11 - Banach *-Algebras

Summary

We have already obtained the main results on Banach *-algebras in Chapter 10. That chapter is devoted to the study of *-algebras that satisfy essentially algebraic hypotheses which imply properties similar to those enjoyed by Banach *-algebras. Theorems 10.2.8 and 10.3.10 show that Banach *-algebras are Sq*-algebras and hence T*-algebras, BG*-algebras, G*-algebras, U*-algebras and S*-algebras. Since Banach *-algebras satisfy all of the hypotheses investigated in Sections 10.1 through 10.3, all of the results from those sections are available for our present investigations. Sections 10.4, 10.5 and 10.6 are devoted to hypotheses that hold for hermitian Banach *-algebras, *-regular Banach *-algebras and Banach *-algebras with a large supply of minimal ideals, respectively. We will restate and sometimes even re-derive some of the most basic consequences of these assertions.

A reader who wishes to begin the book with this chapter should peruse Section 9.1, which gives the basic terminology of *-algebras along with many important examples of Banach *-algebras. For instance, a *-homomorphism φ: A → B between *-algebras A and B is an algebra homomorphism satisfying φ (a*) = φ(a)* for all a ∈ A. Volume I contains all of the results on Banach algebras that we will need. Much of this material is standard, but for material on spectral semi-norms, spectral algebras and spectral subalgebras, Volume I is essentially the only source, particularly Chapter 2. In general, such a reader should carefully check the references to earlier chapters that we give here; even when we repeat a definition we seldom repeat comments and explanations that have already appeared.