A note on latticeability and algebrability

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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) (ℓp\(∪q<pq))∪{0} contains a sublattice which is dense in ℓp, 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 languageEnglish (US)
Article number19800
Pages (from-to)523-537
Number of pages15
JournalJournal of Mathematical Analysis and Applications
Issue number1
StatePublished - Feb 1 2016


  • Algebrability
  • Banach lattices
  • Latticeability
  • Spaceability

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics


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