The Stone-Weierstrass theorem has an analog for the algebras of smooth functions, called  

> **Naсhbin's theorem**: An involutive subalgebra $A$ in the algebra ${\mathcal C}^\infty(M)$ of smooth functions on a smooth manifold $M$ is dense in ${\mathcal C}^\infty(M)$ if and only if $A$ separates the points and the tangent vectors of $M$.

(See details in: "L.Nachbin. Sur les algèbres denses de fonctions diffèrentiables sur une variètè, C.R. Acad. Sci. Paris 228 (1949) 1549-1551", or in  [J.G.Llavona's monograph][1], or [here][2].) 

This is strange, I can't find an analog for the algebras of holomorphic functions (on complex manifolds). Did anybody think about this?

> **Question**: let $A$ be a subalgebra in the algebra ${\mathcal O}(M)$ of holomorphic functions on a complex manifold $M$ (as a first approximation, we can think that $M$ is just an open subset in ${\mathbb C}^n$). Which conditions should $A$ satisfy for being dense in ${\mathcal O}(M)$? 


  [1]: https://books.google.ru/books/about/Approximation_of_Continuously_Differenti.html?id=C7auJPRK5nAC&redir_esc=y
  [2]: http://arxiv.org/abs/1303.2424