It is shown that if a Banach space X is not isomorphic to a Hilbert space then the spaces ℓ₂(X) and Rad(X) contain a subspace Z without local unconditional structure, and therefore without an unconditional basis. Moreover, if X is of cotype r < ∞, then a subspace Z of ℓ₂(X) can be constructed without local unconditional structure but with 2-dimensional unconditional decomposition, hence also with basis.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm149-1-1,
author = {Ryszard A. Komorowski and Nicole Tomczak-Jaegermann},
title = {Subspaces of l2(X) and Rad(X) without local unconditional structure},
journal = {Studia Mathematica},
volume = {151},
year = {2002},
pages = {1-21},
zbl = {0998.46010},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm149-1-1}
}
Ryszard A. Komorowski; Nicole Tomczak-Jaegermann. Subspaces of ℓ₂(X) and Rad(X) without local unconditional structure. Studia Mathematica, Tome 151 (2002) pp. 1-21. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm149-1-1/