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 X_AH which has very few operators and a certain subspace Y of X_AH. The key property of Y is that every bounded operator from Y into X_AH is the sum of a
scalar multiple of the inclusion map and a compact operator.non-zer