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I am starting to study noncommutative geometry and C^* algebras so my question is

Does anyone know a good reference on this subject?

I would like a basic book with intuitions for definitions and this kind of things. I come from algebraic geometry so if it talks a bit about the relation with it would be highly appreciated.

I have already taken a look to Conne's book but I find it too hard and I'm currently studying Landi's but it lacks lots of proofs (he refers to several papers)

Thanks in advance

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Varilly's "An Introduction to Noncommutative Geometry" goes often beyond operator algebras into topology and other surrounding fields so the intuition settles well. On the other hand, it is fairly complete in basic technical detail in operator algebras needed for the noncommutative geometry. Connes' book is more of a survey, at more expert level, with few proofs. – Zoran Skoda Feb 3 '11 at 13:42
up vote 11 down vote accepted

First of all, let me mention that functional analysis plays a similar role in noncommutative geometry that commutative algebra plays in algebraic geometry, and it pays off to at least have a reference handy. To that end, I recommend "Banach Algebra Techniques in Operator Theory" by Ronald Douglas: it develops the essentials of elementary C* algebra theory completely from scratch (the first chapter is on Banach Spaces). From there, I recommend starting with Higson and Roe's "Analytic K-homology". The book offers a self-contained introduction to C*-algebra theory and operator K-theory and it culminates in a very detailed exposition of the K-homological proof of the Atiyah-Singer index theorem. This is all foundational material in noncommutative geometry in the sense that much of the rest of the subject is organized around these tools.

For example, Connes developed cyclic homology so that he could generalize the Chern character map from topological K-theory (AKA K-theory for commutative C* algebras) to K-theory for noncommutative C*-algebras. Likewise, the inspiration for the the notion of a spectral triple came from the index pairing between K-theory and K-homology: a spectral triple consists of a representation of a C*-algebra on Hilbert space together with an unbounded operator which is compatible with the representation, while a K-homology class is represented by a representation of a C*-algebra on Hilbert space together with a bounded operator which is compatible with the representation. In both cases, the point is to capture some feature of the Atiyah-Singer index theorem and generalize it to the noncommutative setting.

So once you've assimilated enough of Analytic K-homology, it probably wouldn't be quite so hard to go back and tackle some of the literature. Connes' book is of course great with the right background, but you might find his very well written paper "Noncommutative Differential Geometry" easier to tackle. At that point you will have to decide where you want to go: one can dig deeper into noncommutative geometry proper, or one can pursue interactions between noncommutative geometry and conventional geometry and topology. There is also "noncommutative measure theory" built around Von Neumann algebras, but I know much less about that side of things.

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These all sound like good suggestions (I never read much of Higson & Roe despite buying it as a student to try and learn some of this stuff, but what I did read I remember as being well written and well organised). My impression from looking at bits and pieces of Connes' NCDG paper (the IHES one?) is that it helps to be comfortable with all the "DG" before moving on to the "NC", but perhaps that just reflects holes in my own learing. – Yemon Choi Feb 3 '11 at 19:36

For $C^{\star}$-algebras I would suggest in addition the lovely book of Gerard Murphy "$C^\star$-algebras and operator theory".

If you are interested in K-Theory Blackadar's "K-Theory for Operator algebras" might be a good choice after Murphy's book.

For Noncommutative Geometry, there are nice little introductory books by Varilly "An Introduction to Noncommutative Geometry" and Khalkhali "Basic Noncommutative Geometry". They are maybe more "beginner friendly" than Connes' master piece.

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I'd recommend Khalkhali's book if you already know some Operator Algebra theory, but not much geometry. Varilly's book (in my opinion) is the other way round! – Matthew Daws Feb 3 '11 at 12:28

A while ago Lieven Le Bruyn put together a list called "Top 10 noncommutative geometry books for newbies", with a short description/review of each book. Check this link.

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For $C^*$-algebras in general, there are many textbooks. Famous names are e.g. the two volume book by Kadison&Ringrose or the (by now somehow old but still very nice) book by Sakai. I also enjoyed the encyclopedic book by Blackadar as well as the three volumes by Takesaki (mainly von Neumann algebras, though). For more physics applications, the two volume book by Bratteli&Robinson is nice (though not many proofs). For many reasons you also might want to take a look into Rudin's two books on Real&Complex Analysis and Functional analysis.

More NCG like things you can find in Gracia-Bondía, Várilly, and Figueroa. Though there is certainly no way around Connes' book :)

I guess that is enough for a first reading, hope that helps

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I don't recommend Kadison&Ringrose! – Martin Brandenburg Feb 4 '11 at 4:58
@Martin: maybe I'm reading the wrong books, but I found the proofs in K&R quite helpful. In particular, they seem to be reliable, which is not always the case with other textbooks of $C^*$ algebras :) Of course, this one is not so much about NCG but $C^+$ algebras/von Neumann algebras in general. – Stefan Waldmann Feb 4 '11 at 8:10

Let me add some new-ish books to the mix that I liked and deal with some topics in $C^*$-algebras that have picked up some steam in recent years:

If you want an introduction to the state of the art in the theory of $C^*$-algebras then the book $C^*$-algebras and Finite-Dimensional Approximations by Brown and Ozawa is what you want to read. If you always wanted to know what completely positive maps are and how they are used to define nuclearity or if you want to remind yourself about the intricacies of tensor products of $C^*$-algebras, if you want to know everything about group $C^*$-algebras and exactness or if you work in von Neumann algebras and want to see if any of the things you do still work in the $C^*$-algebra case, it is all in this book! Apart from that it contains a lot of humorous footnotes and is a great read. Gosh, I sound like a car salesman, but I am a huge fan of this book.

Next up is Classification of Nuclear $C^*$-algebras. Entropy in Operator Algebras by Rørdam and Størmer. This is a book about more advanced topics, but if you are looking for everything related to the classification program of nuclear, simple $C^*$-algebras, which has had some recent breakthroughs, then this is your book. It gives a good description of the purely infinite case for example, so if you always wanted to read up on what makes the Cuntz algebras $\mathcal{O}_2$ and $\mathcal{O}_{\infty}$ so special, then give it a shot. The reason why I am neglecting the entropy part in this description is entirely due to my own laziness: It is still on my reading list.

If you want to know how algebraic topology, in particular sheaf cohomology, can be used to classify continuous trace $C^*$-algebras or if you want to learn about Morita equivalences and Brauer groups, then Morita Equivalence and Continuous-Trace $C^*$-Algebras by Raeburn and Williams is the book of your choice. It contains a proof of the Imprimitivity Theorem and that stuff you always wanted to know about the whole induction and restriction business.

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