Rosenthal operator spaces

M. Junge ¹   ; N. J. Nielsen ²   ; T. Oikhberg ³

¹ Department of Mathematics University of Illinois at Urbana-Champaign 1409 W. Green Street Urbana, IL 61801, U.S.A.
² Department of Mathematics and Computer Science University of Southern Denmark Campusvej 55 DK-5230 Odense M, Denmark
³ Department of Mathematics University of California, Irvine 103 MSTB Irvine, CA 92697-3875, U.S.A.

Studia Mathematica, Tome 188 (2008) no. 1, pp. 17-55

Résumé

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.

Détail

DOI : 10.4064/sm188-1-2

Keywords: lindenstrauss rosenthal showed banach space isomorphic complemented subspace p space either p space isomorphic hilbert space motivation paper where study non hilbertian complemented operator subspaces non commutative p spaces class much richer commutative investigate local properties classes operator spaces every infty which considered operator space analogues rosenthal sequence spaces banach space theory constructed under usual conditions defining sequence sigma prove these spaces operator p spaces completely isomorphic previously known spaces however turns out column row versions spaces operator p spaces have rather complicated local structure which implies lindenstrauss rosenthal alternative does carry non commutative