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Rosenthal operator spaces

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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.
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  • Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, IL 61801, U.S.A.
  • Department of Mathematics and Computer Science, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark
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  • Department of Mathematics, University of California, Irvine, 103 MSTB, Irvine, CA 92697-3875, U.S.A.
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bwmeta1.element.bwnjournal-article-doi-10_4064-sm188-1-2
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