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Derivations mapping into the radical

Authors: Ndipingwi, Christopher Mbuh
Date: 2010-05-27T06:03:08Z
Subjects: Banach algebras , Radical theory
Type: Thesis
Identifier: http://ujcontent.uj.ac.za8080/10210/379789 , uj:6842 , http://hdl.handle.net/10210/3274
Description: M.Sc. , One of the earliest results (1955) in the theory of derivations is the celebrated theorem of I. M. Singer and J. Wermer [14] which asserts that every bounded derivation on a commutative Banach algebra has range contained in the radical. However, they immediately conjectured that their result will still hold if the boundedness condition was dropped. This conjecture of Singer and Wermer was confirmed only in 1988, by M. P. Thomas [23], when he showed that every derivation (bounded or unbounded) on a commutative Banach algebra has range contained in the radical. But it is not known whether an analogue of the Kleinecke-Shirokov Theorem holds for everywhere defined unbounded derivation.
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Uniqueness of the norm topology in Banach algebras

Authors: Cawdery, John Alexander
Date: 2012-06-07
Subjects: Banach algebras , Topology
Type: Thesis
Identifier: uj:8733 , http://hdl.handle.net/10210/5083
Description: M.Sc. , The aim of this dissertation will be an investigation into a classical result which asserts the uniqueness of the norm topology on a semi-simple Banach algebra. For a commutative semi-simple Banach algebra, say A, it is relatively simple matter, with the aid of the Closed Graph Theorem, to show that all Banach algebra norms on A must be equivalent. The same result for non-commutative Banach algebras was conjectured by I. Kaplansky in the 1950’s and solved more then a decade later, in 1967, by B E Johnson. However, Johnson’s proof was difficult and relied heavily on representation theory. As a result, the problem remained unsolved for the more difficult situation of Jordan Banach algebras. Fifteen years later in 1982, B. Aupetit succeeded in proving Johnson’s result, using a subharmonic method that was independent of algebra representations. Moreover he could, using these techniques, also settle the problem in the Jordan Banach algebra case. A while later, in 1989, T. Ransford provided a shorter algebraic proof of Johnson’s result using the well-known spectral radius formula. This dissertation will be a comparative study of the three different approaches on the problem for Banach algebras.
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