Model theory and the QWEP conjecture

Abstract

We observe that Kirchberg’s QWEP conjecture is equivalent to the statement that C∗(F) is elementarily equivalent to a QWEP C∗ algebra. We also make a few other model-theoretic remarks about WEP and LLP C∗ algebras. For the sake of simplicity, all C∗ algebras in this note are assumed to be unital. Suppose that B is a C∗ algebra and A is a subalgebra. We say that A is relatively weakly injective in B if there is a u.c.p. map φ : B → A∗∗ such that φ|B = idA; such a map is referred to as a weak conditional expectation. (We view A as canonically embedded in its double dual.) A C∗ algebra A is said to have the weak expectation property (or be WEP) if it is relatively weakly injective in every extension and A is said to be QWEP if it is the quotient of a WEP algebra. Kirchberg’s QWEP Conjecture states that every separable C∗ algebra is QWEP. In the seminal paper [9], Kirchberbg proved that the QWEP Conjecture is equivalent to the Connes Embedding Problem (CEP), namely that every finite von Neumann algebra embeds into an ultrapower of the hyperfinite II1 factor. If F is the free group on countably many generators, then using the fact that C∗(F) is surjectively universal, we see that the QWEP conjecture is equivalent to the statement that C∗(F) is QWEP. The main point of this note is to point to an a priori weaker equivalent statement of the QWEP conjecture: Theorem 1. The QWEP conjecture is equivalent to the statement that C∗(F) is elementarily equivalent to a QWEP C∗ algebra. Here, two C∗ algebras A and B are elementarily equivalent if they have the same first-order theories in the sense of model theory. (Here, we work in the signature for unital C∗ algebras.) The next lemma is probably well known to the experts but since we could not locate it in the literature we include a proof here. Proposition 2. Let A be a C∗ algebra and ω a nonprincipal ultrafilter on some (possibly uncountable) index set. The work here was partially supported by NSF CAREER grant DMS-1349399.