This volume is based on a collection of lectures intended for graduate students and others with a basic knowledge of functional analysis. It surveys several areas of current research interest, and is designed to be suitable preparatory reading for those embarking on graduate work. The volume consists of five parts, which are based on separate sets of lectures, each by different authors. Each part provides an overview of the subject that will also be useful to mathematicians working in related areas. The chapters were originally presented as lectures at instructional conferences for graduate students, and we have maintained the styles of these lectures.

The sets of lectures are an introduction to their subjects, intended to convey the flavour of certain topics, and to give some basic definitions and motivating examples: they are certainly not comprehensive accounts. References are given to sources in the literature where more details can be found.

The chapters in Part I are by H. G. Dales. These are an introduction to the general theory of Banach algebras, and a description of the most important examples: B(E), the algebra of all bounded linear operators on a Banach space E; L1(G), the group algebra of a locally compact group G, taken with the convolution product; commutative Banach algebras, including Banach algebras of functions on compact sets in ℂ and radical Banach algebras. Chapters 3–6 cover Gelfand theory for commutative Banach algebras, the analytic functional calculus, and, in a chapter on ‘automatic continuity’, the lovely results that show the intimate connection between the algebraic and topological structures of a Banach algebra.