  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 a
script 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 script 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 script
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.
