Journal of Operator Theory

Volume 92, Issue 2, Autumn 2024 pp. 349-362.

The undecidability of having the QWEP

Authors: Jananan Arulseelan (1), Isaac Goldbring (2), and Bradd Hart (3)
Author institution: (1) Department of Mathematics and Statistics, McMaster University, Hamilton ON, L8S 4K1, Canada
(2) Department of Mathematics, University of California, Irvine, Irvine, CA, 92697-3875, U.S.A.
(3) Department of Mathematics and Statistics, McMaster University, Hamilton ON, L8S 4K1, Canada

Summary: We show that neither the class of $C^\ast$-algebras with Kirchberg's QWEP property nor the class of $W^\ast$-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 $0\lambda\in (0,1)$, when equipped with their canonical Powers states, do not have computable universal theory. Our results allow us to conclude the existence of a family of $C^\ast$-algebras (respectively a family of $W^\ast$-probability spaces), none of which have QWEP, but for which some ultraproduct of the family does have QWEP.

DOI: http://dx.doi.org/10.7900/jot.2022oct21.2416
Keywords: continuous logic, $W^*$-probability spaces, QWEP $C^*$-algebras, axiomatizations, computability, modular automorphism group, states, ultraproducts

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