Rosenthal operator spaces

Tom 188 / 2008

M. Junge, N. J. Nielsen, T. Oikhberg 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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