# Modern introduction to $C^*$-algebras

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.

• I haven't read it (partially because I prefer Murphy's book) but have you taken a look at Kadison and Ringrose vol. 1? – Aweygan Feb 19 '19 at 21:16
• @Aweygan Kadison-Ringrose cannot really be called "modern", can it? Anyway, they certainly don't use categorical language. – MaoWao Feb 20 '19 at 9:00

• 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.
• I guess I have found the first one and it looks good, there are a lot of commutative diagrams. By the way, I really enjoy by my advisor's approach: he says that linear operators should be thought of as a representations of some kind. For example a normal operator is the same as a representation of $C(X)$ algebra and so on (in his lectures he defines a functional calculus as a suitable algebra homomorphism, not the usual "we can kind of evaluate $f(T)$ you know"). Unfortunately, I have never seen such a great algebraic point of view in books on the subject. – Vladislav Feb 20 '19 at 20:48