About the notion of eigenstates for $C^{\ast }$-algebras and some application in Quantum Mechanics

This work is concerned with the notion of {eigenstates} for $C^{\ast }$-algebras. After reviewing some basic and structural results, we explore the possibility of reinterpreting certain typical concepts of quantum mechanics (\eg dynamical equilibrium states, ground states, gapped states, Fermi surfaces) in terms of (algebraic) eigenstates.