The Gelfrand duality says that the category of locally compact Hausdorff spaces (with proper continuous functions) is equivalent to the category of commutative $C^*$ algebras (with proper $*$-homomorphisms). For instance, of $X$ is such a topological space, then $C_0(X) = \{f: X\to \mathbb{C}, f$ is continuous and $f$ vanishes at $\infty \}$ is its related $C^*$ algebra. We can go the other way by looking at a $C^*$ algebra $\mathcal{A}$ and taking its set of characters $\text{Hom}(\mathcal{A},\mathbb{C})$ under pointwise convergence to recover the topological space.
So say one has a commutative $C^*$ algebra $\mathcal{A}$, how does one recover topological invariants, like say, the number of connected components of $\text{Hom}(\mathcal{A},\mathbb{C})=X$, from $\mathcal{A}$ itself?
Is this even possible? Or am I wrongly asserting that equivalence of categories says something about the individual objects?
edit many responses focus on the number of connected components, which I appreciate, but that was only meant as an example of the sort qualitative info I would like to recover. Can we recover the singular homology of $X$ from $\mathcal{A}$ ? The fundamental group? Is X metrizable?
The Boolean algebra of connected components is equivalent to the projections (the elements with $p^2=p$) in the algebra of functions, with multiplication of functions representing intersection and $(p_1,p_2) \to p_1 + p_2 - p_1p_2$ being the union of sets of components.
I think there is a version of algebraic K-theory for topological algebras whose value on $C_0(X)$ is the topological $K$-theory of $X$.
Connes' book on NCG has more of the dictionary but also omits many basic things.