We previously defined noncommutative probability spaces as a *-algebra together with a nondegenerate state satisfying a completeness property. Justification for the stated definition was twofold. First, an argument similar to the construction of measurable random variables on classical probability spaces was used, by taking all possible limits for which an expectation can reasonably be defined. Second, I stated various natural mathematical properties of this construction, including the existence of completions and their functorial property, which allows us to pass from preprobability spaces, and homomorphisms between these, to the NC probability spaces which they generate. However, the statements were given without proof, so the purpose of the current post is to establish these results. Specifically, I will give proofs of each of the theorems stated in the post on noncommutative probability spaces, with the exception of the two theorems relating commutative *-probability spaces to their classical counterpart (theorems 2 and 10), which will be looked at in a later post. Continue reading “Completions of *-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
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 ,
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 .