What are the difference between $L^2(S^2) $ and $L^2(T^2)$?

Question

$S^2$ is 2-dim sphere , $T^2$ is 2-dim torus. $L^2$ is Lebesgue spaces. What is the difference of $L^2(S^2)$ and $L^2(T^2)$ ?

In fact, I don't know how to define the difference ,because isomorphism of Hilbert space can't distinguish them. But I feel there should be some difference.

This is a loose question, I am sorry for this , but I really don't know how to ask it exactly.

Answer 1

This doesn't apply to Hilbert spaces, but you might be interested in the fact that if you shift your attention to Banach algebras you have the Gelfand Representation, which I think of as a way to go from a function/operator space back to the underlying topological space. I think this addresses the underlying spirit of your 'loose' question.

If that scratches your itch, you could look into Noncommutative Geometry, which takes the ideas further.

Answer 2

Naturally, the first Hilbert space is a representation of $SO(3)$, and splits explicitely as an orthonormal Hilbert sum of finite dimensional irreducible representation of $SO(3)$ (theory of spherical harmonics), whereas the second is a representation of the abelian group $R/Z^2$ and splits as the direct sum of (one dimensional) representation of this group (Fourier series).

But of course as separable Hilbert spaces they are isomorphic.