Given a commutative $C^{*}$ algebra $A$, by Gelfrand duality, $A$ is homeomorphic to a locally compact Hausdorff space $X$. Then can we know the topological dimension of $X$ from $A$?and how?
My first thought is the Krull dimension, by the maximal length of chain of prime ideal of $A$. But I don't think that is the case.