Asymptotic Grothendieck groups and c.b.l.f. positive dg-algebras

We introduce the notion of asymptotic Grothendieck groups for abelian and triangulated categories that are both AB4 and AB4*. We study when the asymptotic Grothendieck group of the heart of a triangulated category with a t-structure is isomorphic to the asymptotic Grothendieck group of the triangulated category itself. We also explain a connexion with the notion of topological Grothendieck group from Achar-Stroppel. Finally, we compute the asymptotic Grothendieck group of some positive (dg-)algebras having a cone bounded, locally finite dimension.