
> 
> Let $X$ be a compact Hausdorff space and $\mathcal{A}$ be a closed self-adjoint subalgebra of $C(X)$ which contains the constants. Then $\mathcal{A}$ is the collection of continuous functions on $X$ which are constant on the sets of $\prod\_\mathcal{A}$ where
>  $$
> \prod\_\mathcal{A}=X/\sim
> $$
>  with $x\sim y$ iff $f(x)=f(y)$ for all $f\in\mathcal{A}$. 
> 
> 
> 


Could anyone tell me with cited references that who gave the generalized Stone-Weierstrass Theorem above?



Who gave the generalized Stone-Weierstrass Theorem?
This (for real-valued rather than complex-valued functions) was in [Stone's original paper](http://www.ams.org/journals/tran/1937-041-03/S0002-9947-1937-1501905-7/home.html) that proved the Stone-Weierstrass theorem, as Theorem 84. The statement is a bit funny, since he defines the equivalence relation $x\sim y$ not in the obvious way but as "$x$ is in the intersection of all sets $f^{-1}(U)$ such that $f\in\mathcal{A}$, $U\subset\mathbb{R}$ is an open interval, and $y\in f^{-1}(U)$" (this definition appears in the statement of Theorem 81).


