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

Abstract

We construct a Banach space $Z$ such that the lattice of closed two-sided ideals of the Banach algebra $\mathscr{B}(Z)$ of bounded operators on $Z$ is as follows: $$ \{0\}\subset \mathscr{K}(Z)\subset\mathscr{E}(Z) \raisebox{-.5ex}% {\ensuremath{\overset{\begin{turn}{30}$\subset$\end{turn}}% {\begin{turn}{-30}$\subset$\end{turn}}}}\!\!% \begin{array}{c}\mathscr{M}_1\\[1mm]\mathscr{M}_2\end{array}\!\!\!% \raisebox{-1.25ex}% {\ensuremath{\overset{\raisebox{1.25ex}{\ensuremath{\begin{turn}{-30}$\subset$\end{turn}}}}% {\raisebox{-.25ex}{\ensuremath{\begin{turn}{30}$\subset$\end{turn}}}}}}\,\mathscr{B}(Z) $$ We then determine which kinds of approximate identities (bounded/left/right), if any, each of the four non-trivial closed ideals of $\mathscr{B}(Z)$ contain, and we show that the maximal ideal $\mathscr{M}_1$ 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 (\emph{Studia Math.} 2013). In contrast, the other maximal ideal $\mathscr{M}_2$ is not finitely generated as a left ideal. The Banach space $Z$ is the direct sum of Argyros and Haydon's Banach space $X_{\text{AH}}$ which has very few operators and a certain subspace $Y$ of $X_{\text{AH}}$. The key property of~$Y$ is that every bounded operator from $Y$ into $X_{\text{AH}}$ is the sum of a scalar multiple of the inclusion mapping and a compact operator.