Introduction

In any mathematical theory it is important to compare the general abstract structures which arise with comparatively well understood concrete examples. Representation theory is a systematic attempt to do this. We have already seen one example of a representation theory in the Gelfand theory for commutative spectral algebras given in Chapter 3. Any commutative spectral algebra modulo its Gelfand radical is isomorphic to a spectral subalgebra of the algebra of all continuous functions vanishing at infinity on a locally compact Hausdorff space. Since algebras of continuous functions are comparatively concrete and can perhaps be somewhat better understood a priori than general commutative spectral algebras, this is an important (and characteristic) representation theory. Note that the class of all commutative spectral algebras is too large to be successfully represented. Only the semisimple commutative spectral algebras can be faithfully represented. Fortunately the pathology of the non-semisimple algebras can be neatly excised by dividing out the Gelfand radical.

In this work we will study several other representation theories in considerable detail. This chapter deals with the representation of general algebras and part of the second volume is devoted to the representation theory of *-algebras. In both these cases the phenomena noted above occur. Not all of the objects (algebras or *-algebras) can be faithfully represented, but the pathological part can be divided out.