Algebraic properties of the algebra of continuous functions on a manifold. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T07:57:48Z http://mathoverflow.net/feeds/question/22065 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/22065/algebraic-properties-of-the-algebra-of-continuous-functions-on-a-manifold Algebraic properties of the algebra of continuous functions on a manifold. Eric A. Bunch 2010-04-21T15:18:42Z 2010-04-25T01:02:16Z <blockquote> <p>Does the algebra of continuous functions from a compact manifold to \$\mathbb{C}\$ satisfy any specific algebraic property?</p> </blockquote> <p>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.</p> <p>I'm really hoping for some necessary and sufficient condition, but if anybody knows of any sort of condition, that would be appreciated.</p> http://mathoverflow.net/questions/22065/algebraic-properties-of-the-algebra-of-continuous-functions-on-a-manifold/22151#22151 Answer by Jonas Meyer for Algebraic properties of the algebra of continuous functions on a manifold. Jonas Meyer 2010-04-22T05:30:15Z 2010-04-25T01:02:16Z <p>I found a reference for a necessary property that might be called algebraic.</p> <p>Browder <a href="http://projecteuclid.org/euclid.bams/1183524331" rel="nofollow">proved a theorem</a> 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 <a href="http://www.ams.org/mathscinet-getitem?mr=130580" rel="nofollow">here's the MR review</a>.</p> <hr> <p>Just some comments, added later:</p> <p>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 <a href="http://en.wikipedia.org/wiki/Separable_space#Properties" rel="nofollow">is equivalent to</a> \$A\$ being separable in the norm topology; that's not algebraic, but might be considered more intrinsic to the C*-algebra.</p> <p>But this only leads to another, more specific question: Is there a useful or interesting (C*-)algebraic characterization of <code>\$C_0(\mathbb{R}^n)\$?</code></p>