I want to proof the following theorem:
Let A be a commutative unital C*-Algebra and $a_1, ..., a_n\in A_{sa}$. Then $C^*(1_A, a_1, ..., a_n)\subseteq A$ is *-isomorphic to $C(\Omega )$ for a compact subset $\Omega \subseteq \mathbb{R}^n$.
What may I say about the case $a_1, ..., a_n \in A_+$?
What if $a_1, ..., a_n \in A$ arbitrary?
Let $X$ denote the set of non-zero multiplicative linear functionals on $B := C^{\ast}(1_A, a_1, a_2,\ldots, a_n)$, and let $$ Y := \prod_{i=1}^n \sigma(a_i) $$ Since each $a_i$ is self adjoint, $Y \subset \mathbb{R}^n$. Now consider the map $$ \mu : X \to Y \text{ given by } \tau \mapsto (\tau(a_1), \tau(a_2),\ldots , \tau(a_n)) $$ Since $\tau(1) = 1$ for all $\tau \in X$, it follows that $\mu$ is injective (Why?). It is also clearly continuous (note that $X$ is equipped with the weak-$\ast$ topology). Since $X$ is compact and $Y$ is Hausdorff, $\mu$ induces a homeomorphism $$ X \to \mu(X) =: \Omega $$ Hence, $$ B \cong C(X) \cong C(\Omega) $$ So clearly if each $a_i \in A_+$, then $\Omega \subset (\mathbb{R}_+)^n$