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Ideal structure of the algebra of bounded operators acting on a Banach space

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<mark>Journal publication date</mark>2017
<mark>Journal</mark>Indiana University Mathematics Journal
Issue number3
Number of pages25
Pages (from-to)1019-1043
Publication StatusPublished
<mark>Original language</mark>English


We construct a Banach space Z such that the Banach algebra B(Z) of bounded operators on Z contains exactly four non-zero, proper closed ideals, including two maximal ideals.
We then determine which kinds of approximate identities (bounded/left/right), if any, each of these four ideals contains, and we show that one of the two maximal ideals is generated as a left ideal by two operators, but not by a single operator, thus answering a question left open in our collaboration with Dales, Kochanek and Koszmider (Studia Math. 2013). In contrast, the other maximal ideal is not finitely generated as a left ideal.
The Banach space Z is the direct sum of Argyros and Haydon's Banach space XAH which has very few operators and a certain subspace Y of XAH. The key property of Y is that every bounded operator from Y into XAH is the sum of a
scalar multiple of the inclusion map and a compact operator.non-zer