I've been searching google and scholar google, but i only have come upon orderings and Hermitian forms on \*-fields.


Has real algebraic geometry been carried over to \*-rings? \*-rings are rings with an involution. For example are there Positivstellensätze and characterizations of sums of squares by total positivity etc.


If anyone has references, it would be of great help. Books, papers or online resources are all great. thanks in advance.



Doing Real Algebraic Geometry on *-Rings
Konrad Schmüdgen has set up a programme to develop analogues of the basic results in Real Algebraic Geometry in a setup of $\star$-algebras. See [arxiv.org/abs/0709.3170](http://arxiv.org/abs/0709.3170) and for example [arxiv.org/abs/0903.2708](http://arxiv.org/abs/0903.2708). There is also interesting recent work by Jaka Cimpric on that topic, [arxiv.org/abs/0807.5020](http://arxiv.org/abs/0807.5020). Looking up the references in those articles will give you a lot more interesting sources of information.


There is also a reformulation of the Connes embedding conjecture in terms of sums of squares, which is due to Hadwin, Radulescu and Klep-Schweighofer in different variations.


