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 $T:X \rightarrow \mathfrak {X}$ 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 $\| |\cdot \| |$ on $\mathfrak {X}$ such that for every non-separable subspace $X$ of $\mathfrak {X}$ there exist $x,y\in S_X$ such that $\| |x\| |/\| |y\| |\ge \lambda$.