## The probability space as a fiction

In a sequel to the blog post just mentioned Tao continues to discuss free probability. This is a kind of non-commutative extension of ordinary probability. It is a subject I do not feel I have to learn at this moment but I do think that it would be useful to have an idea how it reduces to ordinary probability in the commutative case. There is an analogy between this and non-commutative geometry. The latter subject is one which fascinated me sufficiently at the time I was at IHES to motivate me to attend a lecture course of Alain Connes at the College de France. The common idea is to first replace a space (in some sense) by the algebra of (suitably regular) functions on that space with pointwise operations. In practise this is usually done in the context of complex functions so that we have a * operation defined by complex conjugation. This then means that continuous functions on a compact topological space define a commutative $C^*$-algebra. The space can be reconstructed from the algebra. This leads to the idea that a $C^*$-algebra can be thought of as a non-commutative topological space. I came into contact with these things as an undergraduate through my honours project, supervised by Ian Craw. Non-commutative geometry has to do with extending this to replace the topological space by a manifold. Coming back to the original subject, this procedure has an analogue for probability theory. Here we replace the continuous functions by $L^\infty$ functions, which also form an algebra under pointwise operations. In fact, as discussed in Tao’s notes, it may be necessary to replace this by a restricted class of $L^\infty$ functions which are in particular in $L^1$. The reason for this is that a key structure on the algebra of functions (random variables) is the expectation. In this case the * operation is also important. The non-commutative analogue of a probability space is then a $W^*$-algebra (von Neumann algebra). Comparing with the start of this discussion, the connection here is that while the probability space fades into the background the random variables (elements of the algebra) become central.