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

© P. K. Jakobsen and V. V. Lychagin

Per K. Jakobsen and Valentin V. Lychagin

OPERATOR VALUED PROBABILITY THEORY

ABSTRACT. We outline an extention of probability theory based on positive operator valued measures. We generalize the main notions from probability theory such as random variables, conditional expectations, densities and mappings. We introduce a product of extended probability spaces and mappings, and show that the resulting structure is a monoidal category, just as in the classical theory.

Contents

1. Introduction

2. Extended probability spaces

3. Random vectors

3.1. The space of random vectors

3.2. The expectation of random vectors

3.3. Conditional expectation

4. Densities and random operators

4.1. The Hilbert module of half densities

4.2. Random operators

5. The category of extended probability spaces

5.1. Morphisms of extended probability spaces

5.2. The Naimark functor

6. Monoidal structure on the category of extended probability spaces

6.1. Product of extended probability spaces and morphisms

6.2. The monoidal structure

References

1. Introduction

2. Extended probability spaces

3. Random vectors

3.1. The space of random vectors

3.2. The expectation of random vectors

3.3. Conditional expectation

4. Densities and random operators

4.1. The Hilbert module of half densities

4.2. Random operators

5. The category of extended probability spaces

5.1. Morphisms of extended probability spaces

5.2. The Naimark functor

6. Monoidal structure on the category of extended probability spaces

6.1. Product of extended probability spaces and morphisms

6.2. The monoidal structure

References

In this paper we present an extension of standard probability theory. An extended probability space is defined to be a normalized positive operator valued measure defined on a measurable space of events. This notion of extended probability space includes probability spaces and spectral measures as important special cases. The use of the word probability in this context is justified by showing that extended probability spaces enjoy properties analog to all the basic properties of classical probability spaces. Random vectors are defined as a generalization of the usual Hilbert space of square integrable functions. This generalization is well known in the literature and was first described by Naimark. Expectation and conditional expectation is defined for extended probability spaces by orthogonal projections in complete analogy with probability spaces.

The introduction of probability densities presents special problems in the context of extended probability spaces. For the case of probability spaces a probability density is any normalized positive integrable function, whereas for the case of extended probability spaces it turns out that the right notion is not a density but a half density. These half densities are elements in a Hilbert module of length one. Special cases of such half densities are well known in quantum mechanics where they are called wave functions. We define a random operator to be a linear operator on the space of half densities. The expectation of random operators are operators acting on the Hilbert space underlying the extended probability space. For the case of probability spaces the notion of random vectors and random operators coincide.

We introduce mappings or morphisms of extended probability spaces through a generalization of the notion of absolute continuity in probability theory. Half densities plays a pivotal role in this generalization. We show that the morphisms can be composed and that extended probability spaces and morphisms forms a category just as for probability spaces. The Naimark construction extends to morphisms and in fact defines a functor on the category of extended probability spaces.

Extended probability spaces can be multiplied and we furthermore show that this multiplication can be extended to morphisms in such a way that it defines a monoidal structure on the category of extended probability spaces. This is in complete analogy with the case of probability spaces and testify strongly to the naturalness of our constructions.

We do not in this paper attempt to give any interpretation of extended probabilities beyond the one implied by the strong structural analogies that we have shown to exists between the categories of probability spaces and extended probability spaces. It is well known that the interpretation of the classical Kolmogorov formalism for standard probability theory is not without controversy as the old debate between frequentists and Bayesians, among others, clearly demonstrate. Our theory of extended probability spaces is evidently a generalization of the Kolmogorov framework and it might be hoped that this enlarged framework will put some of the controversy in a different light. As a case in point note that extended probabilities are in general only partially ordered. The notion of partially ordered probabilities has been discussed and argued over for a very long time. In our theory of extended probability spaces, ordered and partially ordered probabilities lives side by side and enjoy the same formal categorical properties.

In this section we will make some technical assumptions that will assumed to hold throughout this paper. These assumptions are not necessarily the most general ones possible.

A measurable space [5] is a pair

Let now

where the supremum is taken with respect to the usual partial ordering of
self adjoint operators. The supremum always exists since the sequence

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

We are now ready to define our first main object

Definition 1. A extended probability space

Note that a probability space

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

where

where

for all

The last identity follows from the fact the

Since

We also have a spectral measure

where

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

As our first example let

For this case we have

where for any

As our second example let

where

Any function

Similar expressions for the inner product in

3.2. The expectation of random vectors.
Recall that we have a isometric embedding

Note that the image

Definition 2. The expectation of a random vector

The following result is a immediate consequence of the definition

Proposition 3. The expectation is a surjective continuous linear map

Note that adjointness condition uniquely determines the expectation. In
fact we could define the expectation to be the adjoint of the embedding

Using this proposition it is easy to verify that the expectation of a simple function
element

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

Note that it is natural to put the differential

Let

3.3. Conditional expectation.
Let

Definition 4. The conditional expectation of a element

It is evident that

Proof. Let

for all

Let

Clearly

Let

Thus

Therefore

The assumption in the proposition holds for example if

Let

for any

This systems does not have a unique solution in

When dim

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

If we try to generalize this formula directly to the case of POV measures we run into problems.

Let

There is nothing inconsistent in this definition,
the only problem is that it is very limited. In fact if

Now if the numbers

where

There is however a natural way out of these problems. It is very simple to verify that if

| (1) |

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

| (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

where

This action clearly makes

where

Proposition 6. The following properties

hold.

Thus the product is like a Hermitian product where the role
of complex numbers are played by the elements of the real

Note that the product we have constructed is not positive
definite. In fact, since the sum of positive operators in a real

These identities can easily be satisfied for nonzero operators

For any element

where

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

hold.

Proof. A positive linear functional,

For any given state

Therefore

Dividing by

The first inequality now follows since this numerical inequality holds for all
states

and this proves the second inequality. □

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

Let

For any operator

Thus

Elements in

and as a consequence of this

This product enjoy the same properties as the product on

We have now made sense of equation
(

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

This map is clearly a

Proposition 9. The following properties

hold.

The last property shows that if

The last property shows also that the map

Let now

Clearly

Proof. Let first

It is a fact [1] that the sequence

for all

Let

By continuity we know that

for all sets

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

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

4.2. Random operators.
Recall [2] that a map

for all elements

The expectation of a random operator

The expectation of a random operator with respect to a density

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

Let us assume that the real Hilbert space underlying the extended probability
space

The half density

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

There are several other possibilities for morphisms of probability spaces [11]. We could
have required

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

where

Note that we have

Definition 12. Let

Let us assume that the real Hilbert spaces underlying the extended probability
spaces

This is of course the condition for

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

The map

Define a linear map

where

Proposition 14. The map

Proof. Let

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

and

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

Using this proposition we can extend the map

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

is a mapping of extended probability spaces

Proof. In order to show that

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

Let

Since the identity holds on a dense subset is also holds for all elements in

We now can use this Theorem to define composition of mappings

Definition 16. Let

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

Theorem 17. Let

Proof. Clearly we have

Extended probability spaces and mappings of extended probability spaces does unfortunately not form a category, we will in general not have unit morphisms.

For a given extended probability space

For this mapping it is easy to show that

Thus the mapping is not a unit morphism in the categorical sense unless

Recall that for any mapping

Definition 19. Two mappings

If

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

Proof. For any

Therefore by uniqueness

Proof. We only need to verify the identity on the dense subset

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

Proposition 22. Let

Proof. We only need to prove that

□

Definition 23. A morphism between extended probability spaces

In order to keep the notation simple we will always denote a morphism

We can now formulate the main result of this subsection.

Proof. We know that composition is well defined and associative. For any object

because

We know that the category of probability spaces[11] has a terminal object,

Let

Using the definition of

and clearly this identity will be satisfied by many choices of

5.2. The Naimark functor. In probability theory there is a certain functor that plays a major role in the theory. We will now review the construction of this functor and show that a analog functor is defined on the category of extended probability spaces. The existence of this functor testify to the naturalness of our constructions. The functor will be called the Naimark functor since the Naimark dilatation construction plays a major role in its construction.

Let us start with a review of the functor for the
case of probability spaces. For any probability space

Define a mapping

It is easy to verify, using the Radon Nikodym theorem, that

Let

It is easy to see that the mapping

Recall that if

where

Proof. Let

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

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

Proposition 26. Let

and that satisfy

Proof. We will start by showing that

By continuity

The assumption

For the last part of the Theorem let

Note that

We have

The last statement of the theorem is verified by a trivial calculation. □

We are now finally ready to prove the existence of the Naimark functor.

Theorem 27.

Proof. We only need to prove that

since we can bring it to this form by the same construction
as in lemma 29. The equivalence then amounts to

The case for general densities follows by continuity. □

The Naimark functor

Theorem 28. For any extended probability space

For the case of probability spaces the Hilbert module

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

For all objects

These are the MacLane coherence conditions. The naturality
conditions are expressed as follows. For any arrows

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

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

The product measure acts on the Hilbert space

Recall that for any extended probability space

is the real linear space of simple

For a pair of extended probability spaces define a map

where

For this map we have the following

Proof. We evidently have

where for any set

Therefore

But then we have for any

This show that

But

Using the lemma we have a well linear map, also denoted by

The map

Proof. Any

We can now state and prove the main property of

Proposition 31. There exists an injective morphism of Hilbert modules

Proof. Let

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

Proof. For

By continuity and density we can conclude that the identity

Let now

where

Proof. We need to prove that

where we have used the previous lemma. This proves that

Having proved that

Proof. By continuity we only need to prove the identity on the dense subset

We can now prove our first main result in this section

Theorem 35. The operation

Proof. The unit property is trivial to verify and for the first identity we only need to
prove that

6.2. The monoidal structure.
Showing that

The only reasonable candidate for a unit object is clearly the extended probability space

where

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.

and are the components of natural isomorphisms. Furthermore

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

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

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

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

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

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

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

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

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

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

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

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

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

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