About the QWEP conjecture

Model theory and the QWEP conjecture

We observe that Kirchberg's QWEP conjecture is equivalent to the statement that $C^*(\mathbb{F})$ is elementarily equivalent to a QWEP C$^*$ algebra. We also make a few other model-theoretic remarks about WEP and LLP C$^*$ algebras.

Tsirelson's problem and Kirchberg's conjecture

Tsirelson's problem asks whether the set of nonlocal quantum correlations with a tensor product structure for the Hilbert space coincides with the one where only commutativity between observables located at different sites is assumed. Here it is shown that Kirchberg's QWEP conjecture on tensor products of C*-algebras would imply a positive answer to this question for all bipartite scenarios. This remains true also if one considers not only spatial correlations, but also spatiotemporal correlations, where each party is allowed to apply their measurements in temporal succession; we provide an ex…

About the Connes Embedding Conjecture---Algebraic approaches---

This is an expanded lecture note for "Masterclass on sofic groups and applications to operator algebras" (University of Copenhagen, 5-9 November 2012). It is about algebraic aspects of the Connes Embedding Conjecture. It contains new proofs of equivalence of the Connes Embedding Conjecture, Positivstellensatze for trace positive polynomials, Kirchberg's Conjecture, and Tsirelson's Problem.

About the difficulty to prove the Baum Connes conjecture without coefficient for a non-cocompact lattice in $Sp_4$ in a local field

We introduce property $(T_{Schur},G,K)$ and prove it for some non-cocompact lattice in $Sp_4$ in a local field of finite characteristic. We show that property $(T_{Schur},G,K)$ for a non-cocompact lattice $Γ$ in a higher rank almost simple algebraic group in a local field is an obstacle to proving the Baum Connes conjecture without coefficient for $Γ$ with known methods, and this is stronger than the well-known fact that $Γ$ does not have the property of rapid decay (property (RD)). It is the first example (as announced in [Laf10a]) for which all known methods for proving the Baum Connes conje…

关于正压缩算子乘积的Paszkiewicz猜想的证明

Proof of the Paszkiewicz's conjecture about a product of positive contractions

本文完整证明了一般情形下关于正压缩算子乘积强收敛的Paszkiewicz猜想，并进一步证明其广义版本在部分情形下成立。