2 - Preliminaries

Summary

In this chapter we shall introduce all the basic materials and preliminary notions needed later on in this book.

C^* and von Neumann algebras

For the details on the material of this section, the reader may be referred to [125], [40] and [76].

C^*-algebras

An abstract normed *-algebra A is said to be a pre C^*-algebra if it satisfies the C^*-property : ‖x^*x‖ = ‖x‖². If A is furthermore complete under the norm topology, one says that A is a C^*-algebra. The famous structure theorem due to Gelfand, Naimark and Segal (GNS) asserts that every abstract C^*-algebra can be embedded as a norm-closed *-subalgebra of B(H) (the set of all bounded linear operators on some Hilbert space H). In view of this, we shall fix a complex Hilbert space H and consider a concrete C^*-algebra A inside B(H). The algebra A is said to be unital or nonunital depending on whether it has an identity or not. However, even any nonunital C^*-algebra always has a net (sequence in case the algebra is separable in the norm topology) of approximate identity, that is, an nondecreasing net eμ of positive elements such that eμa → a for all a ∈ A. Note that the set of compact operators on an infinite dimensional Hilbert space H, to be denoted by K(H), is an example of nonunital C^*-algebra.

We now briefly discuss some of the important aspects of C^*-algebra theory. First of all, let us mention the following remarkable characterization of commutative C^*-algebras.