I am looking for an example of a state (positive linear functional of norm 1) on the space $\displaystyle C(\mathbb{T}^2)$ of continuous functions on the torus $\displaystyle \mathbb{T}^2$.

My first guess would be something like

$\displaystyle \omega(f)=\int^{2\pi}_{0}\int^{2\pi}_{0}f(x,y)dxdy$

with norm

$\displaystyle \|\omega\|=\sup_{f\in C(\mathbb{T}^2)}\left|\int^{2\pi}_{0}\int^{2\pi}_{ 0}f(x,y)dxdy\right|$

with $\displaystyle f\in C(\mathbb{T}^2)$. However, if we take the function $\displaystyle h$ to map any element of the torus identically to 1 the above definition would imply that

$\displaystyle \omega(h)=\int^{2\pi}_{0}\int^{2\pi}_{0}h(x,y)dxdy$

$\displaystyle =\int^{2\pi}_{0}\int^{2\pi}_{0}dxdy$

$\displaystyle =4\pi^2$

so this idea does not work. Any ideas?