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.