On Subprojectivity and Superprojectivity of Banach Spaces

Abstract

We obtain some results for and further examples of subprojective and superprojective Banach spaces. We also give several conditions providing examples of non-reflexive superprojective spaces; one of these conditions is stable under c 0 -sums and projective tensor products.

Complete latticeability in vector-valued sequence spaces

“…Let d(w; 1) be the Lorentz space of [13,Theorem 5.4] and denote by d * (w; 1) its predual. The sequence of canonical unit vectors (e j ) ∞ j=1 is a 1-unconditional basis for d(w; 1) (see [1]) and the sequence of coordinate functionals (e * j ) ∞ j=1 is an unconditional basis for d * (w; 1) (see [19]), hence it is a 1-unconditional basis (see [22, p. 19]). We consider d * (w; 1) as a Banach lattice with the order given by its 1-unconditional basis and d(w; 1) with its dual structure (which coincides, by the way, with the order given by the 1-unconditional basis (e j ) ∞ j=1 ).…”

Complete latticeability in vector‐valued sequence spaces

“…Proof Let $$d(w;1)$$ be the Lorentz space of [21, Theorem 5.4] and denote by $$d_*(w;1)$$ its predual. The sequence of canonical unit vectors $$(e_j)_{j=1}^\infty$$ is a 1‐unconditional basis for $$d(w;1)$$ (see [1]) and the sequence of coordinate functionals $$(e^*_j)_{j=1}^\infty$$ is an unconditional basis for $$d_*(w;1)$$ (see [27]), hence it is a 1‐unconditional basis (see [33, I, p. 19]). We consider $$d_*(w;1)$$ as a Banach lattice with the order given by its 1‐unconditional basis and $$d(w;1)$$ with its dual structure (which coincides, by the way, with the order given by the 1‐unconditional basis $$(e_j)_{j=1}^\infty$$).…”

The Strong Property (B) for $$L_p$$ Spaces

“…Martínez-Abejón MJOM Let us recall that a Banach space X is said to be subprojective if every closed, infinite-dimensional subspace Y of X contains a closed, infinite-dimensional subspace Z complemented in X. For more information about subprojective spaces, see [16] and [28].…”