This is a continuation of a previous post where, to start the discussions, I quoted a portion of a recent interview with Alain (incidentally the full interview is now available and I highly recommend it to the readers of this blog). After a couple of busy weeks now I have some time to follow up on this important question. My aim is to gradually explore some of the themes discussed in the introduction to Alain's 1985 paper: "Noncommutative Differential Geometry". Without any doubts this will take many posts.
When the question ``what is a noncommutative space?" is put to a mathematician you may hear one of the following answers, among others:
1. A noncommutative space is just an associative algebra which may or may not be commutative,
2. A noncommutative space is just a C*-algebra which may or may not be commutative,
3. A noncommutative space is a von Neumann algebra which may or may not be commutative,
4. A noncommutative space is a `spectral triple',
5. A noncommutative space is an abelian category, possibly with some extra structure,
6. No one knows what a noncommutative space is.
I don't agree with item 6 and I would like to add that items 1-5 all have some elements of truth in them. The problem is that of course the very classic question " what is a space ?" is not so well defined and its answer depends on the context. So the best answer would be something like this: a space is a set endowed with some extra structure like a topology, a measure, a smooth structure, a metric, a sheaf, etc. etc...
A similar working definition can be adopted to noncommutative geometry and in fact this turns out to be essential for the whole subject. To start let us agree to call a not necessarily commutative associative algebra a noncommutative algebra. Then we can adopt the thesis:
``A noncommutative space is a noncommutative algebra possibly endowed with some extra structure"
This is a very powerful idea and can be fully justified as we shall see a bit later. For the moment
we should note that examples 1-4 in the above list all fall within the scope of this definition. For example a C* algebra is an algebra over the field of complex numbers equipped with an involution and a norm satisfying some conditions.
Proposal 5 however looks very different from the rest. This becomes relevant when we need to discuss morphisms between noncommutative spaces. Definitions 1-4 suggest that a morphism should be something like an algebra homomorphism. This is however too rigid and there may not be enough morphisms between noncommutative spaces in this rigid sense. This is in fact a purely noncommutative phenomenon to which we shall return later. For the moment it suffices to say that given an algebra A we can pass to the abelian category A-mod of representations of A. There is less information in A-mod than in A and in fact A-mod characterizes A only up to Morita equivalence but the topological information in A is not lost in this process. In many cases, like Hochschild homology, cyclic homology, and K-theory, it can be fully defined in terms of A-mod. More on this later.
Talking about a good definition of a noncommutative space inevitably brings up the issue of the role of definitions in the development of mathematics. The Orthodox view of mathematics as an exact science (generally held by non-mathematicians!) assumes that first come precise definitions of objects of study and then their main properties are explored by proving theorems
about them. In reality, however, it often happens that the main results of a subject, up to a certain point, are proved before a general agreement is reached on foundational issues. One can give so many examples of this but the development of differential geometry along with the notion of a manifold from Riemann to Weyl and Whitney is a case in point. Up to the early 1960's one of the main textbooks on differential geometry was Élie Cartan's classic, Geometry of Riemannian Spaces. Even in its second enlarged 1946 French edition Cartan says
'' The general notion of manifold is quite difficult to define with precision"
(see also this) and simply refrains from giving a general definition. He had much more important and interesting things to say!
Back to the main topic of this post, what we should do next is to justify our notion of a noncommutative space: why this is a reasonable idea and in what sense it extends the classical idea of a geometric space. This I will do in the next post.