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The automorphism group $Aut(E_1)$ of the $E_1$ operad is the cyclic group of order 2, $C_2$, and thus $C_2$ acts on any category of algebras (by reversing the multiplication). The seeming coincidence that $Gal(\mathbb C/\mathbb R) = C_2$ as well allows one to define such notions as $C^\ast$ algebras, where one asks for an appropriate interaction between the actions of $Gal(\mathbb C / \mathbb R)$ and $Aut(E_1)$.

It goes without saying that by the law of small numbers, many mathematical objects will have automorphism group $C_2$, and in general one shouldn't expect to read too much into such coincidences. But the simple fact that $C^\ast$-algebras and other sorts of $\mathbb C$-algebras with involution turn out to be so incredibly useful in mathematics (and physics, for that matter) suggests to me that maybe in this case, it's not a coincidence that $Gal(\mathbb C/\mathbb R) = Aut(E_1)$. However, I can't really think of a "good reason" that these two groups are isomorphic. So I'll ask

Questions:

  1. Is there some "good reason" why $Gal(\mathbb C / \mathbb R)$ acts on the category of $\mathbb C$-algebras in two different ways -- both canonically and by reversing multiplication?

  2. Even better, is there a "good reason" why $Gal(\mathbb C / \mathbb R)$ acts on the $E_1$ operad?

  3. Is there some generalization of the notion of algebra with involution that makes sense for other Galois extensions? Or is there another direction of generalization which sheds light on the usefulness of $\mathbb C$-algebras with involution?

  4. If there's not really a "good reason" for all of this, then is there some other way to explain the usefulness of the notion of a $C^\ast$ algebra, or a $\mathbb C$-algebra with involution?

In (1) and (2), roughly by a "good reason" for an action I mean: could you construct this action without having done the computation to show that $Gal(\mathbb C / \mathbb R) = C_2$.

If I recall correctly, the fact that $Gal(\mathbb C / \mathbb R) = C_2$ could be chalked up to the fact (I think?) that any finite-index subfield of an algebraically closed field is of index 2. But (perhaps since I'm not familiar with the proof of this fact) I'm not sure how to relate this to the idea that one can reverse the multiplication on an algebra to get another algebra.

I'm tempted to frame this in the more general context of the notion of duality in mathematics, and to ask "Which mathematical dualities are "related", and which ones are "unrelated"?" -- but I think I may need to sharpen the question a bit before asking something like that here on MO.

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    $\begingroup$ The Artin-Schreier theorem is actually even stronger. It says that if an algebraically closed field $F$ has a finite index subfield $R$, then $R$ must be a real closed field and $F=R[\sqrt{-1}]$. In particular $\mathrm{char}\,F=0$. That said I would argue that this is a case of the law of small numbers. $\endgroup$
    – Denis Nardin
    Commented Sep 26, 2019 at 17:17
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    $\begingroup$ You might find Theo’s paper interesting, though I’m not sure it really conclusively gives an answer to your question it has a lot of interesting things to say about it: arxiv.org/abs/1507.06297 $\endgroup$
    – Noah Snyder
    Commented Sep 26, 2019 at 17:47
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    $\begingroup$ Another vague answer is that the E2 operad is canonically a Z/2-equivariant operad via complex conjugation, which a very natural structure to consider for "real homotopy theory" reasons (for example the configuration spaces of n points in the plane are complex points of varieties defined over Q). The (non-equivariant) E1 operad is the fixed point sub-operad. Combining this conjucation equivariance structure with the natural cyclic Z/2 action on E2 should give a Z/2 equivariant E1 operad as some sort of twisted fixed points. $\endgroup$
    – Dmitry Vaintrob
    Commented Sep 26, 2019 at 19:33
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    $\begingroup$ This may be slightly orthogonal to what you're seeking, but I just want to say that the depth/importance of Cstar algebras, if one believes in such, is much much much much much much much more to do with the Cstar condition on the norm. Banach star-algebras, let alone ${\mathbb C}$-algebras with involution, have far fewer of the magical properties of ${\rm C}^*$-algebras $\endgroup$
    – Yemon Choi
    Commented Sep 26, 2019 at 21:35
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    $\begingroup$ One can also take this backward: the Notion of C* algebra exists and is so nice essentially because of this coincidence (which by itself is just as you say an example of the law of small number). An evidence for this is that while one can perfectly do Galois theory over p-adic field and p-adic functional analysis with no problem, there are no really nice $p$-adic analogue of the notion C*-algebra. (And I know several people, me included, that have looked for one). I feel like your question is closely related to that problem by the way... $\endgroup$
    – Simon Henry
    Commented Sep 26, 2019 at 22:52

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