Why C*-algebras is not as popular as other areas of pure mathematics? I am applying for graduate school in pure mathematics and I recently got very interested in C*-algebra.
I am definitely wrong but I get the feeling that C*-algebras is not as popular as other areas of pure mathematics like number theory, analysis, algebraic geometry, etc. It also seems that most top ranked universities like MIT, Harvard, Stanford, Princeton, etc do not have any active research group in C*-algebras.
If my observations is right, then what is the reason? Is it because C*-algebra is harder than other areas of pure mathematics or is it because it is still a young area of pure mathematics?
Given that I am interested in C*-algebras and most top ranked universities are not active in this area, where can I apply? Also, will doing graduate work in C*-algebras instead other more popular areas of pure mathematics have a negative effect on my academic career?
 A: One way to tell how active a field is is by looking at what's appearing on the arXiv in that area. I think that will show you that operator algebra is a robust subject with a lot of activity.
In the comments, MaoWao points out that UC Berkeley and UCLA have very strong operator algebra groups, and Ulrich Pennig mentions groups in Münster, Göttingen, Erlangen, and Glasgow as places with substantial groups.  Copenhagen is another good example.
On the other hand, the OP's observation that most top schools don't have an operator algebra group is quite correct.  I would think that simply has to do with the size of the field --- there aren't enough C*-algebraists to populate that many departments. The two Fields Medals in the subject (to Connes and Jones) show that people in other areas do respect the field, I think.
The question is partly about career advice. All I can do there is report my impression that C*-algebraists don't seem to have more trouble finding employment than mathematicians of equal ability in other areas.
You ask which schools you should apply to --- in the comments MaoWao gave this site, which I was not aware of, which lists operator algebraists worldwide. It looks pretty complete to me.
However, my last comment is that as a prospective graduate student, you are at a very early stage to be settling on a specialty. Not saying you shouldn't, but I think most of us would recommend putting your main effort into getting a broad education during the first two years of grad school.
Related MathOverflow questions:
What are the applications of operator algebras to other areas?
States in C*-algebras and their origin in physics?
Quantum functional analysis
applications of C*-algebras in the field of PDEs
