Lobachevskii Journal of Mathematics http://ljm.ksu.ru Vol. 16, 2004, 17 – 56
© P. K. Jakobsen and V. V. Lychagin
Per K. Jakobsen and Valentin V. Lychagin
OPERATOR VALUED PROBABILITY THEORY
ABSTRACT. We outline an extention of probability theory based on positive operator valued measures. We generalize the main notions from probability theory such as random variables, conditional expectations, densities and mappings. We introduce a product of extended probability spaces and mappings, and show that the resulting structure is a monoidal category, just as in the classical theory.
In this paper we present an extension of standard probability theory. An extended probability space is defined to be a normalized positive operator valued measure defined on a measurable space of events. This notion of extended probability space includes probability spaces and spectral measures as important special cases. The use of the word probability in this context is justified by showing that extended probability spaces enjoy properties analog to all the basic properties of classical probability spaces. Random vectors are defined as a generalization of the usual Hilbert space of square integrable functions. This generalization is well known in the literature and was first described by Naimark. Expectation and conditional expectation is defined for extended probability spaces by orthogonal 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 a probability density is any normalized positive integrable function, whereas for the case of extended probability spaces it turns out that the right notion is not a density but a half density. These half densities are elements in a Hilbert module of length one. Special cases of such half densities are well known in quantum mechanics where they are called wave functions. We define a random operator to be a linear operator on the space of half densities. The expectation of random operators are operators acting on the Hilbert space underlying the extended probability space. For the case of probability spaces the notion of random vectors and random operators coincide.
We introduce mappings or morphisms of extended probability spaces through a generalization of the notion of absolute continuity in probability theory. Half densities plays a pivotal role in this generalization. We show that the morphisms can be composed and that extended probability spaces and morphisms forms a category just as for probability spaces. The Naimark construction extends to morphisms and in fact defines a functor on the category of extended probability spaces.
Extended probability spaces can be multiplied and we furthermore show that this multiplication can be extended to morphisms in such a way that it defines a monoidal structure on the category of extended probability spaces. This is in complete analogy with the case of probability spaces and testify strongly to the naturalness of our constructions.
We do not in this paper attempt to give any interpretation of extended probabilities beyond the one implied by the strong structural analogies that we have shown to exists between the categories of probability spaces and extended probability spaces. It is well known that the interpretation of the classical Kolmogorov formalism for standard probability theory is not without controversy as the old debate between frequentists and Bayesians, among others, clearly demonstrate. Our theory of extended probability spaces is evidently a generalization of the Kolmogorov framework and it might be hoped that this enlarged framework will put some of the controversy in a different light. As a case in point note that extended probabilities are in general only partially ordered. The notion of partially ordered probabilities has been discussed and argued over for a very long time. In our theory of extended probability spaces, ordered and partially ordered probabilities lives side by side and enjoy the same formal categorical properties.
In this section we will make some technical assumptions that will assumed to hold throughout this paper. These assumptions are not necessarily the most general ones possible.
A measurable space  is a pair where is a set and is a -algebra on . A measurable map is a map of sets such that for all . Let be a set and let be a topology on . In this paper the term topology is taken to mean a second countable,locally compact Hausdorff topology . Note that any such space is metrizable,Polish and -compact. The Borel structure corresponding to a topology is the smallest -algebra containing the topology and is denoted by . A Borel space is a measurable space where the -algebra is a Borel structure. Any continuous map is measurable with respect to the Borel structures and . Borel sets are the observable events to which we must assign probabilities.
Let now be a Borel space and let be the real algebra  of bounded operators on the real Hilbert space . A positive operator valued measure (POV)  defined on is a map from to such that ,. The map is assumed to be finitely additive on disjoint union of sets and for any increasing sequence of sets satisfy the following continuity condition
where the supremum is taken with respect to the usual partial ordering of self adjoint operators. The supremum always exists since the sequence is increasing and bounded above by . The continuity condition implies that is additive on countable disjoint unions.
where the sum converges in the strong operator topology, that is, pointwise convergence in norm.
A positive operator valued measure is a spectral measure if is a projector for all . A necessary and sufficient condition for a POV, ,to be a spectral measure is that it is multiplicative
We are now ready to define our first main object
Note that a probability space can be identified with a extended probability space in many different ways. In fact for any given Hilbert space we can identify the probability space with a extended probability space where .
In standard probability theory quadratic integrable random variables and their expectation plays an important role. We will now review the classical Naimark construction of the analog of such random variables for the case of extended probability spaces. We will call such random variables random vectors. The space of random vectors forms a Hilbert spaces and we use this structure to define expectation and conditional expectation by orthogonal projections in complete analogy with the standard case.
3.1. The space of random vectors. Let be a extended probability space and let be the linear space of simple measurable functions . The linear structure is defined through pointwise operations as usual. Elements in can be written as finite sums of characteristic functions.
where is a -measurable partition of the set . We define a pseudo inner product on by
where , and is the inner product in the Hilbert space . The product is not definite. In fact we have
The last identity follows from the fact the is a positive operator. So for any simple function we have if and only if for all . This is of course true if is of measure zero but it can also be true if but is in the kernel of .
Since is a pseudo inner product the set of elements of length zero, , form a linear subspace and we can divide by this subspace. and thereby get a, in general, incomplete inner product space. The completion of this space with respect to the associated norm is by definition the space of random vectors and is a Hilbert space. We will use the notation or just for this space in analogy with the classical notation . The set of equivalence classes of simple functions evidently form a dense set in . Denote this dense subspace by . We have a well defined isometric embedding of into defined by
We also have a spectral measure . On the dense set the spectral measure is given by
In fact the existence of this spectral measure is the whole point of the Naimark construction. It show that by extending the Hilbert space one can turn any POV into a spectral measure. This idea has been generalized by 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 first example let be a measure on the measurable space and let be a Hilbert space. Define a positive operator valued measure on acting on by
For this case we have
where for any valued functions we define . Thus for this case our space will be the space of valued function elements such that . When the space turns into the space of square integrable complex valued functions .
As our second example let be two dimensional and let a basis be given. With respect to this basis we have
where and and are signed measures. In order for to be positive for all it is easy to see that and must be positive measures and that the following inequality must hold
Any function determines a pair of real valued functions through . The inner product in is given in terms of the measures , and as
Similar expressions for the inner product in exists for any finite dimensional Hilbert space .
3.2. The expectation of random vectors. Recall that we have a isometric embedding defined by
Note that the image is a closed subspace and therefore the orthogonal projection onto exists. Let be this orthogonal projection.
The following result is a immediate consequence of the definition
Note that adjointness condition uniquely determines the expectation. In fact we could define 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 where is given by
This example makes it natural to introduce a integral inspired notation for the expectation
Note that it is natural to put the differential in front of to emphasize the fact that is a operator valued measure that acts on the function valued of .
Let be an orthonormal basis for . For general elements the following formula holds
3.3. Conditional expectation. Let be a -subalgebra. We can restrict the POV to and will in this way get the Hilbert space of measurable random vectors. We obviously have a isometric embedding of into . Thus can be identified with a closed subspace of and therefore the orthogonal projection is defined. In complete analogy with the classical case we now define
It is evident that is isomorphic to when and that for this case we have . Let us consider the next simplest case when is generated by a partition where and when . We need the following result
Proof. Let be a Cauchy sequence in the inner product space . This means that when and goes to infinity. But and since are positive operators we get
Let be the range of and let be the orthogonal complement of . We have and since by assumption is a closed subspace we have the decomposition . Write with and . We then have by orthogonality
Clearly is a positive, bounded, injective and surjective map.
Let be the square root of this operator. It is also a positive bounded injective and surjective map and therefore has a bounded inverse. From the previous limit we can conclude that
Thus is a Cauchy sequence in and since is closed there exists a element such that . From the previous remarks the element exists and . If we let we have
Therefore is complete. □
The assumption in the proposition holds for example if is finite dimensional or if is infinite dimensional but all the are orthogonal projectors or isomorphisms. For the classical measure case and the proposition is true.
Let be a simple function in . Then by the previous proposition the conditional expectation must be of the form . It is uniquely determined by the conditions for all and . These conditions give us the following systems of equations for the unknown vectors :
This systems does not have a unique solution in but all solutions represents the same element in . For the special case we get the simplified system
When dim and we get the usual classical expression for the conditional expectation of given .
Densities are important for most applications of probability theory. For us they will make their appearance when we seek to generalize the relation of absolute continuity between measures to the context of positive operator valued measures. This generalization will play a pivotal role when we define maps between extended probability spaces. The generalization of the notion of density to the case of operator measures turns out to be surprisingly subtle.
4.1. The Hilbert module of half densities. Let be a measure. A density is a positive measurable function such that . Using this density we can define a new measure
If we try to generalize this formula directly to the case of POV measures we run into problems.
Let be a POV defined on a measurable space and let be a function as above. Then we can certainly define a new POV measure by the following formula
There is nothing inconsistent in this definition, the only problem is that it is very limited. In fact if is a finite set then any POV measure on is given by a finite set of positive operators between zero and the identity with the single condition . If is the new POV determined by the above formula then we have for some set of numbers . Thus each is proportional to .
Now if the numbers were changed into positive operators we could produce a much more general starting from a given . We would thus be considering a formula like
where is a positive operator valued function. However even if we could make sense of the proposed integral we would have problems. This is because the product of positive operators is positive if and only if they commute. This would put a highly nontrivial constraint on the allowed densities, constraints it would be difficult to verify and keep track of.
There is however a natural way out of these problems. It is very simple to verify that if is a POV measure acting on and a operator, then is a new POV measure. This suggest that we consider a density to be a operator valued function such that
We could then use this density to define a new POV measure by
On a formal level this now looks fine, the only remaining problem is to make sense of the proposed integrals. We will now proceed to do this.
where form a measurable partition of . These are simple measurable operator valued functions. The set is a real linear space through pointwise operations as usual. We can define a left action of on in the following way
This action clearly makes into a left module over the real - algebra . Define an valued product on through
where and . This product is clearly bilinear over the real numbers.
Thus the product is like a Hermitian product where the role of complex numbers are played by the elements of the real -algebra . Such structures have been known and studied for a long time. They leads, as we will see, in a natural way to the idea that probability densities for operator measures are elements in a Hilbert module. Our main sources for the theory of Hilbert modules are the paper  and the book . Chapters on Hilbert modules can also be found in the books  and .
Note that the product we have constructed is not positive definite. In fact, since the sum of positive operators in a real -algebras is zero only if each operator is zero, the identity holds if and only if
These identities can easily be satisfied for nonzero operators . In fact if are projectors and are projectors orthogonal to then the equations are clearly satisfied. In order to make the product definite we will need to divide out by the set of simple functions whose square is zero . In order to do this we will need the analog of the Cauchy-Swartz inequality.
For any element we know that and therefore there exists a positive operator such that . Denote this operator by . Thus we have . Also for any element define a real number by
where is the operator norm of the positive operator . With these definitions at hand we can now state the following Cauchy Swartz inequalities for . The proof of this proposition is an adaption of the proof in  to the case of real algebras.
Proof. A positive linear functional, ,on is a real valued linear functional such that whenever . A state on is a positive linear functional such that and . The main property that makes states useful in algebra theory is that if there exists a state such that . From this it follows immediately that if for all states then and this implies that if for all states then . In this way verification of inequalities in a algebra is reduced to the verification of numerical inequalities. Also recall that in any real -algebra the following important inequality holds 
For any given state define . It is evident that is a pseudo inner product on . It therefore satisfy the Cauchy-Swartz inequality . Define . We clearly have
Dividing by we find
The first inequality now follows since this numerical inequality holds for all states . As for the second inequality recall that in any real -algebra we have and for any pair of operators we have . Using this we have
and this proves the second inequality. □
From the second inequality we can in the usual way conclude that the triangle inequality holds for .
Let be the subset of elements in of pseudonorm zero.
For any operator and a pair of elements and in we now have
Thus is a submodule and we can therefore define a quotient module
Elements in are equivalent classes of simple operator valued functions denoted by . Note that for any elements with we have
and as a consequence of this . We therefore have a well defined operator valued product on defined through
This product enjoy the same properties as the product on and is in addition positive definite. Thus with this product is a pre-Hilbert module with a norm defined on the underlying real vector space. In general this vector space is not complete with respect to the norm. We can however complete the vector space with respect to the norm. The resulting structure is a Hilbert module over the real -algebra . We will call it the Hilbert module corresponding to the extended probability space . With the analogy with Hilbert spaces in mind we will consider to the the square length of . Note that for a general Hilbert module the length is a positive operator, not a positive number. Also note that in order to simplify the notation we use the same symbol for the norm on and for the operator norm on . This is the sense of the formula .
We have now made sense of equation (. It just state that should be a element in the Hilbert module of length .
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-Nikodym theorem.
For any define a map by
This map is clearly a module morphism.
The last property shows that if then . Therefore induce a well defined map, also denoted by , on through
The last property shows also that the map is bounded on . It therefore extends to a unique bounded linear map on . This map clearly also enjoy the properties listed in the previous proposition.
Let now be a element in the Hilbert module of unit length For each set define a operator on the Hilbert space by
Clearly and for all . It is also evident from the previous proposition that is finitely additive on disjoint sets. It is in fact also countably additive as we now show.
Proof. Let first be a element in with and let be a increasing sequence of sets with limit . The set of operators is a increasing sequence of positive operators. The supremum of this sequence exists . Denote the supremum by . In order to show that is a positive operator valued measure we only need to show that
It is a fact  that the sequence converges strongly to the limit . Since the strong limit is unique when it exists we must only show that for all elements . We know that is a positive operator valued measure so strongly. But then since all are bounded operators we have
for all This proves that is a POV. Next for any element in we define . It is trivial to verify that so that the previous proof show that is a POV. Finally let be a arbitrary element in . Then there exists a sequence of elements in such that . Since is a POV we know that for all is a measure.
Let be the positive set function defined by
By continuity we know that in the uniform norm and thus strongly. But then by continuity of the inner product on we can conclude that
We have now made sense of equation (2) and are now ready to define the symbolic expressions occurring in equation (1) and (2).
We define the integrals and as follows:
We have thus found that probability densities for operator valued measures are not functions but elements in a Hilbert module. They should in fact not be thought of as densities but as half densities, their square is a density in the above sense. This is a startling conclusion. Half densities are however not unfamiliar to anyone that has been exposed to quantum mechanics. Wave functions are half densities. In fact wave functions appear naturally in this scheme. If is a positive operator valued measure acting on a real two dimensional Hilbert space we are lead to define densities as functions whose values are operators on the plane. The complex numbers are isomorphic to a special subalgebra of operators on the plane (the conformal operators). Thus a large class of densities can be identified with complex valued functions of length one. Since self-adjoint operators are now naturally identified with real numbers the length can be considered to be a number. What we are describing are of course wave functions. Thus densities for positive operator valued measures acting on a two-dimensional plane are wave functions.
4.2. Random operators. Recall  that a map is said to be adjointable if there exists a map denoted by such that
for all elements and in . A map is self-adjoint if . It follows directly from the algebraic properties of the inner product and the completeness of the underlying real vector space that any adjointable map is a bounded module morphism. In fact the set of all adjointable maps form a abstract real -algebra that we denote by . We will call the elements in random operators.
The expectation of a random operator with respect to a density is by definition given by
The expectation of a random operator with respect to a density is thus a operator on . We can also use the density to define a POV acting on as we have seen. Note that the expectation of self-adjoint random operators is a self-adjoint operator in .
Returning to the two dimensional example discussed above we see that in that case for complex valued densities the expectation of self-adjoint random operators can be identified with real numbers and thus the expectation of random operators can be thought of as numbers. In higher dimensions and for more general densities no such identification with real numbers is possible. Furthermore no such reduction should be expected. After all, the self-adjoint elements in a real -algebra are the right analog of real numbers.
Let us assume that the real Hilbert space underlying the extended probability space is one dimensional. If we choose a basis we can identify the Hilbert space with and the Hilbert module with the real Hilbert space of square integrable functions on . A positive operator valued measure is through the basis identified with a probability measure and therefore for a half density the formula turns into
The half density is of course not uniquely determined by the probability measures 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 density we choose, they will all produce the same expectation. Thus by restricting to random vectors as our observables the difference between the various half densities are not observable. However there is really no rational reason to restrict to this class of observables. If we include random operators in our observables the difference between the half densities are readily observable.
In classical probability theory the notion of morphisms of probability spaces plays a role at least as important as the notion of a probability space. In fact from the Categorical point of view morphisms are the most important element in any theory construction. All other entities should be defined in terms of the morphisms. In this section we review the notion of a morphism in the context of probability spaces and then define the corresponding notion for extended probability spaces. The naturalness of our definition is verified by proving that extended probability spaces and morphisms forms a category. We also show that just as for the case of probability spaces we get a functor mapping the category of extended probability spaces into the category of Hilbert spaces. The existence of this functor is a verification of the naturalness of our constructions.
Let and be probability spaces. A morphism is a measurable map such that is absolutely continuous with respect to the push forward of the measure by , . By the Radon-Nikodym theorem this means that there exists a probability density such that
There are several other possibilities for morphisms of probability spaces . We could have required or . They can all be composed and lead to a category structure. However the only possibility that generalize well to extended probability spaces is the first one .
5.1. Morphisms of extended probability spaces. In this section we will introduce the notion of mapping between extended probability spaces and will then use mappings to define morphisms. This distinction between mappings and morphisms does not exist for probability spaces.
In order to define what a mapping is in the context of extended probability spaces, we must first generalize the notions of absolute continuity and push forward to positive operator valued measures. We will do this by combining them into a single entity.
Definition 11. Let be a extended probability space, a measurable space and the 3 tuple where is a measurable map, is a isometry and is a element in the Hilbert module corresponding to . Then the push forward of by is the positive operator valued measure,, defined on the measurable space by
where is the adjoint of .
Note that we have where is the orthogonal projection onto the closed subspace and therefore and .We can now define mappings between extended probability spaces using push forward in a very simple way.
Let us assume that the real Hilbert spaces underlying the extended probability spaces and are one dimensional. If we choose basis for these two spaces we can identify the Hilbert spaces with , the positive operator valued measures with probability measures and and the half density with a real valued function on . We must have and the condition for to be a mapping is
This is of course the condition for to be a mapping between the probability spaces and if we identify the classical density with .
Our first goal is to show that the proposed mappings can be composed. In order to do this we must first define a certain pullback of half densities induced by a mapping. Let therefore mappings and of extended probability spaces be given. Let us first define a measurable map , a isometry and a linear map by
The map has the following easily verifiable properties
Define a linear map by
where . The map has the following important properties
Proof. Let and . Then it is easy to verify that form a partition of and that . But then we have
This proves the second statement. For the third statement we have
proves the fourth statement. The first and last statement in the proposition follows from the fourth. Finally□
Using this proposition we can extend the map to a continuous linear map from to . This map is given on the dense set by
All the properties in the proposition holds for the extension. We are now ready to prove that our mappings can be composed
is a mapping of extended probability spaces and we have
Proof. In order to show that is a mapping we must prove that . But doing this is now a straight forward calculation if we use the previous proposition.
The last statement in the theorem is also proved by direct calculation.
Let . Then we have
Since the identity holds on a dense subset is also holds for all elements in and this proves the theorem. □
We now can use this Theorem to define composition of mappings
It is now straight forward to prove that composition of mappings is associative.
Proof. Clearly we have and . And from the previous theorem we have□
Extended probability spaces and mappings of extended probability spaces does unfortunately not form a category, we will in general not have unit morphisms.
For a given extended probability space the only reasonable candidate for a unit morphism is
For this mapping it is easy to show that
Thus the mapping is not a unit morphism in the categorical sense unless is a isomorphism. It is for this reason that we distinguish between mappings and the yet to be defined morphisms. Morphisms will be defined in terms of a equivalence relation on mappings.
Recall that for any mapping , is the orthogonal projection on the closed subspace .
If and are equivalent we will write .
The defined relation is a equivalence relation. In order to define morphisms we must show that composition of mappings extends to equivalence classes of mappings. For this we need the following two lemmas.
Proof. For any , is the unique vector in such that is orthogonal to . But for any in we have
Therefore by uniqueness . □
Proof. We only need to verify the identity on the dense subset . But for any with we have□
We can now prove that composition is well defined on classes.
Proof. We only need to prove that . But using the previous two lemmas we have
In order to keep the notation simple we will always denote a morphism by a representative mapping . Thus when we speak of a morphism we mean the class . The meaning will always be clear, we just have to make sure that any operations involving morphisms does not depend on choice of representative.
We can now formulate the main result of this subsection.
Proof. We know that composition is well defined and associative. For any object let the unit mapping be . From proposition 18 we have for any morphisms
because is a projection. □
We know that the category of probability spaces has a terminal object, ,in the categorical sense, there is a unique morphism from any probability space to . Here with , and the only possible probability measure on . The existence of makes it possible to define points in probability spaces categorically. We will now see that the category of extended probability spaces does not have a terminal object and thus extended probability spaces will not have points in the categorical sense, but only generalized points. The only possible candidate for a terminal object in the category of extended probability spaces is the object where is the only possible positive operator valued measure, . We will now show that is in fact not a terminal object.
Let be any morphism of extended probability spaces. We have and clearly is unique. The map is a isometry and is therefore determined by a vector where and . The vector and element must satisfies the single condition
Using the definition of we find that the following identity must be satisfied
and clearly this identity will be satisfied by many choices of and . Thus the morphism is not uniquely determined and therefore is not a terminal object.
5.2. The Naimark functor. In probability theory there is a certain functor that plays a major role in the theory. We will now review the construction of this functor and show that a analog functor is defined on the category of extended probability spaces. The existence of this functor testify to the naturalness of our constructions. The functor will be called the Naimark functor since the Naimark dilatation construction plays a major role in its construction.
Let us start with a review of the functor for the case of probability spaces. For any probability space define a Hilbert space,denoted by , by . Let and be two probability spaces and let be a morphism of probability spaces in the sense that
Define a mapping by
It is easy to verify, using the Radon Nikodym theorem, that is in fact a isometry and moreover that is a functor from the category of probability spaces to the category of Hilbert spaces. We will now show that it is possible to define a functor, also denoted by , from the category of extended probability spaces to the category of Hilbert spaces that for probability spaces reduce to the functor discussed above.
Let and be extended probability spaces and let and be the corresponding Hilbert spaces of random vectors. Informally to any morphism of extended probability spaces we will define a isometry by the formula
It is easy to see that the mapping is a special case of this general formula. Of course we can not use this formula to actually define since elements in are not vector functions and elements in are not operator valued functions. The action of elements in on implied by the formula must also be made sense of and since morphisms are classes of mappings we need to prove independence of representative.. We will now prove that the map exists and that it defines a functor.
Recall that if denote the space of simple valued functions with inner product then is the closure of where iff . For any extended probability space, is the linear space of simple operator valued functions occurring in the construction of the Hilbert module . For a measurable map ,a isometry and a element define a linear map by
Proof. Let and . Then we have
In the last line we used the Cauchy-Swartz inequality and the definition of the norm in the Hilbert module. □
This lemma implies that if then and therefore we can extend to a bounded linear map . It is defined on the dense subset by .
The following proposition sets the stage for proving the existence of the Naimark functor.
and that satisfy
Proof. We will start by showing that only depends on the class of . Let be a sequence of elements in converging to . For each we can define a positive operator valued measure on acting on the Hilbert space by
By continuity strongly and thus weakly. But then we have
The assumption means that , so depends only on the class of . Therefore is well defined on the dense subset and the argument just given show that it is a isometry. It therefore extends to a isometry from to .
For the last part of the Theorem let and be sequences in and converging to and . Here and . For with we have by continuity of all maps involved that if we define by then we have
The last statement of the theorem is verified by a trivial calculation. □
We are now finally ready to prove the existence of the Naimark functor.
Proof. We only need to prove that is well defined for a given morphism . The functorial properties follows from the previous proposition. Assume . Let us first assume that the densities of and are and . We can without loss of generality assume that and are of the form
since we can bring it to this form by the same construction as in lemma 29. The equivalence then amounts to for all . Then on the dense subset we have for that
The case for general densities follows by continuity. □
The Naimark functor is not the only functor occurring in this theory. In fact if we recall the properties of the pullback operation defined earlier in this section we can define a second functor.
Theorem 28. For any extended probability space , define a Hilbert module and for any morphism of extended probability spaces define a morphism of Hilbert modules . Then is a functor from the category of extended probability spaces to the category of Hilbert modules.
For the case of probability spaces the Hilbert module and the space of random vectors are both isomorphic to the Hilbert space of square integrable real valued function. This is why random variables and densities appear to be taken from the same space in probability theory. But this is a very special situation. If the underlying Hilbert space is not one dimensional but two dimensional the densities and random vectors start to reveal their different nature. As we have discussed previously for this case a important subclass of densities are the one whose values are contained in the conformal group of the plane. These densities form a sub-Hilbert module that is actually a isomorphic to the complex Hilbert space of complex valued functions.
In probability theory the notion of product measures and product densities play a major role. It is through these that dependence and independence for random variables are defined. From a categorical point of view the situation is summarized by saying that the category of probability spaces supports a monoidal structure. We will now show that the category of extended probability spaces also supports a monoidal structures and that as a consequence the notions of dependence and independence can be defined.
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 is associative up to natural isomorphism and has a unit object up to natural isomorphism. What this means is that if , and are objects in the category and if the product is denoted by then we require that there exists a isomorphism . Similarly if is the unit object we require that there exists isomorphisms and . The isomorphisms can not be arbitrarily chosen for different objects, they must form the components of a natural transformation. In addition they must satisfies a set of equations known as the MacLane coherence conditions. These equations ensure that associativity and unit isomorphisms can be extended consistently to products of finitely many objects. The conditions that must be satisfied by , and are the following.
For all objects ,, and we must have
These are the MacLane coherence conditions. The naturality conditions are expressed as follows. For any arrows , and we must have
In general such equations are difficult to solve, there is a very large number of variables and equations. However in some simple situations the naturality conditions can be used to reduce the system of equations to a much smaller set.
The reader not familiar with categories,natural transformations and Coherence conditions might want to consult the book  for a elementary introduction to the categorical view of mathematics, a more advanced introduction can be found in the book 
The notion of product measures in probability theory has of course been known for a long time. The corresponding monoidal structure in the category of probability spaces is described in detail in . The main features are as follows. For two probability spaces and their product is the probability space , where is the product measure. The product of two morphisms and is a morphism where is just the Cartesian product of the maps and . The associativity and unit isomorphisms are just the usual one from the category of sets. , and . For the category of probability spaces this choice of , and are the only possible ones as we show in . 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 define the product of extended probability spaces and morphisms and show that this product is a bifunctor on the category of extended probability spaces.
Let and be two extended probability spaces. The product of the two positive operator valued measures and always exists and is uniquely determined  by its value on measurable boxes by
The product measure acts on the Hilbert space . The tensor product is the Hilbert tensor product. We now need to extend the product to morphisms and show that it is a bifunctor. Before we do this we must specify the relationship between the Hilbert modules and . We will show that, as expected, we can map the first into the second using a continuous injective module morphism. We will start by constructing this morphism.
Recall that for any extended probability space , is the completion of the dense subspace and
is the real linear space of simple valued measurable functions on .
For a pair of extended probability spaces define a map by
where and .
For this map we have the following
Proof. We evidently have for all real numbers . Let and be two elements in . Define a new sequence of sets where for and for and let . Let be the set of all valued functions on the index set . The set is a index set for a new partition, of the set defined by
where for any set we define and , the complement of . We evidently have
But then we have for any that
This show that is bilinear. For the second part of the statement in the lemma we have
But implies that and the identity just derived then implies that and therefore by definition . □
Using the lemma we have a well linear map, also denoted by , from to
The map satisfy the following important identity
Proof. Any is of the form where and . But then we have□
We can now state and prove the main property of . First we will recall some facts about (external) tensor products of Hilbert modules. Let denote the tensor product of and ,as real vector spaces, with topology determined by the norm induced from the operator valued inner product . The completion of is the external tensor product  of the Hilbert modules and and will be denoted by . It is a module over the spatial tensor product  of the represented algebras and .
is a dense subspace of and on this dense subspace is given by
Proof. Let and be the projective tensor products  of the underlying real vector spaces. Note that the tensor product spaces have not been completed with respect to the projective norm. The embedding is know to exist and be dense . The norm on and induced by the operator valued inner product is evidently a cross norm and it is know that the projective norm is the largest possible cross norm. Therefore we can conclude that is a dense subspace of and thus by completion in . By the previous lemma is bounded and therefore extends uniquely to a bounded map . The first identity in the statement of the proposition follows from the previous lemma and the continuity of the operator valued inner product. □
In order to introduce tensor product of morphisms between extended probability spaces we need the previous proposition and the following lemma
Proof. For and we have
By continuity and density we can conclude that the identity holds on . □
Let now and be morphisms of extended probability spaces. We thus have and where and . Define a 3-tuple by
where , and . Then we have
Proof. We need to prove that . But this is true because
where we have used the previous lemma. This proves that is a mapping of extended probability spaces. In order to show that it is also a morphism we must show that it is independent of choice of representatives. Thus assume that and . We need to show that and this amounts to proving that . But from the identity we have and the rest of the proof is a simple calculation. □
Having proved that is a morphism our next goal is to prove that it behaves as a functor under composition. For this we need the following lemma.
Proof. By continuity we only need to prove the identity on the dense subset . But on this subset we have□
We can now prove our first main result in this section
Proof. The unit property is trivial to verify and for the first identity we only need to prove that . But using the previous lemma we have□
6.2. The monoidal structure. Showing that exists and is a bifunctor is the only hard part in proving that there is a monoidal structure on the category of extended probability spaces.
The only reasonable candidate for a unit object is clearly the extended probability space discussed previously. For any objects and define
These are obviously the simplest choices we can make and it is a tedious but simple exercise prove the following theorem. This is the second main result of this section.
and are the components of natural isomorphisms. Furthermore is a monoidal structure on the category of extended probability spaces.
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Received October 1, 2004