Let $K$ be a skew-field, infinite dimensional over its center $F$. 


From Kaplansky's PI-theorem it then follows that $K$ cannot satisfy a polynomial identity (the theorem says that primitive PI-algebras have finite dimension over their center).


There, a GPI (generalized polynomial identity) has coefficients from the center $F$. If the coefficients are arbitrary, one has a GI (generalized identity), and a theorem by Amitsur describing the possible structure of $K$.


In my research, I now came upon skew-fields which might satisfy a "skew" GPI in the following sense: 
Let $\sigma$ be an involutory antiautomorphism of $K$ and $\gamma$ an automorphism of $K$ of order 1 or 2. For simplicity, let's restrict to the case that $\sigma$ and $\gamma$ commute.



> 
> Is it possible that $K$ satisfy an "skew" GPI, meaning that
>  coefficients are arbitrary from $K$, and the GPI contains not
>  just $x$, but also $x^\gamma$, $x^\sigma$ and $x^{\gamma\sigma}$ ? The case with a single unknown is all that interests me.
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> 
> 


I actually kinda hope these skew fields don't exist respectively must be finite dimensional over their center. 



(Non-)existence of skew fields satisfying a SGPI (=skew generalized polynomial identity)
I recently came back to this question after leaving it aside for a few years, and this time my digging seems to have yielded an answer: Chen-Lian Chuang, "Differential identities with automorphisms and antiautomorphisms. I", Journal of Algebra 149, 371-404 (1992), proved the following theorem:



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> A prime ring satisfying a nontrivial differential identity with automorphisms and antiautomorphisms must also satisfy a nontrivial ordinary generalized polynomial identity (without derivations, automorphisms, and antiautomorphisms).
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> 


This applies in particular to division rings; in the language I used in the questions: A "skew GPI" implies the existence of a GPI.


As a corollary, any skew field admitting a non-trivial "skew GPI" must be finite-dimensional over its center.


