Model theory and the QWEP conjecture (Q327414)

The author rephrases Kirchberg's QWEP conjecture for \(C^*\)-algebras, which is known to be equivalent to Connes' embedding problem fo \(II_1\)-factors, in terms of continuous model theory.NEWLINENEWLINEA unital \(C^*\)-algebra \(A\) is WEP if the inclusion into its double dual factors through \(B(H)\) via completely positive and contractive maps which, when composed, are the identity on \(A\), and a \(C^*\)-algebra is said QWEP if it is the quotient of a WEP algebra. Kirchberg's QWEP conjecture asserts that every \(C^*\)-algebra has the QWEP, and this is known to be equivalent to the fact that \(C^*(\mathbb F_{\infty})\), the full \(C^*\)-algebra of the free group on countably many generators, does so.NEWLINENEWLINEThe author, working in the ambience of continuous model theory, shows that the class of algebras with QWEP is axiomatizable (in fact, \(\forall\exists\)-axiomatizable), and deduces that, to get a positive answer to the QWEP conjecture, it is sufficient to show that \(C^*(\mathbb F_{\infty})\) is elementary equivalent to a QWEP \(C^*\)-algebra. He also shows that, even though limits of WEP algebras are WEP, being WEP is not axiomatizable, and at the end of the paper he offers a second, shorter proof of the fact that the theory of unital \(C^*\)-algebras does not have a model companion, which was a result of \textit{C. Eagle} et al. [``Quantifier elimination in \(C^\ast\) algebras'', Preprint, \url{arXiv:1502.00573}].