This is related to my previous question on math.se .
I'm trying to learn "the basics" of this maximal ideal space, let's denote it by $\Delta(\mathcal{M}(\mathbb{T}))$. What lives on $\Delta(\mathcal{M}(\mathbb{T}))$? I've learnt from Y. Katznelson's book 'An introduction to harmonic analysis' that $$h_n\colon\mathcal{M}(\mathbb{T})\to\mathbb{C},\ h_n(\mu):=\hat{\mu}(n),$$ are elements of $\Delta(\mathcal{M}(\mathbb{T}))$. Here $\hat{\mu}(n)$ is the $n$-th Fourier-Stieltjes coefficient of $\mu\in\mathcal{M}(\mathbb{T})$. Identifying $n\leftrightarrow h_n$ we can see $\mathbb{Z}$ as a subset of $\Delta(\mathcal{M}(\mathbb{T}))$.
Why there are more elements in $\Delta(\mathcal{M}(\mathbb{T}))$ than $\mathbb{Z}$? From Tomek Kania's answer in my previous post, clearly one answer is because $\Delta(\mathcal{M}(\mathbb{T}))$ is non-separable. However, this is a hard result. The book from Y. Katznelson answers this question by showing that $\Delta(\mathcal{M}(\mathbb{T}))$ is uncontable (section 9.1, p. 259). He argues appealing to the compactness of $\Delta(\mathcal{M}(\mathbb{T}))$:
"Since $\Delta(\mathcal{M}(\mathbb{T}))$ is compact, the range of every $\hat{\mu}$ on $\Delta(\mathcal{M}(\mathbb{T}))$ is compact and therefore contains the closure of the sequence $\{\hat{\mu}(n)\}_{n\in\mathbb{Z}}$ , which may well be uncountable (e.g., if $\hat{\mu}(n) = \cos n,\ n\in\mathbb{Z}$). Thus $\Delta(\mathcal{M}(\mathbb{T}))$ is uncountable and is therefore much bigger than $\mathbb{Z}$".
(I've adjusted the notation.) The meaning of this has eluded me so far; $\hat{\mu}$ is the Fourier-Stieltjes transform of $\mu$ and therefore a function from $\mathbb{R}$ to $\mathbb{C}$ so why would it have its image on $\Delta(\mathcal{M}(\mathbb{T}))$? Also, why would $\Delta(\mathcal{M}(\mathbb{T}))$ contain a closure of a sequence in $\mathbb{C}$?
I suppose I am missing something obvious here cause I don't see any sense in this. Can someone please help me understand why this proves that $\Delta(\mathcal{M}(\mathbb{T}))$ is uncontable? (Another accesible argument proving $\Delta(\mathcal{M}(\mathbb{T}))\setminus\mathbb{Z}\neq\phi$ would be well-received too.)
