12 - Locally Compact Groups and their *-Algebras pages 1422 to 1490

Summary

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.