Operator algebra is a very young subject, which was built by Mr. Neumann for the axiomatization of quantum mechanics. Vonnemann algebra comes first, followed by c* algebra. They belong to the same branch, but the problems they consider and the application methods are completely different. C* algebra now has three directions: application in physics, application in other branches of mathematics (mainly geometric topology), and internal meaningful problems. I don't know much about the physical application, but @ Qianben does. Part of my own problems focus on the classification of c* algebras. To learn this knowledge, if you take the new path of winter or Lin Huaxin, you mainly need algebraic and analytical skills. It is teacher Lin's motto to realize the insight of algebra with analysis. Algebra is mainly group ring theory, homology algebra, algebraic K theory and so on. Harmonic analysis and function theory are best used in analysis. If we follow the path of teacher Gong Guihua, we need a deep knowledge of topology. No matter what kind of classification, it is inseparable from K theory. The three people mentioned above are the three most representative people in the current classification theory. Regarding the nature and internal structure of c* algebra, there is not much demand for other knowledge, but the level basically depends on your "extracurricular knowledge", especially operators. Its application in other branches of mathematics mainly benefits from K theory and Allen Connes' noncommutative geometry theory. The most popular aspect now is the popularization of Atia-Singer index theorem led by Yu Guoliang, a professor in China. If you want to learn from this aspect, it will take about three or four years to get started, and you need a solid foundation of functional analysis, a solid foundation of topology, basic differential geometry and K theory. Then you can read a book about c* algebra, have a certain understanding of c* algebra itself and its classification, and then you can learn Atiyah-Singer index theory (17+ only), geometric group theory and topological dynamic system knowledge as appropriate. C* algebra can be regarded as a topological theory, and it is natural to learn topology. Von Neumann algebra can be regarded as a measure theory, which is essentially an analysis. So there is a big difference between the two. The basic knowledge of analysis is extremely thick, and the main things to learn are measure theory, harmonic analysis, entropy theory, ergodicity theory, group theory (here group theory is very important) and so on. At present, China mathematician Li Hanfeng is an outstanding figure. Voicu Lescu also put forward a theory of free measure, which is very difficult and has never been touched by anyone. Look at fate, and you may know something about the knot theory and the work of the Fields Prize! That's about it. In fact, I'm really an academic scum. So far, I haven't formed a mature and complete overall view on this field. All the above are pieced together by a few words I usually hear. There are many Chinese names mentioned above, and they are really the most influential people in this field. But in terms of overall strength, the strongest places are Japan and Canada. Japan has a profound tradition and many scholars. For Canada, operator algebra is a universal learning. France, Germany and the Nordic countries all have outstanding figures. Of course, everything will be fine in America. Operator algebra is not a mainstream subject. My teacher said: there are no scholars in our field. It's not Princeton's problem, it's our problem.