Why can we classify the W*algebra?

Question

Many operator algebra books discuss the classifiation of W*algebra(von Neumann algebra),but not the C*algebra,why?

I think a direct reason is that we have the projection comparison theorem in the W*algebra,so we can compare projections in the the factor of W*algebra.

But I want know some basic reason,going back to original definition,from which part, the W*algebra is more rich than the C*algebra,so it can be classified.

Answer 1

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.