reference request – Stone-Weierstrass theorem for modules of non-self-adjoint subalgebras


In “Weierstrass-Stone, the Theorem” by Joao Prolla, there is a Stone-Weierstrass theorem for modules, stated as the following:

Let $mathcal{A}$ be a subalegebra of $C(X, mathbb{R})$ and $(E, |cdot|)$ be a normed space over $mathbb{R}$. Let $Wsubset C(X, E)$ be a vector subspace which is an $mathcal{A}$-module. For each $fin C(X, E)$ and $epsilon>0$, there exists $gin W$ such that $|f-g|<epsilon$ if and only if for each $xin X$, there exists $g_xin W$ such that $|f(t) – g_x(t)| < epsilon$ for all $tin (x)_{mathcal{A}}$, where $(x)_mathcal{A}$ is the equivalent class of $x$ under $mathcal{A}$.

I know that the above theorem can be extended to $mathcal{A}subset C(X, mathbb{C})$ with $mathcal{A}$ being a self-adjoint subalgebra. I wonder whether there are some similar results for modules of non-self-adjoint algebras.

I’m interested in generalizing the above theorem into the following case. Let $mathcal{S}$ be a finite subset of $C((0, 1), E)$, denoted as $S:={s_1, ldots, s_m}$, and $mathcal{A}subset C((0, 1), mathbb{C})$ be a subalgebra (not necessarily self-adjoint). Then $W := mathrm{span}{as : ain mathcal{A}, sin mathcal{S}}$ is a vector subspace which is an $mathcal{A}$-module. Shall we still claim that $fin overline{W}$ if and only if $fbigvert_{(x)_{mathcal{A}}} in overline{W}bigvert_{(x)_{mathcal{A}}}$? Or is there any counter-example to this statement?