Discontinuous homomorphisms from Banach algebras of operators
The relationship between a Banach space X and its Banach algebra of bounded operators B(X) is rich and complex; this is especially so for non-classical Banach spaces. In this thesis we consider questions of the following form: does there exist a Banach space X such that B(X) has a particular (Banach...
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ndltd-bl.uk-oai-ethos.bl.uk-6844912018-10-03T03:21:49ZDiscontinuous homomorphisms from Banach algebras of operatorsSkillicorn, RichardLaustsen, Niels2016The relationship between a Banach space X and its Banach algebra of bounded operators B(X) is rich and complex; this is especially so for non-classical Banach spaces. In this thesis we consider questions of the following form: does there exist a Banach space X such that B(X) has a particular (Banach algebra) property? If not, is there a quotient of B(X) with the property? The first of these is the uniqueness-of-norm problem for Calkin algebras: does there exist a Banach space whose Calkin algebra lacks a unique complete norm? We show that there does indeed exist such a space, answering a classical open question [101]. Secondly, we turn our attention to splittings of extensions of Banach algebras. Work of Bade, Dales and Lykova [12] inspired the problem of whether there exist a Banach space X and an extension of B(X) which splits algebraically but not strongly; this asks for a special type of discontinuous homomorphism from B(X). Using the categorical notion of a pullback we obtain, jointly with N. J. Laustsen [71], new general results about extensions and prove that such a space exists. The same space is used to answer our third question, which goes back to Helemskii, in the positive: is there a Banach space X such that B(X) has homological bidimension at least two? The proof uses techniques developed (with N. J. Laustsen [71]) during the solution to the second question. We use two main Banach spaces to answer our questions. One is due to Read [90], the other to Argyros and Motakis [8]; the former plays a much more prominent role. Together with Laustsen [72], we prove a major original result about Read’s space which allows for the new applications. The conclusion of the thesis examines a class of operators on Banach spaces which have previously received little attention; these are a weak analogue of inessential operators.512Lancaster Universityhttps://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.684491http://eprints.lancs.ac.uk/79356/Electronic Thesis or Dissertation |
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512 Skillicorn, Richard Discontinuous homomorphisms from Banach algebras of operators |
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The relationship between a Banach space X and its Banach algebra of bounded operators B(X) is rich and complex; this is especially so for non-classical Banach spaces. In this thesis we consider questions of the following form: does there exist a Banach space X such that B(X) has a particular (Banach algebra) property? If not, is there a quotient of B(X) with the property? The first of these is the uniqueness-of-norm problem for Calkin algebras: does there exist a Banach space whose Calkin algebra lacks a unique complete norm? We show that there does indeed exist such a space, answering a classical open question [101]. Secondly, we turn our attention to splittings of extensions of Banach algebras. Work of Bade, Dales and Lykova [12] inspired the problem of whether there exist a Banach space X and an extension of B(X) which splits algebraically but not strongly; this asks for a special type of discontinuous homomorphism from B(X). Using the categorical notion of a pullback we obtain, jointly with N. J. Laustsen [71], new general results about extensions and prove that such a space exists. The same space is used to answer our third question, which goes back to Helemskii, in the positive: is there a Banach space X such that B(X) has homological bidimension at least two? The proof uses techniques developed (with N. J. Laustsen [71]) during the solution to the second question. We use two main Banach spaces to answer our questions. One is due to Read [90], the other to Argyros and Motakis [8]; the former plays a much more prominent role. Together with Laustsen [72], we prove a major original result about Read’s space which allows for the new applications. The conclusion of the thesis examines a class of operators on Banach spaces which have previously received little attention; these are a weak analogue of inessential operators. |
author2 |
Laustsen, Niels |
author_facet |
Laustsen, Niels Skillicorn, Richard |
author |
Skillicorn, Richard |
author_sort |
Skillicorn, Richard |
title |
Discontinuous homomorphisms from Banach algebras of operators |
title_short |
Discontinuous homomorphisms from Banach algebras of operators |
title_full |
Discontinuous homomorphisms from Banach algebras of operators |
title_fullStr |
Discontinuous homomorphisms from Banach algebras of operators |
title_full_unstemmed |
Discontinuous homomorphisms from Banach algebras of operators |
title_sort |
discontinuous homomorphisms from banach algebras of operators |
publisher |
Lancaster University |
publishDate |
2016 |
url |
https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.684491 |
work_keys_str_mv |
AT skillicornrichard discontinuoushomomorphismsfrombanachalgebrasofoperators |
_version_ |
1718757993593438208 |