Modern introduction to $C^*$-algebras

Question

Well, I am asking for the references on the subject for those who can't stand the Murphy's book. I have a background in functional analysis (including Banach algebras, functional calculus, Gelfand theorem for commutative $C^*$-algebras and so on) and prefer categorical language. I hope there is something but Murphy which isn't an encyclopedia and I would appreciate any help finding it.

Answer 1

There is a couple of lecture notes I enjoyed:

• Landsman, $C^\ast$-algebras and K-Theory. It's a two-part lecture series. The first part covers the basics of $C^\ast$-algebras (given youur background, it might be to basic for you), while the second focuses on K-theory. It uses categorical language, but I feel not quite to the extent it could in the second part (there is a much nicer proof of Bott periodicity). Unfortunately, I am not sure if you can still find these notes online.
• Warner, $C^\ast$-algebras. Does not look too modern, but don't let that shy you away. These notes are far more extensive than the ones by Landsman and probably closer to the level you're looking for (they are explicitly "addressed to those readers who are already familiar with the elements of the theory but wish to go further"). The author not only uses categorical language, but also discusses $C^\ast$-categories. However, I feel that the notes lack a little when it comes to connections to current research (maybe that's just my ignorance showing).
• Rieffel, $C^\ast$-algebras and more $C^\ast$-algebras. These are two iterations of the same course at UC Berkeley. At places these notes are rather sketchy, but the students who took the notes are (or were?) regulars on MSE and MO, so you should have a good chance to get all your questions answered. These notes probably go deepest among all the ones mentioned with a special focus on group actions and noncommutative dynamics. As far as I remember, there is no categorical language used here.

Beyond that there are some classics like the three books by Takesaki and the two books by Bratelli and Robinson. I did not use them to learn the basics of C*-algebras, so I don't really know how good they are for that, but they are the best places I know to start learning about von Neumann algebras and the connections to quantum mechanics.