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

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?

• I don't know the answer to your question, but my understanding of schemes was surely influenced by knowing the Gelfand-Naimark theory. – Andrea Ferretti Apr 4 '10 at 1:38
• And Grothendieck, who started out in functional analysis, must have also been familiar with the Gelfand theory before he moved to algebraic geometry? – Kevin H. Lin Apr 4 '10 at 7:24
• @Kevin: I agree, and I would be interested to see if he ever mentioned an influence of the former on the latter. – Jonas Meyer Apr 4 '10 at 8:11

## 4 Answers

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.

• Here's a quote from Gel'fand concerning his general point of view on things. It's taken from ng.ru/science/2009-10-28/14_gelfand.html. When he was 15 years old, he saw the power series for the sine function and his previous idea that math has two separate parts, algebra and geometry, completely changed: "Mathematics was represented before me in its unity. And since that time I understood that the different areas of mathematics along with mathematical physics form a single whole." He needed no inspiration from alg. geometry to use abstract algebra in functional analysis. – KConrad Apr 4 '10 at 8:39
• @KConrad: seeing as I wrote (the original version of) the part of that Wikipedia entry which deals with the Gelfand representation on general CBAs, I'm not sure that I completely agree with that interpretation. The conceptual proof of the 1/f lemma was one of the first striking applications; I don't know if it was one of the original motivations. – Yemon Choi Apr 4 '10 at 19:14
• On the 6th slide of the talk math.albany.edu/boris_talk.pdf it is written that Wiener's Tauberian theorem was "the main motivation for Gelfand's theory of normed rings". Email the speaker and ask how he knows that was the main motivation! In Dolgachev's book review of Eisenbud and Harris's book on schemes (ams.org/bull/2001-38-04/S0273-0979-01-00911-9/…), the first section has a history of the ideas on the path to schemes. There was work of Stone, Gelfand, Jacobson, Zariski, and so on. – KConrad Apr 5 '10 at 4:33
• Professor Kehe Zhu kindly replies (quoted with permission): "The remark quoted in your message below was actually from Professor Boris Korenblum himself who knew Gelfand pretty well. I am not sure if he has any reference for it." – Jonas Meyer Apr 5 '10 at 20:17
• The way you start the first paragraph, the exact same thing could be said about Stone's work on Boolean algebras, recast as Boolean rings (that he had to devise an abstract space on which the ring is realized as a ring of functions), and which I would regard as a forerunner of the idea of affine scheme (actually, it's a special case). That was about 1936-1937. I wonder whether that might have had any influence on Gelfand's thinking? (I don't have firm dates to hand, but it was my impression that the Gelfand representation theory for commutative Banach algebras came a few years later.) – Todd Trimble Mar 29 '15 at 20:09

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.

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".

• More precisely, Cartier says on p. 398 with respect to Gelfand's work (and earlier work by von Neumann and in quantum mechanics): " From that point on the evolution of the meaning of the word spectrum can be understood" This is somewhat weaker than that it was the motivation, but probably means something similar. At least, Cartier knows it probably as well as anybody as he was around then. – Lennart Meier Jan 13 '14 at 12:04

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.)

• Thanks, that's very interesting and informative. I had feared that your first sentence would be true, but I'm hoping that someone who was around at the time might have gone on record at some point about the influence (if any) of Gelfand's spectrum. – Jonas Meyer Apr 4 '10 at 8:08
• A side comment: in the introduction to his collected works, Zariski writes of studying Krull's work, and understanding how it might be applied to algebraic geometry. (In particular, how to apply concepts like integrally closed and higher rank valuation theory.) This provides evidence for your assertion that Krull was not himself writing as part of the algebraic geometry community (or, at least, was not in the centre of that community). – Emerton Apr 5 '10 at 2:51