Rosenthal operator spaces

@article{M2008,
abstract = {In 1969 Lindenstrauss and Rosenthal showed that if a Banach space is isomorphic to a complemented subspace of an $L_\{p\}$-space, then it is either an $L_\{p\}$-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 $L_\{p\}$-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 $L_\{p\}$-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 $L_\{p\}$-spaces and have a rather complicated local structure which implies that the Lindenstrauss-Rosenthal alternative does not carry over to the non-commutative case.},
author = {M. Junge, N. J. Nielsen, T. Oikhberg},
journal = {Studia Mathematica},
keywords = {Rosenthal spaces; noncommutative -spaces; -spaces},
language = {eng},
number = {1},
pages = {17-55},
title = {Rosenthal operator spaces},
url = {http://eudml.org/doc/284945},
volume = {188},
year = {2008},
}

TY - JOUR
AU - M. Junge
AU - N. J. Nielsen
AU - T. Oikhberg
TI - Rosenthal operator spaces
JO - Studia Mathematica
PY - 2008
VL - 188
IS - 1
SP - 17
EP - 55
AB - In 1969 Lindenstrauss and Rosenthal showed that if a Banach space is isomorphic to a complemented subspace of an $L_{p}$-space, then it is either an $L_{p}$-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 $L_{p}$-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 $L_{p}$-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 $L_{p}$-spaces and have a rather complicated local structure which implies that the Lindenstrauss-Rosenthal alternative does not carry over to the non-commutative case.
LA - eng
KW - Rosenthal spaces; noncommutative -spaces; -spaces
UR - http://eudml.org/doc/284945
ER -