10 - Special *-Algebras

Summary

In this chapter we study various essentially algebraic hypotheses on *-algebras, most of which are satisfied by Banach *-algebras and all of which are satisfied by hermitian Banach *-algebras. Virtually all known results on Banach *-algebras and hermitian Banach *-algebras (that are not explicitly properties of the complete norm) are obtained in this more general setting. We study these classes of *-algebras partly for their own interest, but mainly because they lend themselves to particularly simple proofs of the theorems we wish to establish. Furthermore, we can define categories which include all Banach *-algebras (or all hermitian Banach *-algebras) among their objects but which are much better behaved than the awkward categories of Banach *-algebras. These more inclusive categories facilitate constructions and proofs.

As we have seen in Chapter 9, the *-representations of an arbitrary *-algebra A endow A with a topology. This *-representation topology is defined entirely in terms of the *-algebraic structure of the *-algebra. The closure of zero is the reducing ideal. In this chapter we will consider several classes of *-algebras in which a geometrical structure arises from the *-algebraic structure. In each case we find some quantitative notion of boundedness and use it to define a semi-norm.