A $c\_0$-saturated Banach space with no long unconditional basic sequences
| dc.creator | Abad, Jordi Lopez | |
| dc.creator | Todorcevic, Stevo | |
| dc.date | 2006-10-18 | |
| dc.date.accessioned | 2026-07-07T07:39:46Z | |
| dc.date.available | 2026-07-07T07:39:46Z | |
| dc.description | We present a Banach space $\mathfrak X$ with a Schauder basis of length $ω\_1$ 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 $ω\_1$-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 $ω\_1$-distortable in the sense that for every $λ>1$ there is an equivalent norm $\||\cdot \||$ on $\mathfrak{X}$ such that for every non-separable subspace $X$ of $\mathfrak{X}$ there are $x,y\in S\_X$ such that $\||\cdot \|| / \||\cdot \||\ge \la$. | |
| dc.identifier | https://arxiv.org/abs/math/0610562 | |
| dc.identifier | http://arxiv.org/abs/math/0610562 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/121576 | |
| dc.subject | Functional Analysis | |
| dc.subject | Logic | |
| dc.subject | 46B20, 03E02, 46B26, 46B28 | |
| dc.title | A $c\_0$-saturated Banach space with no long unconditional basic sequences | |
| dc.type | text |