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(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 algebraalgebras. So in some from this experience, it seemed seems natural to topologize the dual of an algebra in Algebraic geometry as welland 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.
show/hide this revision's text 3 deleted 3 characters in body; added 169 characters in body

(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 algebra. So in some from this experience, it seemed natural to topologize the dual of an algebra in Algebraic geometry as well.

Pontryagin duality:

The Gelfand Naimark theorem can be specialized 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(?)geometry. Similar things are happening with spectral triples in Arakelov theory now, I guess.
show/hide this revision's text 2 added 291 characters in body

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

Gelfand Naimark theorem.

For a commutative $C^*$ C^\star$ algebra $A$, the spectrum of $A$ are 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 of between locally compact Hausdorff spaces with commutative $C^*$ C^\star$ algebras. This equivalence generalizes to so called to sober spaces.

Pontryagin duality:

The Gelfand Naimark theorem can be specialized 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.
show/hide this revision's text 1