In 1969 Lindenstrauss and Rosenthal showed that if a Banach space is isomorphic to a complemented subspace of an -space, then it is either an -space or isomorphic to a Hilbert space. This is the motivation of this paper where we study non-Hilbertian complemented operator subspaces of non-commutative -spaces and show that this class is much richer than in the commutative case. We investigate the local properties of some new classes of operator spaces for every 2 < p < ∞ which can be considered as operator space analogues of the Rosenthal sequence spaces from Banach space theory, constructed in 1970. Under the usual conditions on the defining sequence σ we prove that most of these spaces are operator -spaces, not completely isomorphic to previously known such spaces. However, it turns out that some column and row versions of our spaces are not operator -spaces and have a rather complicated local structure which implies that the Lindenstrauss-Rosenthal alternative does not carry over to the non-commutative case.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm188-1-2,
author = {M. Junge and N. J. Nielsen and T. Oikhberg},
title = {Rosenthal operator spaces},
journal = {Studia Mathematica},
volume = {187},
year = {2008},
pages = {17-55},
zbl = {1187.46043},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm188-1-2}
}
M. Junge; N. J. Nielsen; T. Oikhberg. Rosenthal operator spaces. Studia Mathematica, Tome 187 (2008) pp. 17-55. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm188-1-2/