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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?

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$C(X)$ is contained in the full $C^*$-algebra almost-periodic functions so, by Gelfand duality, $X$ is the continuous image of the Bohr compactification of $\mathbb{R}$. Since $bohr(\mathbb{R})$ can be expressed as the group of all homomorphisms $\mathbb{R} \to \mathbb{T}$ under the topology of pointwise convergence, it may be reasonable to guess that $X$ can be expressed as the group of all homomorphisms $\mathbb{Z} + \theta \mathbb{Z} \to \mathbb{T}$ with the surjection $bohr(\mathbb{R}) \to X$ being the restriction map. –  Michael Jun 6 at 19:08

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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.

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