Q:

Algebraic properties of the algebra of continuous functions on a manifold.

Does the algebra of continuous
  functions from a compact manifold to
  $\mathbb{C}$ satisfy any specific
  algebraic property?

I'm not sure what kind of algebraic property I expect, but I feel that because of the Gel'fand transform, it may not be unreasonable to expect something. We can drop the compactness condition if we switch to continuous functions to $\mathbb{C}$ that vanish at infinity.
I'm really hoping for some necessary and sufficient condition, but if anybody knows of any sort of condition, that would be appreciated.

A:

I found a reference for a necessary property that might be called algebraic.
Browder proved a theorem relating the number of generators of a complex commutative Banach algebra to the Čech cohomology with complex coefficients of the maximal ideal space, and as a corollary concluded that if $M$ is a compact orientable $n$-dimensional manifold, then $C(M)$ cannot be generated as a Banach algebra by fewer than $n+1$ elements. The paper is very short, but for an even shorter summary here's the MR review.

Just some comments, added later:
One obtains the compact Hausdorff space $X$ (up to homeomorphism) from $C(X)$ by considering the maximal ideal space of $C(X)$ with Gelfand topology, but clearly you want something less tautological than "the maximal ideal space is a manifold."  A small step in this direction would be to try to formulate the topological properties of $X$ in terms of the closed ideals of $C(X)$.  As alluded to in Qiaochu's comment, there is an analogue of Nullstellensatz: each closed ideal in $C(X)$ consists of all functions vanishing on a (uniquely determined) closed subset of $X$.  So for example, the locally Euclidean property could be reformulated for a commutative C*-algebra $A$ as follows: There is an $n$ such that for every maximal ideal $M$ of $A$ there is a closed ideal $I$ of $A$ such that $I$ is not contained in $M$ and $I$ is $*$-isomorphic to $C_0(\mathbb{R}^n)$.  Second countability of the maximal ideal space is equivalent to $A$ being separable in the norm topology; that's not algebraic, but might be considered more intrinsic to the C*-algebra.
But this only leads to another, more specific question: Is there a useful or interesting (C*-)algebraic characterization of $C_0(\mathbb{R}^n)$?`

