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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 Now I want to find an alternative book for Vol 2, because I once heard that the content in this book is somewhat out of date and the theories are developed in a rather slow pace.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?

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Community Wiki? –  Matthew Daws Oct 18 '10 at 18:30
    
Hmm, we seem to have lost the original questioner from this page. It also currently gives the impression that Dmitri has asked this question, which is I'm guessing a very misleading impression. –  Yemon Choi Oct 19 '10 at 3:52
    
There is no book that covers all "basic" aspects of operator algebras because the topic is much too broad for this. At least chapters 6 to 9 of Kadison/Ringrose still consists of material everyone in operator algebras should know, and it is explained very lucid and in detail, so I would recommend that you read those, even if the book as a whole is considered to be out of date now. –  Tim van Beek Oct 20 '10 at 12:06
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4 Answers

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.

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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.

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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.

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Good points. I forgot about Sinclair-Smith. That's a very nice book, but if I remember correctly you'll have to look somewhere else for the basic in vN algebras. –  Martin Argerami Oct 20 '10 at 23:52
    
You're right, Martin. What you suggest about Sunder etc. is great for this. Another good place to play to get feelings on this stuff is in Jones's online notes for his von Neumann algebras course. –  Jon Bannon Oct 21 '10 at 17:56
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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.

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My guess would be that the book is more inclined towards operator theory than operator algebras (of course the two areas overlap, but there seems to my inexpert eye to often be a difference in emphasis and motivation). –  Yemon Choi Oct 19 '10 at 3:51
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Yeah, this book is not on the subject asked for. –  Jonas Meyer Dec 5 '10 at 8:07
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