Why can we classify the W*algebra?

There are probably several answers to this. Here's my take.

Two things make the classification of von Neumann algebras interesting and useful, in my view:

1. After you define the types, the abundance of projections allows you to show that any von Neumann algebra is a direct sum of subalgebras of some of the types.

2. There are many cases where the type information on its own tells you a big deal about the algebra: I'm thinking of results like:

• Type I factors can be completely classified;
• Type II$_1$ factors always carry a faithful normal tracial state;
• Type II$_\infty$ factors are always a tensor product of a II$_1$ and a I$_\infty$;
• Type III factors are a crossed product of a II$_\infty$ and an action of $\mathbb R$.
• AFD factors can be completely characterized for all types.

For C $\!\!^*$-algebras, one can try to play the same game (for example, "simple" could play the role of "factor", Type I C $\!\!^*$-algebras, purely infinite versus finite, AFD, etc.), but one is immediately hampered by the (eventual) lack of projections, that forbids to always have a C $\!\!^*$-algebra as a direct sum of simpler ones.

As a final word, "classification" is also used as in Elliott's Classification Program. In this setting, it is not clear at all that von Neumann algebras are on better footing that C$^*$-algebras. Of course type I von Neumann algebras can be completely classified, and rather easily; but, for example, a complete classification of all II$_1$ factors is considered completely hopeless by all experts.

This is somewhere between an answer and a comment. Just think about the dichotomy between the two in the abelian case. In this case classifying von Neumann algebras is fairly simple. However, classifying all $C^*$-algebras essentially encompasses the entire field of topology.

From this point of view $C^*$-classification is hopeless.

Also a final note, which isn't an aspect that make proof easier but might be viewed as an extension of the above observation to the noncommutative case. In some sense there just are fewer von Neumann algebras so they are easier to classify. What I mean by this is that given a single von Neumann algebra there might be many weakly dense $C^*$-subalgebras that are not isomorphic.