Lobachevskii Journal of Mathematics http://ljm.ksu.ru Vol. 16, 2004, 17 – 56

© P. K. Jakobsen and V. V. Lychagin

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.

Contents

1. Introduction

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.

2. Extended probability spaces

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 [5] is a pair X = 〈ΩX,BX〉 where ΩX is a set and BX is a σ-algebra on ΩX . A measurable map f : X → Y is a map of sets ΩX → ΩY such that f−1(A) ∈B X for all A ∈BY . Let Ω be a set and let τ be a topology on Ω. In this paper the term topology is taken to mean a second countable,locally compact Hausdorff topology [3]. Note that any such space is metrizable,Polish and σ-compact. The Borel structure corresponding to a topology τ is the smallest σ-algebra containing the topology τ and is denoted by B(τ). A Borel space is a measurable space where the σ-algebra is a Borel structure. Any continuous map f : 〈ΩX,τX〉→〈ΩY ,τY 〉 is measurable with respect to the Borel structures B(τX) and B(τY ). Borel sets are the observable events to which we must assign probabilities.

Let now 〈ΩX,B(τX)〉 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 operator valued measure (POV) [1] defined on 〈ΩX,B(τX)〉 is a map FX from B(τX) to O(HX) such that F X (∅) = 0,FX(ΩX) = 1. The map FX is assumed to be finitely additive on disjoint union of sets and for any increasing sequence of sets {V i} 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 {F X(V i)} is increasing and bounded above by FX(lim i→∞V i). The continuity condition implies that FX 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 F X (V ) is a projector for all V ∈B. A necessary and sufficient condition for a POV, F X ,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 X = 〈ΩX,B(τX),μX〉 can be identified with a extended probability space in many different ways. In fact for any given Hilbert space HX we can identify the probability space with a extended probability space X = 〈ΩX,B(τX),FX〉 where F X (V ) = μX(V )IHX.

3. Random vectors

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 〈Ω,B,F〉 be a extended probability space and let S be the linear space of simple measurable functions v : Ω → H. The linear structure is defined through pointwise operations as usual. Elements in S can be written as finite sums of characteristic functions.

where {V i} is a B -measurable partition of the set Ω. We define a pseudo inner product on S by

where v = ∑ iξiθV i, w = ∑ jηjθWj and 〈 〉H is the inner product in the Hilbert space H. The product is not definite. In fact we have

for all i.

The last identity follows from the fact the F (V i) is a positive operator. So for any simple function v = ∑ ξiθV i we have 〈v, v〉 = 0 if and only if F(V i)ξi ⊥ ξi for all i. This is of course true if V i is of F measure zero but it can also be true if F(V i)≠0 but ξi is in the kernel of F(V i).

Since 〈 〉 is a pseudo inner product the set of elements of length zero, 〈v, v〉 = 0, form a linear subspace and we can divide S 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 L2 (B,F) or just L2(F) for this space in analogy with the classical notation L2 (μ). The set of equivalence classes of simple functions [v] evidently form a dense set in L2(F). Denote this dense subspace by T(F). We have a well defined isometric embedding π of H into L2 (F) defined by

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

where v = ∑ ξiθV i.

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 μ be a measure on the measurable space 〈Ω,B〉 and let H be a Hilbert space. Define a positive operator valued measure on 〈Ω,B〉 acting on H by

For this case we have

where for any H valued functions f,g we define 〈f,g〉H(x) = 〈f(x),f(x)〉H. Thus for this case our space L2(F) will be the space of H valued function elements such that ∫ 〈f,f〉Hdμ < ∞. When H = ℂ the space L2(F) turns into the space of square integrable complex valued functions L2 (μ).

As our second example let H be two dimensional and let a basis {ξ1,ξ2} be given. With respect to this basis we have

where μ and ν and ω are signed measures. In order for F(U) to be positive for all U it is easy to see that μ and ν must be positive measures and that the following inequality must hold

Any function f : Ω → H determines a pair of real valued functions {f1,f2} through f(x) = f1(x)ξ1 + f2(x)ξ2. The inner product in L2 (F) is given in terms of the measures μ,ν and ω as

Similar expressions for the inner product in L2 (F) exists for any finite dimensional Hilbert space H.

3.2. The expectation of random vectors. Recall that we have a isometric embedding π : H → L2(F) defined by

Note that the image π(H) ⊂ L2(F) is a closed subspace and therefore the orthogonal projection onto π(H) exists. Let QH 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 π.

Using this proposition it is easy to verify that the expectation of a simple function element [v] where v = ∑ ξiθV i 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 dF in front of f to emphasize the fact that F is a operator valued measure that acts on the function valued of f.

Let {ξi} be an orthonormal basis for H. For general elements f the following formula holds

3.3. Conditional expectation. Let A⊂B be a σ-subalgebra. We can restrict the POV F to A and will in 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). Thus L2(A,F) can be identified with a closed subspace of L2 (B,F) and therefore the orthogonal projection QA : L2(B,F) → L2(A,F) is defined. In complete analogy with the classical case we now define

It is evident that L2(A,F) is isomorphic to H when A = {Ω,∅} and that for this case we have EA(f) = π(E(f)). Let us consider the next simplest case when A is generated by a partition {A1...An} where Ω = ∪Ai and Ai ∩ Aj = ∅ when i≠j. We need the following result

The assumption in the proposition holds for example if H is finite dimensional or if H is infinite dimensional but all the F(Ai) are orthogonal projectors or isomorphisms. For the classical measure case H ≈ ℝ and the proposition is true.

Let v = ∑ ξjθV j be a simple function in L2(B,F). Then by the previous proposition the conditional expectation must be of the form QA (v) = ∑ ηiθAi. It is uniquely determined by the conditions 〈v − QA(v),ξθAj〉H = 0 for all ξ ∈ H and j = 1..n. These conditions give us the following systems of equations for the unknown vectors ηi:

for any  i.

This systems does not have a unique solution in H but all solutions represents the same element in L2 (A,F) = T(A,F). For the special case v = ξ0θC we get the simplified system

When dim H = 1 and F(Ai) = μ(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 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 ν be a measure. A density is a positive measurable function ρ such that ∫ ρdν = 1. 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 F be a POV defined on a measurable space 〈Ω,B(τ)〉 and let ρ 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 Ω is a finite set then any POV measure on Ω is given by a finite set {Fi} of positive operators between zero and the identity with the single condition ∑ Fi = 1. If E is the new POV determined by the above formula then we have Ei = ρiFi for some set of numbers {ρi}. Thus each Ei is proportional to Fi.

Now if the numbers ρi were changed into positive operators we could produce a much more general E starting from a given F. We would thus be considering a formula like

where ρ 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 F is a POV measure acting on H and Q a operator, then QFQ∗ is a new POV measure. This suggest that we consider a density to be a operator valued function ϕ 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.

Let

where {V i} form a measurable partition of Ω. These are simple measurable operator valued functions. The set V is a real linear space through pointwise operations as usual. We can define a left action of O(H) on V in the following way

This action clearly makes V into a left module over the real C∗- algebra O(H). Define an O(H) valued product on V through

where s = ∑ siθV i and t = ∑ tjθWj. 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 C∗ -algebra O(H). 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 [10] and the book [2]. Chapters on Hilbert modules can also be found in the books [7] and [13].

Note that the product we have constructed is not positive definite. In fact, since the sum of positive operators in a real C∗ -algebras is zero only if each operator is zero, the identity 〈s, s〉 = 0 holds if and only if

These identities can easily be satisfied for nonzero operators si . In fact if F (V i) are projectors and si are projectors orthogonal to F(V i) 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 〈s, s〉 = 0. In order to do this we will need the analog of the Cauchy-Swartz inequality.

For any element s ∈ V we know that 〈s,s〉≥ 0 and therefore there exists a positive operator h such that h2 = 〈s,s〉. Denote this operator by ∣s∣. Thus we have ∣s∣2 = 〈s,s〉. Also for any element s ∈ V define a real number ∣∣s∣∣ by

where ∣∣〈s,s〉∣∣ is the operator norm of the positive operator 〈s, s〉. With these definitions at hand we can now state the following Cauchy Swartz inequalities for V . The proof of this proposition is an adaption of the proof in [13] to the case of real C∗ algebras.

From the second inequality we can in the usual way conclude that the triangle inequality holds for ∣∣ ∣∣.

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

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

Thus N is a submodule and we can therefore define a quotient module

Elements in H˜ are equivalent classes of simple operator valued functions denoted by [s]. Note that for any elements [s], [t] ∈H˜ with [s] = 0 we have

and as a consequence of this 〈s,t〉 = 0. We therefore have a well defined operator valued product on H ˜ defined through

This product enjoy the same properties as the product on V and is in addition positive definite. Thus H˜ 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 C∗ -algebra O(H). We will call it the Hilbert module corresponding to the extended probability space 〈Ω,B(τ),F〉. With the analogy with Hilbert spaces in mind we will consider 〈ϕ,ϕ〉 to the the square length of ϕ. 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 H and for the operator norm on O(H). This is the sense of the formula ∣∣ϕ∣∣2 = ∣∣〈ϕ,ϕ〉∣∣.

We have now made sense of equation ( 1). It just state that ϕ 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-Nikodym theorem.

For any U ∈B(τ) define a map PU : V → V by

This map is clearly a O(H) module morphism.

The last property shows that if ∣∣s∣∣ = 0 then ∣∣PU(s)∣∣ = 0. Therefore P U induce a well defined map, also denoted by PU, on H˜ through

The last property shows also that the map P U is bounded on H˜. It therefore extends to a unique bounded linear map on H. This map clearly also enjoy the properties listed in the previous proposition.

Let now ϕ be a element in the Hilbert module H of unit length 〈ϕ,ϕ〉 = 1. For each set U ∈B(τ) define a operator Eϕ(U) on the Hilbert space H by

Clearly Eϕ(Ω) = 1 and Eϕ(U) ≥ 0 for all U. It is also evident from the previous proposition that Eϕ is finitely additive on disjoint sets. It is in fact also countably additive as we now show.

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 ∫ ϕdFψ∗and ∫ V ϕdFϕ∗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 F 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 [2] that a map A : H→H is said to be adjointable if there exists a map denoted by A∗ : H→H such that

for all elements ϕ and ψ in H. A map is self-adjoint if A∗ = A. 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 O(H) module morphism. In fact the set of all adjointable maps form a abstract real C∗ -algebra that we denote by A. We will call the elements in A random operators.