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

2010-04-21T15:18:42Z

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.

Answer by Jonas Meyer for Algebraic properties of the algebra of continuous functions on a manifold.

2010-04-22T05:30:15Z

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 $C_0(\mathbb{R}^n)$?

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

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

Answer by Jonas Meyer for Algebraic properties of the algebra of continuous functions on a manifold.