## Abstract

Suppose A is a subset of a Banach lattice (Banach algebra) X. We look for "large" sublattices (resp. subalgebras) of A. If X is a Banach lattice, we prove: (1) If Y is a closed subspace of X of codimension at least n, then (X\Y)∪{0} contains a sublattice of dimension n. (2) If Y is a closed infinite codimensional ideal in X, then (X\Y)∪{0} contains a closed infinite dimensional sublattice. (3) If the order in X is induced by a 1-unconditional basis, and Y is a closed infinite codimensional subspace of X, then (X\Y)∪{0} contains a closed infinite dimensional ideal. Further, we show that (4) (ℓ_{p}\(∪_{q<p}ℓ_{q}))∪{0} contains a sublattice which is dense in ℓ_{p}, and that (5) the sets L1(T)\(∪p>1Lp(T))∪{0} and S∞\(∪p<∞Sp)∪{0} contain a dense subalgebra with a continuum of free generators (here Sp denotes the Schatten p-space).

Original language | English (US) |
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Article number | 19800 |

Pages (from-to) | 523-537 |

Number of pages | 15 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 434 |

Issue number | 1 |

DOIs | |

State | Published - Feb 1 2016 |

## Keywords

- Algebrability
- Banach lattices
- Latticeability
- Spaceability

## ASJC Scopus subject areas

- Analysis
- Applied Mathematics