The undecidability of having the QWEP

TL;DR

This paper demonstrates that the classes of C*-algebras and W*-probability spaces with the QWEP property are not effectively axiomatizable, highlighting fundamental limits in their logical descriptions and computability.

Contribution

It proves the non-axiomatizability of QWEP classes and the non-computability of universal theories for certain hyperfinite factors in operator algebra logic.

Findings

QWEP classes are not effectively axiomatizable.

Hyperfinite III$_1$ factor lacks a computable universal theory.

Powers' factors $ _l$ do not have computable universal theory.

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.