Algebraic Tools

Abstract

This chapter introduces an abstract algebraic language unifying the description of classical systems, finite quantum systems, and infinite ones, the last appearing in particle and statistical physics. In the first part, the general theory of C*-algebras is presented and illustrated by the following examples: general finite dimensional algebras, Abelian algebras and Gelfand's theorem, UHF-algebras, and algebras generated by group representations such as the CCR-algebra arising from the Heisenberg group. Then the theory of states on C*-algebras leading to the GNS-representation in terms of operators on Hilbert spaces is outlined. The basic notion of algebraic dynamical system is given in terms of automorphisms on a C*-algebra of observables and the link to the Hilbert space formalism based on unitary operators is provided by the theory of von Neumann algebras. The examples of the Koopman formalism and the rotation algebra are worked out.