A category theoretic approach to extensions of Banach algebras

Menez, Christopher and Choi, Yemon (2018) A category theoretic approach to extensions of Banach algebras. PhD thesis, Lancaster University.

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Abstract

This thesis aims to generalise Busby’s framework for studying extensions of C∗-algebras, to the Banach-algebraic setting, without requiring admissibility assumptions on extensions. In the case where the canonical embedding ιJ of a faithful Banach algebra J into its multiplier algebra MJ has closed range, we classify all extensions of an arbitrary Banach algebra B by J. This is done by presenting two categories, one of extensions and another of Busby maps, and proving that these categories are equivalent. We then consider cases where the canonical embedding of J need not have closed range, and provide some partial results in such cases under the extra assumption of a bounded linear lift for a given Busby map. These results are then applied to several examples, where we also compute explicit multiplier norms for MJ when J is a maximal ideal in Ck([−1, 1]). To go further, we study the quotient MJ/ιJ (J) not as a seminormed space but as an object in a suitable derived category. To lay the necessary foundations, the derived category construction of Grothendieck and Verdier is applied to the category of Banach spaces and bounded linear maps. Using this framework, we introduce a class of “Q-Busby maps” from an arbitrary Banach algebra B into MJ/ιJ (J), and obtain a restricted version of Busby’s original correspondence, applicable whenever J is a faithful Banach algebra.