The undecidability of having the QWEP

Abstract

We show that neither the class of C*-algebras with Kirchberg's QWEP property nor the class of W*-probability spaces with the QWEP property are effectively axiomatizable (in the appropriate languages). The latter result follows from a more general result, namely that the hyperfinite III${}_1$ factor does not have a computable universal theory in the language of W*-probability spaces. We also prove that the Powers' factors ${\mathcal{R}}_{\lambda }$, for $0<\lambda <1$, when equipped with their canonical Powers' states, do not have computable universal theory.

14 pages; a minor mistake in Theorem 2.1 has been fixed, leading to a stronger conclusion. Final version, to appear in Journal of Operator Theory