This chapter provides the definitions and many of the basic properties of the objects of abstract harmonic analysis, including locally compact groups, their representations, and various algebras associated with both the groups and their representations. Besides providing the notational conventions used throughout the book, the necessary concepts are organized in a manner useful for the development of the theory of induced representations. For most of the propositions and theorems, we do not provide proofs as these are generally known and accessible in existing monographs. In Section 1.8, we provide a brief guide to the existing literature for the reader who seeks a more comprehensive treatment of a topic. However, we do provide full proofs in Section 1.3, which contains the tools for analysis on coset spaces that may not be so well known but are essential in defining induced representations and proving many of the key theorems.

Locally compact groups

A topological group G is a set with the structure of both a group and a topological space such that the group product is a continuous map from G × G into G and the group inverse is continuous on G. The group product of x and y in G will be denoted multiplicatively as xy and the inverse of x is x−1 except in a few specific cases such as the group of integers or the real numbers.