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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?

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    $\begingroup$ To add an amusing anecdote to a serious question, I recall Marshall Stone himself remarking decades ago that he found it comical when some of his students asked him about his "collaboration" with Weierstrass (who was to Stone an entirely historical person, as Stone himself is now to most mathematicians). $\endgroup$
    – Jim Humphreys
    Commented Jan 26, 2016 at 19:41

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This (for real-valued rather than complex-valued functions) was in Stone's original paper 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).

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