K-theory for the $C^*-$algebra of the continuous functions on the $2-$torus and the Bott projection

Question

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.

Answer 1

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 $$ p(\phi,\theta) = \left[\begin{array}{cc} f(\phi) & g(\phi)+h(\phi)e^{i\theta}\\ g(\phi)+h(\phi)e^{-i\theta} & 1-f(\phi) \end{array}\right] . $$

Answer 2

I think you may have mixed up the K-theory of the 2-torus with the K-theory 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 $$p(x,y,z) = \begin{pmatrix} \frac{1+x}{2} & \frac{y+iz}{2}\\ \frac{y-iz}{2} & \frac{1-x}{2}\\ \end{pmatrix}$$ 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 K-theory and its Applications."

EDIT: Since you do want to know about the K-theory of the 2-torus, 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 K-theory 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.

Answer 3

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 K-theory 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 one-point 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 well-adapted to your formula.