Title:

Ideals in Banach algebras and notions of amenability

This thesis investigates Banach algebras, mainly focusing on the area of amenability of Banach algebras,
and particularly studying the properties of approximately amenable Banach algebras. Before this is
discussed, we generalise a result of Read [14] to do with prime ideals of Banach algebras.
Chapter 1 contains the necessary background material we shall need from the areas of Banach
spaces, Banach algebras and locally convex algebras. We look at tensor products of Banach and
Fn5chet algebras, and outline the basic theory of amenability of Banach algebras.
  _..._  .__._._
The work in the second chapter appears in my paper [9], and generalises the result of Read [14] that
any prime ideal of a Banach algebra which is generated by a single element is automaticaliy closed. In
particular, this new result shows that the same is true for any finitely generated prime ideal. We also
discuss the difficulties in generalising this result further.
In chapter 3, we outline the theory of approximately amenable Banach algebras, introduced by
Ghahramani and Loy [5], and discuss the aims and possible limitations of this theory.
In chapter 4 we present joint work with Charles Read from ,the paper [10] where we formulate
approximate amenability for Frechet algebras, for which the obstacles encountered in the Banach algebra
case seem to be much easier to deal with. In particular, we exhibit approximately amenable
Frechet algebras without bounded approximate identities, something that we have so far failed to achieve
in the Banach algebra setting.
In chapter 5, we discuss the paper of Dales [2], which shows that lP is not approximately amenable
for any p ~ 1. We generalise this result to all Banach algebras except Co whose underlying Banach
spaces have unconditional bases, and whose multiplication is pointwise with respect to the basis elements.
At the same time, this provides a new proof of the original result.
