advice on PhD proposal

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I am a PhD student in pure mathematics, and I want to work in the field of $C^*$-algebras and operator algebras I am looking for a relatively comprehensive view of this subject. For example,

  1. what are the mainstream areas?
  2. Who works on essential topics in the field?
  3. What are the main questions in these areas?
  4. ...?

Of course, I like the geometric representation of these $C^*$-algebras and their applications in quantum physics, but by consulting with others, it's hard for a doctorate. I'm glad if anyone can introduce a resource or something that will increase my information.

PS : This question is my first question; maybe I have not entirely complied with the rules