Suppose we take, for example, the $C^\*$-algebra which is the sup norm closure of the exponentials $e^{2 \pi i ax}$ where $a \in \mathbb{Z} + \theta \mathbb{Z}$ for $\theta$ an irrational number. This is the commutative $C^\*$-algebra of almost periodic functions whose spectrum lies in $\mathbb{Z} + \theta \mathbb{Z}.$ 


1.What is the spectrum of this $C^\*$-algebra? For what space $X$ is it isomorphic to $C(X)?$ 


2.Can we compute the (Cech) cohomology or the K-theory of this space? 


3.Same questions, but instead take the analogous $C^\*$-algebra generated by any finitely generated subgroup of $\mathbb{R}^n.$ Can we do any computations for these spaces?



C* algebras of Almost Periodic Functions
To answer my own question: One can see that the $C^\*$-algebra of almost periodic functions with spectrum in $\mathbb{Z}+ \theta \mathbb{Z}$ is isomorphic to the $C^\*$-algebra of a torus. It is generated by two commuting unitaries $e^{2\pi i x}$ and $e^{2 \pi i \theta x}.$ This same argument shows that for any finitely generated subgroup of $\mathbb{R}^n,$ the corresponding $C^\*$-algebra of almost periodic functions is isomorphic to the $C^\*$-algebra of a torus with dimension equal to the rank of the subgroup.


