(In some sense, there is some overlap with Ralph's answer)

Gelfand Naimark theorem.

For a commutative $C^\star$ algebra $A$, the spectrum of $A$ is the set of primitive ideals (=kernel of functionals). With the Zariski topology ($C^\star$ algebraist prefer the notion Jacobson topology/hull-kernel topology), they become a topological space $X$ and we have $C_0(X) \cong A$. This yields an anti equivalence between locally compact Hausdorff spaces with commutative $C^\star$ algebras. This equivalence generalizes to so called to sober spaces, where the dual objects are complete Heyting algebras. So from this experience, it seems natural to topologize the dual of an algebra and see how much is encoded.

Pontryagin duality:

The Gelfand Naimark theorem can be enhanced to the Pontryagin duality of locally compact abelian groups.

Note that the Gelfand Naimark theorem was first, and probably inspired some of the constructions in algebraic geometry. Similar things are happening with spectral triples in Arakelov theory now, I guess.