A note on latticeability and algebrability

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\(∪q1Lp(T))∪{0} and S∞\(∪p<∞Sp)∪{0} contain a dense subalgebra with a continuum of free generators (here Sp denotes the Schatten p-space).