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I have read few textbooks and papers about the $K$-theory groups $K_{0}$ and $K_{1}$ of (reduced) $C^{*}$-algebra and most of them didn't give a clear simple way to define these groups.

Just wondering if anybody can give me good sources for that?

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".. and most of them didn't give a clear simple way to define these groups." Well, what do you expect? What do you miss exactly in the definitions? And which sources have you considered? I downvote because this is not a specific question. The answers are general and just contain notes, articles, books etc. which can be found by some google research. – Martin Brandenburg May 14 '12 at 14:52
up vote 4 down vote accepted

I also think this question would benefit from some more detail, however...

  1. A very down-to-earth account of K-theory for operator algebras is in the book "K-Theory and C*-algebras" by N.E. Wegge-Olsen.
  2. A more sophisticated approach is in the book "Analytic K-Homology" by Nigel Higson and John Roe.

If you prefer getting your hands dirty, I recommend the first book. The approach is somewhat pedestrian (as the author writes in the introduction) but you will get a good sense of how things work, and there are lots of great exercises and examples. There are lots of arguments involving explicit homotopies of loops of projections and unitaries.

The second book I would recommend more if you prefer a functorial approach to things. In particular, Higson and Roe emphasize that Bott periodicity will be true for any sequence of half-exact functors satisfying similar properties to the K-functors.

I think the following is a "clear and simple" way to define the $K_0$ group: let $A$ be a unital C*-algebra. Define an equivalence relation (stable equivalence) on the class of finitely generated projective $A$-modules by saying that $P_1 \sim P_2$ if there is a finitely generated free module $A^{n}$ such that $$ P_1 \oplus A^{n} \simeq P_2 \oplus A^{n}.$$

Then denote by $V$ the set of stable equivalence classes of finitely generated projective modules. $V$ is an abelian semigroup with respect to the direct sum operation. Finally, $K_0(A)$ is defined to be the Grothendieck group of $V$. This means that you have to add in formal inverses to elements of $V$ in order to make it a group. (For example, the Grothendieck group of the abelian semigroup $\mathbb{N}$ is $\mathbb{Z}$.)

Of course this is not so easy to calculate. But the definition itself I think is not that complicated.

Also, these things admit a pretty concrete description. Finitely generated projective modules correspond to projections in matrix algebras over $A$, and stable equivalence corresponds to the two projections being connected by a continuous path of projections in some (possibly larger) matrix algebra over $A$. So you can really get in there and work with things if you have a good description of the algebra. In particular, projective modules over noncommutative tori can be described concretely in this fashion. This is shown in the book Elements of Noncommutative Geometry by Varilly, Figueroa, and Gracia-Bondia.

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There should be $n_1 = n_2$ in your definition of stable equivalence. – Johannes Ebert May 16 '12 at 8:44
Fixed, thanks a lot! – MTS May 16 '12 at 18:40
Thanks MTC that is so clear now for $K_{0}$. – Dragon Jun 14 '12 at 7:14

I found the notes of Timothy Gowers an easy read:

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Thanks Mrc that was so helpful. – Dragon May 14 '12 at 13:20

I quite liked Bruce Magurn's "An algebraic introduction to $K$-theory": it starts low enough, and pushes high enough (at least for my taste!).

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