A $c\_0$-saturated Banach space with no long unconditional basic sequences

Abstract

We present a Banach space $\mathfrak{X}$ with a Schauder basis of length $\omega \_1$ which is saturated by copies of $c\_0$ and such that for every closed decomposition of a closed subspace $X=X\_0\oplus X\_1$, either $X\_0$ or $X\_1$ has to be separable. This can be considered as the non-separable counterpart of the notion of hereditarily indecomposable space. Indeed, the subspaces of $\mathfrak{X}$ have ``few operators'' in the sense that every bounded operator $T:X\to \mathfrak{X}$ from a subspace $X$ of $\mathfrak{X}$ into $\mathfrak{X}$ is the sum of a multiple of the inclusion and a $\omega \_1$-singular operator, i.e., an operator $S$ which is not an isomorphism on any non-separable subspace of $X$. We also show that while $\mathfrak{X}$ is not distortable (being $c\_0$-saturated), it is arbitrarily $\omega \_1$-distortable in the sense that for every $\lambda >1$ there is an equivalent norm $\Vert |\cdot \Vert |$ on $\mathfrak{X}$ such that for every non-separable subspace $X$ of $\mathfrak{X}$ there are $x,y\in S\_X$ such that $\||\cdot \|| / \||\cdot \||\ge \la$.