Non-embeddable II_1 factors resembling the hyperfinite II_1 factor
aa r X i v : . [ m a t h . OA ] J a n NON-EMBEDDABLE II FACTORS RESEMBLING THE HYPERFINITEII FACTOR
ISAAC GOLDBRINGAbstract. We consider various statements that characterize the hyperfinite II factors amongst embeddable II factors in the non-embeddable situation. Inparticular, we show that “generically” a II factor has the Jung property (whichstates that every embedding of itself into its ultrapower is unitarily conjugate tothe diagonal embedding) if and only if it is self-tracially stable (which says thatevery such embedding has an approximate lifting). We prove that the enforce-able factor, should it exist, has these equivalent properties. Our techniques aremodel-theoretic in nature. We also show how these techniques can be used togive new proofs that the hyperfinite II factor has the aforementioned proper-ties. introduction In [10], Murray and von Neumann proved that there exists a unique (up toisomorphism) separable hyperfinite II factor. This unique factor, henceforthdenoted by R , plays a crucial role in the theory of finite von Neumann algebras.By Connes’ seminal work in [4], we know that R is also the unique separable II factor possessing any of the following properties: injectivity, semidiscreteness,and amenability.In this article, our focus will be on some statements that characterize R amongstthe class of separable embeddable II factors, where a separable tracial von Neu-mann algebra is embeddable if it embeds into some (equivalently, any) ultra-power R U of R with U a nonprincipal ultrafilter on N . For example, as proven byJung in [9], any embedding of R into R U is unitarily conjugate to the diagonalembedding. In [3], the authors say that a separable II factor M has the Jungproperty if and only if any embedding of M into M U is unitarily conjugate tothe diagonal embedding. In [2] (see also [3, Theorem 3.1.3]), the authors showthat R is the unique separable embeddable II factor with the Jung property.In [1], the author defines a separable tracial von Neumann algebra M to be self-tracially stable if any embedding of M into M U has an “approximate lifting.”(See the next section for a precise definition.) It is easy to see that any II factor The author was partially supported by NSF CAREER grant DMS-1349399. with the Jung property is self-tracially stable (see [3, Proposition 3.3.14] for aproof). It follows that R is self-tracially stable. The fact that R is the uniqueseparable embeddable self-tracially stable II factor is the content of [2, Theorem2.4].Recall that the Connes Embedding Problem (CEP) asks whether or not ev-ery separable tracial von Neumann algebra is embeddable. As announced inthe recent landmark paper [8], the Connes Embedding Problem has a negativeanswer. It thus makes sense to ask whether or not there are separable non-embeddable II factors that have the Jung property or are self-tracially stable.(See [3, Question 3.3.12] for an explicit mention of the former question.) Our first main result is that “generically” these are the same question. To ex-plain this, recall that a tracial von Neumann algebra M is existentially closed (or e.c. for short) if: whenever M is contained in the tracial von Neumann al-gebra N , then there is an embedding of N into M U that restricts to the diagonalembedding of M . The notion of e.c. tracial von Neumann algebras comes frommodel theory and has proven useful in applications of model theory to operatoralgebras. Much is known about the class of e.c. tracial von Neumann algebras:they must be McDuff II factors with only approximately inner automorphisms(see [5] for more on this class). The generic separable tracial von Neumannalgebra is e.c. in the sense that in a natural Polish topology on the space of sepa-rable tracial von Neumann algebras, the e.c. algebras form a comeager set. Thenotion of e.c. factor can be relativized to the class of embeddable factors, inwhich case R is an e.c. embeddable factor. We show the following:
Theorem 1. If M is a separable e.c. factor, then any embedding of M into itself isapproximately inner. From this theorem, it follows fairly quickly that if M is separable, e.c., and self-tracially stable, then M has the Jung property; see Proposition 6 below.We next turn to two model theoretic characterizations of R amongst embed-dable factors. We call an e.c. (embeddable) factor enforceable if it embeds into Separability is not an issue here if one allows ultrafilters over larger index sets. We should mention that in [3, Theorem 3.2.5] it was shown that R is the unique separableembeddable II factor with the generalized Jung property , meaning that any two embeddingsof itself into its ultrapower are conjugate by some (not necessarily inner) automorphism of theultrapower; [3, Theorem 3.3.1] shows that there are non-embeddable factors with this property. Again, this definition makes sense for not necessarily separable factors using ultrfilters onlarger index sets. Alternatively, one can give a purely syntactical, model-theoretic, definitionwhich makes it clear that density character is irrelevant. This follows immediately from the fact that R has the Jung property, but we will discussanother proof at the end of this paper. ON-EMBEDDABLE II FACTORS RESEMBLING THE HYPERFINITE II FACTOR 3 all other e.c. (embeddable) factors. Should the enforceable (embeddable) fac-tor exist, it is automatically unique. In [7, Theorems 5.1 and 5.2], it is shownthat R is the enforceable embeddable factor and that the CEP is equivalent to R being the enforceable factor. Due to the negative solution of the CEP, we see that R is not the enforceable factor. This does not, however, preclude the existence ofthe enforceable factor. We view the problem of the existence of the enforceablefactor to be one of the central problems in the model theory of II factors, forif the enforceable factor exists, then it is a canonical object deserving of furtherstudy, whereas any proof that it does not exist yields a stronger refutation ofCEP.In this paper, we prove: Theorem 2.
If the enforceable factor exists, then it has the Jung property.
It is worth noting that by [6, Theorem 2.14], if M is a II factor with the Jungproperty and M is elementarily equivalent to M ⊗ M , then M ∼ = R . Conse-quently, if the enforceable factor E exists, then we have that E is not elementarilyequivalent to E ⊗ E . Our final result concerns the finite forcing companion . A finitely generic fac-tor is a particular kind of e.c. II factor with the generalized Jung property. (See[7, Definition 5.3] for a precise definition. Alternatively, [7, Propsition 3.10]presents a more workable version of the notion.) These factors always exist andany two of them are elementarily equivalent; the common first-order theory ofthe finitely generic factors is known as the finite forcing companion, denoted T f .In the embeddable situation, R is a finitely generic embeddable factor (see [7,Corollary 3.14]), whence the finite forcing companion is simply the completetheory of R . Since R embeds in any model of its theory (due to the axioma-tizability of being McDuff), the fact that R has the generalized Jung propertyimplies that it is the prime model of its theory. It is currently unknown whether or not T f has a prime model. However, if itdoes, then it is a non-embeddable factor with the Jung property: This is not the original definition given in [7], but is equivalent by the results in Section 6 ofthat paper. Here, two II factors are elementarily equivalent if they have the same first-order theory. Bythe Keisler-Shelah Theorem, we can equivalently say that they have isomorphic ultrapowers.The reference [6] actually only deals with strongly self-absoring C ∗ -algebras, but the proof thereimplies the result that we mention above. By the work in [7, Remark 5.8], the failure of CEP already implies that E , should it exist,could not be isomorphic to E ⊗ E . In general, the prime model of a theory is a model which elementarily embeds into any othermodel of the theory and, if it exists, it is automatically unique.
ISAAC GOLDBRING
Theorem 3. If T f has a prime model M , then M has the Jung property. In the final section, we revisit the embeddable situation and give model-theoreticproofs that R has the Jung property and is self-tracially stable that might be ofindependent interest.In order to keep this note fairly short, we will freely use model-theoretic lan-guage when necessary. The reader is advised to consult [3, Section 2] for amore thorough introduction.We would like to thank Scott Atkinson and Srivatsav Kunnawalkam Eyavalli forhelpful discussions in preparing this paper.2. Proofs of theorems
We first prove Theorem 1. In fact, the following yields an even stronger result:
Theorem 4.
Suppose that M is an e.c. factor with subalgebra N . Then any embeddingof N in M is approximately unitarily conjugate to the inclusion map.Proof. Let f : N → M be an embedding. Let P be the HNN extension obtainedfrom M and f ; we note that P is finite and there is a trace on P such that theinclusion M ⊆ P is trace-preserving (see [11, Corollary 4.2]). In particular,there is a unitary u ∈ P such that uf ( x ) u ∗ = x for all x ∈ N . Since M is e.c., thisimplies that for any finite F ⊆ N and ǫ > 0 , there is a unitary v ∈ M such that k vf ( x ) v ∗ − x k < ǫ for all x ∈ F , as desired. (cid:3) Remark 5 (For the model theorists) . Theorem 4 implies that, in any e.c. II fac-tor, the quantifier-free type of a tuple implies its complete type. It would beinteresting to see if one could leverage this fact to gain any further insight intothe class of e.c. factors.In connection with Theorem 1, we say that a II factor M has the weak Jungproperty if every endomorphism of M is approximately inner. Proposition 6.
A separable II factor has the Jung property if and only if it has theweak Jung property and is self-tracially stable.Proof. First suppose that M has the Jung property and that f : M → M is anendomorphism. By viewing f as taking values in M U , there is a unitary u ∈ M U such that uf ( x ) u ∗ = x for all x ∈ M . In particular, given any finite F ⊆ M and ǫ > 0 , there is a unitary v ∈ M such that k vf ( x ) v ∗ − x k < ǫ for all x ∈ F , whence f is approximately inner. As mentioned in the introduction, any II factor withthe Jung property is self-tracially stable.The converse is clear. (cid:3) ON-EMBEDDABLE II FACTORS RESEMBLING THE HYPERFINITE II FACTOR 5
Given the fact that R is the unique separable embeddable factor with either theJung property or the property of being self-tracially stable, the following ques-tion seems natural : Question 7.
Must there be at most one non-embeddable factor with the Jungproperty? That is self-tracially stable?We now move on to Theorem 2, which will follow from an alternative character-ization of the enforceable factor. First, recall from [1] that if C is a class of tracialvon Neumann algebras, then a tracial von Neumann algebra M is said to be C -tracially stable if whenever f : M → Q U N i is an embedding with U a nonprin-cipal ultrafilter on N and each N i belongs to C , then there are *-homomorphisms f i : M → N i such that f ( x ) = ( f i ( x )) U for all x ∈ M . (We refer to the sequence ( f i ) i ∈ N as an “approximate lifting” of f .) In particular, M is self-tracially stableif and only if M is { M } -tracially stable.We let E denote the class of e.c. factors. The following theorem immediatelyimplies Theorem 2: Theorem 8.
The II factor M is the enforceable factor if and only if it is e.c. and E -tracially stable. In order to prove Theorem 8, we need to recall a few model-theoretic facts from[7]. First, if M is an e.c. factor and a is a tuple from M , the existential type of a in M , denoted etp M ( a ) , is the collection of existential formulae ϕ ( x ) such that ϕ M ( a ) = . Such an existential type is called isolated if, given any ǫ > 0 , thereis an existential formula ϕ ( x ) and δ > 0 such that ϕ M ( a ) = and whenever N is an e.c. factor with a tuple b ∈ N such that ϕ N ( b ) < δ , then there is c ∈ N suchthat k b − c k < ǫ and for which etp M ( a ) = etp N ( c ) . The e.c. factor M is called e-atomic if the existential types of all finite tuples are isolated. It is shown in[7, Section 6] that an e.c. factor M is e-atomic if and only if it is the enforceablefactor (in which case it is unique). Proof of Theorem 8.
First suppose that M is the enforceable factor. We must showthat M is E -tracially stable. Towards this end, fix an embedding f : M → Q U N i with each N i e.c. Since M is e.c., it is McDuff, whence singly generated. Fix agenerator a of M and write f ( a ) = ( a i ) U . Fix ǫ > 0 and let ϕ ( x ) and δ > 0 be asin the definition of isolated existential type for a and ǫ . Since M is e.c., f is anexistential embedding, meaning that ϕ Q U N i ( f ( a )) = and thus ϕ N i ( a i ) < δ for U -almost all i . For these i , there is b i ∈ N i such that k a i − b i k < ǫ and for whichthe map a → b i extends to an isomorphism between M and the subalgebra of N i generated by b i . Thus, f has an approximate lifting. On the other hand, there may be more than one non-embeddable factor with the generalizedJung property; see [3, Corollary 3.3.5].
ISAAC GOLDBRING
Conversely, suppose that M is e.c. and E -tracially stable. It follows that M em-beds into every e.c. II factor, whence M is enforceable. (cid:3) Finally, we prove Theorem 3.
Proof of Theorem 3.
Let M be the prime model of T f . To show that M has theJung property, we show that M has the weak Jung property and is self-traciallystable.Fix a finitely generic factor N . Since M is the prime model of T f , we have that M embeds elementarily in N . Thus, by [7, Corollary 3.12], M itself is finitelygeneric. In particular, M is e.c. and thus has the weak Jung property.It remains to show that M is self-tracially stable. The argument for showingthis is similar to that showing that the enforceable factor is self-tracially stable.Indeed, fix an embedding f : M → M U . This time, given any a ∈ M , thecomplete type of a in M , denoted tp M ( a ) , is isolated . Fix a generator a of M and write f ( a ) = ( a i ) U . Given ǫ > 0 , there is some formula ϕ ( x ) and δ > 0 such that ϕ M ( a ) = and such that, given any model N of T f and any b ∈ N with ϕ N ( b ) < δ , there is c ∈ N such that tp M ( a ) = tp N ( c ) and k b − c k < ǫ .Since M is finitely generic, f is an elementary map, whence ϕ M U ( f ( a )) = andthus ϕ M ( a i ) < δ for U -almost all i . As before, for these i , this guarantees theexistence of b i ∈ M such that k a i − b i k < ǫ and such that the map a → b i extends to an embedding of M into itself. Thus, f has an approximate lift. (cid:3) Remark 9.
It is not clear if there is any relationship between the existence of theenforceable factor and the existence of the prime model of T f .3. Revisiting the embeddable situation
In this section, we show how our techniques from above can yield differentproofs that R has the Jung property and is self-tracially similar.Recall from the introduction that R is the enforceable embeddable factor. Be-sides the model theory behind building models by games, the two main operator-algebraic ingredients in the proof are: • Being hyperfinite is ∀ W ∃ -axiomatizable , whence being hyperfinite isan enforceable property for embeddable factors. • R is the unique separable hyperfinite factor. This follows from the fact that prime models of theories are atomic models. Morally speaking, one just axiomatizes the property that any finite tuple is within any pos-itive tolerance of a copy of some matrix algebra.
ON-EMBEDDABLE II FACTORS RESEMBLING THE HYPERFINITE II FACTOR 7
Noting that our proof from the previous section that the enforceable factor (shouldit exist) is self-tracially stable relativizes immediately to the embeddable situa-tion, we obtain the fact that R is self-tracially stable, without resorting to the factthat R has the Jung property .Unfortunately, our proof in the previous section that the enforceable factor (shouldit exist) has the weak Jung property does not necessarily relativize to the em-beddable situation as the following seems to be an open question: Question 10.
Is the class of embeddable tracial von Neumann algebras closedunder HNN extensions?If the answer to the previous question is positive, then we learn that all e.c. em-beddable factors (and thus, in particular, R itself) have the weak Jung property.Nevertheless, we can give a proof that is similar in spirit that does relativizeto the embeddable situation. Indeed, fix an endomorphism f : R → R ; weshow that f is approximately inner. Let a be a generator of R . Since R is afinitely generic embeddable factor , we have tp R ( a ) = tp R ( f ( a )) . Consequently,there is an elementary extension N of R and an automorphism σ of N such that σ ( f ( a )) = a . Using that R ⊆ N ⋊ σ Z and the class of embeddable factors is closedunder crossed products by Z (and, more generally, by any amenable group), wehave that, given any ǫ > 0 , there is a unitary u ∈ R such that k uf ( a ) u ∗ − a k < ǫ .Consequently, f is approximately inner.Combining these proofs gives a new proof that R has the Jung property.We end with the following natural question which, to the best of our knowledge,is open: Question 11. Is R the unique embeddable factor with the weak Jung property?As mentioned above, if the class of embeddable tracial von Neumann algebras isclosed under HNN extensions, then there are a plethora of embeddable factorswith the weak Jung property. References [1] S. Atkinson,
Some results on tracial stability and graph products , to appear in Indiana Univ.Math. J.[2] S. Atkinson and S. Kunnawalkam Elayavalli,
On ultraproduct embeddings and amenability fortracial von Neumann algebras , to appear in Int. Math. Res. Not.[3] S. Atkinson, I. Goldbring, and S. Kunnawalkam Elayavalli,
Factorial relative commutants andthe generalized Jung property for II factors , preprint. arXiv 2004.02293. This follows from being the enforceable factor.
ISAAC GOLDBRING [4] A. Connes,
Classification of injective factors. Cases II , II ∞ , III λ , λ = , Ann. of Math. (1976), 73–115.[5] I. Farah, I. Goldbring, B. Hart, and D. Sherman, Existentially closed II factors , FundamentaMathematicae (2016), 173-196.[6] I. Farah, B. Hart, A. Tikuisis, and M. Rørdam, Relative commutants of strongly self-absorbing C ∗ -algebras , Selecta Math. (2017) 363-387.[7] I. Goldbring, Enforceable operator algebras , to appear in the Journal of the Institute of Math-ematics of Jussieu.[8] Z. Ji, A. Natarajan, T. Vidick, J. Wright and H. Yuen, MIP* = RE, preprint, arxiv 2001.04383.[9] K. Jung,
Amenability, tubularity, and embeddings into R ω , Math. Ann. (2007), 241–248.[10] F.J. Murray and J. von Neumann, On rings of operators IV , Ann. of Math. (1943), 716-808.[11] Y. Ueda, HNN extensions of von Neumann algebras , J. Funct. Anal. (2005), 383-426.
Department of Mathematics, University of California, Irvine, 340 Rowland Hall (Bldg.
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