# 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 $$|f-g| if and only if for each $$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?