My question concerns a construction presented in the proof of theorem 12.21 of Rudin's Functional Analysis. Rudin defines a *-isomorphism $\Psi:A_1\to A_2$ between Banach algebras $A_1,A_2$ to be an isomorphism such that $\forall a_1\in A_1:\Psi(\overline{a}_1) = \Psi(a_1)^*$ with $\Psi^*$ being the adjoint of $\Psi$.
Assume that $H$ is a Hilbert space and $\mathfrak{M}$ is a $\sigma$-algebra on a set $\Omega\subset H$. Rudin begins as follows:
To begin with, let $\{\omega_1,\dots,\omega_n\}$ be a partition of $\Omega$, with $\omega_i\in\mathfrak{M}$, and let $s$ be a simple function, such that $s = \alpha_i$ on $\omega_i$. Define $\Psi(s)\in \mathscr{B}(H)$ by $\Psi(s) = \sum_{i=1}^n\alpha_iE(\omega_i).$
Here $E(.)$ is a resolution of the identity, that is a mapping $E:\mathfrak{M}\to \mathscr{B}(H)$ with the properties
1.) $E(\varnothing) = 0, E(\Omega) = I$.
2.) Each $E(\omega)$ is a self-adjoined projection.
3.) $E(\omega\cap\omega') = E(\omega)E(\omega')$.
4.) If $\omega\cap \omega' = \varnothing$, then $E(\omega\cup\omega') = E(\omega') + E(\omega)$.
5.) For every $x, y\in H$, the set function $E_{x,y}$ defined by $E_{x,y}(\omega) = (E(\omega)x,y)$ is a complex measure on $\mathfrak{M}$.
6.) If in addition $\mathfrak{M}$ is the $\sigma$-algebra of all Borel sets on a compact or locally compact Hausdorff space, then each $E_{x,y}$ is a Borel regular measure.
Question: At this point of the proof, exactly what is the domain of $\Psi$ and in what, if in any, way is $\Psi$ defined w.r.t. the basic elements of its domain? Is $\Psi$'s domain, for example, the set of all simple functions $s$ over all finite partitions of $\Omega$, that is if
$$S_n := \{\sum_{i=1}^n \alpha_i\chi_{\omega_i}\mid \forall i,j=1,\dots,n:\text{$\alpha_i\in\mathbb{C}$ and $\omega_i\in\mathfrak{M}$ and $\Omega = \bigcup_{i=1}^n\omega_i$ and $\omega_i\cap\omega_j = \varnothing\leftrightarrow i\neq j$}\}, n\in\mathbb{N}_+$$
then $\mathrm{Dom}(f) = \bigcup_{n=1}^\infty S_n$. And if this is the case, then, again, what should the basic elements be? I am asking this because whenever you "just" define a mapping in this sort of setting, it can get quite messy to know what should be mapped to what.
I must note that Rudin does not explain the definition of $\Psi$ any further, but rather starts to construct the necessary properties for $\Psi$ to be a *-isomorphism.