STANDARD GENERALIZED VECTORS IN THE SPACE OF HILBERT-SCHMIDT OPERATORS

Given an O-algebra N acting in a Hilbert space K, standard generalize d vectors for N are a possible tool for setting up a Tomita-Takesaki t heory of modular automorphisms oil N, and thus for constructing KMS qu asi-weights on N. If N is the observable algebra of a physical system, these quasi-weights may be interpreted as equilibrium states of the s ystem. In this paper, We consider the case where K is the space of Hil bert-Schmidt operators on some Hilbert space H and N the natural repre sentation pi (M) on that space of a self-adjoint O-algebra M acting i n H. We show that every positive Hilbert-Schmidt operator on H, and mo re generally every positive self-adjoint unbounded operator on H, dete rmines a standard generalized vector for pi (M). Then we apply this ma chinery to several physical examples.