categorization of $C^*$ algebra

Question

There are three types of Von Neumann algebras,namely,Type $I,II,III$ VNA .I wonder whether the $C^*$ algebra can be categorized completely?

Answer 1

Well, you need to be more careful on what do you mean by categorize. $C^\ast$-algebras include locally compact topological spaces whose classification by homeomorphism is undecidable. Indeed, just differentiable manifolds of dimension higher than $5$ are unclassifiable (in the sense that its classification problem would be equivalent to the Halting problem).

That result follows from the fact that finitely presented groups have a non-decidable isomorphism problem and, using a surgery-like procedure, it is possible to construct 5-manifolds with any given finitely presented group as fundamental group [Ma].

Nevertheless there is a "classification program" for nuclear, separable and simple $C^\ast$ algebras (if you will, the $C^\ast$-algebra version of hyperfinite factors) using $K$ theory.

[Ma] Markov, A., The insolubility of the problem of homeomorphy, Dokl. Akad. Nauk SSSR 121, 218-220 (1958). ZBL0092.00702.