concept of the classification of $C^\ast$-algebras, introduction/overview

Question

I don't have a specific mathematical problem at the moment but nevertheless I hope, my question is suitable for math.stackexchange. I'm interested in $C^\ast$-algebras and I would like to begin with the study of classification of $C^\ast$-algebras soon. I think, this is a huge field and I heard that K-Theory for operator algebras is an important tool for classification (but I still don't know something about K-Theory). My question is: What exactly is meant by classification of $C^\ast$-algebras? What are interesting properties which $C^\ast$-algebras do have in common? Could you give me a short overview or do you know good literature for beginners, which gives a good overview or introduction of classification of $C^\ast$-algebras?

I still know that every $C^\ast$-algebra could be identified with a sub-$C^\ast$-algebra of one of these three $C^\ast$-algebras: $C_0(X)$ (X localcompact, Hausdorff space), $C(X)$ (X compact, Hausdorff space) or $L(H)$ ($H$ is a Hilbert space). But it seems that classification means something different in this case, maybe something similar as in algebraic topology, if you consider homology of topological spaces for example. In the field of algebraic topology, you can consider homology or cohomology of topological spaces to distinguish between the spaces.

Greetings

Answer 1

In short, the objective of the classification program is to give a (relatively) simple set of invariants that distinguish between the isomorphism classes of $C^\ast$-algebras, that is, if the invariants are isomorphic, then the $C^\ast$-algebras are isomorphic. It is thus stronger than what homology and cohomology theories yield for topological spaces: Non-homeomorphic spaces may have isomorphic homology and cohomology groups.

By now, this goal is far too ambitious, so one restricts the interest to certain subclasses of $C^\ast$-algebras. In Elliott's classification program, these are the simple, separable, nuclear, unital $C^\ast$-algebras, and even for them, the program has not succeeded yet (to my knowledge).

As you mention, every $C^\ast$-algebra is isomorphic to a closed subalgebra of $L(H)$ for some Hilbert space $H$, but the closed subalgebras of $L(H)$ do not meet the condition of being a simple invariant - there is no general method to decide whether two of them are isomorphic.

The main tool for Elliott's classification program is, as you say, K-Theory of $C^\ast$-algebras. It is quite similar to (co-)homology theories for topological spaces. When restricted to commutative $C^\ast$-algebras, it gives a generalized (not satisfying the dimension axiom) cohomology theory for locally compact spaces, called (topological) K-Theory. Roughly speaking, K-Theory is the set of projection in matrix algebras over a given $C^\ast$-algebra, endowed with the direct sum of projections as group operation (for commutative $C^\ast$-algebras, the projections correspond to vector bundles over the base space).

Two good introductory treatments of the classification program are:

Structure and classification of $C^\ast$-algebras

Classification of $C^\ast$-algebras

As for K-theory, I originally learned it from lecture notes by Landsman, which I cannot find at the moment but they should be somewhere on the net. The proof of Bott periodicity (the central theorem in K-Theory) given there is very similar to the original one by Atiyah for topological K-theory. I personnally like Cuntz' proof better - you should also find it somewhere on the net, I learned it in a lecture and have no public source.

If you have further questions, just go ahead - but I should mention that I am far from being an expert in this field.