Standard generalized vectors in the space of Hilbert-Schmidt operators

Antoine, Jean-Pierre [UCL] Ogi, H. Inoue, A. Trapani, C.

Given an O*-algebra N acting in a Hilbert space K, standard generalized vectors for N are a possible tool for setting up a Tomita-Takesaki theory of modular automorphisms oil N, and thus for constructing KMS quasi-weights on N. If N is the observable algebra of a physical system, these quasi-weights may be interpreted as equilibrium states of the system. In this paper, We consider the case where K is the space of Hilbert-Schmidt operators on some Hilbert space H and N the natural representation pi (M) on that space of a self-adjoint O*-algebra M acting in H. We show that every positive Hilbert-Schmidt operator on H, and more generally every positive self-adjoint unbounded operator on H, determines a standard generalized vector for pi (M). Then we apply this machinery to several physical examples.