Some algebraic structures related to a quantum system with infinite degrees of freedom

In this short note, we give a brief study of the *-algebra R ⊗ ∞ which is given by certain “inductive limit process” of the algebraic tensor product R ⊗ n of the C*-algebra R representing quantum mechanics with one particle. In particular, we will study “feasible pure states” of R ⊗ ∞ and their GNS constructions.