ed with aextended probability space in many di¤erent ways. In fact for any given Hilbertspace HX we can identify the probability space with a extended probabilityspace X = hX ;B(�X); FXi where FX(V ) = �X(V )IHX

.

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3 Random vectors

In standard probability theory quaderatic integrable random variables and theirexpectation plays an important role. We will now review the classical Naimarkconstruction of the analog of such random variables for the case of extendedprobability spaces. We will call such random variables random vectors. Thespace of random vectors forms a Hilbert spaces and we use this structure to de-�ne expectation and conditional expectation by orthogonal projections in com-plete analogy with the standard case.

3.1 The space of random vectors

Let h;B; F i be a extended probability space and let S be the linear space ofsimple measurable functions v : ! H. The linear structure is de�ned throughpointwise operations as usual. Elements in S can be written as �nite sums ofcharacteristic functions.

v =Xi

�i�Vi

where fVig is a B -measurable partition of the set . We de�ne a pseudoinnerproduct on S by

hv; wi =Xi;j

hF (Vi \Wj)�i; �jiH

where v =X

i�i�Vi , w =

Xj�j�Wj and h iH is the inner product in the

Hilbert space H. The product is not de�nite. In fact we have

hv; vi = 0

mXi

hF (Vi)�i; �iiH = 0

mhF (Vi)�i; �ii = 0 8i

The last identity follows from the fact the F (Vi) is a positive operator. Sofor any simple function v =

P�i�Vi we have hv; vi = 0 if and only if F (Vi)�i ?

�i for all i. This is of course true if Vi is of F measure zero but it can also betrue if F (Vi) 6= 0 but �i is in the kernel of F (Vi).

Since h i is a pseudo inner product the set of elements of length zero,hv; vi = 0, form a linear subspace and we can divide S by this subspace. andthereby get a, in general, incomplete inner product space. The completion ofthis space with respect to the associated norm is by de�nition the space ofrandom vectors and is a Hilbert space. We will use the notation L2(B; F ) orjust L2(F ) for this space in analogy with the classical notation L2(�). The

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set of equivalens classes of simple functions [v] evidently form a dense set inL2(F ). Denote this dense subspace by T (F ). We have a well de�ned isometricembedding � of H into L2(F ) de�ned by

�(�) = [��]

We also have a spectral measure P : B ! O(L2(F )). On the dense set T (F )the spectral measure is given by

P (�)[v] = [Xi

�i�Vi\�]

where v =P�i�Vi . In fact the existence of this spectral measure is the whole

point of the Naimark construction. It show that by extending the Hilbert spaceone can turn any POV into a spectral measure. This idea has been generalizedby Sz.-Nagy and J. Arveson into a theory for generating representations of ��-semigroups but we will not need any of these generalization in our work.As our �rst example let � be a measure on the measurable space h;Bi and

let H be a Hilbert space. De�ne a positive operator valued measure on h;Biacting on H by

F (U) = �(U)1H

For this case we have

h[v]; [w]i =Xi;j

h�(Vi \Wj)�i; �jiH =Xi;j

h�i; �jiH�(Vi \Wj) =

Zhv; wiHd�

where for any H valued functions f; g we de�ne hf; giH(x) = hf(x); f(x)iH .Thus for this case our space L2(F ) will be the space of H valued function

elements such thatZhf; fiHd� < 1. When H = C the space L2(F ) turns

into the space of square integrable complex valued functions L2(�).As our second example let H be two dimensional and let a basis f�1; �2g be

given. With respect to this basis we have

F (U) =

��(U) !(U)!(U) �(U)

�where � and � and ! are signed measures. In order for F (U) to be positive

for all U it is easy to see that � and � must be positive measures and that thefollowing inequality must hold

!(U)2 � �(U)�(U)

Any function f : ! H determines a pair of real valued functions ff1; f2gthrough f(x) = f1(x)�1+f2(x)�2. The inner product in L2(F ) is given in termsof the measures �,� and ! as

h(f1; f2); (g1; g2)i

=

Zf1g1d�+

Zf2g2d� +

Z(f1g2 + f2g1)d!

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Similar expressions for the inner product in L2(F ) exists for any �nite di-mensional Hilbert space H.

3.2 The expectation of random vectors

Recall that we have a isometric embedding � : H ! L2(F ) de�ned by

�(�) = [��]

Note that the image �(H) � L2(F ) is a closed subspace and therefore theorthogonal projection onto �(H) exists. Let QH be this orthogonal projection.

De�nition 2 The expectation of a random vector f 2 L2(F ) is the uniqueelement E(f) 2 H such that

�(E(f)) = QH(f)

The following result is a immediate consequence of the de�nition

Proposition 3 The expectation is a surjective continuous linear map : L2(F )!H and is the adjoint of the embedding �

hf; �(�)i = hE(f); �i 8� 2 H

Note that adjointness condition uniquely determines the expectation. In factwe could de�ne the expectation to be the adjoint of the embedding �.Using this proposition it is easy to verify that the expectation of a simple

function element [v] where v =P�i�Vi is given by

E([v]) =Xi

F (Vi)(�i)

This example makes it natural to introduce a integral inspired notation forthe expectation

E(f) =

ZdFf

Note that it is natural to put the di¤erential dF in front of f to emphasizethe fact that F is a operator valued measure that acts on the function valuedof f .Let f�ig be a orthonormal basis for H. For general elements f the following

formula holds

E(f) =Xi

hf; �(�i)i�i

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3.3 Conditional expectation

Let A � B be a subsigma algebra. We can restrict the POV F to A and willin this way get the Hilbert space L2(A; F ) of A measurable random vectors.We obviously have a isometric embedding of L2(A; F ) into L2(B; F ). ThusL2(A; F ) can be identi�ed with a closed subspace of L2(B; F ) and thereforethe orthogonal projection QA : L2(B; F ) ! L2(A; F ) is de�ned. In completeanalogy with the classical case we now de�ne

De�nition 4 The conditional expectation of a element f 2 L2(B; F ) is givenby

EA(f) = QA(f) 2 L2(A; F )

It is evident that L2(A; F ) is isomorphic to H when A = f; ;g and thatfor this case we have EA(f) = �(E(f)). Let us consider the next simplest casewhen A is generated by a partition fA1:::Ang where = UiAiand Ai \Aj = ;when i 6= j. We need the following result

Proposition 5 Let F (Ai) for i = 1::n have closed range. Then L2(A; F ) =T (A; F )

Proof. Let [vn] be a cauchy sequence in the inner product space T (A; F ).This means that jj[vn] � [vm]jj2 ! 0 when m and n goes to in�nity. Butvn =

Xi�ni �Ai and thusX

i

hF (Ai)(�ni � �mi ); �ni � �mi i ! 0

+hF (Ai)(�ni � �mi ); �ni � �mi i ! 0 8i

since F (Ai) are positive operators. Let Li = F (Ai)(H) be the range of F (Ai)and let L? be the orthogonal complement of Li. We have L?i = Ker(F (Ai))and since Li by assumption is a closed subspace we have the decompositionH = Li�L?i . Write �

ni = rni + t

ni with r

ni 2 L?i and tni 2 Li. We then have by

orthogonalityhF (Ai)(tni � tmi ); tni � tmi i ! 0

Clearly F (Ai)jLi : Li ! Li is a positive, bounded, injective and surjectivemap. Let Ti : Li ! Li be the square root of this operator. It is also a positivebounded injective and surjective map and therefore has a bounded inverse. Fromthe previous limit we can conclude that

hTi(tni � tmi ); Ti(tni � tmi )i ! 0

Thus fTi(tni )g is a cauchy sequence in Li and since Li is closed there existsa element yi 2 Li such that Ti(tni ) ! yi. From the previous remarks the

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element �i = T�1i (yi) 2 Li exists and limn!1 tni = limn!1 T�1i (Ti(tni )) =

T�1i (limn!1 Ti(tni )) = T�1i (yi) = �i. If we let v =

P�i�Ai

we have

jj[vn]� [v]jj2

=Xi

hF (Ai)(�ni � �i); �ni � �ii

=Xi

hTi(tni � �i); Ti(tni � �i)i

=Xi

hTi(tni )� yi; Ti(tni )� yii

=Xi

jjTi(tni )� yijj ! 0

and therefore T (A; F ) is complete.The assumption in the proposition holds for example if H is �nite dimen-

sional or if H is in�nite dimensional but all the F (Ai) are orthogonal projectorsor isomorphisms. For the classical measure case H � R and the proposition istrue. Let v =

P�j�Vj be a simple function in L2(B; F ). Then by the previous

proposition the conditional expectation must be of the form QA(v) =P�i�Ai

.It is uniquely determined by the conditions hv�QA(v); ��Aj

iH = 0 for all � 2 Hand j = 1::n. These conditions give us the following systems of equations forthe unknown vectors �i.

F (Ai)�i =Xk

F (Vk \Ai)�k 8i

This systems does not have a unique solution inH but all solutions representsthe same element in L2(A; F ) = T (A; F ). For the special case v = �0�C we getthe simpli�ed system

F (Ai)�i = F (C \Ai)�0When dim H = 1 and F (Ai) = �(Ai) we get the usual classical expression

for the conditional expectation of C given A.

4 Densities and random operators

Densities are important for most applications of probability theory. For us theywill make their apperance when we seek to generalize the relation of absolutecontinuity between measures to the context of positive operator valued mea-sures. This generalization will play a pivotal role when we de�ne maps betweenextended probability spaces. The generalization of the notion of density to thecase of operator measures turns out to be surprisingly subtle.

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4.1 The Hilbert module of half densities

Let � be a measure. A density is a positive measurable function � such thatZ�d� = 1. Using this density we can de�ne a new measure

�(V ) =

ZV

�d�

If we try to generalize this formula directly to the case of POV measures werun into problems.Let F be a POV de�ned on a measurable space h;B(�)i and let � be a

function as above. Then we can certainly de�ne a new POV measure by thefollowing formula

E(V ) =

ZV

�dF

There is nothing inconsistent in this de�nition, the only problem is that itis very limited. In fact if is a �nite set then any POV measure on is givenby a �nite set fFig of positive operators between zero and the identity withthe single condition

PFi = 1. If E is the new POV determined by the above

formula then we have Ei = �iFi for some set of numbers f�ig. Thus each Ei isproportional to Fi.Now if the numbers �i were changed into positive operators we could produce

a much more general E starting from a given F . We would thus be consideringa formula like

E(V ) =

ZV

�dF

were � is a positive operator valued function. However even if we couldmake sense of the proposed integral we would have problems. This is becausethe product of positive operators is positive only if they commute. This wouldput a highly nontrivial constraint on the allowed densities, constraints it wouldbe di¢ cult to verify and keep track of.There is however a natural way out of these problems. It is very simple to

verify that if F is a POV measure acting on H and Q a operator, then QFQ� isa new POV measure. This suggest that we consider a density to be a operatorvalued function ' such that Z

'dF'� = 1 (1)

We could then use this density to de�ne a new POV measure by

E(V ) =

ZV

'dF'� (2)

On a formal level this now looks �ne, the only remaining problem is to makesense of the proposed integrals. We will now proceed to do this.

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LetV = fs =

Xi

si�Vi j si 2 O(H) Vi 2 B(�)g

where fVig form a measurable partition of . These are simple measurableoperator valued functions. The set V is a real linear space through pointwiseoperations as usual. We can de�ne a left action of O(H) on V in the followingway

as =Xi

(asi)�Vi

This action clearly makes V into a left module over the real C�- algebraO(H). De�ne a O(H) valued product on V through

hs; ti =Xi;j

siF (Vi \Wj)t�j

where s =Psi�Vi and t =

Ptj�Wj . This product is clearly bilinear over

the real numbers. The following properties of the product are easy to verify.

Proposition 6

hs; si � 0

has; ti = ahs; tihs; ti = ht; si�

hs; ati = hs; tia�

Thus the product is like a hermitean product where the role of complexnumbers are played by the elements of the real C�-algebra O(H). Such struc-tures have been known and studiet for a long time. They leads, as we will see,in a natural way to the idea that probability densities for operator measuresare elements in a Hilbert module. Our main sources for the theory of Hilbertmodules are the paper [10] and the book [2]. Chapters on Hilbert modules canalso be found in the books [7] and [13].Note that the product we have constructed is not positive de�nite. In fact,

since the sum of positive operators in a real C�-algebras is zero only if eachoperator is zero, the identity hs; si = 0 holds if and only if

siF (Vi)s�i = 0 8i

and these identities can easily be satis�ed for nonzero operators si. In fact ifF (Vi) are projectors and si are projectors orthogonal to F (Vi) then the equationsare clearly satis�ed. In order to make the product de�nite we will need to divideout by the set of simple functions whose square is zero hs; si = 0. In order todo this we will need the analog of the Cauchy-Swartz inequality.For any element s 2 V we know that hs; si � 0 and therefore there exists a

positive operator h such that h2 = hs; si. Denote this operator by jsj. Thus wehave jsj2 = hs; si. Also for any element s 2 V de�ne a real number jjsjj by

jjsjj2 = jjhs; sijj

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where jjhs; sijj is the operator norm of the positive operator hs; si. With thesede�nitions at hand we can now state the following Cauchy Swartz inequalitiesfor V . The proof of this proposition is a adaption of the proof in [13] to thecase of real C� algebras.

Proposition 7

hs; tiht; si � jsj2jjtjj2

jjhs; tijj � jjsjj jjtjj

Proof. A positive linear functional, ! ,onO(H) is a real valued linear functionalsuch that !(a) � 0 whenever a � 0. A state on O(H) is a positive linearfunctional such that !(1) = 1 and !(a) = !(a�). The main property thatmakes states useful in C� algebra theory is that if a 6= 0 there exists a statesuch that !(a) = jjajj. From this it follows immediately that if !(a) = 0 forall states ! then a = 0 and this implies that if !(a) � !(b) for all states thena � b. In this way veri�cation of inequalities in a C� algebra is reduced to theveri�cation of numerical inequalities. Also recall that in any real C� -algebrathe following important inequality holds [4]

!(a�b�ba) � jjb�bjj!(a�a)

For any given state ! de�ne (s; t)! = !(hs; ti). It is evident that ( ; )! isa pseudo inner product on V . It therefore satisfy the Cauch-Swartz inequality(s; t)2! � (s; s)!(t; t)!. De�ne a = hs; ti. We clearly have !(aa�) = !(aht; si) =!(hat; si) = (at; s)!. Therefore

!(aa�) � [(at; at)!(s; s)!]12

= [!(aht; tia�)(s; s)!]12

= [!(ajtj2a�)(s; s)!]12

� jjht; tijj 12!(aa�) 12!(hs; si) 12

Dividing by !(aa�)12 we �nd

!(aa�)12 � jjtjj!(hs; si) 12 = !(jjtjjhs; si)

The �rst inequality now follows since this numerical inequality holds for allstates !. As for the second inequality recal that in a real C� algebra we havejjaa�jj = jjajj2 and for any pair of operators 0 � a � b we have jjajj � jjbjj.Using this we have

jjhs; tijj2 = jjhs; tihs; ti�jj = jjhs; tiht; sijj � jj jsj2jjtjj2jj = jjsjj2jjtj2

and this proves the second inequality.From the second inequality we can in the usual way conclude that the triangle

inequality holds for jj jj.

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Corollary 8 jj jj is a pseudo norm on V .

Let N be the subset of elements in V of pseudonorm zero.

N = fs j jjsjj = 0g

For any operator a 2 O(H) and a pair of elements s and t in N we now have

jjasjj2 = jjhas; asijj = jjahs; sia�jj � jjajj jjsjj2 jja�jj = 0jjs+ tjj � jjsjj+ jjtjj = 0

Thus N is a submodule and we can therefore de�ne a quotient moduleeH = V=N

Elements in eH are equivalent classes of simple operator valued functionsdenoted by [s]. Note that for any elements [s]; [t] 2 eH with [s] = 0 we have

jjhs; tijj � jjsjj jjtjj = 0

an as a consequence of this hs; ti = 0. We therefore have a well de�nedoperator valued product on eH de�ned through

h[s]; [t]i = hs; ti

This product enjoy the same properties as the product on V and is in ad-dition positive de�nit. Thus eH with this product is a pre-Hilbert module witha norm jj jj de�ned on the underlying real vectorspace. In general this vec-torspace is not complete with respect to the norm. We can however completethe vectorspace with respect to the norm. The resulting structure is a Hilbertmodule over the real C�-algebra O(H). We will call it the Hilbert module cor-responding to the extended probability space h;B(�); F i. With the analogywith Hilbert spaces in mind we will consider h';'i to the the square length of'. Note that for a general Hilbert module the length is a positive operator, nota positive number. Also note that in order to simplify the notation we use thesame symbol jj jj for the norm on H and for the operator norm on O(H). Thisis the sense of the formula jj'jj2 = jjh';'ijj.We have now made sense of equation (1). It just state that ' should be a

element in the Hilbert module H of length 1.We will next proceed to make sense of equation (2). Note that what we do

is in fact to prove the analog of the easy part of the classical Radon-Nikodymtheorem.For any U 2 B(�) de�ne a map PU : V ! V by

PU (s) =Xi

si�Vi\U

This map is clearly a O(H) module morphis ·m. The following properties ofPU are easily veri�ed.

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Proposition 9

PU � PU = PU

PU (as) = aPU (s) 8a 2 O(H)PU\V = PU � PV

hPU (s); ti = hs; PU (t)ihs; PU (s)i � 0

PV + PW = PV [W V \W = ;hPU (s); PU (s)i � hs; si

jjPU (s)jj � jjsjj

The last property show that if jjsjj = 0 then jjPU (s)jj = 0. Therefore PUinduce a well de�ned map, also denoted by PU , on eH through

PU ([s]) = [PU (s)]

The last property show that the map PU is bounded on eH. It thereforeextends to a unique bounded linear map on H. This map clearly also enjoy theproperties listed in the previous proposition.Let now ' be a element in the Hilbert module H of unit length h';'i = 1:

For each set U 2 B(�) de�ne a operator E'(U) on the Hilbert space H by

E'(U) = h'; PU (')i

Clearly E'() = 1 and E'(U) � 0 for all U . It is also evident from theprevious proposition that E' is �nitely additive on disjoint sets. It is in factalso contably additive as we now show.

Theorem 10 E' : B(�)! O(H) is a positive operator valued measure

Proof. Let �rst s =X

isi�Vi be a element in V with hs; si = 1 and let fTjg

be a increasing sequence of sets with limit T = [jTj . The set of operatorsfEs(Tj)g is a increasing sequence of positive operators. The supremum of thissequence exists [1]. Denote the supremum by SupfEs(Tj)g. In order to showthat Es is a positive operator valued measure we only need to show that

Es([jTj) = SupfEs(Tj)g

It is a fact [1] that the sequence Es(Tj) converges strongly to the limitSupfEs(Tj)g. Since the strong limit is unique when it exists we must only showthat Es(Tj)(x) ! Es([jTj)(x) for all elements x 2 H. We know that F is apositive operator valued measure so F (Tj \Vi)! F (T \Vi) strongly. But then

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since all si are bounded operators we have for all x 2 H

siF (Tj \ Vi)s�i (x) ! siF (T \ Vi)s�i (x)+X

i

siF (Tj \ Vi)s�i (x) !Xi

siF (T \ Vi)s�i (x)

+Es(Tj)(x) ! Es(T )(x)

This proves that Es is a POV. Next for any element [s] in eH we de�neE[s](U) = h[s]; PU ([s])i. It is trivial to verify that E[s] = Es so that the previousproof show that E[s] is a POV. Finally let ' be a arbitrary element in H. Thenthere exists a sequence of elements [sn] in H such that [sn]! '. Since E[sn] isa POV we know that for all x 2 H

�nx(U) = hE[sn](U)x; xiH

is a measure. Let �x be the positive set function de�ned by

�x(U) = hE'(U)x; xiH

By continuity we know that E[sn](U)! E'(U) in the uniform norm and thusstrongly. But then by continuity of the inner product on H we can concludethat

limn!1

�nx(U) = �X(U)

for all sets U 2 B(�). This implies through the Vitali-Hahn-Saks teorem [5]that �x is a measure and then it follows [1] that E' is a POV.We have now made sense of equation (2) and are now ready to de�ne the

symbolic expressions occuring in equation (1) and (2).

De�nition 11 Z'dF � = h'; iZ

V

'dF'� = h'; PV (')i

We have thus found that probability densities for operator valued measuresare not functions but elements in a Hilbert module. They should in fact not bethought of as densities but as half densities, their square is a density in the abovesense. This is a startling conclusion. Half densities are however not unfamiliarto anyone that has ben exposed to quantum mechanics. Wavefunctions arehalf densities. In fact wavefunctions appear naturally in this scheeme. If Fis a positive operator valued measure acting on a real two dimensional Hilbertspace we are lead to de�ne densities as functions whose values are operatorson the plane. The complex numbers are isomorphic to a special subalgebra of

14

operators on the plane (the conformal operators). Thus a large class of densitiescan be identi�ed with complex valued functions of length one. Since selfadjointoperators are now naturally identi�ed with real numbers the length can beconsidered to be a number. What we are describing are of course wavefunctions.Thus densities for positive operator valued measures acting on a twodimensionalplane are wavefunctions.

4.2 Random operators

Recall [2] that a map A : H ! H is said to be adjointable if there exists a mapdenoted by A� : H ! H such that

hA�'; i = h';A i

for all elements ' and in H. A map is selfadjoint if A� = A. It followsdirectly from the algebraic properties of the inner product and the completnessof the underlying real vector space that any adjointable map is a bounded O(H)module morphism. In fact the set of all adjointable maps form a abstractreal C�-algebra that we denote by A. We will call the elements in A randomoperators. The expectation of a random operator A with respect to a density 'is by de�nition given by

hAi = h';A'i

The expectation of a random operator with respect to a density ' is thus aoperator on H. We can also use the density to de�ne a POV acting on H aswe have seen. Note that the expectation of selfadjoint random operators is aselfadjoint operator in O(H).Returning to the two dimensional example discussed above we see that in

that case for complex valued densities the expectation of selfadjoint random op-erators can be identi�ed with real numbers and thus the expectation of randomoperators can be thought of as numbers. In higher dimensions and for moregeneral densities no such identi�cation with real numbers is possible. Further-more no such reduction should be expected. After all, the selfadjoint elementsin a real C�-algebra are the analog of real numbers.Let us assume that the real Hilbert space underlying the extended probability

space X is one dimensional. If we choose a basis we can identify the Hilbertspace with R and the Hilbert module HX with the real Hilbert space of squareintegrable functions on R. A positive operator valued measure is through thebasis identi�ed with a probability measure and therefore for a half density ' 2HX the formula E(V ) = h'; PV 'i turns into

�(V ) =

Z'2d�

The half density ' is of course not uniquely determined by the probabilitymeasures � and � unless we by convention always take the positive square root.If all our observables are random vectors then it does not matter which half

15

density we choose, they will all produce the same expectation. Thus by restrict-ing to random vectors as our observables the di¤erence between the varioushalf densities ' are not obervable. However there is really no rational reasonto restrict to this class of observables. If we include random operators in ourobservables the di¤erence between the half densities are readily observable.

5 The category of extended probability spaces

In classical probability theory the notion of morphisms of probability spacesplays a role at least as important as the notion of a probability space. In factfrom the Categorical point of view morphisms are the most important elementin any theory construction. All other entities should be de�ned in terms of themorphisms. In this section we review the notion of a morphism in the contextof probability spaces and then de�ne the corresponding notion for extendedprobability spaces. The naturalness of our de�nition is veri�ed by proving thatextended probability spaces and morphisms forms a category. We also show thatjust as for the case of probability spaces we get a functor mapping the categoryof extended probability spaces into the category of Hilbert spaces. The existenceof this functor is a veri�cation of the naturalness of our constructions.Let X = hX ;B(�X); �Xi and Y = hY ;B(�Y ); �Y i be probability spaces.

A morphism f : X ! Y is a measurable map f : X ! Y such that �Y isabsolutely continous with respect to the push forward of the measure �X by f ,�Y � f��X . By the Radon-Nikodym theorem this means that there exists aprobability density � : Y ! R such that

�Y (V ) =

Zf�1(V )

�d�X

There are several other possibilities for morphisms of probability spaces. Wecould have required f��X � �Y or f��X � �Y . They can all be composed andlead to a category structure. However the only possibility that generalize wellto extended probability spaces is the �rst one �Y � f��X .

5.1 Morphisms of extended probability spaces

In this section we will introduce the notion of mapping between extended prob-ability spaces and will then use mappings to de�ne morphisms. This distinctionbetween mappings and morphisms does not exist for probability spaces.In order to de�ne what a mapping is in the context of extended probability

spaces, we must �rst generalize the notions of absolute contiuity and push for-ward to positive operator valued measures. We will do this by combining theminto a single entity.

De�nition 12 Let X = hX ;B(�X); FXi be a extended probability space, Y =hY ;B(�Y )i a measurable space and h the 3 tuple h = hfh; gh; ' hi where fh :

16

X ! Y is a measurable map,gh : HY ! HX is a isometry and 'h 2 HX is aelement in the Hilbert module corresponding to X. Then the push forward of FXby h is the positive operator valued measure,h�FX , de�ned on the measurablespace Y by

h�FX(V ) = g�h � h'h; Pf�1h (V )'hi � gh

where g�h is the adjoint of gh.

Note that we have g�h = g�1h � Qh where Qh is the orthogonal projectiononto the closed subspace gh(HY ) � HX and therefore g�h � gh = 1 and gh � g�h =Qh.We can now de�ne mappings between extended probability spaces usingpush forward in a very simple way.

De�nition 13 Let X = hX ;B(�X); FXi and Y = hY ;B(�Y ); FY i be ex-tended probability spaces. A mapping h : X ! Y is a 3 tuple,h, as in theprevious de�nition such that

h�FX = FY

Let us assume that the real Hilbert spaces underlying the extended probabil-ity spaces X and Y are one dimensional. If we choose basis for these two spaceswe can identify the Hilbert spaces with R, the positive operator valued measureswith probability measures � and � and the half density ' with a real valuedfunction on X . We must have gh = 1 and the condition for h = hfh; 1; 'hi tobe a mapping is

�(V ) =

Zf�1h (V )

'2hd�

This is of course the condition for fh to be a mapping between the probabilityspaces hX ;B(�X); �i and hX ;B(�X); �i if we identify the classical densitywith '2h.Our �rst goal is to show that the proposed mappings can be composed.

In order to do this we must �rst de�ne a certain pullback of half densitiesinduced by a mapping. Let therefore mappings h : X ! Y and k : Y ! Zof extended probability spaces be given. Let us �rst de�ne a measurable mapfk�h, a isometry gk�h and a linear map h� by

fk�h = fk � fh : X ! Z

gk�h = gh � gk : HZ ! HX

h�(a) = gh � a � g�h : O(HY )! O(HX)

The map h� has the following easily veri�able properties

Proposition 14 The map h� is bounded and

h�(a+ b) = h�(a) + h�(b)

h�(ab) = h�(a)h�(b)

17

De�ne a linear map h� : VY ! HX by

h�(s) =Xj

h�(sj)Pf�1h (Vj)('h)

where s =Psj�Vj . The map h

� has the following important properties

Proposition 15 The map h� is bounded and

h�(s+ t) = h�(s) + h�(t)

h�(as) = h�(a)h�(s)

hh�(s); h�(t)i = h�(hs; ti)[s] = 0 =) [h�(s)] = 0

h�(PV (s)) = Pf�1h (V )(h�(s))

Proof. Let s =Psi�Vi and t =

Ptj�Wj

. Then it is easy to verify thatfVi \Wjg form a partition of Y and that s+ t =

P(si + tj)�Vi\Wj

. But thenwe have

h�(s+ t) =Xi;j

�h�(si + tj)Pf�1h (Vi\Wj)('h)

=Xi;j

h�(si)Pf�1h (Vi)\f�1h (Wj)('h) +

Xi;j

h�(tj)Pf�1h (Vi)\f�1h (Wj)('h)

=Xi

h�(si)Pf�1h (Vi)('h) +

Xj

h�(tj)Pf�1h (Wj)('h) = h�(s) + h�(t)

This proves the second statement. For the third statement we have

h�(as) = h�(Xi

asi�Vi) =Xi

h�(asi)Pf�1h (Vi)('h)

=X

h�(a)h�(si)Pf�1h (Vi)('h) = h�(a)h�(s)

18

and

hh�(s); h�(t)i =Xi;j

hh�(si)Pf�1h (Vi)('h); h

�(tj)Pf�1h (Wj)('h)i

=Xi;j

h�(si) � h'h; Pf�1h (Vi\Wj)('h)i � h�(tj)�

= gh �

0@Xi;j

si � g�h � h'h; Pf�1h (Vi\Wj)('h)i � gh � t�j

1A � g�h= gh �

0@Xi;j

si � h�FX(Vi \Wj) � t�j

1A � g�h= gh �

0@Xi;j

si � FY (Vi \Wj) � t�j

1A � g�h= gh � hs; ti � g�h = h�(hs; ti)

proves the fourth statement. The �rst and last statement in the propositionfollows from the fourth. Finally

h�(PV (s)) = h�(Xi

si�V \Vi) =Xi

h�(si)Pf�1h (V \Vi)(')

=Xi

h�(si)Pf�1h (V )(Pf�1h (Vi)('))) = Pf�1h (V )(h

�(s))

Using this proposition we can extend the map h� to a continous linear mapfrom HY to HX . This map is given on the dense setgHY by

h�([s]) = h�(s)

All the properties in the proposition holds for the extension. We are nowready to prove that our mappings can be composed

Theorem 16 Let h : X ! Y and k : Y ! Z be mappings of extended probabil-ity spaces. De�ne 'k�h 2 HX by 'k�h = h�('k). Then

k � h = hfk�h; gk�h; 'k�hi

is a mapping of extended probability spaces k � h : X ! Z and we have

(k � h)� = h� � k�

Proof. In order to show that k�h is a mapping we must prove that (k�h)�FX =FZ . But doing this is now a straight forward calculation if we use the previous

19

proposition.

(k � h)�FX(V )= g�k�h � h'k�h; Pf�1k�h(V )

('k�h)i � gk�h= g�k � g�h � hh�('k); Pf�1h (f�1k (V ))(h

�('k))i � gh � gk= g�k � g�h � hh�('k); h�(Pf�1k (V )('k))i � gh � gk= g�k � g�h � gh � h'kPf�1k (V )('k)i � g

�h � gh � gk

= g�k � h'kPf�1k (V )('k)i � gk = FZ(V )

The last statement in the theorem is also proved by direct calculation. Lets =

Psj�Vj 2 VZ . Then we have

(k � h)�([s])=

Xj

(k � h)�(sj)Pf�1k�h(Vj)('k�h)

=Xj

h�(k�(sj))Pf�1h (f�1k (Vj))(h�('k))

=Xj

h�(k�(sj))h�(Pf�1k (Vj)

('k))

= h�(Xj

k�(sj)Pf�1k (Vj)('k)) = h�(k�(s))

Since the identity holds on a dense subset is also holds for all elements inHZ and this proves the theorem.We now can use this Theorem to de�ne composition of mappings

De�nition 17 Let h : X ! Y and k : Y ! Z be mappings of extended proba-bility spaces. Then k � h is the composition of k and h.

It is now straight forward to prove that composition of mappings is associa-tive.

Theorem 18 Let h : X ! Y , k : Y ! Z and r : Z ! T be mappings ofextended probability spaces. Then we have

r � (k � h) = (r � k) � h

Proof. Clearly we have fr�(k�h) = f(r�k)�h and gr�(k�h) = g(r�k)�h. And fromthe previous Theorem we have

'r�(k�h) = (k � h)�('r) = h�(k�('r))

'(r�k)�h = h�('r�k) = h�(k�('r))

20

Extended probability spaces and mappings of extended probability spacesdoes unfortunately not form a category, we will in general not have unit mor-phisms.For a given extended probability space X = hX ;B(�X); FXi the only rea-

sonable candidat for a unit morphism is

1X = h1X ; 1HX; 1HX

�X i

For this mapping it is easy to show that

Proposition 19

k � 1X = k

1Y � h = hfh; gh; Qh'hi

Thus the mapping is not a unit morphism in the categorical sense unless ghis a isomorphism. It is for this reason that we distinguise between mappingsand the yet to be de�ned morphisms. Morphisms will be de�ned in terms of aequivalence relation on mappings.Recall that for any mapping h : X ! Y , Qh : HX ! gh(HY ) is the

orthogonal projection on the closed subspace gh(HY ).

De�nition 20 Two mappings h; k : X ! Y of extended probability spaces areequivalent if

fh = fk

gh = gk

Qh'h = Qk'k

If h and k are equivalent we will write h � k.

The de�ned relation is a equivalence relation. In order to de�ne morphismswe must show that composition of mappings extends to equivalence classes ofmappings. For this we need the following two lemmas.

Lemma 21 Let h : X ! Y and k : Y ! Z be mappings of extended probabilityspaces. Then

Qk�h = h�(Qk)

Proof. For any � 2 HX , Qk�h(�) is the unique vector in gh(gk(HZ)) suchthat � � Qk�h(�) is orthogonal to gh(gk(HZ)). But for any � = gh(gk(�)) ingh(gk(HZ)) we have

h� � h�(Qk)(�); �i= h� � (gh �Qk � g�h)(�); gh(gk(�)))i= hg�k(g�h(�))� (g�k � g�h � gh �Qk � g�h)(�); �i= hg�k(g�h(�))� g�k(g�h(�)); �i = 0

Therefore by uniqueness Qk�h(�) = h�(Qk)(�).

21

Lemma 22 Let h; h0 : X ! Y be equivalent. Then

h� = h0�

Proof. We only need to verify the identity on the dense subsetgHX � HX . Butfor any [s] 2gHX with s =

Psi�Vi we have

h0�([s]) =Xi

h0�(si)Pfh0 (Vi)('h0)

=Xi

(gh0 � si � g�1h0 �Qh0)Pfh0 (Vi)('h0)

=Xi

(gh � si � g�1h )Qh0Pfh(Vi)('h0)

=Xi

(gh � si � g�1h )Pfh(Vi)(Qh0'h0)

=Xi

(gh � si � g�1h )Pfh(Vi)(Qh'h) = h�([s])

We can now prove that composition is well de�ned on classes.

Proposition 23 Let h; h0 : X ! Y be equivalent and k; k0 : Y ! Z be equiva-lent. Then

k � h � k0 � h0

Proof. We only need to prove that 'k�h = 'k0�h0 . But using the previous twolemmas we have

Qk�h'k�h = h�(Qk)h�('k) = h�(Qk'k) = h0�(Qk0'k0) = Qk0�h0('k0�h0)

De�nition 24 A morphism between extended probability spaces X and Y is aequivalence class, [h];of mappings h : X ! Y .

In order to keep the notation simple we will always denote a morphism [h] bya representative mapping h. Thus when we speak of a morphism h we mean theclass [h]. The meaning will always be clear, we just have to make sure that anyoperations involving morphisms does not depend on choise of representative.We can now formulate the main result of this subsection.

Theorem 25 Extended probability spaces and morphisms form a category.

Proof. We know that composition is well de�ned and associative. For anyobject X; let the unit mapping be 1X = h1X ; 1Hx

; 1HX�X i. From proposition

19 we have for any morphisms h : X ! Y

h � 1X � h

1Y � h = hfh; gh; Qh'hi � h

22

because Qh is a projection.We know that the category of probability spaces[11] has a terminal object,

T ,in the categorical sense, there is a unique morphism from any probabilityspace X to T . Here T = hT ;BT ; �T i with T = f�g , BT = f;; f�gg and�T the only possible probability measure on BT . The existense of T makes itpossible to de�ne points in probability spaces categorical. We will now see thatthe category of extended probability spaces does not have a terminal object andthus extended probability spaces will not have points in the categorical sense,only generalized points. The only possible candidate for a terminal object in thecategory of extended probability spaces is the object T = hT ;BT ; FT i whereFT : BT ! O(R) � R is the only possible positive operator valued measure,FT (T ) = 1R. We will now show that T is in fact not a terminal object.Let h : X ! T be any morphism of extended probability spaces. We have

h = hfh; gh; 'hi and clearly fh : X ! T = f�g is unique. The map gh :R! HX is a isometry and is therefore determined by a vector �h 2 HX whereh�h; �hi = 1 and gh(1) = �h. The vector �h and element 'h 2 HX must satis�ethe single condition

h�FX(T ) = FT (T ) = 1R

Using the de�nition of h� we �nd that the following identity must be satis�ed

hh'h; 'hi(�h); �hi = 1

and clearly this identity will be satis�ed by many choises of 'h and �h. Thusthe morphism h is not uniquely determined and therefore T is not a terminalobject.

5.2 The Naimark functor

In probability theory there is a certain functor that plays a major role in the the-ory. We will now review the construction of this functor and show that a analogfunctor is de�ned on the category of extended probability spaces. The existenseof this functor testify to the naturalness of our constructions. The functor willbe called the Naimark functor since the Naimark dilitation construction playsa major role in its construction.Let us start with a review of the functor for the case of probability spaces. For

any probability space X = hX ;B(�X); �Xi de�ne a Hilbert space,denoted byL2(X), by L2(X) = L2(�X). LetX = hX ;B(�X); �Xi and Y = hY ;B(�Y ); �Y ibe two probability spaces and let f : X ! Y be a morphism of probabilityspaces in the sense that

�Y (V ) =

Zf�1(V )

�d�X

De�ne a mapping L2(f) : L2(Y )! L2(X) by

L2(f)(�) =p�(� � f)

23

It is easy to verify,using the Radon Nikodym theorem, that L2(f) is in facta isometry and moreover that L2 is a functor from the category of probabilityspaces to the category of Hilbert spaces. We will now show that it is possible tode�ne a functor, also denoted by L2, from the category of extended probabilityspaces to the category of Hilbert spaces that for probability spaces reduce tothe functor discussed above.Let X and Y be extended probability spaces and let L2(X) and L2(Y ) be the

corresponding Hilbert spaces of random vectors. Informally to any morphismh : X ! Y of extended probability spaces we will de�ne a isometry L2(h) :L2(Y )! L2(X) by the formula

L2(h)(�)(x) = '�h(x)(gh((� � fh)(x)))

It is easy to see that the mapping L2(f) is a special case of this generalformula. Of course we can not use this formula to actually de�ne L2(h) sinceelements in L2(Y ) are not vector functions and elements in HX are not operatorvalued functions. The action of elements inHX on L2(X) implied by the formulamust also be made sense of and since morphisms are classes of mappings we needto prove independence of representative.. We will now prove that the map L2(h)exists and that it de�nes a functor.Recall that if SY denote the space of simple HY valued functions with inner

product hv; wi =P

i;jhFY (Vi \ Tj)�i; �jiHYthen L2(Y ) is the closure of TY =

f[v] j v 2 SY g where [v] = 0 i¤ hv; vi = 0. For any extended probabilityspace, VX is the linear space of simple operator valued functions occuring in theconstruction of the Hilbert module HX . For a measurable map f : X ! Y ,aisometry g : HY ! HX and a element v =

Pi �i�Vi 2 SY de�ne a linear map

tf;gv : VX ! L2(X) by

tf;gv (s) = [Xi;j

s�j (g(�i))�f�1(Vi)\Wj]

where s =P

j sj�Wj2 VX .The linear map tf;gv enjoy the following important

property

Lemma 26htf;gv (s); tf;gv (s)i � cv;gjjsjj2

Proof. Let v =P

i �i�Vi and s =P

j sj�Wj. Then we have

htf;gv (s); tf;gv (s)i=

Xi;j

hFX(Wj \ f�1(Vi))s�j (g(�i)); s�j (g(�i))iHX

=Xi;j

h(sjFX(Wj \ f�1(Vi))s�j )(g(� i)); g(�i)iHX

=Xi

hhs; Pf�1(Vi)(s)i(g(� i)); g(�i)iHX

�Xi

hhs; si(g(� i)); g(�i)iHX� cv;gjjsjj2

24

where in the last line we used Cauch-Swartz and the de�nition of the normin the Hilbert module.This lemma implies that if [s] = 0 then [tf;gv (s)] = 0 and therefore we can

extend tf;gv to a bounded linear map tf;gv : HX ! L2(X). It is de�ned on thedense subset gHX by tf;gv ([s]) = [tf;gv (s)].The following proposition sets the stage for proving the existence of the

Naimark functor.

Proposition 27 Let h : X ! Y be a mapping of extended probability spaces.Then there exists a isometry L2(h) : L2(Y ) ! L2(X) that is de�ned on thedense subset TY by

L2(h)([v]) = tfh;ghv ('h)

and that satisfy

L2(k � h) = L2(h) � L2(k)L2(1X) = 1L2(X)

Proof. We will start by showing that tfh;ghv only depends on the class of v.Let fsng be a sequence of elements in HX converging to 'h. For each n we cande�ne a positive operator valued measure on hY ;B(�Y )i acting on the Hilbertspace HY by FnY (V ) = g��hsn; Pf�1(V )(sn)i�g. By continuity FnY (V )! FY (V )strongly and thus weakly. But then we have

htfh;ghv ('h); tfh;ghv ('h)i

= limn!1

htfh;ghv (sn); tfh;ghv (sn)i

= limn!1

Xi

hhsn; Pf�1(Vi)(sn)i(g(� i)); g(�i)iHX

= limn!1

Xi

h(g� � hsn; Pf�1(Vi)(sn)i � g)(� i)); �iiHY

= limn!1

Xi

hFnY (Vi)(�i); �iiHY

=Xi

hFY (Vi)�i; �iiHY= hv; vi

And the assumption [v] = 0 means that hv; vi = 0, so tfh;ghv depends onlyon the class of v. Therefore L2(h) is well de�ned on the dense subset TY andthe argument just given show that it is a isometry. It therefore extends to aisometry from L2(Y ) to L2(X).For the last part of the Theorem let [sn] and [tm] be sequences in HX and

HY converging to 'h and 'k. Here sn =P

l snl�Wnland tm =

Pj tmj�Tmj

. For[v] 2 TZ � L2(Z) with v =

Pi �i�Vi we have by continuity of all maps involved

that if we de�ne [u] 2 TY � L2(Y ) by um =X

i;jt�mj(gk(�i))�f�1k (Vi)\Tmj

then

25

we have

L2(h) � L2(k)([v])= L2(h)(t

fk;gkv ('k)) = L2(h)(t

fk;gkv ( lim

m!1[tm]))

= limm!1

L2(h)(tfk;gkv ([tm]))

= limm!1

L2(h)(Xi;j

t�mj(gk(�i))�f�1k (Vi)\Tmj) = lim

m!1L2(h)([um])

= limm!1

tfh;ghum ('h) = limm!1

limn!1

tfh;ghum ([sn])

= limm!1

limn!1

Xi;j;l

(s�nl � gh � t�mj � gk)(�i)�f�1h (f�1k (Vi)\Tmj)\Wnl

Note that

h�([tm])

=Xj

h�(tmj)Pf�1h (Tmj)('h)

=Xj

h�(tmj)Pf�1h (Tmj)( limn!1

[sn])

= limn!1

Xj;l

h�(tmj)snl�f�1h (Tmj)\Wnl

we have

L2(k � h)([v]) = tfk�h;gk�hv ('k�h) = tfk�h;gk�hv (h�('k))

= tfk�h;gk�hv (h�( limm!1

[tm])) = limm!1

tfk�h;gk�hv (h�([tm]))

= limm!1

tfk�h;gk�hv ( limn!1

Xj;l

h�(tmj)snl�f�1h (Tmj)\Wnl)

= limm!1

limn!1

Xi;j;l

(h�(tmj)snl)�(gk�h(�i))�f�1k�h(Vi)\f

�1h (Tmj)\Wnl

= limm!1

limn!1

Xi;j;l

(s�nl � gh � t�mj � g�h � gh � gk)(�i)�f�1h (f�1k (Vi)\Tmj)\Wnl

= limm!1

limn!1

Xi;j;l

(s�nl � gh � t�mj � gk)(�i)�f�1h (f�1k (Vi)\Tmj)\Wnl

The last statement of the Theorem is veri�ed by a trivial calculation.We are now �nally ready to prove the existense of the Naimark functor.

Theorem 28 L2(h) is a well de�ned functor from the category of extendedprobability spaces to the category of Hilbert spaces.

Proof. We only need to prove that L2(h) is well de�ned for a given morphismh. The functorial properties follows from the previous proposition. Assume

26

h � h0. Let us �rst assume that the densities of h and h0 are [s] and [s0]. Wecan without loss of generality assume that s and s0 are of the form

s =Xi

si�Wi

s0 =Xi

s0i�Wi

since we can bring it to this form by the same construction as in lemma 30.The equivalence then amounts to Qhsi = Qh0s

0i for all i. Then on the dense

subset TY � L2(Y ) we have for v =P�i�Vi that

L2(h)([v]) =Xi;j

s�j (gh(�i))�f�1h (Vi)\Wj

=Xi;j

(s�j �Qh � gh)(�i)�f�1h (Vi)\Wj

=Xi;j

(s0�j �Qh0 � gh0)(�i)�f�1h0 (Vi)\Wj

= L2(h0)([v])

The case for general densities follows by continuity.The Naimark functor L2 is not the only functor occuring in this theory. In

fact if we recall the properties of the pullback operation h! h� de�ned earlierin this section we can de�ne a second functor.

Theorem 29 For any extended probability space X, de�ne a Hilbert moduleH(X) = HX and for any morphism h : X ! Y of extended probability spacesde�ne a morphism of Hilbert modules H(h) = h�. Then H is a functor from thecategory of extended probability spaces to the category of Hilbert modules.

For the case of probability spaces the Hilbert module H(X) and the spaceof random vectors L2(X) are both isomorphic to the Hilbert space of squareintegrable real valued function. This is why random variables and densitiesappear to be taken from the same space in probability theory. But this is a veryspecial situation. If the underlying Hilbert space is not one dimensional buttwo dimensional the densities and random vectors start to reveal their di¤erentnature. As we have discussed previously for this case a important subclass ofdensities are the one whose values are contained in the conformal group of theplane. These densities form a sub-Hilbert module that is actually a isomorphicto the complex Hilbert space of complex valued functions.

6 Monoidal structure on the category of extendedprobability spaces

In probability theory the notion of product measures and product densitiesplay a major role. It is through these that dependence and independence for

27

random variables are de�ned. From a categorical point of view the situation issummarized by saying that the category of probability spaces supports a monidalstructure. We will now show that the category of extended probability spacesalso supports a monoidal structures and that as a consequence the notions ofdependence and independence can be de�ned.Let us start by reviewing the notion of a monoidal structure for a category.

A monoidal structure in a category is basically a product in the category that isassociative up to natural isomorphism and has a unit object up to natural iso-morphism. What this means is that ifX,Y and Z are objects in the category andif the product is denoted by then we require that there exists a isomorphism�XY Z : X (Y Z)! (XY )Z. Similarly if I is the unit object we requirethat there exists isomorphisms �X : IX ! X and X : X I ! X. The iso-morphisms can not be arbitrarily choosen for di¤erent objects, they must formthe components of a natural transformation. In addition they must satis�e aset of equations known as the MacLane Coherence Conditions. These equationsensure that associativity and unit isomorphisms can be extended consistentlyto products of �nitely many objects. The conditions that must be satis�ed by�, and � are the following.For all objects X,Y ,Z and T we must have

�XY;Z;T � �X;Y;ZT = (�X;Y;Z 1T ) � �X;YZ;T � (1X �Y;Z;T )( X 1Y ) � �X;I;Y = 1X �Y

I = �I

These are the MacLane coherence conditions. The naturality conditions areexpressed as follows. For any arrows f : X ! X 0,g : Y ! Y 0 and h : Z ! Z 0

we must have

((f g) h) � �X;Y;Z = (f (g h)) � �X0;Y 0;Z0

f � �X = �X0 � (1I f)f � X = X0 � (f 1I)

In general such equations are di¢ cult to solve, there is a very large numberof variables and equations. However in some simple situations the naturalityconditions can be used to reduce the system of equations to a much smaller set.The reader not familiar with categories,natural transformations and Coher-

ence conditions might want to consult the book [8] for a elementary introductionto the categorical view of mathematics, a more advanced introduction can befound in the book [9]The notion of product measures in probability theory has of course been

known for a long time. The corresponding monoidal structure in the category ofprobability spaces is described in detail in [11]. The main features are as follows.For two probability spaces X = hX ;B(�X); �Xi and Y = hY ;B(�Y ); �Y itheir product is the probability space X Y = h;X � Y ;B(�X �Y ); �X �Y i, where �X �Y is the product measure. The product of two morphismsf : X ! Y amd g : X 0 ! Y 0 is a morphism f g : X X 0 ! Y Y 0

28

where f g = f � g is just the cartesian product of the maps f and g. Theassociativity and unit isomorphisms are just the usual one from the category ofsets. �XY Z((x; (y; z))) = ((x; y); z); �X((�; x)) = x and X((x; �)) = x. For thecategory of probability spaces this choise of �, � and are the only possibleones as we show in [11]. The unit object for the monoidal structure is the trivial,one point probability space.

6.1 Product of extended probability spaces and morphisms

We will now de�ne the product of extended probability spaces and morphismsand show that this product is a bifunctor on the category of extended probabilityspaces.Let X = hX ;B(�X); FXi and Y = hY ;B(�Y ); FY i be two extended prob-

ability spaces. The product of the two positive operator valued measures FXand FY always exists and is uniquely determined [1] by its value on measurableboxes by

(FX FY )(C �D) = FX(D) FY (D)

The product measure acts on the Hilbert space HX HY . The tensorproduct is the Hilbert tensor product. We now need to extend the product tomorphisms and show that it is a bifunctor. Before we do this we must specifythe relationship between the Hilbert modules HX HY and HXY . We willshow that, as expected, we can map the �rst into the second using a continousinjective module morphism. We will start by constructing this morphism.Recall that for any extended probability space X, HX is the completion of

the dense subspace gHX = f[s] j s 2 VXg and

VX = fs =Xi

si�Vi jsi 2 O(HX); fVig is a B(�X) measurable partition of Xg

is the real linear space of simple O(HX) valued measurable functions on X .For a pair of extended probability spaces de�ne a map XY : VX � VY !by

XY (s; t) =Xi;j

(si tj)�Vi�Wj

where s =Psi�Vi and t =

Ptj�Wj

. For this map we have the following

Lemma 30 The map is bilinear and if [s] = 0 or [t] = 0 then [ (s; t)] = 0.

Proof. We evidently have XY (as; t) = XY (s; at) for all real numbers a.Let s =

Pni=1 si�Vi and r =

Pmk=1 rk�Ck be two elements in VX . De�ne a

new sequence of sets fAlg where Al = Vl for l = 1::n and Al = Cl�n forl = n+1; ::::n+m and let L = f1; 2; :::n+mg. Let S = f� : L! Z2g be the setof all Z2 = f�1;+1g valued functions on the index set L. The set S is a indexset for a new partition, fT �g�2S of the set X de�ned by

T � = \l2LA�(l)l

29

where for any set U we de�ne U+1 = U and U�1 = U c, the complement ofU . We evidently have

Vi = [f�j�(i)=1gT �

Ck = [f�j�(n+k)=1gT �

Therefore

s+ r =X�

0@ Xfij�(i)=1g

si +X

fkj�(k+n)=1g

rk

1A �T�

But then we have for any t =Ptj�Wj 2 VY that

XY (s+ r; t)

=X�;j

0@0@ Xfij�(i)=1g

si +X

fkj�(k+n)=1g

rk

1A tj1A �T��Wj

=X�;j

Xfij�(i)=1g

(si tj)�T��Wj+X�;j

Xfkj�(k+n)=1g

(rk tj)�T��Wj

=Xi;j

(si tj)X

f�j�(i)=1g

�T��Wj+Xk;j

(rk tj)X

f�j�(n+k)=1g

�T��Wj

=Xi;j

(si tj)�Vi�Wj+Xk;j

(rk tj)�Ck�Wj= XY (s; t) + XY (r; t)

This show that is bilinear. For the second part of the statement in thelemma we have

h XY (s; t); XY (s; t)i=

Xi;j;k;l

(si tj)FXY ((Vi �Wj) \ (Vk �Wl))(sk tl)�

=Xi;j;k;l

(si tj)(FX(Vi \ Vk) FY (Wj \Wl))(s�k t�l )

=Xi;j

(si tj)(FX(Vi) FY (Wj))(s�i t�j )

=

Xi

siFX(Vi)s�i

!

0@Xj

tjFY (Wj)t�j

1A = hs; si ht; ti

But [s] = 0 implies that hs; si = 0 and the identity just derived then impliesthat h XY (s; t); XY (s; t)i = 0 and therefore by de�nition [ XY (s; t)] = 0.Using the lemma we have a well linear map, also denoted by XY , fromgHX gHY to HXY

XY ([s] [t]) = [ XY (s; t)]The map XY satisfy the following important identity

30

Lemma 31h XY (v); XY (v)i = hv; vi

Proof. Any v 2gHX gHY is of the form v =X

isi ti where si =

Xjsij�Vij

and ti =X

ktik�Wik

. But then we have

h XY (v); XY (v)i=

Xi;j;k;l;m;n

(sij tik)FXY ((Vij �Wik) \ (Vlm �Wln))(slm tln)�

=X

i;j;k;l;m;n

(sij tik)(FX(Vij \ Vlm) FY (Wik \Wln))(s�lm t�ln)

=Xi;l

0@Xj;m

sijFX(Vij \ Vlm)s�lm

1A0@Xk;n

tikFY (Wik \Wln)t�ln

1A=

Xi;l

hsi; sli hti; tli =Xi;l

hsi ti; sl tli = hv; vi

We can now state and prove the main property of XY . First we will recallsome facts about (external) tensor products of Hilbert modules. Let HXHHY

denote the tensor product of HX and HY ,as real vector spaces, with topologydetermined by the norm induced from the operator valued inner product h' ;'0 0i = h';'0i h ; 0i. The completion of HX H HY is the externaltensor product [2] of the Hilbert modules HX and HY and will be denoted byHX HY . It is a module over the spatial tensor product O(HX)O(HY ) [12]of the represented C�� algebras O(HX) and O(HX).

Proposition 32 There exists a injective morphism of Hilbert modules XY :HX HY ! HXY such that

h XY (v); XY (v)i = hv; vigHX H gHY is a dense subspace of HX HY and on this dense subspace XY is given by

XY ([s] [t]) = [ XY (s; t)]

Proof. Let gHX �gHY and HX � HY be the projective tensor products [6] ofthe underlying real vector spaces. Note that the tensor product spaces have notbeen completed with respect to the projective norm. The embeddinggHX�gHY

,! HX � HY is know to exist and be dense [6]. The norm on gHX HgHY andHX H HY induced by the operator valued inner product is evidently a crossnorm and it is know that the projective norm is the largest possible cross norm.Therefore we can conclude that gHX HgHY is a dense subspace of HX HHY

and thus by completion in HX HY . By the previous lemma XY is boundedand therefore extends uniquely to a bounded map XY : HX HY ! HXY .

31

The �rst identity in the statement of the proposition follows from the previouslemma and the continuity of the operator valued inner product.In order to introduce tensor product of morphisms between extended prob-

ability spaces we need the previous proposition and the following lemma

Lemma 33 For any measurable sets C 2 B(�X) and D 2 B(�Y ) we have theidentity

XY � (PC PD) = PC�D � XY

Proof. For C 2 B(�X) and D 2 B(�Y ) we have

( XY � (PC PD))([s] [t])= XY ([PC(s)] [PD(t)])=

Xi;j

(si tj)�(Vi\C)�(Wj\D)

=Xi;j

(si tj)�(Vi�Wj)\(C�D) = PC�D( XY ([s] [t])

By continuity and density we can conclude that the identity XY � (PC PD) = PC�D � XY holds on HX HY .Let now h : X ! Y and k : X 0 ! Y 0 be morphisms of extended probability

spaces. We thus have h = hfh; gh; 'hi and k = hfk; gk; 'ki where 'h 2 HX and'k 2 HX0 . De�ne a 3 tuple h k by

h k = hfhh; ghk; 'hki

where fhk = fh � fk , ghk = gh gk and 'hk = XX0('h 'k). Thenwe have

Proposition 34 hk : XX 0 ! Y Y 0 is a morphism of extended probabilityspaces.

Proof. We need to prove that (hk)�FXX0 = FYY 0 . But this is true because

(h k)�FXX0(C �D)= g�hk � h'hk; Pf�1hk(C�D)

('hk)i � ghk= (gh gk)� � h XX0('h 'k); (Pf�1hk(C�D)

� XX0)('h 'k))i � (gh gk)

= (g�h g�k) � h XX0('h 'k); ( XX0 � (Pf�1h (C) Pf�1k (D)))('h 'k)i � (gh gk)= (g�h g�k) � h'h 'k; Pf�1h (C)('h) Pf�1k (D)('k))i � (gh gk)= (g�h � h'h; Pf�1h (C)('h)i � gh) (g

�k � h'k; Pf�1k (D)('k)i � gk)

= (h�FX)(C) (k�FX0)(D) = FY (C) FY 0(D) = FYY 0(C �D)

where we have used the previous lemma. This proves that hk is a mappingof extended probability spaces. In order to show that it is also a morphism we

32

must show that it is independent of choise of representatives. Thus assumethat h � h0 and k � k0. We need to show that h k � h0 k0 and thisamounts to proving that Qhk'hk = Qh0k0'h0k0 . But from the identity(gh gk)(HX HX0) = gh(HX) gk(HX0) we have Qhk = Qh Qk and therest of the proof is a simple calculation.Having proved that h k is a morphism our next goal is to prove that it

behaves as a functor under composition. For this we need the following lemma.

Lemma 35 XX0 � (h� k�) = (h k)� � Y Y 0

Proof. By continuity we only need to prove the identity on the dense subsetgHY H gHY 0 � HY HY 0 . But on this subset we have

((h k)� � Y Y 0)([s] [t])= (h k)�( Y Y 0(s; t))

=Xi;j

(h k)�(si tj)P(fh�fk)�1(Vi�Wj)('hk)

=Xi;j

(h�(si) k�(tj))(P(fh�fk)�1(Vi�Wj) � XX0)('h 'k)

=Xi;j

(h�(si) k�(tj))( XX0 � (Pf�1h (Vi) Pf�1k (Wj)

))('h 'k)

= XX0(Xi;j

(h�(si)Pf�1h (Vi)('h)) (k�(tj)Pf�1k (Wj)

('k)))

= ( XX0 � (h� k�))([s] [t])

We can now prove our �rst main result in this section

Theorem 36 The operation is a bifunctor on the category of extended prob-ability spaces.

(h0 k0) � (h k) = (h0 � h) (k0 � k)1X 1Y = 1XY

Proof. The unit property is trivial to verify and for the �rst identity we onlyneed to prove that XX0('k�h 'k0�h0) = (h h0)�('kk0). But using theprevious lemma we have

XX0('k�h 'k0�h0)= XX0(h�('k) h0�('k0))= ( XX0 � (h� h0�))('k 'k0)= ((h� h0�) � Y Y 0)('k 'k0)= (h� h0�)('kk0)

33

6.2 The monoidal structure

Showing that exists and is a bifunctor is the only hard part in proving thatthere is a monoidal structure on the category of extended probability spaces.The only reasonable candidate for a unit object is clearly the extended prob-

ability space T discussed previously. For any objects X;Y and Z de�ne

�X = hf�X ; g�X ; '�X i X = hf X ; g X ; ' X i

�XY Z = hf�XYZ; h�XYZ

; '�XYZi

where

f�X (�; x) = f X (x; �) = x

f�XYZ((x; (y; z)) = ((x; y); z)

g�X (�) = 1 �g X (�) = � 1

g�XYZ(� (�0 �00)) = (� �0) �00

'�X = 1HTX

' X = 1HXT

'�XYZ= 1HX(YZ)

These are obviously the simplest choises we can make and it is a tedious butsimpel exercise prove the following Theorem. This is the second main result ofthis section.

Theorem 37 �X ; X and �XY Z are morphisms of extended probability spaces

�X : T X ! X

X : X T ! X

�XY Z : X (Y Z)! (X Y ) Z

and are the components of natural isomorphisms. Furthermore h; T; �; ; �iis a monoidal structure on the category of extended probability spaces.

References

[1] Sterling K. Berberian. Notes on Spectral Theory. Van Nostrand, 1966.

[2] E. C.Lance. Hilbert C*-Modules: A Toolkit for Operator Algebraists. Uni-versity press, 1995.

[3] Karl Stromberg Edwin Hewitt. Real and Abstract Analysis. Springer Verlag,1969.

34

[4] K. R. Goodearl. Notes on Real and Complex C*-Algebras. Shiva PublishingLimited, 1982.

[5] Konrad Jacobs. Measure and Integral. Academic Press, 1978.

[6] G. Köthe. Topological Vector Spaces, volume II. Springer Verlag, 1979.

[7] N. P. Landsman. Mathematical Topics Between Classical and QuantumMechanics. Springer Verlag, 1998.

[8] F. W. Lawere and S. H. Schanuel. Conceptual Mathematics. Cambridge,1997.

[9] S. Mac Lane. Categories for the Working Mathematician. Springer, 1998.

[10] William L. Paschke. Inner product modules over b*-algebras. Transactionsof the American Mathematical Society, 182:443�468, August 1973.

[11] Valentin Lychagin Per Jakobsen. Relations and quantizations in the cate-gory of probabilistic bundles. Acta Applicandae Mathematicae, 82(3):269�308, 2004.

[12] John R. Ringrose Richard V. Kadison. Fundamentals of the Theory ofOperator Algebras, volume II. Academic Press, 1986.

[13] Nik Weaver. Mathematical Quantization. Chapman and Hall/CRC, 2001.

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