I am trying to understand the K-theory 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. 



K-theory for the $C^*-$algebra of the continuous functions on the $2-$torus and the Bott projection
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
$$
g=\begin{cases}
0 & x\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\\
\sqrt{f-f^{2}} & x\notin\left[-\frac{\pi}{2},\frac{\pi}{2}\right]
\end{cases}
$$
and
$$
h=\begin{cases}
\sqrt{f-f^{2}} & x\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\\
0 & x\notin\left[-\frac{\pi}{2},\frac{\pi}{2}\right]
\end{cases} .
$$
The projection then is then
$$
p(\phi,\theta) = \left[\begin{array}{cc}
f(\phi) & g(\phi)+h(\phi)e^{i\theta}\\
g(\phi)+h(\phi)e^{-i\theta} & 1-f(\phi)
\end{array}\right] .
$$


