95 research outputs found
Multi-indexed p-orthogonal sums in non-commutative Lebesgue spaces
In this paper we extend a recent Pisier's inequality for p-orthogonal sums in
non-commutative Lebesgue spaces. To that purpose, we generalize the notion of
p-orthogonality to the class of multi-indexed families of operators. This kind
of families appear naturally in certain non-commutative Khintchine type
inequalities associated with free groups. Other p-orthogonal families are given
by the homogeneous operator-valued polynomials in the Rademacher variables or
the multi-indexed martingale difference sequences. As in Pisier's result, our
tools are mainly combinatorial.Comment: To appear in Indiana Univ. Math. J. 14 page
Rosenthal's theorem for subspaces of noncommutative Lp
We show that a reflexive subspace of the predual of a von Neumann algebra
embeds into a noncommutative Lp space for some p>1. This is a noncommutative
version of Rosenthal's result for commutative Lp spaces. Similarly for 1 < q <
2, an infinite dimensional subspace X of a noncommutative Lq space either
contains lq or embeds in Lp for some q < p < 2. The novelty in the
noncommutative setting is a double sided change of density.Comment: 42 page
Non-commutative Khintchine type inequalities associated with free groups
Let \Free_n denote the free group with generators .
Let stand for the left regular representation of \Free_n and let
be the standard trace associated to . Given any positive
integer , we study the operator space structure of the subspace
\Word_p(n,d) of generated by the family of operators
with . Moreover, our
description of this operator space holds up to a constant which does not depend
on or , so that our result remains valid for infinitely many generators.
We also consider the subspace of generated by the image under
of the set of reduced words of length . Our result extends to any
exponent a previous result of Buchholz for the space
\Word_{\infty}(n,d). The main application is a certain interpolation theorem,
valid for any degree (extending a result of the second author restricted to
). In the simplest case , our theorem can be stated as follows:
consider the space formed of all block matrices
with entries in the Schatten class , such that is in relative to
and moreover such that and both belong to . We equip
with the maximum of the three corresponding norms. Then, for we have with .Comment: 20 page
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