In many areas of mathematics, one can usually assign a commutative algebraic structure to the “space” that one is studying. For example, instead of studying an (affine) algebraic variety, one can study algebraic functions on that variety, which gives rise to a commutative ring. Another example is that one can study the space of complex-valued continuous functions on a topological space (with some “nice” topology) instead of the topological space itself, and this space of complex-valued continuous functions is a commutative C*-algebra. These algebraic structures have their noncommutative analogues (namely noncommutative rings and noncommutative C*-algebras), and these might correspond to some “noncommutative varieties” or “noncommutative topological space”. This is probably one of the motivations of the area noncommutative geometry.

This abstraction process also applies to probability spaces, because there is a very natural algebra on a probability space. Given a probability space , one can define the space of (essentially) bounded random variables , and a linear functional on this space, namely the expected value. So we may study the algebra instead of the probability space itself.

In fact, it is very natural to study only random variables but not the probability space in probability theory. Usually (but not always!), the role of the sample space is not important at all, but the random variables and their associated -algebras are more important.

This leads to the following definition/abstraction.

A (noncommutative) probability space is a pair , where is a complex algebra with a unit, and is a linear map such that .Defintion.

The requirement can be seen as an abstraction of the sample space has probability . To see that this is not a meaningless generalization, we can look at the following examples.

Examples.

- As we have seen, for a given probability space , is a noncommutative probability space.
- In many cases, we also care about random variables that are not bounded, for instance Gaussian random variables. So it might be more natural to study with .
- with .
- , fix a unit vector with and define . Then is also a probability space.

In classical probability, we care about the distribution of a random variable, which is the push forward of the probability measure on by the random variable. We don’t have a natural measure in noncommutative context, but we can still define a distribution of a random variable using moments.

Let be a probability space. We say has a distribution , where is a probability measure on , if for all .Definition.

Note that even in the classical case, may not be uniquely determined by the moments, so it might actually be better to view a distribution as a sequence of moments instead of a probability measure in this case.

ExampleConsider , where . Let be self-adjoint. Then there exist an orthonormal basis in and such that , . We have.,

where . That is, has distribution . We call the empirical eigenvalue distribution of .

We can construct the space of random matrices as follows. Given a probability space and , we define

.

For , are random variables with for all . Define by . Then is a probability space.

Suppose that for all . Then

,

where is the empirical eigenvalue distribution of . We can see that a distribution of is given by ““.

We will now consider another example of probability space. Let be a group and let be the group algebra of . One can define a norm and consider the completion of under this norm:

.

One can show that for all , . Note that the unit of is also a multiplicative unit in and . We define (or ) by

.

Consider a particular case, where is isomorphic to . Write . Then one can associate with a function (in fact, Fourier series) on , the dual group of . Then the expected value in this case can be computed by .

Let’s see an application, namely random walks on a group . Let . Define a sequence of random variables by , and

for all . Of interest are the numbers , the return probabilities.

Define by . Then we have for all , because if we write , then

,

and hence

.

It is not difficult to see that the right hand side is exactly (by simply writing down the definition of ).

Let’s consider a simpler case, being isomorphic to . Let with . As we have seen before, we can associate with . Then is given by , which implies , the return probability of simple random walk on .

Next time we will talk about a more interesting and important topic: independence in noncommutative probability.