Chapter 13 Semi-Groups of Superoperators

Abstract

This chapter discusses the semi-groups of superoperators. Unitary superoperator groups play a very important role in quantum theory. It is conventional to describe the symmetries of quantum systems by superoperator groups on the kinematical set. The basic theorem regarding unitary group is presented in the chapter. The Stone's theorem establishes the general form of a continuous unitary superoperator. If M be an operator Banach space, a set of superoperators on M together with a single binary operation is called a “superoperator groupoid.” A groupoid satisfying the associative law is called a “semi-group.” The unital semigroups are discussed in the chapter, which are semi-groups with an identity superoperator L I . The chapter also discusses the basic results of the superoperator semi-group theory that may be considered as a generalization of the Stone's theorem for one parameter superoperator group on a Hilbert operator space.