Properties of Banach algebras and their extensions

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Banach algebras can have many different properties. Two contrasting examples are the algebra of all continuous complex-valued functions on the closed unit disc, and the subalgebra of this algebra consisting of those functions which are continuous on the closed disc and analytic on the interior of the disc. In the second of these algebras, any function which is zero throughout some non-empty open set must be constantly zero. This is very much not the case in the bigger algebra: indeed Urysohn's lemma shows that for any two disjoint closed subsets of the closed disc, there is a continuous, complex-valued function defined on the disc which is constantly 0 on one closed set and constantly 1 on the other (algebras of this type are called regular algebras).Most commutative Banach algebras have some features in common with one or the other of these two algebras. The aim of this project is to investigate a variety of conditions (including regularity conditions) for Banach algebras, especially Banach function algebras, to relate these conditions to each other, and to other important conditions that Banach algebras may satisfy, and to investigate the preservation or introduction of these conditions when you form various types of extension of the algebras (especially 'algebraic' extensions such as Arens-Hoffman or Cole extensions).