Are these strengthenings of Serre-Swan and Gelfand-Naimark true? Let $X$ be a compact Hausdorff space.


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*The Serre-Swan theorem allows us to identify complex vector bundles with projective finitely generated modules over the ring $C(X;\mathbb{C})$ of complex-valued functions on $X$.

*The Gelfand-Naimark theorem tells us that $X$ is determined up to homeomorphism by the C*-algebra $C(X;\mathbb{C})$.

*The Gelfand-Kolmogorov theorem says that $X$ is determined up to homeomorphism by the $\mathbb{R}$-algebra $C(X;\mathbb{R})$.


There are several things that I'm bothered with.


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*Considering just how many functions a given space admits, and considering that any function is determined by its local behaviour at every point, I find it difficult to believe that compactness is really necessary.

*The asymmetry between $\mathbb{R}$ and $\mathbb{C}$ is surprising. Why is there no Serre-Swan theorem for real vector bundles? And why does $C(X;\mathbb{C})$ need more algebraic structure to recover $X$ than $C(X;\mathbb{R})$ does? A search for a real Serre-Swan theorem or a complex Gelfand-Kolmogorov theorem gave me nothing.


The above considerations bring me to the following conjectures. Let $X$ be a locally compact Hausdorff space. 


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*Conjecture 1. Complex vector bundles correspond to projective modules over the ring $C(X;\mathbb{C})$, and real vector bundles correspond to projective modules over the ring $C(X;\mathbb{R})$.

*Conjecture 2. The space $X$ is determined up to homeomorphism by the $\mathbb{C}$-algebra $C(X;\mathbb{C})$, as well as by the $\mathbb{R}$-algebra $C(X;\mathbb{R})$. Perhaps even the ring structure suffices.


My question is of course whether the conjectures are true or not.
 A: Take $X=\omega_1$, in the order topology. Any real or complex valued function on it is bounded so $C(X)$ and $C(\beta X)$ (here $\beta X \simeq \omega_1 +1$, of course) are isomorphic as rings (and as (real) algebras as well, an isomorphism of rings $C(X)$ is always an $\Bbb R$-algebra isomorphism too; I haven't studied the complex case well enough, but the real case is classical.) So in general locally compact Hausdorff spaces you cannot distinguish a space from its (C-S) compactification on ring or algebra structure alone...
A: > Conjecture 1 is true.
The main technical content of the proof is that over a compact Hausdorff space, every real or complex vector bundle is a direct summand of a trivial real or complex vector bundle; see this math.SE question (which doesn't have a proof but does have a comment with a link to a proof). 

When $X$ is compact Hausdorff, the C*-algebra structure of $C(X, \mathbb{C})$ can be recovered from its $\mathbb{C}$-algebra structure, so the $\mathbb{C}$-algebra structure determines $X$.

To see this, first observe that the range of a function $f : X \to \mathbb{C}$ can be recovered by considering its spectrum $\sigma(f)$ (the set of $\lambda \in \mathbb{C}$ such that $f - \lambda$ fails to be invertible), which only depends on the $\mathbb{C}$-algebra structure. From the spectrum we can recover the norm as the spectral radius
$$\| f \| = \sup_{\lambda \in \sigma(f)} \| \lambda \|$$
so the norm can be recovered from the $\mathbb{C}$-algebra structure. Next, we can also get the *-structure by considering the subspaces of $C(X, \mathbb{C})$ consisting of elements with purely real resp. purely imaginary spectrum; $C(X, \mathbb{C})$ is always the direct sum of these, and the *-structure acts by the identity on the first bit and by $-1$ on the second bit.
