II1 factors with nonisomorphic ultrapowers

Abstract

Abstract. We prove that there exist uncountably many separable II1 factors whose ultrapowers (with respect to arbitrary ultrafilters) are non-isomorphic. In fact, we prove that the families of non-isomorphic II1 factors originally introduced by McDuff [MD69a,MD69b] are such examples. This entails the existence of a continuum of non-elementarily equivalent II1 factors, thus settling a well-known open problem in the continuous model theory of operator algebras.

Explicit sentences distinguishing Mcduff’s II1 factors

“…The following definition, implicit in [1] and made explicitly in [3], is central to our work in this paper.…”

“…Note that in the version of [1] currently available, the lemma only allows for unitaries in L(Γ n+1 ) rather than L(Γ n+1 ⊗ Q). However, the proof readily adapts to this more general situation and, indeed, the lemma is used in this more general form in the proof of [1,Lemma 4.4].…”

“…Towsner's work was partially supported by NSF grant DMS-1600263. 1 distinguishing all of the McDuff examples is elucidated. We note that the concrete sentences given here that distinguish examples at "level" k also have 5k + 3 alternations of quantifiers, agreeing with the game-theoretic bounds predicted in [3].…”

“…In [3], the techniques in [1] were dissected in order to give some information about the sentences distinguishing the McDuff examples. Indeed, if we enumerate the McDuff examples by M α for α ∈ 2 ω and k ∈ ω is the least digit such that α(k) = β(k), then it was shown that there must be a sentence θ with at most 5k + 3 alternations of quantifiers such that θ Mα = θ M β .…”

“…In this paper, an even finer analysis of the work in [1] is carried out in order to obtain concrete sentences that distinguish McDuff examples that differ at the first digit; this analysis appears in Section 3. In Section 4, the details of the plan outlined in [3,Section 4.2] are given and the inductive construction of sentences Goldbring's work was partially supported by NSF CAREER grant DMS-1349399.…”