# Functional Interpretation of Variety?

I am simultaneously taking courses in functional analysis and commutative algebra. In doing so, I found that there is, at least heuristically, some similarity between the notion of an algebraic variety (essentially, the zero locus of a family of polynomials) and the annihilator of a subspace of a Banach space (the collection of continuous functionals that "kill" the subspace). In doing some reading, I came upon the Banach-Stone theorem, which implies that the algebra of scalars is an analogue of the structure sheaf of a ring, and is evidently an essential ingredient in non-commutative geometry (thank you, Wikipedia).

One finds a one-to-one correspondence between finitely generated nilpotent free k-algebras and affine varieties by first asking what algebraic structure (finitely generated nilpotent free k-algebra) corresponds to the geometric object (affine variety). Then, inspecting what sort of geometric object gives the correspondence with the more general commutative rings, we find schemes.

This analogue of a structure sheaf makes me think that there might be analogous sort of correspondence in the Banach algebra setting. Is this the case? If so, can anybody provide some references, interesting papers, etc?

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I found the following article by Cartier interesting, which contains a short biography of Grothendieck, along with some musings about these sorts of ideas: ams.org/journals/bull/2001-38-04/S0273-0979-01-00913-2 – Andrew Nov 13 '12 at 5:53
That is not what the Banach-Stone theorem says. – Qiaochu Yuan Nov 13 '12 at 6:11
@QiaochuYuan Ah, you are right. Thank you. I was conflating the algebra of scalars and the dual of Banach space. – John Martin Nov 13 '12 at 6:21