# Wydawnictwa / Czasopisma IMPAN / Studia Mathematica / Wszystkie zeszyty

## Rosenthal operator spaces

### Tom 188 / 2008

Studia Mathematica 188 (2008), 17-55 MSC: 46B20, 46L07, 46L52. DOI: 10.4064/sm188-1-2

#### Streszczenie

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 $2< p< \infty$ 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 $\sigma$ 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.

#### Autorzy

• M. JungeDepartment of Mathematics
University of Illinois at Urbana-Champaign
1409 W. Green Street
Urbana, IL 61801, U.S.A.
e-mail
• N. J. NielsenDepartment of Mathematics
and Computer Science
University of Southern Denmark
Campusvej 55
DK-5230 Odense M, Denmark
e-mail
• T. OikhbergDepartment of Mathematics
University of California, Irvine
103 MSTB
Irvine, CA 92697-3875, U.S.A.
e-mail

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