In this chapter we open the discussion of one of our main objectives: Comparison algebras are certain C*-algebras of linear operators on an L2-space. In essence their operators may be explicitly represented either as Cauchy-type singular integral operators or else as pseudo-differential operators of order 0. However, we never use (or even discuss) such explicit representations.

Except for a short excursion into Fredholm results for differential operators on compact manifolds in VI,3 and VI,4, in ch's V, VI and VII we will entirely focus on the discussion of the structure of such C*-algebras. The reader should keep in mind, however, that the results obtained will be used later on (ch.X) to discuss the properties of differential operators. For certain differential tial expressions (said to be ‘within reach’ of a comparison algebra C) a realization will be defined (cf. def.6.2 below) In ch.X we will focus on the spectral (and Fredholm) properties of such realizations.

While we have discussed abstract spectral theory in ch.I, we refer to [C1], app.A2, regarding an abstract theory of Fredholm operators. Also we will use some results of abstract spectral theory not discussed in detail in earlier chapters, such as the discussion of the essential spectrum of a differential operator. (Note that at least a survey of some such abstract results is made in ch.X.)

We work in the Hilbert space H = L2(Ω, dμ), on a C∞-manifold Ω, with a positive C∞-measure dμ, as introduced in ch.3.