Rieffel projections and 2-by-2 matrices

For a compact space $Y$, we view $C(Y\times S^1)$ as the crossed product $C(Y)\rtimes\mathbb{Z}$, with $\mathbb{Z}$ acting trivially. This allows us to study Rieffel projections in $M_2(C(Y\times S^1))$: we characterize them and compute their image under the projection $\partial_0:K_0(C(Y\times S^1))\rightarrow K_1(C(Y))$. We provide a new Rieffel projection in $M_2(C(\mathbb{T}^2))$, different from Loring's one, and involving only trigonometric polynomials plus the square root of $2-e^{2\pi i\theta}-e^{-2\pi i\theta}$. We give applications of this projection, e.g. explicit generators for the K-theory of $C(\mathbb{T}^3)$. Finally, we prove that, if a Banach algebra completion $\mathcal{B}$ of $\mathbb{C}[\mathbb{Z}^n]$ is continuously contained in $C(\mathbb{T}^n)$ and such that the Fourier series of $(2-e^{2\pi i\theta_j}-e^{-2\pi i\theta_j})^{1/2}\;(j=1,...,n)$ converges in $\mathcal{B}$, then the inclusion $\mathcal{B}\hookrightarrow C(\mathbb{T}^n)$ induces isomorphisms in K-theory.