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 .
Previously, I defined a *-probability space to be a pair consisting of a *-algebra and a state. This is, by itself, too simplistic to serve as a definition in keeping with the classical, commutative, theory. As an example, consider the commutative algebra of complex polynomials in a single variable and define a state by,
Then, is a representation of the standard Lebesgue or uniform measure on the unit interval. However, there are many other choices of algebras that could have been used including,
- linear combinations of exponentials, for nonegative , in which case the state is given by the Laplace transform.
- linear combinations of complex exponentials, for real , in which case the state is given by the Fourier transform.
- the space of continuous complex-valued functions on the unit interval.
- the space of bounded complex-valued measurable functions on the unit interval.
The state represents the same uniform distribution in all these cases, but this is not transparent from the algebras, no two of which are isomorphic. Other than , all of the algebras above are too small to correspond to the space of bounded random variables. The probability of an event is given, in terms of the state, by , with being the indicator function. However, of the algebras above, only contains the indicator functions. Furthermore, homomorphisms to from another *-probability space are given by *-homomorphisms . We should choose as large as possible in order to include the image of all such homomorphisms. The best choice of algebra to represent the uniform distribution is, therefore, the space .
According to the considerations above, if we start with a *-algebra and state then, in order to capture the concept of the space of bounded random variables, the algebra should be enlarged as far as possible. This can be done by taking limits of sequences or, more generally, of nets to complete . So consider a net . If possible, we should take the limit and add this to our algebra. For the state to be extended to the larger algebra, we would need to converge. Moreover, for elements , the value of can be defined as the limit of , so long as this converges.
Making these ideas precise, use the weak topology on , defined as the weakest topology making the maps
continuous for each . A net is weakly Cauchy convergent if and only if converges in , for all , and tends to a limit if . To avoid the kinds of pathologies that occur for unbounded sequences of random variables, we will restrict consideration to uniformly bounded sequences or nets. The seminorm on is . Next, the seminorm of is the operator norm of the left-multiplication map ,
An element is (uniformly) bounded iff is finite. So, we would want weakly Cauchy nets with uniformly bounded to have a weak limit in . By scaling, it is enough to consider limits of nets in the unit ball,
So, should be weakly complete. Also, as the algebra is to reflect the concept of bounded random variables, we should insist that is finite for every . Finally, any with are effectively equal as far as the state is concerned, so should be identified. This corresponds to the process of identifying almost surely equal random variables in classical probability theory. So, whenever or, equivalently, if for all . States with this property are called nondegenerate.
The considerations above lead to the definition of noncommutative probability spaces. Recall that for unitial algebra , a state is a positive linear map satisfying the normalisation . As we do not require algebras to contain a unit , this normalisation is not well-defined in general, so the alternative normalisation condition is used. In fact, although it is not explicitly required by the definition, we will see that all NC probability spaces are unitial.
Definition 1 A W*-probability space (or NC probability space) is a pair , where is a *-algebra and is a nondegenerate state, with respect to which is weakly complete and every is bounded.
This definition was arrived at by the argument outlined above. In the commutative case it can be shown to correspond, up to isomorphism, with classical probability spaces. In the noncommutative case, it is sufficiently general to apply to quantum theory. NC probability spaces according to this definition also come with nice mathematical properties which I will describe. The term `W*-probability space’ is used, as we can show that is a von Neumann algebra, also known as a W*-algebra.
Comparing with the definitions of noncommutative probability spaces in the literature, however, there does not seem to be a consistent standard. Different authors use different definitions, although it is usual to start with a *-algebra and a state fitting into one of the following cases:
- is unitial, with no further restriction.
- is a (possibly nonunitial) C*-algebra.
- is a von Neumann algebra and is normal.
In addition, it is common to require the state to satisfy one or more of the following.
- is tracial, so that for all .
- is faithful, so that whenever .
- is nondegenerate, so that whenever .
The tracial and faithful conditions are much too strong for our considerations — for example, a pure state on the algebra of bounded linear operators on a Hilbert space is neither tracial nor faithful. Nondegeneracy of the state is a considerably weaker requirement than being faithful and, as we saw previously, it is always possible to pass to a nondegenerate state by quotienting out by the ideal of with .
Other than the explanation given above, there are plenty of reasons to adopt definition 1, since the resulting category of W*-probability spaces satisfies various desirable properties. I will state some of these properties here, but leave the proofs for later posts.
First, the definition covers classical probability spaces. Given any such space , use (or for short) to denote the space of bounded and complex-valued random variables identified up to almost sure equivalence. This forms a *-algebra, and the expectation operator (denoted by or ) is a state. Then, is a W*-probability space. In fact, the converse is true, and commutative W*-probability spaces correspond to classical probability spaces.
Theorem 2 Every commutative W*-probability space is isomorphic to for some classical probability space .
More generally, for noncommutative spaces, definition 1 corresponds with the von Neumann algebra construction of NC probability spaces. Recall that a von Neumann algebra on a Hilbert space is a *-subalgebra of the space of bounded linear operators which contains the identity and is closed under the operator topologies. It does not matter which of the weak, strong, ultraweak or ultrastrong topologies is used, as the property of a *-algebra to be closed turns out to be an equivalent statement for each of them. A linear map is normal if it is ultraweakly or ultrastrongly continuous, or if it is weakly or strongly continuous on the unit ball (again, these are all equivalent conditions). More generally, an abstract von Neumann algebra is *-isomorphic to a von Neumann algebra on some Hilbert space.
Theorem 3 The pair is a W*-probability space if and only if is a von Neumann algebra and is a nondegenerate normal state.
It could be argued that this characterisation is a little unsatisfactory, since the definition of a normal state given above depends on the choice of representation of as an algebra of operators on a Hilbert space, which will not be unique. However, it is well-known that the normality of a state can be defined algebraically in terms of its behaviour on projections. An element is a projection iff , and a pair of projections are orthogonal if . Then, a state is normal if and only if
for all maximal collections of pairwise orthogonal projections.
The state need not be nondegenerate in order to define a W*-probability space. We just need to factor through the quotient , with being the *-ideal of elements satisfying . I denote the extension of to the quotient by , so that for all .
Theorem 4 Let be a von Neumann algebra and be a normal state. Then, is a W*-probability space.
Theorems 3 and 4 reduce W*-probability to the study of von Neumann algebras and normal states. An immediate consequence is that W*-probability spaces are unitial, even though this was not explicitly required in the definition.
Corollary 5 If is a W*-probability space, then is unitial.
In these notes I refer to a pair consisting of a *-algebra and a state, with no further restrictions, as a *-probability space. The state directly gives the and seminorms, and is said to be bounded if is finite. I say that is bounded to mean that every is bounded. As noted above and in previous posts, being a *-probability space is too weak a property to provide a good generalisation of classical probability spaces, so definition 1 above is used instead. However, bounded *-probability spaces generate W*-probability spaces in an essentially unique way.
Theorem 6 Let be a bounded *-probability space. Then, there exists a W*-probability space and homomorphism
such that is a weakly dense subset of .
This is unique up to isomorphism in the following sense. For any other homomorphism from to a W*-probability space with a weakly dense subset of , then there exists a unique isomorphism such that .
I will refer to and the homomorphism as the W*-completion of . Normal homomorphisms between bounded *-probability spaces lift uniquely to homomorphisms between their W*-completions.
Theorem 7 Let be a normal homomorphism between bounded *-probability spaces and . Let the W*-completions be
Then, there is a unique normal homomorphism such that . Furthermore, is an isomorphism iff is weakly dense in
That is, W*-completion is functorial, as it defines a functor from the category of bounded *-probability spaces with normal homomorphisms to the category of W*-probability spaces. With the notation of theorem 7, is the unique normal homomorphism making the following diagram commute.
W*-probability spaces and W*-completions are described conveniently by the GNS representation. In the following, is the von Neumann algebra on generated by , which is equivalently defined as its closure under any of the weak, strong, ultraweak, or ultrastrong topologies.
Theorem 8 Let be a bounded *-probability space, and be its GNS representation. Let be the weak closure of and define by
- is a W*-probability space.
- is the W*-completion of .
- is a W*-probability space if and only if is an isomorphism between and .
An alternative type of noncommutative probability space to those described above can be arrived at by using convergence in place of the weak topology. In this case, we do not necessarily have a von Neumann algebra, and is not really a noncommutative generalisation of classical measurable random variables. Instead, we end up with a C*-algebra being a noncommutative extension of the concept of continuous complex-valued functions on a topological space. As such, these C*-probability spaces have topological content and give a generalisation of locally compact spaces. The definition and first properties are very similar to the W*-probability spaces considered above.
Definition 9 A C*-probability space is a pair , where is a *-algebra and is a nondegenerate state, with respect to which is -complete and every is bounded.
Unlike definition 1 above, I am not specifically applying the completeness requirement to the unit ball . However, for normed spaces, completeness of the unit ball is equivalent to completeness of the space, so the distinction is irrelevant here. It should be noted that every W*-probability space is also a C*-probability space, although the converse is not true. In fact, C*-probability spaces need not even be unitial.
For a Hausdorff locally compact space , the collection of continuous functions vanishing at infinity is a commutative *-algebra, which is unitial iff is compact. The algebra operations of addition and multiplication are defined pointwise, and involution is pointwise complex conjugation. A Borel probability measure on defines a state (also denoted by ) by the integral, or expectation,
on . It is usual to consider regular measures, so that equals the supremum of taken over compact , for all measurable . When is second countable, all Borel probability measures are regular, so this is a minor technical constraint. For any such regular measure, its support is the smallest closed subset of whose complement has zero measure. It can be seen that the state defined by is nondegenerate if and only if its support is the whole of — i.e., if has full support. In that case, defines a C*-probability space. The converse statement holds. Every commutative C*-probability space is, up to isomorphism, given by a regular probability measure with full support on a locally compact space. This is the C*-algebra version of theorem 2 stated above for W*-probability spaces.
Theorem 10 Every commutative C*-probability space is isomorphic to for some regular probability measure with full support on a locally compact space .
The locally compact space is uniquely defined up to homeomorphism and, by the Gelfand representation, can be constructed explicitly as the spectrum of . Then, the regular measure is uniquely defined by the Riesz-Markov representation theorem.
Theorem 3 also has a C* version. As previously explained, the seminorm on a bounded *-probability space satisfies the C*-identity . The seminorm is positive definite if is nondegenerate so, if is also -complete, then it is a C*-algebra.
Theorem 11 The pair is a C*-probability space if and only if is a C*-algebra and is a nondegenerate state.
As for W*-probability spaces, it is not required that the state is nondegenerate upfront, so long as we factor through the quotient , where is the *-ideal of elements for which .
Theorem 12 Let be a C*-algebra and be a state. Then, is a C*-probability space.
The previous two theorems reduce C*-probability to the study of states on C*-algebras.
The C*-completion of *-probability spaces can be constructed along the same lines as for W*-probability spaces, as stated above by theorem 6. The situation here is simpler, as seminormed spaces always have a completion, and the state has a unique continuous linear extension.
Theorem 13 Let be a bounded *-probability space. Then, there exists a C*-probability space and homomorphism
such that is a uniformly dense subset of .
This is unique up to isomorphism in the following sense. For any other homomorphism from to a C*-probability space with a uniformly dense subset of , then there exists a unique isomorphism such that .
As we should expect, C*-completions are functorial.
Theorem 14 Let be an -continuous homomorphism between bounded *-probability spaces and . Let the C*-completions be
Then, there is a unique homomorphism such that . Furthermore, is an isomorphism iff is -dense in .
This compares with theorem 7 above stating the analogous functorial property of W*-completions. Rather than normality of the homomorphisms, the weaker property of -continuity is sufficient here. Furthermore, homomorphisms of C*-probability spaces are automatically -continuous, so this requirement need not be stated explicitly.
C*-completions can be constructed via the GNS representation in precisely the same way as in theorem 8 for W*-completions. The only change is that is now the norm-closure of rather than its weak closure.
Theorem 15 Let be a bounded *-probability space, and be its GNS representation. Let be the norm-closure of and define by
- is a C*-probability space.
- is the C*-completion of .
- is a C*-probability space if and only if is an isomorphism between and .
I conclude this post by noting that the C*-completion of a bounded *-probability space is always contained in the W*-completion. We have the following commutative diagram.
Here, denotes the C*-completion and denote the W*-completions. This follows directly from theorem 7, as is dense in , the W*-completions of and are isomorphic.