Authors: | Lopez-Abad, Jorge Todorčević, Stevo |

Title: | A c0-saturated banach space with no long unconditional basic sequences |

Journal: | Transactions of the American Mathematical Society |

Volume: | 361 |

Issue: | 9 |

First page: | 4541 |

Last page: | 4560 |

Issue Date: | 1-Sep-2009 |

Rank: | M21 |

ISSN: | 0002-9947 |

DOI: | 10.1090/S0002-9947-09-04858-2 |

Abstract: | We present a Banach space X with a Schauder basis of length ω1 which is saturatedby copies of c0 and such that for every closed decomposition of a closedsubspace X = X0⊕ X1, either X0 or X1 has to be separable. This can be considered as the non-separable counterpart of the notion of hereditarily indecomposable space. Indeed, the subspaces of X have "few operators" in the sense that every bounded operator T: R → X from a subspace X of X into X is the sum of a multiple of the inclusion and a ω1- singular operator, i.e., an operator S which is not an isomorphism on any non-separable subspace of X. We also show that while X is not distortable (being c0-saturated), it is arbitrarily ω1-distortable in the sense that for every λ > 1 there is an equivalent norm |||. ||| on X such that for every non-separable subspace X of X there exist x, y ∈ SX suchthat |||x|||/|||y||| ≥ λ. |

Publisher: | American Mathematical Society |

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