Theory of C* algebras over other fields than the complex numbers 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. 
 A: 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).
