The asymptotic Fubini-Study operator over general non-Archimedean fields

Given an ample line bundle L over a projective \({{\mathbb {K}}}\)-variety X, with {K}}}\) non-Archimedean field, we study limits of metrics on associated to submultiplicative sequences norms the graded pieces section ring R(X, L). We show that in rather general case, corresponding asymptotic Fubini-Study operator yields one-to-one correspondence between equivalence classes bounded and plurisubharmonic are regularizable from below. This generalizes results Boucksom-Jonsson where this problem has been studied trivially valued case.