Since my initial question (How much algebra is there in Noncommutative Geometry?), I got to study the basics of NCG and I now consider starting a PhD. program in NCG. I read Khalkhali's Basic NCG and to some extent, Landi's An Introduction to noncommutative spaces and their geometries. So I think it's fair to say that I got a bit of a general idea on the concerns of this field.
Now, since Khalkhali's book is more of a survey and most introductory books are like this, how can I continue with something deeper? What I mean is that I think I can easily get lost if I try to read $simultaneously$ some operator algebra, K-theory, differential geometry or any other subject that's included in NCG. I'd like to focus on a particular section, this being the best way to progress, but I can't seem to be able to direct myself.
Of course I'll have this talk with my future PhD. advisor, but your opinions are more than welcome.
I add that my approach is more algebraic, rather than Connes' analytic one. I also read Quillen and Cuntz article - Algebra Extensions and Nonsingularity, some Hochschild and cyclic homology articles and the first half of Kassel's Quantum Groups.
So...how can I bring order to this chaos, by not spreading between so many areas involved? What are exactly the possible algebraic directions? For the analytic/functional ones, it's clear that one has to read a C*-algebra book, for K-theory there are Karoubi, Bass, Rosenberg and others. But what about the algebraic part? What is it, exactly and where to find it?
Thank you very much for your time.