Math Help - State on classical space

1. State on classical space

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

My first guess would be something like

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

with norm

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

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

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

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

$=4\pi^2$

so this idea does not work. Any ideas?

2. Re: State on classical space

What text are you using?

3. Re: State on classical space

Following your ideas, |w(f)| < 4(pi)^2 ||f|| where the last norm is the one in your space so what about v(f)=w(f)/4(pi)^2?

4. Re: State on classical space

I don't think that will work. It will be fine for that particular function, but we can just as well consider a function that maps any element of the torus to say 2, then the integral will have twice that value. I think I have an idea how to construct such a state, will post it if it works.

5. Re: State on classical space

I was assuming there was a typo in your definition of the norm of the functional since by linearity that quantity will never be bounded. So I assumed you meant $\| \omega \| = \sup_{\| f \| =1} |\omega (f)|$ and for this definition my example does work.