## Return to 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?

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# 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?
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