I am trying to understand the Ktheory for the $C^*$algebra of the continuous functions on the $2$dimensional torus $T^2$. In particular I am interested on the $K_0$group. I have read that the generators of the group $K_0(C(T^2))$ are two elements: the unit $[1]$ and the Bott Projection $[Bott]$. Unfortunately, I cannot find the definition of the Bott projection for the torus, I have only seen the definition of the Bott projection for $\mathbb{R}^2$. Can someone tell me the definition of this projection? Or can someone give me another generator (in terms of projections)? I thank you all for the attention and the help.
If you want an explicit projection, you can form a variation of a Rieffel projection as follows. First take any function $f$ from $[\pi/2,\pi/2]$ to $[0,1]$ that sends $\pi/2$ to $1$, dips down to $0$ at $0$ and then goes back up to take value $1$ at $\pi/2$. Now define two more functions $$ g=\begin{cases} 0 & x\in\left[\frac{\pi}{2},\frac{\pi}{2}\right]\\ \sqrt{ff^{2}} & x\notin\left[\frac{\pi}{2},\frac{\pi}{2}\right] \end{cases} $$ and $$ h=\begin{cases} \sqrt{ff^{2}} & x\in\left[\frac{\pi}{2},\frac{\pi}{2}\right]\\ 0 & x\notin\left[\frac{\pi}{2},\frac{\pi}{2}\right] \end{cases} . $$ The projection then is then $$ p(\phi,\theta) = \left[\begin{array}{cc} f(\phi) & g(\phi)+h(\phi)e^{i\theta}\\ g(\phi)+h(\phi)e^{i\theta} & 1f(\phi) \end{array}\right] . $$

$\begingroup$ Thank you very much for the answer. By the way, yesterday I found your work "Torus and Noncommutative topology" and I found it really interesting and helpful, in particular when it is considered the first Chern class to verify that the above projection is a generator. $\endgroup$ – John N. Aug 7 '14 at 12:57

$\begingroup$ I think that $f$ should send $\pi/2$ to one. $\endgroup$ – Alexander Alldridge Feb 1 '19 at 21:54
I think you may have mixed up the Ktheory of the 2torus with the Ktheory of the sphere. Generally, the Bott element is considered as a projection in $K_0(C(S^2)) \cong K_0(C_0(\mathbb{R}^2)) $ which are isomorphic since $S^2$ is the one point compactification of $\mathbb{R}^2.$
If we define $S^2 = \{(x,y,z) \in \mathbb{R}^3 \,  \, x^2+y^2+z^2=1\}$ then $$p(x,y,z) = \begin{pmatrix} \frac{1+x}{2} & \frac{y+iz}{2}\\ \frac{yiz}{2} & \frac{1x}{2}\\ \end{pmatrix}$$ is a projection matrix with values in $C(S^2)$ representing the Bott element in $K_0(C(S^2)).$ This description is taken verbatim from Example 6.2.3 of Rosenberg's book "Algebraic Ktheory and its Applications."
EDIT: Since you do want to know about the Ktheory of the 2torus, here are two ways to understand it. First, as Paul suggested, it is easy to understand $K^0(T^1)$ since it is trivially isomorphic to $\mathbb{Z}.$ Now use the Kunneth Theorem for Ktheory to compute $K^0(T^1 \times T^1)$ and carefully keep track of where everything goes.
Alternatively, from a more algebraic standpoint, one can construct explicit modules over $C(T^2)$ to exhibit the $K_0$ classes. This can be found, for example, in "Projective Multiresolution Analyses for $L^2(\mathbb{R}^2)$ by Packer and Rieffel in Section 4. To construct projections associated to these modules, one needs the tool of standard module frames (as I mentioned in my answer to your other question).
It may help to compare their construction of projective modules over $C(T^2)$ to the classical construction of holomorphic vector bundles over tori in complex analysis, as in Chapter I Section 2 of Mumford's "Abelian Varieties." This should help bridge the gap between the topological/algebraic treatments.

$\begingroup$ @ mkreisel: Thank you for the answer. Unfortunately, I haven't mixed up the Ktheory of the sphere and the $2$torus. I am reading an article (Tom Hadfield, The khomology of $A_\theta$) and he says that the generator of the $K_0$group are $1$ and the Bott projection. I asked this question because as you said, usually the Bott projection is used with $S^2$. However, thanks for the help. $\endgroup$ – John N. Aug 4 '14 at 18:42

$\begingroup$ @ mkreisel: Thank you very much for the help and references. I have been thinking on this question for a while. $\endgroup$ – John N. Aug 4 '14 at 19:14
I'm assuming that by "torus" you mean the $1$torus $T$, i.e. the circle; if you are interested in a higher dimensional torus, you can compute with Ktheory products.
Elements of $K_1(C(T))$ are represented by unitary matrices with entries in $C(T)$, i.e. loops of unitary matrices. Viewing $T$ as the unit circle in the complex plane, the Bott generator is the class in $K_1(C(T))$ represented by the unitary loops $z \mapsto \overline{z}$.
To connect this with your understanding of $K_0(\mathbb{R}^2)$, note that for any C*algebra $A$ we have the suspension isomorphism: $$K_1(A) \cong K_0(C_0(\mathbb{R}) \otimes A)$$ Specializing to the case $A = C(T)$ and noting that $T$ is the onepoint compactification of $\mathbb{R}$, we have: $$K_1(C(T)) \cong K_1(C_0(\mathbb{R})) \cong K_0(C_0(\mathbb{R}^2))$$ To compare the Bott generator that I described above to whatever formula you have for the Bott generator of $K_0(\mathbb{R}^2)$, you simply need to choose an explicit isomorphism between $C(T)$ and the unitalization of $C_0(\mathbb{R})$ and write down an explicit formula for the suspension isomorphism which is welladapted to your formula.

$\begingroup$ Thanks for the answer. I am sorry I meant the $2$dimensional torus $T^2$. I will immediately correct the text. $\endgroup$ – John N. Aug 4 '14 at 16:53

$\begingroup$ @ Paul Siegel: I am sorry. I think I don't understand your answer. I asked for the generators of $K_0(C(T^2))$, but if I have understood correctly you gave me the generator of the $K_1$group. I apology for my description of the problem. $\endgroup$ – John N. Aug 4 '14 at 16:59