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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
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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

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If I'm reading your question correctly (you may be asking several related things and I am not sure exactly what is a question and what is a discussion), the analogous theorem in the Banach algebra setting is the commutative Gelfand-Naimark theorem. The geometric thing is compact Hausdorff spaces and the algebraic thing is commutative unital C*-algebras. The Banach-Stone theorem tells you that you can go from spaces to algebras back to spaces, but the Gelfand-Naimark theorem also tells you that you can go from algebras to spaces back to algebras.

In this way you can think about noncommutative unital C*-algebras as "algebras of functions on noncommutative compact Hausdorff spaces."

I am not sure exactly what you mean by a similarity between the notion of a zero locus and the notion of an annihilator. Here are two things this could mean: they are both given by a suitable equalizer, and both constructions induce a Galois connection between suitable posets.

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In regards to your last paragraph, both of these popped out as connections (no pun intended) to me. –  John Martin Nov 13 '12 at 6:17

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