# Some amenability properties on segal algebras.

## Date

2017

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## Abstract

It has been realized that the definition of amenability given by B. E. Johnson in
his Classical Memoir of American Mathematical Society in 1972 is too restrictive
and does not allow for the development of a rich general theory. For this reason,
by relaxing some of the constraints in the definition of amenability via restricting
the class of bimodules in question or by relaxing the structure of the derivations,
various notions of amenability have been introduced after the pioneering work
of Johnson on amenability in Banach algebras. This dissertation is focused on
six of these notions of amenability in Banach algebras, namely: contractibility,
amenability, weak amenability, generalized amenability, character amenability and
character contractibility. The first five of these notions are studied on arbitrary
Banach algebras and the last two are studied on some classes of Segal algebras.
In particular, results on hereditary properties and several characterizations of
these notions are reviewed and discussed. Indeed, we discussed the equivalent
of these notions with the existence of a bounded approximate diagonal, virtual
diagonal, splitting of exact sequences of Banach bimodules and the existence of a
certain Hahn-Banach extension property. Also, some relations that exist between
these notions of amenability are also established. We show that approximate contractibility
and approximate amenability are equivalent. Some conditions under
which the amenability of the underlying group of a Segal algebra implies the character
amenability of the Segal algebras are also given. Finally, some new results
are obtained which serves as our contribution to knowledge.

## Description

Master of Science in Mathematics, Statistics and Computer Science. University of KwaZulu-Natal, Durban 2017.

## Keywords

Theses - Pure Mathematics.