41 research outputs found
Subspace structure of some operator and Banach spaces
We construct a family of separable Hilbertian operator spaces, such that the
relation of complete isomorphism between the subspaces of each member of this
family is complete \ks. We also investigate some interesting properties of
completely unconditional bases of the spaces from this family. In the Banach
space setting, we construct a space for which the relation of isometry of
subspaces is equivalent to equality of real numbers.Comment: 30 page
Domination of operators in the non-commutative setting
We consider majorization problems in the non-commutative setting. More
specifically, suppose and are ordered normed spaces (not necessarily
lattices), and . If belongs to a certain ideal
(for instance, the ideal of compact or Dunford-Pettis operators), does it
follow that belongs to that ideal as well? We concentrate on the case when
and are -algebras, preduals of von Neumann algebras, or
non-commutative function spaces. In particular, we show that, for
-algebras \A and , the following are equivalent: (1) at
least one of the two conditions holds: (i) \A is scattered, (ii)
is compact; (2) if 0 \leq T \leq S : \A \to {\mathcal{B}},
and is compact, then is compact
On certain extension properties for the space of compact operators
Let be a fixed separable operator space, general separable
operator spaces, and a completely bounded map. is said to have
the Complete Separable Extension Property (CSEP) if every such map admits a
completely bounded extension to ; the Mixed Separable Extension Property
(MSEP) if every such admits a bounded extension to . Finally, is
said to have the Complete Separable Complementation Property (CSCP) if is
locally reflexive and admits a completely bounded extension to provided
is locally reflexive and is a complete surjective isomorphism. Let
denote the space of compact operators on separable Hilbert space and
the sum of {\Cal M}_n's (the space of ``small compact
operators''). It is proved that has the CSCP, using the second
author's previous result that has this property. A new proof is
given for the result (due to E. Kirchberg) that (and hence ) fails the CSEP. It remains an open question if has the MSEP; it
is proved this is equivalent to whether has this property. A new
Banach space concept, Extendable Local Reflexivity (ELR), is introduced to
study this problem. Further complements and open problems are discussed.Comment: 71 pages, AMSTe
Reducing the number of inputs in nonlocal games
In this work we show how a vector-valued version of Schechtman's empirical
method can be used to reduce the number of inputs in a nonlocal game while
preserving the quotient of the quantum over the classical
bias. We apply our method to the Khot-Vishnoi game, with exponentially many
questions per player, to produce another game with polynomially many () questions so that the quantum over the classical bias is