Lobachevskii Journal of Mathematics

http://ljm.ksu.ru

Vol. 16, 2004, 17 – 56 \begin{matrix}c \\ ◯ \end{matrix} 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.

Contents

1. Introduction 18 2. Extended probability spaces 19 3. Random vectors 20 3.1. The space of random vectors 20 3.2. The expectation of random vectors 22 3.3. Conditional expectation 23 4. Densities and random operators 25 4.1. The Hilbert module of half densities 26 4.2. Random operators 32 5. The category of extended probability spaces 34 5.1. Morphisms of extended probability spaces 34 5.2. The Naimark functor 42 6. Monoidal structure on the category of extended probability

spaces 47 6.1. Product of extended probability spaces and morphisms 48 6.2. The monoidal structure 54 References 55

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 = ⟨Ω_{X}, B_{X}⟩ where Ω_{X} is a set and B_{X} is a σ-algebra on Ω_{X}. A measurable map f : X → Y is a map of sets Ω_{X} → Ω_{Y} such that f^{−1}(A) ∈ B_{X} for all A ∈ B_{Y}. 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 : ⟨Ω_{X}, τ_{X}⟩ → ⟨Ω_{Y} , τ_{Y} ⟩ is measurable with respect to the Borel structures B(τ_{X}) and B(τ_{Y}). Borel sets are the observable events to which we must assign probabilities.

Let now ⟨Ω_{X}, B(τ_{X})⟩ be a Borel space and let O(H_{X}) be the real C^{∗} algebra [4] of bounded operators on the real Hilbert space H_{X}. A positive operator valued measure (POV) [1] defined on ⟨Ω_{X}, B(τ_{X})⟩ is a map F_{X} from B(τ_{X}) to O(H_{X}) such that F_{X}(∅) = 0,F_{X}(Ω_{X}) = 1. The map F_{X} 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

F_{X}( lim_{i⟶∞} V_{i}) = sup{F_{X}(V_{i}) | i = 1, 2, 3, ....},

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 F_{X}(lim_{i⟶∞} V_{i}). The continuity condition implies that F_{X} is additive on countable disjoint unions.

F_{X}(U_{i=1}^{∞}V_{i}) = ∑_{i=1}^{∞} F_{X}(V_{i}),

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

F_{X}(V_{1} ∩ V_{2}) = F_{X}(V_{1})F_{X}(V_{2}).

We are now ready to define our first main object

Definition 1. A extended probability space X is a triple X = ⟨Ω_{X}, B(τ_{X}), F_{X}⟩ where F_{X} : B(τ_{X}) ⟶ O(H_{X}) is a positive operator valued measure.

Note that a probability space X = ⟨Ω_{X}, B(τ_{X}), μ_{X}⟩ can be identified with a extended probability space in many different ways. In fact for any given Hilbert space H_{X} we can identify the probability space with a extended probability space X = ⟨Ω_{X}, B(τ_{X}), F_{X}⟩ where F_{X}(V) = μ_{X}(V)I_{H_{X}}.

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.

v = ∑_{i}ξ_{i}θ_{V_{i}} ,

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

⟨v, w⟩ = ∑_{i,j}⟨F (V_{i} ∩ W_{j})ξ_{i}, η_{j}⟩_{H},

where v = ∑_{i}ξ_{i}θ_{V_{i}}, w = ∑_{j}η_{j}θ_{W_{j}} and ⟨ ⟩_{H} is the inner product in the

Hilbert space H. The product is not definite. In fact we have

⟨v, v⟩ = 0
⇕
∑_{i}⟨F (V_{i})ξ_{i}, ξ_{i}⟩_{H} = 0
⇕
⟨F (V_{i})ξ_{i}, ξ_{i}⟩ = 0

for all i.

The last identity follows from the fact the F (V_{i}) is a positive operator. So for any simple function v = ∑ξ_{i}θ_{V_{i}} we have ⟨v, v⟩ = 0 if and only if F (V_{i})ξ_{i} ⊥ ξ_{i} 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 ξ_{i} 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 L_{2}(B, F) or just L_{2}(F) for this space in analogy with the classical notation L_{2}(μ). The set of equivalence classes of simple functions [v] evidently form a dense set in L_{2}(F). Denote this dense subspace by T (F). We have a well defined isometric embedding π of H into L_{2}(F) defined by

π(ξ) = [ξθ_{Ω}].

We also have a spectral measure P : B → O(L_{2}(F)). On the dense set T (F) the spectral measure is given by

P (α)[v] = [∑_{i}ξ_{i}θ_{V_{i}∩α}],

where v = ∑ξ_{i}θ_{V_{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

F (U) = μ(U)1_{H}.

For this case we have

⟨[v], [w]⟩ = ∑_{i,j}⟨μ(V_{i}∩W_{j})ξ_{i}, η_{j}⟩_{H} = ∑_{i,j}⟨ξ_{i}, η_{j}⟩_{H}μ(V_{i}∩W_{j}) = ∫ ⟨v, w⟩_{H}dμ,

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 L_{2}(F) will be the space of H valued function

elements such that ∫ ⟨f, f⟩_{H}dμ < ∞. When H = ℂ the space L_{2}(F)

turns into the space of square integrable complex valued functions L_{2}(μ).

As our second example let H be two dimensional and let a basis {ξ_{1}, ξ_{2}} be given. With respect to this basis we have

F (U) = [ \begin{matrix}μ(U) & ω(U) \\ ω(U) & ν(U) \end{matrix} ] ,

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

ω(U)^{2} ≤ μ(U)ν(U).

Any function f : Ω → H determines a pair of real valued functions {f_{1}, f_{2}} through f(x) = f_{1}(x)ξ_{1} + f_{2}(x)ξ_{2}. The inner product in L_{2}(F) is given in terms of the measures μ,ν and ω as

⟨(f_{1}, f_{2}), (g_{1}, g_{2})⟩
= ∫ f_{1}g_{1}dμ + ∫ f_{2}g_{2}dν + ∫ (f_{1}g_{2} + f_{2}g_{1})dω.

Similar expressions for the inner product in L_{2}(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 → L_{2}(F) defined by

π(ξ) = [ξθ_{Ω}].

Note that the image π(H) ⊂ L_{2}(F) is a closed subspace and therefore the orthogonal projection onto π(H) exists. Let Q_{H} be this orthogonal projection.

Definition 2. The expectation of a random vector f ∈ L_{2}(F) is the unique element E(f) ∈ H such that

π(E(f)) = Q_{H}(f).

The following result is a immediate consequence of the definition

Proposition 3. The expectation is a surjective continuous linear map : L_{2}(F) → H and is the adjoint of the embedding π

⟨f, π(ξ)⟩ = ⟨E(f), ξ⟩ ∀ξ ∈ H.

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 = ∑ξ_{i}θ_{V_{i}} is given by

E([v]) = ∑_{i}F (V_{i})(ξ_{i}).

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

E(f) =^{def} ∫ dF f.

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

E(f) = ∑_{i}⟨f, π(ξ_{i})⟩ξ_{i}.

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 L_{2}(A, F) of A measurable random vectors. We obviously have a isometric embedding of L_{2}(A, F) into L_{2}(B, F). Thus L_{2}(A, F) can be identified with a closed subspace of L_{2}(B, F) and therefore the orthogonal projection Q_{A} : L_{2}(B, F) → L_{2}(A, F) is defined. In complete analogy with the classical case we now define Definition 4. The conditional expectation of a element f ∈ L_{2}(B, F) is given by

E_{A}(f) = Q_{A}(f) ∈ L_{2}(A, F).

It is evident that L_{2}(A, F) is isomorphic to H when A = {Ω, ∅} and that for this case we have E_{A}(f) = π(E(f)). Let us consider the next simplest case when A is generated by a partition {A_{1}...A_{n}} where Ω = ∪ A_{i} and A_{i} ∩ A_{j} = ∅ when i ≠ j. We need the following result

Proposition 5. Let F (A_{i}) for i = 1..n have closed range. Then L_{2}(A, F) = T (A, F).

Proof. Let [v_{n}] be a Cauchy sequence in the inner product space T (A, F). This means that ||[v_{n}] − [v_{m}]||^{2} → 0 when m and n goes to infinity. But v_{n} = ∑_{i}ξ_{i}^{n}θ_{A_{i}} and since F (A_{i}) are positive operators we get

∑_{i}⟨F (A_{i})(ξ_{i}^{n} − ξ_{i}^{m}), ξ_{i}^{n} − ξ_{i}^{m}⟩ → 0
⇓
⟨F (A_{i})(ξ_{i}^{n} − ξ_{i}^{m}), ξ_{i}^{n} − ξ_{i}^{m}⟩ → 0

for all i.

Let L_{i} = F (A_{i})(H) be the range of F (A_{i}) and let L^{⊥} be the orthogonal complement of L_{i}. We have L_{i}^{⊥} = Ker(F (A_{i})) and since L_{i} by assumption is a closed subspace we have the decomposition H = L_{i} ⊕L_{i}^{⊥} . Write ξ_{i}^{n} = r_{i}^{n} + t_{i}^{n} with r_{i}^{n} ∈ L_{i}^{⊥} and t_{i}^{n} ∈ L_{i}. We then have by orthogonality

⟨F (A_{i})(t_{i}^{n} − t_{i}^{m}), t_{i}^{n} − t_{i}^{m}⟩ → 0.

Clearly F (A_{i})|_{L_{i}} : L_{i} → L_{i} is a positive, bounded, injective and surjective map.

Let T_{i} : L_{i} → L_{i} 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

⟨T_{i}(t_{i}^{n} − t_{i}^{m}), T_{i}(t_{i}^{n} − t_{i}^{m})⟩ → 0.

Thus {T_{i}(t_{i}^{n})} is a Cauchy sequence in L_{i} and since L_{i} is closed there exists a element y_{i} ∈ L_{i} such that T_{i}(t_{i}^{n}) → y_{i}. From the previous remarks the element ξ_{i} = T_{i}^{−1}(y_{i}) ∈ L_{i} exists and lim_{n→∞} t_{i}^{n} = lim_{n→∞} T_{i}^{−1}(T_{i}(t_{i}^{n})) = T_{i}^{−1}(lim_{n→∞} T_{i}(t_{i}^{n})) = T_{i}^{−1}(y_{i}) = ξ_{i}. If we let v = ∑ ξ_{i}θ_{A_{i}}we have

||[v_{n}] − [v]||^{2}

= ∑_{i}⟨F (A_{i})(ξ_{i}^{n} − ξ_{i}), ξ_{i}^{n} − ξ_{i}⟩

= ∑_{i}⟨T_{i}(t_{i}^{n} − ξ_{i}), T_{i}(t_{i}^{n} − ξ_{i})⟩

= ∑_{i}⟨T_{i}(t_{i}^{n}) − y_{i}, T_{i}(t_{i}^{n}) − y_{i}⟩

= ∑_{i}||T_{i}(t_{i}^{n}) − y_{i}|| → 0.

Therefore T (A, F) is complete. □

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

Let v = ∑ ξ_{j}θ_{V_{j}} be a simple function in L_{2}(B, F). Then by the previous proposition the conditional expectation must be of the form Q_{A}(v) = ∑ η_{i}θ_{A_{i}}. It is uniquely determined by the conditions ⟨v−Q_{A}(v), ξθ_{A_{j}}⟩_{H} = 0 for all ξ ∈ H and j = 1..n. These conditions give us the following systems of equations for the unknown vectors η_{i}:

F (A_{i})η_{i} = ∑_{k}F (V_{k} ∩ A_{i})ξ_{k}

for any i.

This systems does not have a unique solution in H but all solutions represents the same element in L_{2}(A, F) = T (A, F). For the special case v = ξ_{0}θ_{C} we get the simplified system

F (A_{i})η_{i} = F (C ∩ A_{i})ξ_{0}.

When dim H = 1 and F (A_{i}) = μ(A_{i}) 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

μ(V) = _{∫V} ρdν.

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

E(V) = _{∫V} ρdF.

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 {F_{i}} of positive operators between zero and the identity with the single condition ∑ F_{i} = 1. If E is the new POV determined by the above formula then we have E_{i} = ρ_{i}F_{i} for some set of numbers {ρ_{i}}. Thus each E_{i} is proportional to F_{i}.

Now if the numbers ρ_{i} 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

E(V) = _{∫V} ρdF,

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 QF Q^{∗} is a new POV measure. This suggest that we consider a density to be a operator valued function φ such that

∫ΩφdF φ^{∗} = 1.  (1)

We could then use this density to define a new POV measure by

E(V) = _{∫V} φdF φ^{∗}.  (2)

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

V = {s = ∑_{i}s_{i}θ_{V_{i}} | s_{i} ∈ O(H) V_{i} ∈ B(τ)},

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

as = ∑_{i}(as_{i})θ_{V_{i}}.

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

⟨s, t⟩ = ∑_{i,j}s_{i}F (V_{i} ∩ W_{j})t_{j}^{∗},

where s = ∑ s_{i}θ_{V_{i}} and t = ∑ t_{j}θ_{W_{j}}. This product is clearly bilinear over the real numbers.

Proposition 6. The following properties

⟨s, s⟩ ≥ 0,
⟨as, t⟩ = a⟨s, t⟩,
⟨s, t⟩ = ⟨t, s⟩^{∗},
⟨s, at⟩ = ⟨s, t⟩a^{∗}

hold.

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

s_{i}F (V_{i})s_{i}^{∗} = 0 for all i.

These identities can easily be satisfied for nonzero operators s_{i}. In fact if F (V_{i}) are projectors and s_{i} 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 h^{2} = ⟨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

||s||^{2} = ||⟨s, s⟩||

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.

Proposition 7. The following forms of the Cauchy-Swartz inequality

⟨s, t⟩⟨t, s⟩ ≤ |s|^{2}||t||^{2},
||⟨s, t⟩|| ≤ ||s|| ||t||

hold.

Proof. A positive linear functional, ω ,on O(H) is a real valued linear functional such that ω(a) ≥ 0 whenever a ≥ 0. A state on O(H) is a positive linear functional such that ω(1) = 1 and ω(a) = ω(a^{∗}). The main property that makes states useful in C^{∗} algebra theory is that if a ≠ 0 there exists a state such that ω(a) = ||a||. From this it follows immediately that if ω(a) = 0 for all states ω then a = 0 and this implies that if ω(a) ≤ ω(b) for all states then a ≤ b. In this way verification of inequalities in a C^{∗} algebra is reduced to the verification of numerical inequalities. Also recall that in any real C^{∗} -algebra the following important inequality holds [4]

ω(a^{∗}b^{∗}ba) ≤ ||b^{∗}b||ω(a^{∗}a)

For any given state ω define (s, t)_{ω} = ω(⟨s, t⟩). It is evident that ( ,)_{ω} is a pseudo inner product on V. It therefore satisfy the Cauchy-Swartz inequality (s, t)_{ω}^{2} ≤ (s, s)_{ω}(t, t)_{ω}. Define a = ⟨s, t⟩. We clearly have

ω(aa^{∗}) = ω(a⟨t, s⟩) = ω(⟨at, s⟩) = (at, s)_{ω}.

Therefore

ω(aa^{∗}) ≤ [(at, at)_{ω}(s, s)_{ω}]^{\frac{1}{2}}
= [ω(a⟨t, t⟩a^{∗})(s, s)_{ω}]^{\frac{1}{2}}
= [ω(a|t|^{2}a^{∗})(s, s)_{ω}]^{\frac{1}{2}}
≤ ||⟨t, t⟩||^{\frac{1}{2}} ω(aa^{∗})^{\frac{1}{2}} ω(⟨s, s⟩)^{\frac{1}{2}} .

Dividing by ω(aa^{∗})^{\frac{1}{2}} we find

ω(aa^{∗})^{\frac{1}{2}} ≤ ||t||ω(⟨s, s⟩)^{\frac{1}{2}} = ω(||t||⟨s, s⟩).

The first inequality now follows since this numerical inequality holds for all states ω. As for the second inequality recall that in any real C^{∗}- algebra we have ||aa^{∗}|| = ||a||^{2} and for any pair of operators 0 ≤ a ≤ b we have ||a|| ≤ ||b||. Using this we have

||⟨s, t⟩||^{2} = ||⟨s, t⟩⟨s, t⟩^{∗}|| = ||⟨s, t⟩⟨t, s⟩|| ≤ || |s|^{2}||t||^{2}|| = ||s||^{2}||t|^{2}

and this proves the second inequality. □

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

Corollary 8. || || is a pseudo norm on V.

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

N = {s | ||s|| = 0}.

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

||as||^{2} = ||⟨as, as⟩|| = ||a⟨s, s⟩a^{∗}|| ≤ ||a|| ||s||^{2} ||a^{∗}|| = 0
||s + t|| ≤ ||s|| + ||t|| = 0.

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

H̃ = V/N.

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

||⟨s, t⟩|| ≤ ||s|| ||t|| = 0,

and as a consequence of this ⟨s, t⟩ = 0. We therefore have a well defined operator valued product on H̃ defined through

⟨[s], [t]⟩ = ⟨s, t⟩

This product enjoy the same properties as the product on V and is in addition positive definite. Thus H̃ 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 P_{U} : V → V by

P_{U}(s) = ∑_{i}s_{i}θ_{V_{i}∩U}.

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

Proposition 9. The following properties

P_{U} ∘ P_{U} = P_{U},
P_{U}(as) = aP_{U}(s), ∀a ∈ O(H),
P_{U∩V} = P_{U} ∘ P_{V} ,
⟨P_{U}(s), t⟩ = ⟨s, P_{U}(t)⟩,
⟨s, P_{U}(s)⟩ ≥ 0,
P_{V} + P_{W} = P_{V ∪W} , if V ∩ W = ∅,
⟨P_{U}(s), P_{U}(s)⟩ ≤ ⟨s, s⟩,
||P_{U}(s)|| ≤ ||s||

hold.

The last property shows that if ||s|| = 0 then ||P_{U}(s)|| = 0. Therefore P_{U} induce a well defined map, also denoted by P_{U}, on H̃ through

P_{U}([s]) = [P_{U}(s)].

The last property shows also that the map P_{U} is bounded on H̃. 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 E_{φ}(U) on the Hilbert space H by

E_{φ}(U) = ⟨φ, P_{U}(φ)⟩.

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

Theorem 10. E_{φ} : B(τ) → O(H) is a positive operator valued measure.

Proof. Let first s = ∑_{i}s_{i}θ_{V_{i}} be a element in V with ⟨s, s⟩ = 1 and let {T_{j}}

be a increasing sequence of sets with limit T = ∪_{j}T_{j}. The set of operators {E_{s}(T_{j})} is a increasing sequence of positive operators. The supremum of this sequence exists [1]. Denote the supremum by Sup{E_{s}(T_{j})}. In order to show that E_{s} is a positive operator valued measure we only need to show that

E_{s}(∪_{j}T_{j}) = Sup{E_{s}(T_{j})}.

It is a fact [1] that the sequence E_{s}(T_{j}) converges strongly to the limit Sup{E_{s}(T_{j})}. Since the strong limit is unique when it exists we must only show that E_{s}(T_{j})(x) → E_{s}(∪_{j}T_{j})(x) for all elements x ∈ H. We know that F is a positive operator valued measure so F (T_{j} ∩ V_{i}) → F (T ∩ V_{i}) strongly. But then since all s_{i} are bounded operators we have

s_{i}F (T_{j} ∩ V_{i})s_{i}^{∗}(x) → s_{i}F (T ∩ V_{i})s_{i}^{∗}(x)
⇓
∑_{i}s_{i}F (T_{j} ∩ V_{i})s_{i}^{∗}(x) → ∑_{i}s_{i}F (T ∩ V_{i})s_{i}^{∗}(x)
⇓
E_{s}(T_{j})(x) → E_{s}(T)(x),

for all x ∈ H. This proves that E_{s} is a POV. Next for any element [s] in H̃ we define E_{[s]}(U) = ⟨[s], P_{U}([s])⟩. It is trivial to verify that E_{[s]} = E_{s} so that the previous proof show that E_{[s]} is a POV. Finally let φ be a arbitrary element in H. Then there exists a sequence of elements [s_{n}] in H such that [s_{n}] → φ. Since E_{[s_{n}]} is a POV we know that for all x ∈ H μ_{x}^{n}(U) = ⟨E_{[s_{n}]}(U)x, x⟩_{H} is a measure.

Let μ_{x} be the positive set function defined by

μ_{x}(U) = ⟨E_{φ}(U)x, x⟩_{H}.

By continuity we know that E_{[s_{n}]}(U) → E_{φ}(U) in the uniform norm and thus strongly. But then by continuity of the inner product on H we can conclude that

lim_{n→∞} μ_{x}^{n}(U) = μ_{X}(U),

for all sets U ∈ B(τ). This implies through the Vitali-Hahn-Saks theorem [5] that μ_{x} is a measure and then it follows [1] that E_{φ} is a POV. □

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 ∫ φdF ψ^{∗}and _{∫V} φdF φ^{∗}as follows:

∫ φdF ψ^{∗} =^{def} ⟨φ, ψ⟩,

∫V φdF φ^{∗} =^{def} ⟨φ, P_{V} (φ)⟩.

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

⟨A^{∗}φ, ψ⟩ = ⟨φ, Aψ⟩,

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.

The expectation of a random operator A with respect to a density φ is by definition given by

⟨A⟩ = ⟨φ, Aφ⟩.

The expectation of a random operator with respect to a density φ is thus a operator on H. We can also use the density to define a POV acting on H as we have seen. Note that the expectation of self-adjoint random operators is a self-adjoint 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 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 C^{∗}-algebra are the right 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 Hilbert space with ℝ and the Hilbert module H_{X} with the real Hilbert space of square integrable functions on ℝ. A positive operator valued measure is through the basis identified with a probability measure and therefore for a half density φ ∈ H_{X} the formula E(V) = ⟨φ, P_{V} φ⟩ turns into

μ(V) = ∫ φ^{2}dν.

The half density φ is of course not uniquely determined by the probability measures μ and ν 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 φ 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.

5. The category of extended probability spaces

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 X = ⟨Ω_{X}, B(τ_{X}), μ_{X}⟩ and Y = ⟨Ω_{Y} , B(τ_{Y}), μ_{Y} ⟩ be probability spaces. A morphism f : X → Y is a measurable map f : Ω_{X} → Ω_{Y} such that μ_{Y} is absolutely continuous with respect to the push forward of the measure μ_{X} by f, μ_{Y} ≤ f_{∗}μ_{X}. By the Radon-Nikodym theorem this means that there exists a probability density ρ : Ω_{Y} → ℝ such that

μ_{Y} (V) = _{∫f^{−1}(V)}ρdμ_{X}.

There are several other possibilities for morphisms of probability spaces [11]. We could have required f_{∗}μ_{X} ≤ μ_{Y} or f_{∗}μ_{X} ≈ μ_{Y}. 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 μ_{Y} ≤ f_{∗}μ_{X}.

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 X = ⟨Ω_{X}, B(τ_{X}), F_{X}⟩ be a extended probability space, Y = ⟨Ω_{Y} , B(τ_{Y})⟩ a measurable space and h the 3 tuple h = ⟨f_{h}, g_{h}, φ_{h}⟩ where f_{h} : Ω_{X} → Ω_{Y} is a measurable map,g_{h} : H_{Y} → H_{X} is a isometry and φ_{h} ∈ H_{X} is a element in the Hilbert module corresponding to X. Then the push forward of F_{X} by h is the positive operator valued measure,h_{∗}F_{X}, defined on the measurable space Y by

h_{∗}F_{X}(V) = g_{h}^{∗} ∘ ⟨φ_{h}, P_{f_{h}^{−1}(V)}φ_{h}⟩ ∘ g_{h},

where g_{h}^{∗} is the adjoint of g_{h}.

Note that we have g_{h}^{∗} = g_{h}^{−1} ∘Q_{h} where Q_{h} is the orthogonal projection onto the closed subspace g_{h}(H_{Y}) ⊂ H_{X} and therefore g_{h}^{∗} ∘ g_{h} = 1 and g_{h} ∘ g_{h}^{∗} = Q_{h}.We can now define mappings between extended probability spaces using push forward in a very simple way.

Definition 12. Let X = ⟨Ω_{X}, B(τ_{X}), F_{X}⟩ and Y = ⟨Ω_{Y} , B(τ_{Y}), F_{Y} ⟩ be extended probability spaces. A mapping h : X → Y is a 3 tuple,h, as in the previous definition such that

h_{∗}F_{X} = F_{Y} .

Let us assume that the real Hilbert spaces underlying the extended probability spaces X and Y are one dimensional. If we choose basis for these two spaces we can identify the Hilbert spaces with ℝ, the positive operator valued measures with probability measures μ and ν and the half density φ with a real valued function on Ω_{X}. We must have g_{h} = 1 and the condition for h = ⟨f_{h}, 1, φ_{h}⟩ to be a mapping is

ν(V) = _{∫f_{h}^{−1}(V)}φ_{h}^{2}dμ.

This is of course the condition for f_{h} to be a mapping between the probability spaces ⟨Ω_{X}, B(τ_{X}), μ⟩ and ⟨Ω_{X}, B(τ_{X}), μ⟩ if we identify the classical density with φ_{h}^{2}.

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 h : X → Y and k : Y → Z of extended probability spaces be given. Let us first define a measurable map f_{k∘h}, a isometry g_{k∘h} and a linear map h^{∗} by

f_{k∘h} = f_{k} ∘ f_{h} : Ω_{X} → Ω_{Z},
g_{k∘h} = g_{h} ∘ g_{k} : H_{Z} → H_{X},
h^{∗}(a) = g_{h} ∘ a ∘ g_{h}^{∗} : O(H_{Y}) → O(H_{X}).

The map h^{∗} has the following easily verifiable properties

Proposition 13. The map h^{∗} is bounded and

h^{∗}(a + b) = h^{∗}(a) + h^{∗}(b),
h^{∗}(ab) = h^{∗}(a)h^{∗}(b).

Define a linear map h^{∗} : V_{Y} → H_{X} by

h^{∗}(s) = ∑_{j}h^{∗}(s_{j})P_{f_{h}^{−1}(V_{j})}(φ_{h}),

where s = ∑ s_{j}θ_{V_{j}}. The map h^{∗} has the following important properties

Proposition 14. The map h^{∗} is bounded and

h^{∗}(s + t) = h^{∗}(s) + h^{∗}(t),
h^{∗}(as) = h^{∗}(a)h^{∗}(s),
⟨h^{∗}(s), h^{∗}(t)⟩ = h^{∗}(⟨s, t⟩),
[s] = 0 ⟹ [h^{∗}(s)] = 0,
h^{∗}(P_{V} (s)) = P_{f_{h}^{−1}(V)}(h^{∗}(s)).

Proof. Let s = ∑ s_{i}θ_{V_{i}} and t = ∑ t_{j}θ_{W_{j}}. Then it is easy to verify that {V_{i} ∩ W_{j}} form a partition of Ω_{Y} and that s + t = ∑(s_{i} + t_{j})θ_{V_{i}∩W_{j}}. But then we have

h^{∗}(s + t) = ∑_{i,j}h́^{∗}(s_{i} + t_{j})P_{f_{h}^{−1}(V_{i}∩W_{j})}(φ_{h})

= ∑_{i,j}h^{∗}(s_{i})P_{f_{h}^{−1}(V_{i})∩f_{h}^{−1}(W_{j})}(φ_{h}) + ∑_{i,j}h^{∗}(t_{j})P_{f_{h}^{−1}(V_{i})∩f_{h}^{−1}(W_{j})}(φ_{h})

= ∑_{i}h^{∗}(s_{i})P_{f_{h}^{−1}(V_{i})}(φ_{h}) + ∑_{j}h^{∗}(t_{j})P_{f_{h}^{−1}(W_{j})}(φ_{h}) = h^{∗}(s) + h^{∗}(t).

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

h^{∗}(as) = h^{∗}(∑_{i}as_{i}θ_{V_{i}}) = ∑_{i}h^{∗}(as_{i})P_{f_{h}^{−1}(V_{i})}(φ_{h})
= ∑h^{∗}(a)h^{∗}(s_{i})P_{f_{h}^{−1}(V_{i})}(φ_{h}) = h^{∗}(a)h^{∗}(s),

and

⟨h^{∗}(s), h^{∗}(t)⟩ = ∑_{i,j}⟨h^{∗}(s_{i})P_{f_{h}^{−1}(V_{i})}(φ_{h}), h^{∗}(t_{j})P_{f_{h}^{−1}(W_{j})}(φ_{h})⟩

= ∑_{i,j}h^{∗}(s_{i}) ∘ ⟨φ_{h}, P_{f_{h}^{−1}(V_{i}∩W_{j})}(φ_{h})⟩ ∘ h^{∗}(t_{j})^{∗}

= g_{h} ∘ (∑_{i,j}s_{i} ∘ g_{h}^{∗} ∘ ⟨φ_{h}, P_{f_{h}^{−1}(V_{i}∩W_{j})}(φ_{h})⟩ ∘ g_{h} ∘ t_{j}^{∗}) ∘ g_{h}^{∗}

= g_{h} ∘ (∑_{i,j}s_{i} ∘ h_{∗}F_{X}(V_{i} ∩ W_{j}) ∘ t_{j}^{∗}) ∘ g_{h}^{∗}

= g_{h} ∘ (∑_{i,j}s_{i} ∘ F_{Y} (V_{i} ∩ W_{j}) ∘ t_{j}^{∗}) ∘ g_{h}^{∗}

= g_{h} ∘ ⟨s, t⟩ ∘ g_{h}^{∗} = h^{∗}(⟨s, t⟩)

proves the fourth statement. The first and last statement in the proposition follows from the fourth. Finally

h^{∗}(P_{V} (s)) = h^{∗}(∑_{i}s_{i}θ_{V ∩V_{i}}) = ∑_{i}h^{∗}(s_{i})P_{f_{h}^{−1}(V ∩V_{i})}(φ)

= ∑_{i}h^{∗}(s_{i})P_{f_{h}^{−1}(V)}(P_{f_{h}^{−1}(V_{i})}(φ))) = P_{f_{h}^{−1}(V)}(h^{∗}(s)).

□

Using this proposition we can extend the map h^{∗} to a continuous linear map from H_{Y} to H_{X} . This map is given on the dense set H̃_{Y} by

h^{∗}([s]) = h^{∗}(s).

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

Theorem 15. Let h : X → Y and k : Y → Z be mappings of extended probability spaces. Define φ_{k∘h} ∈ H_{X} by φ_{k∘h} = h^{∗}(φ_{k}). Then

k ∘ h = ⟨f_{k∘h}, g_{k∘h}, φ_{k∘h}⟩

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)_{∗}F_{X} = F_{Z}. But doing this is now a straight forward calculation if we use the previous proposition.

(k ∘ h)_{∗}F_{X}(V)
= g_{k∘h}^{∗} ∘ ⟨φ_{k∘h}, P_{f_{k∘h}^{−1}(V)}(φ_{k∘h})⟩ ∘ g_{k∘h}
= g_{k}^{∗} ∘ g_{h}^{∗} ∘ ⟨h^{∗}(φ_{k}), P_{f_{h}^{−1}(f_{k}^{−1}(V))}(h^{∗}(φ_{k}))⟩ ∘ g_{h} ∘ g_{k}
= g_{k}^{∗} ∘ g_{h}^{∗} ∘ ⟨h^{∗}(φ_{k}), h^{∗}(P_{f_{k}^{−1}(V)}(φ_{k}))⟩ ∘ g_{h} ∘ g_{k}
= g_{k}^{∗} ∘ g_{h}^{∗} ∘ g_{h} ∘ ⟨φ_{k}P_{f_{k}^{−1}(V)}(φ_{k})⟩ ∘ g_{h}^{∗} ∘ g_{h} ∘ g_{k}
= g_{k}^{∗} ∘ ⟨φ_{k}P_{f_{k}^{−1}(V)}(φ_{k})⟩ ∘ g_{k} = F_{Z}(V).

The last statement in the theorem is also proved by direct calculation. Let s = ∑ s_{j}θ_{V_{j}} ∈ V_{Z}. Then we have

(k ∘ h)^{∗}([s])

= ∑_{j}(k ∘ h)^{∗}(s_{j})P_{f_{k∘h}^{−1}(V_{j})}(φ_{k∘h})

= ∑_{j}h^{∗}(k^{∗}(s_{j}))P_{f_{h}^{−1}(f_{k}^{−1}(V_{j}))}(h^{∗}(φ_{k}))

= ∑_{j}h^{∗}(k^{∗}(s_{j}))h^{∗}(P_{f_{k}^{−1}(V_{j})}(φ_{k}))

= h^{∗}(∑_{j}k^{∗}(s_{j})P_{f_{k}^{−1}(V_{j})}(φ_{k})) = h^{∗}(k^{∗}(s)).

Since the identity holds on a dense subset is also holds for all elements in H_{Z} and this proves the theorem. □

We now can use this Theorem to define composition of mappings

Definition 16. Let h : X → Y and k : Y → Z be mappings of extended probability spaces. Then k ∘ h is the composition of k and h.

It is now straight forward to prove that composition of mappings is associative.

Theorem 17. Let h : X → Y , k : Y → Z and r : Z → T be mappings of extended probability spaces. Then we have

r ∘ (k ∘ h) = (r ∘ k) ∘ h.

Proof. Clearly we have f_{r∘(k∘h)} = f_{(r∘k)∘h} and g_{r∘(k∘h)} = g_{(r∘k)∘h}. And from the previous theorem we have

φ_{r∘(k∘h)} = (k ∘ h)^{∗}(φ_{r}) = h^{∗}(k^{∗}(φ_{r}))
φ_{(r∘k)∘h} = h^{∗}(φ_{r∘k}) = h^{∗}(k^{∗}(φ_{r}))
□

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 X = ⟨Ω_{X}, B(τ_{X}), F_{X}⟩ the only reasonable candidate for a unit morphism is

1_{X} = ⟨1_{Ω_{X}}, 1_{H_{X}}, 1_{H_{X}}θ_{Ω_{X}}⟩.

For this mapping it is easy to show that

Proposition 18.

k ∘ 1_{X} = k,
1_{Y} ∘ h = ⟨f_{h}, g_{h}, Q_{h}φ_{h}⟩.

Thus the mapping is not a unit morphism in the categorical sense unless g_{h} 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 h : X → Y , Q_{h} : H_{X} → g_{h}(H_{Y}) is the orthogonal projection on the closed subspace g_{h}(H_{Y}).

Definition 19. Two mappings h, k : X → Y of extended probability spaces are equivalent if

f_{h} = f_{k},
g_{h} = g_{k},
Q_{h}φ_{h} = Q_{k}φ_{k}.

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

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.

Lemma 20. Let h : X → Y and k : Y → Z be mappings of extended probability spaces. Then

Q_{k∘h} = h^{∗}(Q_{k}).

Proof. For any ξ ∈ H_{X} , Q_{k∘h}(ξ) is the unique vector in g_{h}(g_{k}(H_{Z})) such that ξ − Q_{k∘h}(ξ) is orthogonal to g_{h}(g_{k}(H_{Z})). But for any η = g_{h}(g_{k}(α)) in g_{h}(g_{k}(H_{Z})) we have

⟨ξ − h^{∗}(Q_{k})(ξ), η⟩
= ⟨ξ − (g_{h} ∘ Q_{k} ∘ g_{h}^{∗})(ξ), g_{h}(g_{k}(α)))⟩
= ⟨g_{k}^{∗}(g_{h}^{∗}(ξ)) − (g_{k}^{∗} ∘ g_{h}^{∗} ∘ g_{h} ∘ Q_{k} ∘ g_{h}^{∗})(ξ), α⟩
= ⟨g_{k}^{∗}(g_{h}^{∗}(ξ)) − g_{k}^{∗}(g_{h}^{∗}(ξ)), α⟩ = 0.

Therefore by uniqueness Q_{k∘h}(ξ) = h^{∗}(Q_{k})(ξ). □

Lemma 21. Let h, h′ : X → Y be equivalent. Then

h^{∗} = h^{′∗}.

Proof. We only need to verify the identity on the dense subset H̃_{X} ⊂ H_{X}. But for any [s] ∈ H̃_{X} with s = ∑ s_{i}θ_{V_{i}} we have

h^{′∗}([s]) = ∑_{i}h^{′∗}(s_{i})P_{f_{h′}(V_{i})}(φ_{h′})

= ∑_{i}(g_{h′} ∘ s_{i} ∘ g_{h′}^{−1} ∘ Q_{h′})P_{f_{h′}(V_{i})}(φ_{h′})

= ∑_{i}(g_{h} ∘ s_{i} ∘ g_{h}^{−1})Q_{h′}P_{f_{h}(V_{i})}(φ_{h′})

= ∑_{i}(g_{h} ∘ s_{i} ∘ g_{h}^{−1})P_{f_{h}(V_{i})}(Q_{h′}φ_{h′})

= ∑_{i}(g_{h} ∘ s_{i} ∘ g_{h}^{−1})P_{f_{h}(V_{i})}(Q_{h}φ_{h}) = h^{∗}([s]).

□

We can now prove that composition is well defined on classes.

Proposition 22. Let h, h′ : X → Y be equivalent and k, k′ : Y → Z be equivalent. Then

k ∘ h ≈ k′ ∘ h′.

Proof. We only need to prove that φ_{k∘h} = φ_{k′∘h′}. But using the previous two lemmas we have

Q_{k∘h}φ_{k∘h} = h^{∗}(Q_{k})h^{∗}(φ_{k}) = h^{∗}(Q_{k}φ_{k}) = h^{′∗}(Q_{k′}φ_{k′}) = Q_{k′∘h′}(φ_{k′∘h′}).

□

Definition 23. A morphism between extended probability spaces X and Y is a equivalence class, [h],of mappings h : X → Y.

In order to keep the notation simple we will always denote a morphism [h] by a representative mapping h. Thus when we speak of a morphism h we mean the class [h]. 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.

Theorem 24. Extended probability spaces and morphisms form a category.

Proof. We know that composition is well defined and associative. For any object X, let the unit mapping be 1_{X} = ⟨1_{Ω_{X}}, 1_{H_{x}}, 1_{H_{X}}θ_{Ω_{X}}⟩. From proposition 18 we have for any morphisms h : X → Y

h ∘ 1_{X} ≈ h,
1_{Y} ∘ h = ⟨f_{h}, g_{h}, Q_{h}φ_{h}⟩ ≈ h

because Q_{h} 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 probability space X to T. Here T = ⟨Ω_{T} , B_{T} , μ_{T} ⟩ with Ω_{T} = {∗} , B_{T} = {∅, {∗}} and μ_{T} the only possible probability measure on B_{T}. The existence of T 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 T = ⟨Ω_{T} , B_{T} , F_{T} ⟩ where F_{T} : B_{T} → O(ℝ) ≈ ℝ is the only possible positive operator valued measure, F_{T} (Ω_{T}) = 1_{ℝ}. 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 = ⟨f_{h}, g_{h}, φ_{h}⟩ and clearly f_{h} : Ω_{X} → Ω_{T} = {∗} is unique. The map g_{h} : ℝ → H_{X} is a isometry and is therefore determined by a vector ξ_{h} ∈ H_{X} where ⟨ξ_{h}, ξ_{h}⟩ = 1 and g_{h}(1) = ξ_{h}. The vector ξ_{h} and element φ_{h} ∈ H_{X} must satisfies the single condition

h_{∗}F_{X}(Ω_{T}) = F_{T} (Ω_{T}) = 1_{ℝ}.

Using the definition of h_{∗} we find that the following identity must be satisfied

⟨⟨φ_{h}, φ_{h}⟩(ξ_{h}), ξ_{h}⟩ = 1,

and clearly this identity will be satisfied by many choices of φ_{h} and ξ_{h}. Thus the morphism h is not uniquely determined and therefore T 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 X = ⟨Ω_{X}, B(τ_{X}), μ_{X}⟩ define a Hilbert space,denoted by L_{2}(X), by L_{2}(X) = L_{2}(μ_{X}). Let X = ⟨Ω_{X}, B(τ_{X}), μ_{X}⟩ and Y = ⟨Ω_{Y} , B(τ_{Y}), μ_{Y} ⟩ be two probability spaces and let f : Ω_{X} → Ω_{Y} be a morphism of probability spaces in the sense that

μ_{Y} (V) = _{∫f^{−1}(V)}ρdμ_{X}

Define a mapping L_{2}(f) : L_{2}(Y) → L_{2}(X) by

L_{2}(f)(ξ) = \sqrt{ρ}(ξ ∘ f)

It is easy to verify, using the Radon Nikodym theorem, that L_{2}(f) is in fact a isometry and moreover that L_{2} 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 L_{2}, from the category of extended probability spaces to the category of Hilbert spaces that for probability spaces reduce to the functor discussed above.

Let X and Y be extended probability spaces and let L_{2}(X) and L_{2}(Y) be the corresponding Hilbert spaces of random vectors. Informally to any morphism h : X → Y of extended probability spaces we will define a isometry L_{2}(h) : L_{2}(Y) → L_{2}(X) by the formula

L_{2}(h)(ξ)(x) = φ_{h}^{∗}(x)(g_{h}((ξ ∘ f_{h})(x)))

It is easy to see that the mapping L_{2}(f) is a special case of this general formula. Of course we can not use this formula to actually define L_{2}(h) since elements in L_{2}(Y) are not vector functions and elements in H_{X} are not operator valued functions. The action of elements in H_{X} on L_{2}(X) 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 L_{2}(h) exists and that it defines a functor.

Recall that if S_{Y} denote the space of simple H_{Y} valued functions with inner product ⟨v, w⟩ = ∑_{i,j}⟨F_{Y} (V_{i} ∩ T_{j})ξ_{i}, η_{j}⟩_{H_{Y}} then L_{2}(Y) is the closure of T_{Y} = {[v] | v ∈ S_{Y} } where [v] = 0 iff ⟨v, v⟩ = 0. For any extended probability space, V_{X} is the linear space of simple operator valued functions occurring in the construction of the Hilbert module H_{X}. For a measurable map f : Ω_{X} → Ω_{Y},a isometry g : H_{Y} → H_{X} and a element v = ∑_{i} ξ_{i}θ_{V_{i}} ∈ S_{Y} define a linear map t_{v}^{f,g} : V_{X} → L_{2}(X) by

t_{v}^{f,g}(s) = [∑_{i,j}s_{j}^{∗}(g(ξ_{i}))θ_{f^{−1}(V_{i})∩W_{j}}]

where s = ∑_{j} s_{j}θ_{W_{j}} ∈ V_{X}.

Lemma 25. For the linear map t_{v}^{f,g} the following property

⟨t_{v}^{f,g}(s), t_{v}^{f,g}(s)⟩ ≤ c_{v,g}||s||^{2}

holds.

Proof. Let v = ∑_{i} ξ_{i}θ_{V_{i}} and s = ∑_{j} s_{j}θ_{W_{j}}. Then we have

⟨t_{v}^{f,g}(s), t_{v}^{f,g}(s)⟩

= ∑_{i,j}⟨F_{X}(W_{j} ∩ f^{−1}(V_{i}))s_{j}^{∗}(g(ξ_{i})), s_{j}^{∗}(g(ξ_{i}))⟩_{H_{X}}

= ∑_{i,j}⟨(s_{j}F_{X}(W_{j} ∩ f^{−1}(V_{i}))s_{j}^{∗})(g(ξ_{i})), g(ξ_{i})⟩_{H_{X}}

= ∑_{i}⟨⟨s, P_{f^{−1}(V_{i})}(s)⟩(g(ξ_{i})), g(ξ_{i})⟩_{H_{X}}

≤ ∑_{i}⟨⟨s, s⟩(g(ξ_{i})), g(ξ_{i})⟩_{H_{X}} ≤ c_{v,g}||s||^{2}.

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 [s] = 0 then [t_{v}^{f,g}(s)] = 0 and therefore we can extend t_{v}^{f,g} to a bounded linear map t_{v}^{f,g} : H_{X} → L_{2}(X). It is defined on the dense subset H̃_{X} by t_{v}^{f,g}([s]) = [t_{v}^{f,g}(s)].

The following proposition sets the stage for proving the existence of the Naimark functor.

Proposition 26. Let h : X → Y be a mapping of extended probability spaces. Then there exists a isometry L_{2}(h) : L_{2}(Y) → L_{2}(X) that is defined on the dense subset T_{Y} by

L_{2}(h)([v]) = t_{v}^{f_{h},g_{h}}(φ_{h}),

and that satisfy

L_{2}(k ∘ h) = L_{2}(h) ∘ L_{2}(k),
L_{2}(1_{X}) = 1_{L_{2}(X)}.

Proof. We will start by showing that t_{v}^{f_{h},g_{h}} only depends on the class of v. Let {s_{n}} be a sequence of elements in H_{X} converging to φ_{h}. For each n we can define a positive operator valued measure on ⟨Ω_{Y} , B(τ_{Y})⟩ acting on the Hilbert space H_{Y} by

F_{Y}^{n}(V) = g^{∗} ∘ ⟨s_{n}, P_{f^{−1}(V)}(s_{n})⟩ ∘ g.

By continuity F_{Y}^{n}(V) → F_{Y} (V) strongly and thus weakly. But then we have

⟨t_{v}^{f_{h},g_{h}}(φ_{h}), t_{v}^{f_{h},g_{h}}(φ_{h})⟩

= lim_{n→∞}⟨t_{v}^{f_{h},g_{h}}(s_{n}), t_{v}^{f_{h},g_{h}}(s_{n})⟩

= lim_{n→∞} ∑_{i}⟨⟨s_{n}, P_{f^{−1}(V_{i})}(s_{n})⟩(g(ξ_{i})), g(ξ_{i})⟩_{H_{X}}

= lim_{n→∞} ∑_{i}⟨(g^{∗} ∘ ⟨s_{n}, P_{f^{−1}(V_{i})}(s_{n})⟩ ∘ g)(ξ_{i})), ξ_{i}⟩_{H_{Y}}

= lim_{n→∞} ∑_{i}⟨F_{Y}^{n}(V_{i})(ξ_{i}), ξ_{i}⟩_{H_{Y}}

= ∑_{i}⟨F_{Y} (V_{i})ξ_{i}, ξ_{i}⟩_{H_{Y}} = ⟨v, v⟩.

The assumption [v] = 0 means that ⟨v, v⟩ = 0, so t_{v}^{f_{h},g_{h}} depends only on the class of v. Therefore L_{2}(h) is well defined on the dense subset T_{Y} and the argument just given show that it is a isometry. It therefore extends to a isometry from L_{2}(Y) to L_{2}(X).

For the last part of the Theorem let [s_{n}] and [t_{m}] be sequences in H_{X} and H_{Y} converging to φ_{h} and φ_{k}. Here s_{n} = ∑_{l} s_{nl}θ_{W_{nl}} and t_{m} = ∑_{j} t_{mj}θ_{T_{mj}}. For [v] ∈ T_{Z} ⊂ L_{2}(Z) with v = ∑_{i} ξ_{i}θ_{V_{i}} we have by continuity of all maps involved that if we define [u] ∈ T_{Y} ⊂ L_{2}(Y) by

OPERATOR VALUED PROBABILITY THEORY 45

u_{m} = ∑_{i,j}t_{mj}^{∗}(g_{k}(ξ_{i}))θ_{f_{k}^{−1}(V_{i})∩T_{mj}} then we have

L_{2}(h) ∘ L_{2}(k)([v])

= L_{2}(h)(t_{v}^{f_{k},g_{k}}(φ_{k})) = L_{2}(h)(t_{v}^{f_{k},g_{k}}( lim_{m→∞}[t_{m}]))

= lim_{m→∞} L_{2}(h)(t_{v}^{f_{k},g_{k}}([t_{m}]))

= lim_{m→∞} L_{2}(h)(∑_{i,j}t_{mj}^{∗}(g_{k}(ξ_{i}))θ_{f_{k}^{−1}(V_{i})∩T_{mj}}) = lim_{m→∞} L_{2}(h)([u_{m}])

= lim_{m→∞} t_{u_{m}}^{f_{h},g_{h}}(φ_{h}) = lim_{m→∞} lim_{n→∞} t_{u_{m}}^{f_{h},g_{h}}([s_{n}])

= lim_{m→∞} lim_{n→∞} ∑_{i,j,l}(s_{nl}^{∗} ∘ g_{h} ∘ t_{mj}^{∗} ∘ g_{k})(ξ_{i})θ_{f_{h}^{−1}(f_{k}^{−1}(V_{i})∩T_{mj})∩W_{nl}}.

Note that h^{∗}([t_{m}])

= ∑_{j}h^{∗}(t_{mj})P_{f_{h}^{−1}(T_{mj})}(φ_{h})

= ∑_{j}h^{∗}(t_{mj})P_{f_{h}^{−1}(T_{mj})}( lim_{n→∞}[s_{n}])

= lim_{n→∞} ∑_{j,l}h^{∗}(t_{mj})s_{nl}θ_{f_{h}^{−1}(T_{mj})∩W_{nl}}.

We have

L_{2}(k ∘ h)([v]) = t_{v}^{f_{k∘h},g_{k∘h}}(φ_{k∘h}) = t_{v}^{f_{k∘h},g_{k∘h}}(h^{∗}(φ_{k}))

= t_{v}^{f_{k∘h},g_{k∘h}}(h^{∗}( lim_{m→∞}[t_{m}])) = lim_{m→∞} t_{v}^{f_{k∘h},g_{k∘h}}(h^{∗}([t_{m}]))

= lim_{m→∞} t_{v}^{f_{k∘h},g_{k∘h}}( lim_{n→∞} ∑_{j,l}h^{∗}(t_{mj})s_{nl}θ_{f_{h}^{−1}(T_{mj})∩W_{nl}})

= lim_{m→∞} lim_{n→∞} ∑_{i,j,l}(h^{∗}(t_{mj})s_{nl})^{∗}(g_{k∘h}(ξ_{i}))θ_{f_{k∘h}^{−1}(V_{i})∩f_{h}^{−1}(T_{mj})∩W_{nl}}

= lim_{m→∞} lim_{n→∞} ∑_{i,j,l}(s_{nl}^{∗} ∘ g_{h} ∘ t_{mj}^{∗} ∘ g_{h}^{∗} ∘ g_{h} ∘ g_{k})(ξ_{i})θ_{f_{h}^{−1}(f_{k}^{−1}(V_{i})∩T_{mj})∩W_{nl}}

= lim_{m→∞} lim_{n→∞} ∑_{i,j,l}(s_{nl}^{∗} ∘ g_{h} ∘ t_{mj}^{∗} ∘ g_{k})(ξ_{i})θ_{f_{h}^{−1}(f_{k}^{−1}(V_{i})∩T_{mj})∩W_{nl}}.

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.

Theorem 27. L_{2}(h) is a well defined functor from the category of extended probability spaces to the category of Hilbert spaces.

Proof. We only need to prove that L_{2}(h) is well defined for a given morphism h. The functorial properties follows from the previous proposition. Assume h ≈ h′. Let us first assume that the densities of h and h′ are [s] and [s′]. We can without loss of generality assume that s and s′ are of the form

s = ∑_{i}s_{i}θ_{W_{i}},
s′ = ∑_{i}s′_{i}θ_{W_{i}},

since we can bring it to this form by the same construction as in lemma 29. The equivalence then amounts to Q_{h}s_{i} = Q_{h′}s′_{i} for all i. Then on the dense subset T_{Y} ⊂ L_{2}(Y) we have for v = ∑ ξ_{i}θ_{V_{i}} that

L_{2}(h)([v]) = ∑_{i,j}s_{j}^{∗}(g_{h}(ξ_{i}))θ_{f_{h}^{−1}(V_{i})∩W_{j}}

= ∑_{i,j}(s_{j}^{∗} ∘ Q_{h} ∘ g_{h})(ξ_{i})θ_{f_{h}^{−1}(V_{i})∩W_{j}}

= ∑_{i,j}(s_{j}^{′∗} ∘ Q_{h′} ∘ g_{h′})(ξ_{i})θ_{f_{h′}^{−1}(V_{i})∩W_{j}} = L_{2}(h′)([v]).

The case for general densities follows by continuity. □

The Naimark functor L_{2} is not the only functor occurring in this theory. In fact if we recall the properties of the pullback operation h → h^{∗} defined earlier in this section we can define a second functor.

Theorem 28. For any extended probability space X, define a Hilbert module H(X) = H_{X} and for any morphism h : X → Y of extended probability spaces define a morphism of Hilbert modules H(h) = h^{∗}. Then H 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 H(X) and the space of random vectors L_{2}(X) 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.

6. Monoidal structure on the category of extended probability spaces

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 X,Y and Z are objects in the category and if the product is denoted by ⊗ then we require that there exists a isomorphism α_{XY Z} : X ⊗ (Y ⊗ Z) → (X ⊗ Y) ⊗ Z. Similarly if I is the unit object we require that there exists isomorphisms β_{X} : I ⊗ X → X and γ_{X} : X ⊗ I → X. 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 α,γ and β are the following.

For all objects X,Y,Z and T we must have

α_{X⊗Y,Z,T} ∘ α_{X,Y,Z⊗T} = (α_{X,Y,Z} ⊗ 1_{T}) ∘ α_{X,Y ⊗Z,T} ∘ (1_{X} ⊗ α_{Y,Z,T}),
(γ_{X} ⊗ 1_{Y}) ∘ α_{X,I,Y} = 1_{X} ⊗ β_{Y} ,
γ_{I} = β_{I}.

These are the MacLane coherence conditions. The naturality conditions are expressed as follows. For any arrows f : X → X′,g : Y → Y′ and h : Z → Z′ we must have

((f ⊗ g) ⊗ h) ∘ α_{X,Y,Z} = (f ⊗ (g ⊗ h)) ∘ α_{X′,Y′,Z′,}
f ∘ β_{X} = β_{X′} ∘ (1_{I} ⊗ f),
f ∘ γ_{X} = γ_{X′} ∘ (f ⊗ 1_{I}).

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 [8] for a elementary introduction to the categorical view of mathematics, a more advanced introduction can be found 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 of probability spaces is described in detail in [11]. The main features are as follows. For two probability spaces X = ⟨Ω_{X}, B(τ_{X}), μ_{X}⟩ and Y = ⟨Ω_{Y} , B(τ_{Y}), μ_{Y} ⟩ their product is the probability space X ⊗ Y = ⟨, Ω_{X} × Ω_{Y} , B(τ_{X} ⊗ τ_{Y}), μ_{X} ⊗ μ_{Y} ⟩, where μ_{X} ⊗ μ_{Y} is the product measure. The product of two morphisms f : X → Y and g : X′ → Y′ is a morphism f ⊗ g : X ⊗ X′ → Y ⊗ Y′ where f ⊗ g = f × g is just the Cartesian product of the maps f and g. The associativity and unit isomorphisms are just the usual one from the category of sets. α_{XY Z}((x, (y, z))) = ((x, y), z), β_{X}((∗, x)) = x, and γ_{X}((x, ∗)) = x. For the category of probability spaces this choice of α, β and γ are the only possible ones 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 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 X = ⟨Ω_{X}, B(τ_{X}), F_{X}⟩ and Y = ⟨Ω_{Y} , B(τ_{Y}), F_{Y} ⟩ be two extended probability spaces. The product of the two positive operator valued measures F_{X} and F_{Y} always exists and is uniquely determined [1] by its value on measurable boxes by

(F_{X} ⊗ F_{Y})(C × D) = F_{X}(D) ⊗ F_{Y} (D).

The product measure acts on the Hilbert space H_{X} ⊗ H_{Y}. 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 H_{X} ⊗ H_{Y} and H_{X⊗Y}. 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 X, H_{X} is the completion of the dense subspace H̃_{X} = {[s] | s ∈ V_{X}} and

V_{X} = {s = ∑_{i}s_{i}θ_{V_{i}}|s_{i} ∈ O(H_{X}), {V_{i}} is a B(τ_{X}) measurable partition of Ω_{X}}

is the real linear space of simple O(H_{X}) valued measurable functions on Ω_{X}.

For a pair of extended probability spaces define a map γ_{XY} : V_{X}×V_{Y} → V_{X⊗Y} by

γ_{XY} (s, t) = ∑_{i,j}(s_{i} ⊗ t_{j})θ_{V_{i}×W_{j}},

where s = ∑ s_{i}θ_{V_{i}} and t = ∑ t_{j}θ_{W_{j}}.

For this map we have the following

Lemma 29. The map γ is bilinear and if [s] = 0 or [t] = 0 then [γ(s, t)] = 0.

Proof. We evidently have γ_{XY} (as, t) = γ_{XY} (s, at) for all real numbers a. Let s = ∑_{i=1}^{n} s_{i}θ_{V_{i}} and r = ∑_{k=1}^{m} r_{k}θ_{C_{k}} be two elements in V_{X}. Define a new sequence of sets {A_{l}} where A_{l} = V_{l} for l = 1..n and A_{l} = C_{l−n} for l = n + 1, ....n + m and let L = {1, 2, ...n + m}. Let S = {σ : L → ℤ_{2}} be the set of all ℤ_{2} = {−1, +1} valued functions on the index set L. The set S is a index set for a new partition, {T^{σ}}_{σ∈S} of the set Ω_{X} defined by

T^{σ} = ∩_{l∈L}A_{l}^{σ(l)},

where for any set U we define U^{+1} = U and U^{−1} = U^{c}, the complement of U. We evidently have

V_{i} = ∪_{{σ|σ(i)=1}}T^{σ},
C_{k} = ∪_{{σ|σ(n+k)=1}}T^{σ}.

Therefore

s + r = ∑_{σ} ( ∑_{{i|σ(i)=1}}s_{i} + ∑_{{k|σ(k+n)=1}}r_{k}) θ_{T^{σ}}.

But then we have for any t = ∑ t_{j}θ_{W_{j}} ∈ V_{Y} that

γ_{XY} (s + r, t)

= ∑_{σ,j} (( ∑_{{i|σ(i)=1}}s_{i} + ∑_{{k|σ(k+n)=1}}r_{k}) ⊗ t_{j}) θ_{T^{σ}×W_{j}}

= ∑_{σ,j} ∑_{{i|σ(i)=1}}(s_{i} ⊗ t_{j})θ_{T^{σ}×W_{j}} + ∑_{σ,j} ∑_{{k|σ(k+n)=1}}(r_{k} ⊗ t_{j})θ_{T^{σ}×W_{j}}

= ∑_{i,j}(s_{i} ⊗ t_{j}) ∑_{{σ|σ(i)=1}}θ_{T^{σ}×W_{j}} + ∑_{k,j}(r_{k} ⊗ t_{j}) ∑_{{σ|σ(n+k)=1}}θ_{T^{σ}×W_{j}}

= ∑_{i,j}(s_{i} ⊗ t_{j})θ_{V_{i}×W_{j}} + ∑_{k,j}(r_{k} ⊗ t_{j})θ_{C_{k}×W_{j}} = γ_{XY} (s, t) + γ_{XY} (r, t).

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

⟨γ_{XY} (s, t), γ_{XY} (s, t)⟩

= ∑_{i,j,k,l}(s_{i} ⊗ t_{j})F_{X⊗Y} ((V_{i} × W_{j}) ∩ (V_{k} × W_{l}))(s_{k} ⊗ t_{l})^{∗}

= ∑_{i,j,k,l}(s_{i} ⊗ t_{j})(F_{X}(V_{i} ∩ V_{k}) ⊗ F_{Y} (W_{j} ∩ W_{l}))(s_{k}^{∗} ⊗ t_{l}^{∗})

= ∑_{i,j}(s_{i} ⊗ t_{j})(F_{X}(V_{i}) ⊗ F_{Y} (W_{j}))(s_{i}^{∗} ⊗ t_{j}^{∗})

= (∑_{i}s_{i}F_{X}(V_{i})s_{i}^{∗}) ⊗ (∑_{j}t_{j}F_{Y} (W_{j})t_{j}^{∗}) = ⟨s, s⟩ ⊗ ⟨t, t⟩.

But [s] = 0 implies that ⟨s, s⟩ = 0 and the identity just derived then implies that ⟨γ_{XY} (s, t), γ_{XY} (s, t)⟩ = 0 and therefore by definition [γ_{XY} (s, t)] = 0. □

Using the lemma we have a well linear map, also denoted by γ_{XY}, from

H̃_{X} ⊗ H̃_{Y} to { \atop H_{X⊗Y}}

γ_{XY} ([s] ⊗ [t]) = [γ_{XY} (s, t)].

The map γ_{XY} satisfy the following important identity

Lemma 30.

⟨γ_{XY} (v), γ_{XY} (v)⟩ = ⟨v, v⟩.

Proof. Any v ∈ H̃_{X}⊗H̃_{Y} is of the form v = ∑_{i}s_{i}⊗t_{i} where s_{i} = ∑_{j}s_{ij}θ_{V_{ij}}

and t_{i} = ∑_{k}t_{ik}θ_{W_{ik}}. But then we have

⟨γ_{XY} (v), γ_{XY} (v)⟩

= ∑_{i,j,k,l,m,n}(s_{ij} ⊗ t_{ik})F_{X⊗Y} ((V_{ij} × W_{ik}) ∩ (V_{lm} × W_{ln}))(s_{lm} ⊗ t_{ln})^{∗}

= ∑_{i,j,k,l,m,n}(s_{ij} ⊗ t_{ik})(F_{X}(V_{ij} ∩ V_{lm}) ⊗ F_{Y} (W_{ik} ∩ W_{ln}))(s_{lm}^{∗} ⊗ t_{ln}^{∗})

= ∑_{i,l} (∑_{j,m}s_{ij}F_{X}(V_{ij} ∩ V_{lm})s_{lm}^{∗}) ⊗ (∑_{k,n}t_{ik}F_{Y} (W_{ik} ∩ W_{ln})t_{ln}^{∗})

= ∑_{i,l}⟨s_{i}, s_{l}⟩ ⊗ ⟨t_{i}, t_{l}⟩ = ∑_{i,l}⟨s_{i} ⊗ t_{i}, s_{l} ⊗ t_{l}⟩ = ⟨v, v⟩.

□

We can now state and prove the main property of γ_{XY}. First we will recall some facts about (external) tensor products of Hilbert modules. Let H_{X} ⊗_{H} H_{Y} denote the tensor product of H_{X} and H_{Y},as real vector spaces, with topology determined by the norm induced from the operator valued inner product ⟨φ ⊗ ψ, φ′ ⊗ ψ′⟩ = ⟨φ, φ′⟩ ⊗ ⟨ψ, ψ′⟩. The completion of H_{X} ⊗_{H} H_{Y} is the external tensor product [2] of the Hilbert modules H_{X} and H_{Y} and will be denoted by H_{X} ⊗ H_{Y}. It is a module over the spatial tensor product O(H_{X}) ⊗ O(H_{Y}) [12] of the represented C^{∗}− algebras O(H_{X}) and O(H_{X}).

Proposition 31. There exists an injective morphism of Hilbert modules γ_{XY} : H_{X} ⊗ H_{Y} → H_{X⊗Y} such that

⟨γ_{XY} (v), γ_{XY} (v)⟩ = ⟨v, v⟩.

H̃_{X} ⊗_{H} H̃_{Y} is a dense subspace of H_{X} ⊗H_{Y} and on this dense subspace γ_{XY} is given by

γ_{XY} ([s] ⊗ [t]) = [γ_{XY} (s, t)].

Proof. Let H̃_{X} ⊗_{π} H̃_{Y} and H_{X} ⊗_{π} H_{Y} be the projective tensor products [6] of the underlying real vector spaces. Note that the tensor product spaces have not been completed with respect to the projective norm. The embedding H̃_{X} ⊗_{π} H̃_{Y} ↪ H_{X} ⊗_{π} H_{Y} is know to exist and be dense [6]. The norm on H̃_{X} ⊗_{H} H̃_{Y} and H_{X} ⊗_{H} H_{Y} 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 H̃_{X} ⊗_{H} H̃_{Y} is a dense subspace of H_{X} ⊗_{H} H_{Y} and thus by completion in H_{X} ⊗H_{Y}. By the previous lemma γ_{XY} is bounded and therefore extends uniquely to a bounded map γ_{XY} : H_{X} ⊗ H_{Y} → H_{X⊗Y}. 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

Lemma 32. For any measurable sets C ∈ B(τ_{X}) and D ∈ B(τ_{Y}) we have the identity

γ_{XY} ∘ (P_{C} ⊗ P_{D}) = P_{C×D} ∘ γ_{XY}

Proof. For C ∈ B(τ_{X}) and D ∈ B(τ_{Y}) we have

(γ_{XY} ∘ (P_{C} ⊗ P_{D}))([s] ⊗ [t]) = γ_{XY} ([P_{C}(s)] ⊗ [P_{D}(t)])

= ∑_{i,j}(s_{i} ⊗ t_{j})θ_{(V_{i}∩C)×(W_{j}∩D)}

= ∑_{i,j}(s_{i} ⊗ t_{j})θ_{(V_{i}×W_{j})∩(C×D)} = P_{C×D}(γ_{XY} ([s] ⊗ [t]).

By continuity and density we can conclude that the identity γ_{XY} ∘ (P_{C} ⊗ P_{D}) = P_{C×D} ∘ γ_{XY} holds on H_{X} ⊗ H_{Y}. □

Let now h : X → Y and k : X′ → Y′ be morphisms of extended probability spaces. We thus have h = ⟨f_{h}, g_{h}, φ_{h}⟩ and k = ⟨f_{k}, g_{k}, φ_{k}⟩ where φ_{h} ∈ H_{X} and φ_{k} ∈ H_{X′}. Define a 3-tuple h ⊗ k by

h ⊗ k = ⟨f_{h⊗h}, g_{h⊗k}, φ_{h⊗k}⟩,

where f_{h⊗k} = f_{h} × f_{k} , g_{h⊗k} = g_{h} ⊗ g_{k} and φ_{h⊗k} = γ_{XX′}(φ_{h} ⊗ φ_{k}). Then we have

Proposition 33. h ⊗ k : X ⊗ X′ → Y ⊗ Y′ is a morphism of extended probability spaces.

Proof. We need to prove that (h ⊗ k)_{∗}F_{X⊗X′} = F_{Y ⊗Y′}. But this is true because

(h ⊗ k)_{∗}F_{X⊗X′}(C × D) = g_{h⊗k}^{∗} ∘ ⟨φ_{h⊗k}, P_{f_{h⊗k}^{−1}(C×D)}(φ_{h⊗k})⟩ ∘ g_{h⊗k}

= (g_{h} ⊗ g_{k})^{∗} ∘ ⟨γ_{XX′}(φ_{h} ⊗ φ_{k}), (P_{f_{h⊗k}^{−1}(C×D)} ∘ γ_{XX′})(φ_{h} ⊗ φ_{k}))⟩ ∘ (g_{h} ⊗ g_{k})

= (g_{h}^{∗} ⊗ g_{k}^{∗}) ∘ ⟨γ_{XX′}(φ_{h} ⊗ φ_{k}), (γ_{XX′} ∘ (P_{f_{h}^{−1}(C)} ⊗ P_{f_{k}^{−1}(D)}))(φ_{h} ⊗ φ_{k})⟩ ∘ (g_{h} ⊗ g_{k})

= (g_{h}^{∗} ⊗ g_{k}^{∗}) ∘ ⟨φ_{h} ⊗ φ_{k}, P_{f_{h}^{−1}(C)}(φ_{h}) ⊗ P_{f_{k}^{−1}(D)}(φ_{k}))⟩ ∘ (g_{h} ⊗ g_{k})

= (g_{h}^{∗} ∘ ⟨φ_{h}, P_{f_{h}^{−1}(C)}(φ_{h})⟩ ∘ g_{h}) ⊗ (g_{k}^{∗} ∘ ⟨φ_{k}, P_{f_{k}^{−1}(D)}(φ_{k})⟩ ∘ g_{k})

= (h_{∗}F_{X})(C) ⊗ (k_{∗}F_{X′})(D) = F_{Y} (C) ⊗ F_{Y′}(D) = F_{Y ⊗Y′}(C × D),

where we have used the previous lemma. This proves that h ⊗ k 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 h ≈ h′ and k ≈ k′. We need to show that h⊗k ≈ h′⊗k′ and this amounts to proving that Q_{h⊗k}φ_{h⊗k} = Q_{h′⊗k′}φ_{h′⊗k′}. But from the identity (g_{h} ⊗ g_{k})(H_{X} ⊗ H_{X′}) = g_{h}(H_{X}) ⊗ g_{k}(H_{X′}) we have Q_{h⊗k} = Q_{h} ⊗ Q_{k} and the rest 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 34.

γ_{XX′} ∘ (h^{∗} ⊗ k^{∗}) = (h ⊗ k)^{∗} ∘ γ_{Y Y′}

Proof. By continuity we only need to prove the identity on the dense subset H̃_{Y} ⊗_{H} H̃_{Y′} ⊂ H_{Y} ⊗ H_{Y′}. But on this subset we have

((h ⊗ k)^{∗} ∘ γ_{Y Y′})([s] ⊗ [t]) = (h ⊗ k)^{∗}(γ_{Y Y′}(s, t))

= ∑_{i,j}(h ⊗ k)^{∗}(s_{i} ⊗ t_{j})P_{(f_{h}×f_{k})^{−1}(V_{i}×W_{j})}(φ_{h⊗k})

= ∑_{i,j}(h^{∗}(s_{i}) ⊗ k^{∗}(t_{j}))(P_{(f_{h}×f_{k})^{−1}(V_{i}×W_{j})} ∘ γ_{XX′})(φ_{h} ⊗ φ_{k})

= ∑_{i,j}(h^{∗}(s_{i}) ⊗ k^{∗}(t_{j}))(γ_{XX′} ∘ (P_{f_{h}^{−1}(V_{i})} ⊗ P_{f_{k}^{−1}(W_{j})}))(φ_{h} ⊗ φ_{k})

= γ_{XX′}(∑_{i,j}(h^{∗}(s_{i})P_{f_{h}^{−1}(V_{i})}(φ_{h})) ⊗ (k^{∗}(t_{j})P_{f_{k}^{−1}(W_{j})}(φ_{k})))

= (γ_{XX′} ∘ (h^{∗} ⊗ k^{∗}))([s] ⊗ [t]).

□

We can now prove our first main result in this section

Theorem 35. The operation ⊗ is a bifunctor on the category of extended probability spaces.

(h′ ⊗ k′) ∘ (h ⊗ k) = (h′ ∘ h) ⊗ (k′ ∘ k),
1_{X} ⊗ 1_{Y} = 1_{X⊗Y} .

Proof. The unit property is trivial to verify and for the first identity we only need to prove that γ_{XX′}(φ_{k∘h} ⊗ φ_{k′∘h′}) = (h ⊗ h′)^{∗}(φ_{k⊗k′}). But using the previous lemma we have

γ_{XX′}(φ_{k∘h} ⊗ φ_{k′∘h′})
= γ_{XX′}(h^{∗}(φ_{k}) ⊗ h^{′∗}(φ_{k′}))
= (γ_{XX′} ∘ (h^{∗} ⊗ h^{′∗}))(φ_{k} ⊗ φ_{k′})
= ((h^{∗} ⊗ h^{′∗}) ∘ γ_{Y Y′})(φ_{k} ⊗ φ_{k′})
= (h^{∗} ⊗ h^{′∗})(φ_{k⊗k′}).

□

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 T discussed previously. For any objects X, Y and Z define

η_{X} = ⟨f_{η_{X}}, g_{η_{X}}, φ_{η_{X}}⟩,
γ_{X} = ⟨f_{γ_{X}}, g_{γ_{X}}, φ_{γ_{X}}⟩,
α_{XY Z} = ⟨f_{α_{XY Z}}, h_{α_{XY Z}}, φ_{α_{XY Z}}⟩,

OPERATOR VALUED PROBABILITY THEORY 55

where

f_{η_{X}}(∗, x) = f_{γ_{X}}(x, ∗) = x,
f_{α_{XY Z}}((x, (y, z)) = ((x, y), z),
g_{η_{X}}(ξ) = 1 ⊗ ξ,
g_{γ_{X}}(ξ) = ξ ⊗ 1,
g_{α_{XY Z}}(ξ ⊗ (ξ′ ⊗ ξ′′)) = (ξ ⊗ ξ′) ⊗ ξ′′,
φ_{η_{X}} = 1_{H_{T ⊗X}},
φ_{γ_{X}} = 1_{H_{X⊗T}} ,
φ_{α_{XY Z}} = 1_{H_{X⊗(Y ⊗Z)}}.

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.

Theorem 36. η_{X}, γ_{X} and α_{XY Z} are morphisms of extended probability spaces

η_{X} : T ⊗ X → X,
γ_{X} : X ⊗ T → X,
α_{XY Z} : X ⊗ (Y ⊗ Z) → (X ⊗ Y) ⊗ Z,

and are the components of natural isomorphisms. Furthermore ⟨⊗, T, η, γ, α⟩ is a monoidal structure on the category of extended probability spaces.

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56 P. K. JAKOBSEN AND V. V. LYCHAGIN

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

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University of Tromso,9020 Tromso, Norway E-mail address: perj@math.uit.no E-mail address: lychagin@mat-stat.uit.no

Received October 1, 2004