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 $E$ and $F$ are ordered normed spaces (not necessarily lattices), and $0 \leq T \leq S :E \to F$. If $S$ belongs to a certain ideal (for instance, the ideal of compact or Dunford-Pettis operators), does it follow that $T$ belongs to that ideal as well? We concentrate on the case when $E$ and $F$ are $C^*$-algebras, preduals of von Neumann algebras, or non-commutative function spaces. In particular, we show that, for $C^*$-algebras \A and ${\mathcal{B}}$, the following are equivalent: (1) at least one of the two conditions holds: (i) \A is scattered, (ii) ${\mathcal{B}}$ is compact; (2) if 0 \leq T \leq S : \A \to {\mathcal{B}}, and $S$ is compact, then $T$ is compact

On certain extension properties for the space of compact operators

Let $Z$ be a fixed separable operator space, $X\subset Y$ general separable operator spaces, and $T:X\to Z$ a completely bounded map. $Z$ is said to have the Complete Separable Extension Property (CSEP) if every such map admits a completely bounded extension to $Y$; the Mixed Separable Extension Property (MSEP) if every such $T$ admits a bounded extension to $Y$. Finally, $Z$ is said to have the Complete Separable Complementation Property (CSCP) if $Z$ is locally reflexive and $T$ admits a completely bounded extension to $Y$ provided $Y$ is locally reflexive and $T$ is a complete surjective isomorphism. Let ${\bf K}$ denote the space of compact operators on separable Hilbert space and ${\bf K}_0$ the $c_0$ sum of {\Cal M}_n's (the space of ``small compact operators''). It is proved that ${\bf K}$ has the CSCP, using the second author's previous result that ${\bf K}_0$ has this property. A new proof is given for the result (due to E. Kirchberg) that ${\bf K}_0$ (and hence ${\bf K}$) fails the CSEP. It remains an open question if ${\bf K}$ has the MSEP; it is proved this is equivalent to whether ${\bf K}_0$ 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 $G$ while preserving the quotient $\beta^*(G)/\beta(G)$ 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 ($N\approx n^8$) questions so that the quantum over the classical bias is $\Omega (n/\log^2 n)$