Let $A$ be a unital $C^{\*}$-algebra and $a \in A$ be normal, with spectrum $\sigma(a)$. Let $B = C^{\*}(a)$ be the $C^{\*}$-algebra generated by $1$ and $a$, which is abelian. Let $\hat{B}$ be the space of all homomorphisms $\chi: B \to \mathbb{C}$ with the weak\* topology, which makes it a compact Hausdorff topological space.


Gelfand Isomorphism Theorem states that $\varphi: B \to C(\hat{B})$ is an isometric $\*$-isomorphism. Moreover, $\hat{a}: \hat{B} \to \sigma(a) \subset \mathbb{C}$ is a homeomorphism, so that $\hat{B} \cong \sigma(a)$.


On one hand, given a positive functional $\omega: C(\hat{B}) \to \mathbb{C}$, Riesz Representation Theorem states that there exists a unique regular Borel measure $\mu$ on $\sigma(a)$ such that:
$$\omega(f) = \int\_{\hat{B}}f(\chi)d\mu\_{\omega}(\chi).$$


On the other hand, given a function $g \in C(\sigma(a))$, there is a natural identification $g \mapsto g\circ \hat{a}$ from $C(\sigma(a))$ to $C(\hat{B})$. Hence, each $\mu\_{\omega}$ induces a new measure $\nu\_{\omega}$ on $\sigma(a)$ by using $\hat{a}^{-1}$ as a push-forward.


By the abstract change of variables formula, it must hold:
$$\int\_{\hat{B}} (g\circ \hat{a})(\chi)d\mu\_{\omega}(\chi) = \int\_{\sigma(a)}g(z)d\nu\_{\omega}(z) \tag{1}\label{1}$$


My question is: can we extend this analysis to $A$ instead of $B$ (and $\hat{A}$ instead of $\hat{B}$ and so on) when $A$ is abelian? I don't think so because, in this case, $\hat{a}$ is not a homeomorphism between $\hat{A}$ and $\sigma(a)$, but I couldn't tell if a general version of formula (\ref{1}) holds from the literature I know on the topic.



Spectral theorem for unital $C^{*}$-algebras
To see that (1) continues to hold when B is replaced by A,
it suffices (by linearity and density of step functions) to consider only the case when g is the characteristic function of some measurable subset S of σ(a).


Indeed, in this case the right side equals the ν-measure of S, whereas the left side equals the μ-measure of the â-preimage of S. Since ν was defined as the â-pushforward of μ, this equality holds by definition.


