TITLE: Why can we classify the W*algebra?
QUESTION [9 upvotes]: 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.

REPLY [3 votes]: 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.