In classical probability theory, we start with a sample space , a collection of events, which is a sigma-algebra on , and a probability measure on . The triple is a probability space, and the collection of bounded complex-valued random variables on the probability space forms a commutative algebra under pointwise addition and products. The measure defines an expectation, or integral with respect to , which is a linear map
In this post I provide definitions of probability spaces from the algebraic viewpoint. Statements of some of their first properties will be given in order to justify and clarify the definitions, although any proofs will be left until later posts. In the algebraic setting, we begin with a *-algebra , which takes the place of the collection of bounded random variables from the classical theory. It is not necessary for the algebra to be represented as a space of functions from an underlying sample space. Since the individual points constituting the sample space are not required in the theory, this is a pointless approach. By allowing multiplication of `random variables’ in to be noncommutative, we incorporate probability spaces which have no counterpart in the classical setting, such as are used in quantum theory. The second and final ingredient is a state on the algebra, taking the place of the classical expectation operator. This is a linear map satisfying the positivity constraint and, when is unitial, the normalisation condition . Algebraic, or noncommutative probability spaces are completely described by a pair consisting of a *-algebra and a state . Noncommutative examples include the *-algebra of bounded linear operators on a Hilbert space with pure state for a fixed unit vector . Continue reading “Noncommutative Probability Spaces” →
As is well known, the space of bounded linear operators on any Hilbert space forms a *-algebra, and (pure) states on this algebra are defined by unit vectors. Considering a Hilbert space , the space of bounded linear operators is denoted as . This forms an algebra under the usual pointwise addition and scalar multiplication operators, and involution of the algebra is given by the operator adjoint,
for any and all . A unit vector defines a state by .
The Gelfand-Naimark–Segal (GNS) representation allows us to go in the opposite direction and, starting from a state on an abstract *-algebra, realises this as a pure state on a *-subalgebra of for some Hilbert space .
Consider a *-algebra and positive linear map . Recall that this defines a semi-inner product on the *-algebra , given by . The associated seminorm is denoted by , which we refer to as the -seminorm. Also, every defines a linear operator on by left-multiplication, . We use to denote its operator norm, and refer to this as the -seminorm. An element is bounded if is finite, and we say that is bounded if every is bounded.
Theorem 1 Let be a bounded *-probability space. Then, there exists a triple where,
- is a Hilbert space.
- is a *-homomorphism.
- satisfies for all .
- is cyclic for , so that is dense in .
Furthermore, this representation is unique up to isomorphism: if is any other such triple, then there exists a unique invertible linear isometry of Hilbert spaces such that
Continue reading “The GNS Representation” →
Given two *-probability spaces and , we want to consider maps . For example, we can look homomorphisms, which preserve the *-algebra operations, and can also consider restricting to state-preserving maps satisfying . In algebraic probability theory, however, it is often necessary to include a continuity condition, leading to the idea of normal maps, which I look at in this post. In fact, as we will see, all *-homomorphisms between commutative probability spaces which preserve the state are normal, so this concept is most important in the noncommutative setting.
In contrast to the previous few posts on algebraic probability, the current post is a bit of a gear-change. We are still concerned with with the basic concepts of *-algebras and states. However, the main theorem stated below, which reduces to the Radon-Nikodym theorem in the commutative case, is deeper and much more difficult to prove than the relatively simple results with which I have been concerned with so far. Continue reading “Normal Maps” →
I previously introduced the concept of a *-probability space as a pair consisting of a state on a *-algebra . As we noted, this concept is rather too simplistic to properly capture a noncommutative generalisation of classical probability spaces, and I will later give conditions for to be considered as a true probability space. For now, I continue the investigation of these preprobability spaces, and will look at homomorphisms in this post.
A *-homomorphism between *-algebras and is a map preserving the algebra operations,
for all and . The term `*-homomorphism’ is used to distinguish it from the concept of simple algebra homomorphisms which need not preserve the involution (the third identity above). Next, I will say that is a homomorphism of *-probability spaces and if it is a *-homomorphism from to which preserves the state,
for all .
Now, recall that for any *-probability space , we define a semi-inner product on and the associated seminorm, . Homomorphisms of *-probability spaces are clearly -isometries,
For each , the seminorm is defined as the operator norm of the left-multiplication map on , considered as a vector space with the seminorm. Homomorphisms of *-probability spaces do not need to be -isometric.
Lemma 1 If is a homomorphism of *-probability spaces then, for any ,
Continue reading “Homomorphisms of *-Probability Spaces” →
So far, we have been considering positive linear maps on a *-algebra. Taking things a step further, we want to consider positive maps which are normalized so as to correspond to expectations under a probability measure. That is, we require , although this is only defined for unitial algebras. I use the definitions and notation of the previous post on *-algebras.
Definition 1 A state on a unitial *-algebra is a positive linear map satisfying .
Examples 3 and 4 of the previous post can be extended to give states.
Example 1 Let be a probability space, and be the bounded measurable maps . Then, integration w.r.t. defines a state on ,
Example 2 Let be an inner product space, and be a *-algebra of the space of linear maps as in example 2 of the previous post, and including the identity map . Then, any with defines a state on ,
Continue reading “States on *-Algebras” →
After the previous posts motivating the idea of studying probability spaces by looking at states on algebras, I will now make a start on the theory. The idea is that an abstract algebra can represent the collection of bounded, and complex-valued, random variables, with a state on this algebra taking the place of the probability measure. By allowing the algebra to be noncommutative, we also incorporate quantum probability.
I will take very small first steps in this post, considering only the basic definition of a *-algebra and positive maps. To effectively emulate classical probability theory in this context will involve additional technical requirements. However, that is not the aim here. We take a bare-bones approach, to get a feeling for the underlying constructs, and start with the definition of a *-algebra. I use to denote the complex conjugate of a complex number .
Definition 1 An algebra over field is a -vector space together with a binary product satisfying
for all and .
A *-algebra is an algebra over with a unary involution, satisfying
for all and .
An algebra is called unitial if there exists such that
for all . Then, is called the unit or identity of .
Continue reading “*-Algebras” →
We continue the investigation of representing probability spaces as states on algebras. Whereas, previously, I focused attention on the commutative case and on classical probabilities, in the current post I will look at non-commutative quantum probability.
Quantum theory is concerned with computing probabilities of outcomes of measurements of a physical system, as conducted by an observer. The standard approach is to start with a Hilbert space , which is used to represent the states of the system. This is a vector space over the complex numbers, together with an inner product . By definition, this is linear in one argument and anti-linear in the other,
for and . Positive definiteness is required, so that for . I am using the physicists’ convention, where the inner product is linear in the second argument and anti-linear in the first. Furthermore, physicists often use the bra–ket notation , which can be split up into the `bra’ and `ket’ considered as elements of the dual space of and of respectively. For a linear operator , the expression is often expressed as in the physicists’ language. By the Hilbert space definition, is complete with respect to the norm . Continue reading “Algebraic Probability: Quantum Theory” →
Continuing on from the previous post, I look at cases where the abstract concept of states on algebras correspond to classical probability measures. Up until now, we have considered commutative real algebras but, before going further, it will help to look instead at algebras over the complex numbers . In the commutative case, we will see that this is equivalent to using real algebras, but can be more convenient, and in the non-commutative case it is essential. When using complex algebras, we will require the existence of an involution, which can be thought of as a generalisation of complex conjugation.
Recall that, by an algebra over a field , we mean that is a -vector space together with a binary product operation satisfying associativity, distributivity over addition, compatibility with scalars, and which has a multiplicative identity.
Definition 1 A *-algebra is an algebra over together with an involution, which is a unary operator , , satisfying,
- Anti-linearity: .
for all and .
Continue reading “Algebraic Probability (continued)” →
The aim of this post is to motivate the idea of representing probability spaces as states on a commutative algebra. We will consider how this abstract construction relates directly to classical probabilities.
In the standard axiomatization of probability theory, due to Kolmogorov, the central construct is a probability space . This consists of a state space , an event space , which is a sigma-algebra of subsets of , and a probability measure . The measure is defined as a map satisfying countable additivity and normalised as .
A measure space allows us to define integrals of real-valued measurable functions or, in the language of probability, expectations of random variables. We construct the set of all bounded measurable functions . This is a real vector space and, as it is closed under multiplication, is an algebra. Expectation, by definition, is the unique linear map , satisfying for and monotone convergence: if is a nonnegative sequence increasing to a bounded limit , then tends to .
In the opposite direction, any nonnegative linear map satisfying monotone convergence and defines a probability measure by . This is the unique measure with respect to which expectation agrees with the linear map, . So, probability measures are in one-to-one correspondence with such linear maps, and they can be viewed as one and the same thing. The Kolmogorov definition of a probability space can be thought of as representing the expectation on the subset of consisting of indicator functions . In practice, it is often more convenient to start with a different subset of . For example, probability measures on can be defined via their Laplace transform, , which represents the expectation on exponential functions . Generalising to complex-valued random variables, probability measures on are often represented by their characteristic function , which is just the expectation of the complex exponentials . In fact, by the monotone class theorem, we can uniquely represent probability measures on by the expectations on any subset which is closed under taking products and generates the sigma-algebra . Continue reading “Algebraic Probability” →