Noncommutative Riemann Surfaces by Embeddings in R^3

We introduce C-Algebras of compact Riemann surfaces $${\Sigma}$$ as non-commutative analogues of the Poisson algebra of smooth functions on $${\Sigma}$$ . Representations of these algebras give rise to sequences of matrix-algebras for which matrix-commutators converge to Poisson-brackets as N → ∞. For a particular class of surfaces, interpolating between spheres and tori, we completely characterize (even for the intermediate singular surface) all finite dimensional representations of the corresponding C-algebras