*2.1. C*∗*-Algebraic Quantum Theory*

All the statistical aspects of a physical system **S** are registered in a C<sup>∗</sup>-probablity space (X , *<sup>ω</sup>*), a pair of a C∗-algebra X , and a state *ω* on X [21]. Observables of **S** are described by self-adjoint elements of X . On the other hand, the state *ω* is an expectation functional on X and statistically describes a physical situation (or an experimental setting) of **S**. We keep claiming that every quantum system is described in the language of noncommutative (quantum) probability theory (see [22] for an introduction to quantum probability theory). In Appendix A, the basic facts on operator algebras are summarized.
