I am trying to understand the theorem on this Wikipedia article.
It is stated as follows.
Theorem. Let $x$ be a normal element of a $C^*$-algebra A with an identity element e. Let $C$ be the $C^*$-algebra of the bounded continuous functions on the spectrum $\sigma(x)$ of $x$. Then there exists a unique mapping $\pi : C \to A$, where $\pi(f)$ is denoted $f(x)$, such that $\pi$ is a unit-preserving morphism of $C^*$-algebras and $\pi(1) = e$ and $\pi(\text{id}) = x$, where $\text{id}$ denotes the function $z \mapsto z$ on $\sigma(x)$.
I am trying to understand the above theorem in a special example which I now describe.
Let $U$ be a unitary operator on a Hilbert space $H$. Let $A \subseteq \mathscr B(H)$ be the $C^*$-algebra generated by $U$. In notation, $A = C^*(I, U)$.
Let $\sigma$ be the spectrum of $U$ (and we know that $\sigma \subseteq S^1$ since $U$ is unitary).
I am trying to see what the map $\pi: C(\sigma) \to A$ looks like.
Note that if $f = \chi_n$, where $\chi_n$ is the $n$-th character on the circle (restricted to $\sigma$), then clearly $\pi: f\mapsto U^n$. This is because $\pi(\chi_1) = U$ and $\pi$ respects multiplication.
Thus if $f$ is a trigonometric polynomial then we know image of $f$. More precisely, if $f = \sum_{-N\leq n\leq N}a_n \chi_n$, then $f\mapsto \sum_{-N \leq n\leq N}a_n U^n$.
So I guessed that if $f = \sum_{n\in \mathbb Z} a_n \chi_n$ then $f$ must go to $\sum_{n\in \mathbb Z} a_n U^n$.
But the this sum may not be well-defined since if $U$ were the identity map then we would be asking for the series $\sum_{n\in \mathbb Z}\hat f(n)$ to be meaningful, which it it not always.
But if $U$ were the identity operator then the spectrum of $U$ is poor (it has only one point. So this "obstruction" is kind of moot.)
This makes me think that if $\sigma$ were all of $S^1$, then the map should indeed behave like the above guess.
Any and all insights into this are welcome.
Thank you.
If $\sigma(U)=S^1$, $C(S^1;\Bbb{C})$ is isomorphic to $A$. The map $\pi$ is norm preserving so $$\|\pi(f)\|_{H} = \sup_{z\in S^1} |f(z)|$$ By some variant of the Stone-Weierstrass theorem, the trigonometric polynomials are dense in $C(S^1;\Bbb{C})$ in the $\sup$-norm. So any $f\in C(S^1;\Bbb{C})$ has an arbitrarily close trigonometric polynomial $f_\epsilon$ such that $\|f - f_\epsilon\|_\infty < \epsilon$ and then $\|\pi(f)-\pi(f_\epsilon)\|_{H} < \epsilon$, and $\pi(f_\epsilon)$ is just the polynomial $f_\epsilon(U)$ in $U$.
However, that doesn't mean you can approximate $f$ in the $\sup$-norm using the partial sums of its Fourier series. There is a discussion on this Wikipedia page of various forms of convergence of Fourier series and some cases where you can conclude the series converges in the $\sup$-norm.