A topologically cyclic *-representation of a *-algebra A is determined up to unitary equivalence by a certain type of linear functional on A. We will call the linear functional that are associated with topologically cyclic *-representations in this special way, representable positive linear junctionals. From §9.4.2 to §9.4.16 we give a construction, due to Israel Moiseevič Gelfand and Mark Aronovič Naǐmark [1943], of a *-representation Tω from each representable positive linear functional ω. This construction was further developed by Irving E. Segal [1947a]. If ω is associated with the topologically cyclic *-representation T, then Tω is unitarily equivalent to T. Thus our construction gives a representative in each unitary equivalence class of topologically cyclic *-representations. In Section 9.2 we showed that each *-representation is the Hilbert sum of a trivial *-representation and an essential *-representation, and that each essential *-representation is unitarily equivalent to a Hilbert sum of topologically cyclic *-representations. Thus the Gelfand-Naimark construction and the Hilbert sum construction together give a representative in each equivalence class of *-representations.

Thirty years ago, when I first thought of writing a book, the book I wanted to write was the first three chapters of this Volume II. At that time I was rapidly discovering the results that now constitute Chapter 10. For me, at least, these essentially algebraic ideas explain why the subject of Banach *-algebras works so smoothly.

Back then, Jacques Dixmier's book was the only substantial exposition of the subject of C*-algebras. Thus my first manuscript, written between 1970 and 1978, contained many of the results in the present Chapters 9, 10 and 11 plus a complete exposition of C*-algebras. I have always regarded the category of C*-algebras as a purely algebraic category: any *-homomorphism is necessarily contractive. Hence the information coming from the complete norm is geometric, but this information is already completely encoded in the *-algebraic structure. Any of the several purely algebraic formulae for the complete norm provide a Rosetta stone. Not only is the norm determined by the algebraic structure, it is also very rigid. So much so, that one can tell whether a unital Banach algebra (without any involution) is a C*-algebra merely by examining the infinitesimal shape of its unit ball near 1 (cf. Theorem 9.5.9).

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.

A great deal is known about locally compact abelian groups and about compact groups. Frequently the same result has been proved in both cases. For instance, all of the continuous, unitary, irreducible representations are finite-dimensional in both cases. Thus it is natural to look for common generalizations of these two quite different hypotheses: abelian and compact.

In this section we will define 22 important classes of groups (and some less important ones) each of which contains all compact and all locally compact abelian groups. Symbols for these classes and for some other properties (mostly topological) are listed in Table 4, page 1485 at the end of the chapter. A symbol enclosed in square brackets denotes the class of all locally compact groups with the property represented by the symbol.

Each definition in this section (many of which have been given previously) will be accompanied by enough results and remarks (including historical remarks) to orient the reader. Diagrams 1-4, pages 1486,1487 show the inclusion relations among our 22 principal classes for (1) all locally compact groups, (2) all almost connected, locally compact groups, (3) all connected, locally compact groups and (4) all discrete groups. Table 5, pages 1488-1490 relates the classes to various examples given in this work.

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.