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I'm wondering to what extent it might be possible for the theory of $C^*$-algebras to be translated into the $p$-adic context i.e. to define 'p-adic $C^*$-algebras' over some extension of $\mathbb{Q}_p$.

An obvious candidate for a p-adic analogue of the complex numbers would be $\mathbb{C}_p$, but according to Alain M. Robert in A Course in p-adic Analysis, spherical completeness is required for the non-Archimedean analogue of the Hahn-Banach Theorem to work (known as Ingleton's Theorem).

Since spherical completeness is equivalent to maximal completeness in ultrametric fields, it would make sense to look at maximally complete extensions of $\mathbb{Q}_p$. Bjorn Poonen describes in his paper Maximally Complete Fields a maximally complete immediate extension $L$ of $\overline{\mathbb{Q}_p}$ (where $\overline{\mathbb{Q}_p}$ is a fixed algebraic closure of $\mathbb{Q}_p$) which is also Cauchy-complete (in the sense of uniform spaces) and algebraically closed.

A lot of the natural structure present in $\mathbb{C}$ breaks down in a non-Archimedean setting which is what makes me have doubts about whether this is doable. For example, it seems unlikely that there is a natural involution in $L$ and there is no natural choice of ordered subfield analogous to $\mathbb{R}$. Might there be a reasonable way around these obstacles?

For example, if all separable infinite-dimensional complex Hilbert spaces are isomorphic to $\ell^2(\mathbb{C})$, could there be an analogous motivating example in the p-adic case to define 'p-adic Hilbert spaces'?

Any advice would be much appreciated.

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    $\begingroup$ The automorphism group $Aut(L/\mathbb Q_p)$ certainly does not contain an involution - by the Artin-Schreier theorem, its fixed field would have to be a real-closed field. But such a field cannot contain $\mathbb Q_p$, since the latter contains negative integers which are squares ($1-p$ for $p>2$, $-7$ for $p=2$). $\endgroup$
    – Wojowu
    Commented Aug 6, 2021 at 22:08
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    $\begingroup$ A quick Google search reveals some discussion of nonarchimedean $C^*$-algebras and Hilbert spaces. Have you taken a look at those? There is also a related discussion here $\endgroup$
    – Wojowu
    Commented Aug 6, 2021 at 22:15
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    $\begingroup$ Over locally compact non-Archimedean fields, some of the special properties that normally characterize Hilbert spaces (having an orthogonal base, having every closed subspace orthocomplemented, etc.) actually end up characterizing spaces like $c_0$ (i.e., sequences limiting to $0$ with the sup norm). See Chapter 5 of Non-Archimedean Functional Analysis by van Rooij. One major thing you don't have is reflexivity; there are no infinite-dimensional reflexive Banach spaces over such fields. $\endgroup$
    – James E Hanson
    Commented Aug 7, 2021 at 13:12

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