In this chapter we recall basic definitions related to algebras, especially C∗- and W∗-algebras.
Operator algebras are often used in mathematical formulations of quantum theory to describe observables of quantum systems. This is especially useful if we consider infinitely extended systems. They are also convenient to express the Einstein causality properties of relativistic quantum fields.
It is also common to express canonical commutation and anti-commutation relations in terms of algebras. This is especially natural in the case of the CAR. In fact, we will use algebras to treat the CAR in a representation-independent way in Chap. 14. Algebras are less useful in the case of the CCR. We will discuss various choices of CCR algebras ...