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I don't know if this problem is known or not, so any information would be appreciated:

Let $\cal A$ be an $\Bbb{R}$-algebra of (not necessary continuous) real valued functions defined on a topological space. I would like to know what is the necessary and sufficient condition that $\cal A$ is isomorphic (as a ring) to $C(X)$ for some Tychonoff space $X$.

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  • $\begingroup$ Sounds vaguely like the kind of questions that are investigated in the study of natural dualities, a la Clark and Davey ("Natural Dualities for the Working Algebraist", Cambridge Studies in Advanced Mahematics)... $\endgroup$
    – Arturo Magidin
    Commented Sep 21, 2013 at 21:52
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    $\begingroup$ If you restrict "some Tychonoff $X$" to "some compact Hausdorff $X$", then the question is answered by Gelfand duality: there must exist a $C^*$-norm on $\mathcal{A}$ that makes it into a Banach algebra. For more general $X$, I think, the question is more difficult. Also, why present $\mathcal{A}$ as a set of discontinuous functions on a topological space? If the functions are allowed to be discontinuous, then why bother with the topology? $\endgroup$
    – Igor Khavkine
    Commented Sep 21, 2013 at 22:10
  • $\begingroup$ If we say that $\mathcal A$ was functions on $Y$, are you hoping to induce a topology on $Y$ or a quotient to get $X$? $\endgroup$
    – D. Kelleher
    Commented Sep 21, 2013 at 22:21

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In the article Algebraic description of rings of continuous functions you can find an algebraic interpretation of such algebras. the description requires a bit more conditions and it seems better to reference directly to this article.

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