Did Gelfand's theory of commutative Banach algebras influence algebraic geometers?

Question

Guillemin and Sternberg wrote the following in 1987 in a short article called "Some remarks on I.M. Gelfand's works" accompanying Gelfand's Collected Papers, Volume I:

The theory of commutative normed rings [i.e., (complex) Banach algebras], created by Gelfand in the late 1930s, has become today one of the most active areas of functional analysis. The key idea in Gelfand's theory -- that maximal ideals are the underlying "points" of a commutative normed ring -- not only revolutionized harmonic analyis but had an enormous impact in algebraic geometry. (One need only look at the development of the concept of the spectrum of a commutative ring and the concept of scheme in the algebraic geometry of the 1960s and 1970s to see how far beyond the borders of functional analysis Gelfand's ideas penetrated.)

I was skeptical when reading this, which led to the following:

Basic Question: Did Gelfand's theory of commutative Banach algebras have an enormous impact, or any direct influence whatsoever, in algebraic geometry?

I elaborate on the question at the end, after some background and context for my skepticism.

In the late 1930s, Gelfand proved the special case of the Mazur-Gelfand Theorem that says that a Banach division algebra is $\mathbb{C}$. In the commutative case this applies to quotients by maximal ideals, and Gelfand used this fact to consider elements of a (complex, unital) commutative Banach algebra as functions on the maximal ideal space. He gave the maximal ideal space the coarsest topology that makes these functions continuous, which turns out to be a compact Hausdorff topology. The resulting continuous homomorphism from a commutative Banach algebra $A$ with maximal ideal space $\mathfrak{M}$ to the Banach algebra $C(\mathfrak{M})$ of continuous complex-valued functions on $\mathfrak{M}$ with sup norm is now often called the Gelfand transform (sometimes denoted $\Gamma$, short for Гельфанд). It is very useful.

However, it is my understanding that Gelfand wasn't the first to consider elements of a ring as functions on a space of ideals. Hilbert proved that an affine variety can be considered as the set of maximal ideals of its coordinate ring, and thus gave a way to view abstract finitely generated commutative complex algebras without nilpotents as algebras of functions. On the Wikipedia page for scheme I find that Noether and Krull pushed these ideas to some extent in the 1920s and 1930s, respectively, but I don't know a source for this. Another related result is Stone's representation theorem from 1936, and a good summary of this circle of ideas can be found in Varadarajan's Euler book.

Unfortunately, knowing who did what first won't answer my question. I have not been able to find any good source indicating whether algebraic geometers were influenced by Gelfand's theory, or conversely.

Elaborated Question: Were algebraic geometers (say from roughly the 1940s to the 1970s) influenced by Gelfand's theory of commutative Banach algebras as indicated by Guillemin and Sternberg, and if so can anyone provide documentation? Conversely, was Gelfand's theory influenced by algebraic geometry (from before roughly 1938), and if so can anyone provide documentation?

Answer 1

A difference between what Gel'fand did and what the Germans were doing is that in 1930s-style algebraic geometry you had the basic geometric spaces of interest in front of you at the start. Gel'fand, on the other hand, was starting with suitable classes of rings (like commutative Banach algebras) and had to create an associated abstract space on which the ring could be viewed as a ring of functions. And he was very successful in pursuing this idea. For comparison, the Wikipedia reference on schemes says Krull had some early (forgotten?) ideas about spaces of prime ideals, but gave up on them because he didn't have a clear motivation. At least Gel'fand's work showed that the concept of an abstract space of ideals on which a ring becomes a ring of functions was something you could really get mileage out of. It might not have had an enormous influence in algebraic geometry, but it was a basic successful example of the direction from rings to spaces (rather than the other way around) that the leading French algebraic geometers were all aware of.

There is an article by Dieudonne on the history of algebraic geometry in Amer. Math. Monthly 79 (1972), 827--866 (see http://www.jstor.org/stable/pdfplus/2317664.pdf) in which he writes nothing about the work of Gelfand.

There is an article by Kolmogorov in 1951 about Gel'fand's work (for which he was getting the Stalin prize -- whoo hoo!) in which he writes about the task of finding a space on which a ring can be realized as a ring of functions, and while he writes about algebra he says nothing about algebraic geometry. (See http://www.mathnet.ru/php/getFT.phtml?jrnid=rm&paperid=6872&what=fullt&option_lang=rus, but it's in Russian.) An article by Fomin, Kolmogorov, Shilov, and Vishik marking Gel'fand's 50th birthday (see http://www.mathnet.ru/php/getFT.phtmljrnid=rm&paperid=6872&what=fullt&option_lang=rus, more Russian) also says nothing about algebraic geometry.

Is it conceivable Gel'fand did his work without knowing of the role of maximal ideals as points in algebraic geometry? Sure. First of all, the school around Kolmogorov didn't have interests in algebraic geometry. Second of all, Gel'fand's work on commutative Banach algebras had a specific goal that presumably focused his attention on maximal ideals: find a shorter proof of a theorem of Wiener on nonvanishing Fourier series. (Look at http://mat.iitm.ac.in/home/shk/public_html/wiener1.pdf, which is not in Russian. :)) A nonvanishing function is a unit in a ring of functions, and algebraically the units are the elements lying outside any maximal ideal. He probably obtained the idea that a maximal ideal in a ring of functions should be the functions vanishing at one point from some concrete examples.

Answer 2

In conversations and discussions, David Mumford mentioned several times that Gelfand's result about commutative Banach algebras had great influence for the development of schemes. He also mentioned that local coordinate systems on manifolds had influence.

Answer 3

You might take a look at Section 4 ("Toward the concept of spectrum") in the article "A mad day's work: from Grothendieck to Connes and Kontsevich: The evolution of concepts of space and symmetry" by Pierre Cartier:

http://www.ams.org/journals/bull/2001-38-04/S0273-0979-01-00913-2/

He mentions that Gelfand's work provides the motivation for the term "spectrum".

Answer 4

For a definitive answer, this question seems to require someone who was around in the time in question. (I certainly wasn't.) But I'll try anyway:

I think that the notion of prime spectrum of a ring certainly owes some debt to the Gelfand spectrum of a commutative Banach algebra (as well as the Stone space of a Boolean ring, which is in fact a special case). But I gather that the analogy looked rather far-fetched at the time: e.g. if you are used to thinking about compact Hausdorff spaces, the Zariski topology looks quite pathological. Moreover, although nowadays it is often unblinkingly used as motivation that the elements of $R$ are "functions" on $\operatorname{Spec}(R)$, this is of course not literally true -- the residue field, and hence the codomain of the function, is allowed to vary from point to point -- hence I have some suspicions as to whether this was the original motivation of the founders of scheme theory.

As to whether algebraic geometry was a motivation for the Banach algebraists, I rather suspect not: I'm not sure exactly where algebraic geometry was in the 1930's, but I don't think the algebraic geometers were thinking in terms of spectra. (I read somewhere that Krull had ideas in this direction at the time, but was largely ignored. Anyway, Krull was a commutative algebraist, and I think that in the 1930's, that's not as close to being an algebraic geometer as it is today.)

An exception to the above paragraph is the case of finitely generated algebras over $\mathbb{C}$, which was well appreciated since the early 20th century (Hilbert's Nullstellensatz).

More recently, the influence of Gelfand style spectra has increased greatly in the arithmetic geometry community. Namely, in the late 80's, Berkovich defined the Berkovich spectrum of a commutative Banach ring, which in retrospect is such a natural generalization of Gelfand's spectrum that it seems hard to believe that it took so long to catch on. (Of course, if you read Berkovich's book on non-Archimedean spectral theory and analysis, you see that he too had some serious technical difficulties to overcome to develop his theory to the point of recovering, e.g., Serre's GAGA theorems.)