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How much of the theory of C*-algebras holds when the complex numbers are replaced by different (algebraically closed) field (possibly with a distinguished ordered subfield that satisfies the same properties as the reals)?

Examples of some fields for which I would be interested in knowing if there are known analogs include (but aren't limited to):

  • Surreals.
  • Algebraic Closure of Surreals.
  • The uncountable algebraic closed field of characteristic $p$.
  • The p-adics, $\mathbb{Q}_p$.
  • The algebraic closure of the p-adics, $\mathbb{C}_p$.

I realize this is a slightly ill-defined question, however it seems probable that it has none the less been worked out and so I would appreciate any relevant references.

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    $\begingroup$ If you want to do functional analysis over some fields $K$ you might want to assume that $K$ is a complete valued fields, or at least some sort of topological fields. Now there is a all bunch of unique charaterization of $\mathbb{R}$ as the a valued/topological ordered fields. So there is a high chance that your "distinguished ordered subfield that satisfies the same properties as the reals" is actually canonically isomorphic to $\mathbb{R}$ simply because it satisfies one of these characterization. So it might be good to have such an example other than $\mathbb{R}$ and $\mathbb{C}$ to discuss $\endgroup$
    – Simon Henry
    Commented May 6, 2015 at 8:32
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    $\begingroup$ I should perhaps comment that Cstar algebras are much better than arbitrary Banach-star algebras, even if one rules out "silly" involutions, and it seems that the underlying reason is that they have much better positivity properties (and it is a minor miracle how they all flow just from the Cstar identity). I am therefore inclined to think that if one wants analogues of Cstar algebras over non-archimedean fields one has to do more than just transport the axioms; one may have to impose extra axioms to single out the desired behaviour $\endgroup$
    – Yemon Choi
    Commented May 6, 2015 at 11:03
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    $\begingroup$ I would suggest generalising $B(H)$ before generalising the notion of $C^*$-algebra. Thus, the first question is: what do you want the generalisation of a Hilbert space to be (say over $\mathbb C_p$)? A basic property of Hilbert spaces is that there's essentially only one up to isomorphism. Can you reproduce that feature over $\mathbb C_p$? $\endgroup$
    – André Henriques
    Commented May 6, 2015 at 12:14
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    $\begingroup$ The involution is closely related to complex conjugation on $\mathbf C$ being an automorphism of order $2$ with fixed field $\mathbf R$. Although $\mathbf C_p \cong \mathbf C$ as fields, there is no natural subfield of $\mathbf C_p$ (describable in $p$-adic language) over which $\mathbf C_p$ is a quadratic extension. And the algebraic closure of $\mathbf Q_p$ is an infinite-dimensional extension. So basically any naive attempt to define a $C^*$-algebra over the $p$-adics is not likely to succeed. $\endgroup$
    – KConrad
    Commented May 7, 2015 at 2:51
  • $\begingroup$ @AndréHenriques : it seems that the correct notion of "Hilbert space" over a locale non-archimedian field is "Banach space" (with the condition that the norm takes values in the the same set of reals as the valuation). indeed those already satisfy result of existence of "basis" : any banach space over $\mathbb{Q}_p$ is equivalent to the set of sequences indexed by some set $X$ with values in $\mathbb{Q}_p$ that goes to $0$ at infinity (equivalently which are sommable). But then a $C^*$-algebra should be a subalgebra of $B(h)$ stable under $*$ and we don't know what that mean. $\endgroup$
    – Simon Henry
    Commented May 7, 2015 at 8:00

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Nothing is known in the general case. There is a general theory of $C^*$-algebra over the real number which is very satisfying, so it is not necessarily about algebraically closed fields, but no general theory of $C^*$-algebra over let says $\mathbb{Q}_p$ or $\mathbb{C}_p$ has been developed, and no one knows how to do it (and this is probably the simplest case).

What is not clear is what should be the $*$ when we are not over $\mathbb{R}$ or $\mathbb{C}$ ? Is the $*$ involutive because the Galois group of $\mathbb{C} / \mathbb{R}$ is of order two ?

If we consider the commutative case and Gelfand duality, it would be natural to expect that $p$-adic $C^*$-algebras come with an action of the absolute Galois group of $\mathbb{Q}_p$ instead of a $*$ operation, and one can build an axiomatization of such commutative algebras (with such an action) in order to have a form of Gelfand duality, but then it is not clear what this action should satisfies in the non-commutative case...

This being says, there has been occasionally some $p$-adic version of known $C^*$-algebra, or $C^*$-algebraic constructions... The only one I can think of right now is Connes-Consani $p$-adic Bost-Connes system (ArXiv).

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  • $\begingroup$ Do you know if anyone has tried to extend/refine Berkovich's non-archimedean Banach rings ( ncatlab.org/nlab/show/Banach+ring ) to incorporate Cstar-like features? $\endgroup$
    – Yemon Choi
    Commented May 6, 2015 at 11:00
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    $\begingroup$ I never heard of someone succeeding in that anyway. As I said the thing is that it is really not clear what "Cstar-like feature" mean when we are not over the real or complex numbers. The commutative case suggest that the $*$ is a Galois action, but the non-commutative case does not really accommodate with this points of view. $\endgroup$
    – Simon Henry
    Commented May 6, 2015 at 11:59
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    $\begingroup$ I should also point out to this other question of mine : mathoverflow.net/questions/198966/… where we had a short discusion of whether or not the $*$ is a structure of $C^*$-algebra or not. Especially the Vidav-Palmer theorem pointed out by Yemon choi can be translated into a definition of what a p-adic $C^*$-algebra should be (the decomposition H+iH can be replaced by a decomposition over all irreducible representations of the absolute Galois group of $\mathbb{Q}_P$) this gives a quite convincing notion at least in the commutative case. $\endgroup$
    – Simon Henry
    Commented May 6, 2015 at 12:15
  • $\begingroup$ Thanks, this is very helpful. I am also interested in knowing if the theory of C*-algebras generalized for algebraic closures of specific real closed fields (like for example the surreals) $\endgroup$
    – Nate Ackerman
    Commented May 7, 2015 at 1:45
  • $\begingroup$ @NateAckerman, did you have some specific title for an article that you cannot find? I refer to your MSE question math.stackexchange.com/questions/1290667/… $\endgroup$
    – Will Jagy
    Commented May 20, 2015 at 20:09

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