Question

I have read over the book by R. V. Kadison and J. R. Ringrose: Fundamentals of the theory of operator algebras. Vol 1, and have done most of the exercises in it. Now I want to find an alternative book for Vol 2, because I once heard that the content in this book is somewhat out of date and the theories are developed in a rather slow pace. Which book can I choose then?

Answer 1

I guess it depends a little on what you are looking for.

If you want to pay attention to the C*-side, you may want to look at Davidson (very neat presentation of several key topics), or Murphy (maybe more basic, but a favourite in clarity).

For the von Neumann side, an option to get started is Sunder: many things are done in factors, avoiding many complications tackled by Kadison. But if you are going to be into von Neumann algebras at all, I think that you need to have some familiarity with Kadison and with Takesaki. I usually find myself going back to those two fairly often.

Answer 2

I'll add a specific entry for Takesaki's books. I learnt what von Neumann theory I know from these books. Especially volume 2 is a very nice, and concise, guide to an awful lot of the theory around Tomita-Takesaki and Connes' theory of weights.

If you are interested in Tomita-Takesaki, in a gentler fashion, then the old books by Stratila are nice. In particular, they explain the unbounded operator theory somewhat more than Takesaki does.

But it really depends what the original interest is: if you want to get into modern C*-algebra theory, then Takesaki is not the way to go.

Finally, for reference, the recent book by Blakadar is wonderful, and is a great place to look for something, before reading up in more detail somewhere else.

Answer 3

It depends on what you'd like to know. Above answers suggest Davidson and Blackadar for C*-algebras, and Takesaki for some more detailed theory of von Neumann algebras. These are excellent suggestions.

I'd like to add that if you want to learn more about finite von Neumann algebras/factors, you may find it helpful to read chapters 6-9 of Kadison and Ringrose, and then have a look at Finite von Neumann algebras and MASAs by Sinclair and Smith. Then you perhaps should try to read some of Sorin Popa's recent papers and begin backfilling knowledge around these.

Answer 4

A few years ago, a 2 volume advanced text on operator theory (vol. 50: "An invitation to operator theory"; vol. 51: "Problems in operator theory") came out by Y.A. Abramovich and C.D. Aliprantis which is supposed to be excellent and very up to date; it's the only account that gives a detailed treatment of ordered function spaces. A detailed description of the text can be found here:

http://www.ams.org/bookstore?fn=20&arg1=analysis&ikey=GSMSET. 

I haven't done more then browse it and most of it's over my head, but it definitely is worth a look.