Properties expressible in small fragments of the theory of the hyperfinite II_1 factor
aa r X i v : . [ m a t h . OA ] J u l PROPERTIES EXPRESSIBLE IN SMALL FRAGMENTS OF THE THEORYOF THE HYPERFINITE II FACTOR
ISAAC GOLDBRING AND BRADD HARTAbstract. We show that any II factor that has the same 4-quantifier theory asthe hyperfinite II factor R satisfies the conclusion of the Popa Factorial Com-mutant Embedding Problem (FCEP) and has the Brown property. These re-sults improve recent results proving the same conclusions under the strongerassumption that the factor is actually elementarily equivalent to R . In the samespirit, we improve a recent result of the first-named author, who showed thatif (1) the amalgamated free product of embeddable factors over a property (T)base is once again embeddable, and (2) R is an infinitely generic embeddablefactor, then the FCEP is true of all property (T) factors. In this paper, it is shownthat item (2) can be weakened to assume that R has the same 3-quantifier theoryas an infinitely generic embeddable factor. Introduction
The following problem of Popa is the main motivation for the work in this paper:
Problem (Popa’s Factorial Commutant Embedding Problem (FCEP)) . Supposethat M is a separable embeddable factor. Does there exist an embedding i : M ֒ → R U with factorial commutant, that is, such that i ( M ) ′ ∩ R U is a factor? Until recently, very little progress on the FCEP had been made. In [1], the fol-lowing theorem was proven:
Theorem 1. If M is elementarily equivalent to R , then M satisfies the FCEP.Recall that II factors M and N are elementarily equivalent, denoted M ≡ N ,if, for any sentence σ in the language of tracial von Neumann algebras, one has σ M = σ N . A logic-free definition can be given using the Keisler-Shelah Theo-rem: M and N are elementarily equivalent if and only if they have isomorphicultrapowers. By [6, Theorem 4.3], any separable II factor M has continuummany nonisomorphic separable II factors elementarily equivalent to it, whence If one is willing to assume the continuum hypothesis, this can even be improved by sayingthat M and N are elementarily equivalent if and only if M U ∼ = N U for any nonprincipal ultrafilteron N . Theorem 1 gave continuum many new examples of separable II factors satisfy-ing the FCEP.In this paper, we weaken the assumption of the previous theorem and arrive atthe same conclusion. We say that II factors M and N are k -elementarily equiv-alent, denoted M ≡ k N , if they agree on all formulae of quantifier-complexityat most k . (This will be defined precisely in the last section.). The following isan imprecise version of our first main result: Theorem A. If M ≡ R , then M satisfies the FCEP.In another direction, one of the main results of [7] was progress on the FCEPproblem for embeddable property (T) factors: Theorem 2.
Suppose that the following two statements are true:(1) Whenever M and M are embeddable II factors with a common prop-erty (T) subfactor N , then the amalgamated free product M ∗ N M is alsoembeddable.(2) R is an infinitely generic embeddable factor.Then every embeddable property (T) factor satisfies the FCEP.Infinitely generic factors form a large class of “rich” II factors and more infor-mation about them can be found in [5]. In [5], it was claimed that R is an infin-itely generic embeddable factor. However, the proof there is incredibly flawedand settling the question of whether or not R is actually an infinitely genericembeddable factor remains an important open question.Ideally, one would like to remove the model-theoretic assumption (2) in the pre-vious theorem, leaving only the operator-algebraic obstacle (1). Item (2) in theprevious theorem is equivalent to the statement that R is elementarily equiva-lent to an infinitely generic embeddable factor. Consequently, the following the-orem, a consequence of a more general result proven in Section 4, is a strength-ening of the previous result: Theorem B.
Suppose that the following two statements are true:(1) Whenever M and M are embeddable II factors with a common prop-erty (T) subfactor N , then the amalgamated free product M ∗ N M is alsoembeddable.(2’) There is an infinitely generic embeddable factor M such that M ≡ R .Then every embeddable property (T) factor satisfies the FCEP. In this paper, we use the term embeddable as an abbreviation for R U -embeddable. ROPERTIES EXPRESSIBLE IN SMALL FRAGMENTS OF THE THEORY OF THE HYPERFINITE II FACTOR3
It is worth noting that any infinitely generic embeddable factor M satisfies M ≡ R . In Section 4, we also note that the statement that there is an infinitely genericembeddable factor M such that M ≡ R is already known to be “halfway true.”A crucial ingredient to the proof of Theorem 1 above is the following result ofNate Brown [3, Theorem 6.9]: Fact. If N is a separable subfactor of R U , then there is a separable subfactor P of R U with N ⊆ P such that P ′ ∩ R U is a II factor.In [1], we said the II factor M had the Brown property if, for all separablesubfactors N of M U , then there is a separable subfactor P of M U with N ⊆ P such that P ′ ∩ M U is a II factor. It was shown in [1] that any M ≡ R has theBrown property. In the last section of this paper, we prove a strengthening ofthis result: Theorem C. If M ≡ R , then M has the Brown property.An interesting question arises: are these results actually improvements of theirpredecessors? Indeed, perhaps it is the case that there is k ∈ N such that if M ≡ k R , then M ≡ R . If this were to happen, then one would say that Th ( R ) has quantifier simplification . Given recent results showing that the Th ( R ) is very com-plicated from the model-theoretic perspective (see, e.g., [5] and [9]), we stronglybelieve in the following: Conjecture. Th ( R ) does not admit quantifier simplification.For the rest of this paper, we work under the assumption that the previous Con-jecture has a positive solution. In this case, Theorem A yields continuum manyexamples of factors satisfying the FCEP not covered by Theorem 1. Similarly,Theorem C yields continuum many new examples of factors with the Brownproperty.Infinitely generic embeddable factors form a subclass of the more general classof existentially closed embeddable factors . An embeddable factor M is exis-tentially closed (e.c.) if: whenever N is an embeddable factor with M ⊆ N , thereis an embedding N ֒ → M U that restricts to the diagonal embedding M ֒ → M U .It was noted in [5] that R is an e.c. embeddable factor. Existentially closed em-beddable factors have proven very important in applications of model-theoreticideas to the study of II factors. It is a major open question whether or not thereare two non-elementarily equivalent e.c. embeddable factors. If R is not infin-itely generic, then we would have an example of such a pair of e.c. embeddablefactors. However, it could still be the case that all e.c. factors have the same3-quantifier theory, in which case (2’) in Theorem B is actually satisfied. ISAAC GOLDBRING AND BRADD HART
In order to keep this note relatively self-contained, we do not include muchmodel-theoretic or operator-algebraic background. A rather lengthy introduc-tion to model-theoretic ideas as they pertain to problems around factorial com-mutants can be found in [1].In Section 2, we prove the main model-theoretic tools needed in the proof ofTheorem A. In Section 3 we prove Theorem A, in Section 4 we prove TheoremB, and in Section 5 we prove Theorem C.2.
Weak heirs and weak embeddings
In this section, we fix a continuous language L . We say that a formula ϕ is in prenex normal form if it is of the form Q x · · · Q m x m ψ ( x , . . . , x m , ~ y ) , with each Q i ∈ { sup , inf } and with ψ quantifier-free. If the Q i ’s alternate type,then we say that ϕ is ∀ m (respectively ∃ m ) if Q = sup (resp. Q = inf). If aformula is equivalent to a ∀ m or ∃ m formula, we often abuse terminology andrefer to the formula itself as ∀ m or ∃ m .By a fragment of L -formulae, we mean a set ∆ consisting of all ∀ m -formulae orof all ∃ m -formulae for some m . Definition 2.1.
Fix an L -structure M , parameter sets A ⊆ B ⊆ M , and fragments ∆ and ∆ ′ .(1) For c ∈ M , we set tp M∆ ( c/A ) to be the set of all conditions ϕ ( x ) = r , where ϕ ∈ ∆ has parameters from A and ϕ ( c ) M = r .(2) S M∆ ( A ) denotes the set of all tp M∆ ( c/A ) for c ∈ M .(3) For p ∈ S M∆ ( A ) and ϕ ( x ) a formula from ∆ with parameters from A , weset ϕ ( x ) p to be the unique r so that ϕ ( x ) = r belongs to p .(4) For c ∈ M , we set tp M∆,∆ ′ ( c/A, B ) to be the union of tp M∆ ( c/A ) and tp M∆ ′ ( c/B ) .(5) We let S ∆,∆ ′ ( A, B ) denote the set of all tp M∆,∆ ′ ( c/A, B ) for c ∈ M . We ex-tend the notation ϕ ( x ) p to S ∆,∆ ′ ( A, B ) in the obvious way.(6) If p ∈ S ∆ ( A ) , q ∈ S ∆,∆ ′ ( A, B ) , and ∆ ′ ⊆ ∆ , we say that q is an heir of p if,for every b ∈ B , every ϕ ( x, y ) ∈ ∆ ′ , and every ǫ > 0 , there is a ∈ A suchthat | ϕ ( x, a ) p − ϕ ( x, b ) q | < ǫ . Definition 2.2.
Suppose that i : N ֒ → M is an embedding between L -structuresand ∆ is a fragment. We say that i is: Technically we really should be speaking of m − alternations of blocks of quantifiers of thesame length, but we blur this distinction here. ROPERTIES EXPRESSIBLE IN SMALL FRAGMENTS OF THE THEORY OF THE HYPERFINITE II FACTOR5 (1) downward ∆ if, for any nonnegative formula ϕ ( x ) ∈ ∆ and any a ∈ N ,if ϕ ( i ( a )) M = , then ϕ ( a ) N = ;(2) upward ∆ if, for any nonnegative formula ϕ ( x ) ∈ ∆ and any a ∈ N , if ϕ ( a ) N = , then ϕ ( i ( a )) M = .We note one obvious fact: Lemma 2.3.
Given an embedding i : N ֒ → M , we have that i is downwards ∃ m if andonly if i is upwards ∀ m .Proof. Suppose that i is not upwards ∀ m , so there is a nonnegative ∀ m formula ϕ ( x ) and a ∈ N such that ϕ ( a ) N = but ϕ ( i ( a )) M = ǫ > 0 . Then ( ǫ . − ϕ ( i ( a ))) M = and since this formula is equivalent to a ∃ m formula, we havethat ( ǫ . − ϕ ( a )) N = , a contradiction. The other direction is similar. (cid:3) The following is our main technical result concerning the existence of weakheirs. In the remainder of this paper, U denotes a countably incomplete ultra-filter on some index set (unless otherwise specified). Theorem 2.4.
Suppose that M is a separable L -structure. Fix a separable substructure N of M U such that the inclusion N ⊆ M U is downward ∃ m + . Fix also p ∈ S ∀ m ( N ) .Then for any separable parameter set A with N ⊆ A ⊆ M U and any n < m , there is q ∈ S ∀ m , ∀ n ( N, A ) that is an heir of p .Proof. We seek a ∈ M U satisfying the following two kinds of conditions:(1) ψ ( a ) = ψ ( x ) p for any ∀ m -formula ψ ( x ) with parameters from N ;(2) ϕ ( a, c ) M U ≥ ǫ2 for any ∀ n + -formula ϕ ( x, y ) with parameters from A andany ǫ > 0 such that ϕ ( x, b ) p ≥ ǫ for all b ∈ N .Indeed, if a is as above, we claim that q := tp M U ∀ m , ∀ n ( a/A ) is an heir of p . By (1), q is an extension of p . To see that q is an heir, fix a ∀ n -formula ϕ ( x, c ) withparameters from A and set s := ϕ ( x, c ) q = ϕ ( a, c ) M U . Suppose, towards a con-tradiction, that there is ǫ > 0 such that | ϕ ( x, b ) p − s | ≥ ǫ for all b ∈ N . It followsthat | ϕ ( x, b ) − s | p ≥ ǫ for all b ∈ N . Since | ϕ ( x, b ) − s | is logically equivalent toa ∀ n + , whence, by (2), | ϕ ( a, c ) M U − s | ≥ ǫ2 , leading to a contradiction.Suppose now, towards a contradiction, that no such a ∈ M U exists. By countablesaturation, it follows that there are: • a ∀ m -formula ψ ( x ) with parameters from N such that ψ ( x ) p = , • a δ > 0 , and • formulae ϕ ( x, c ) , . . . , ϕ k ( x, c k ) with parameters from A as in (2) ISAAC GOLDBRING AND BRADD HART such that, for any a ∈ M U , if ψ ( a ) < δ , then ϕ i ( a, c i ) < ǫ2 for some i =
1, . . . , k .In other words, (cid:18) sup x min (cid:18) δ − . ψ ( x ) , min ≤ i ≤ k (cid:16) ϕ i ( x, c i ) − . ǫ2 (cid:17)(cid:19)(cid:19) M U = Consequently, (cid:18) inf y · · · inf y k sup x min (cid:18) δ − . ψ ( x ) , min ≤ i ≤ k (cid:16) ϕ i ( x, y i ) − . ǫ2 (cid:17)(cid:19)(cid:19) M U = and thus, since the inclusion N ⊆ M U is downward ∃ m + , we have (cid:18) inf y · · · inf y m sup x min (cid:18) δ − . ψ ( x ) , min ≤ i ≤ m (cid:16) ϕ i ( x, y i ) − . ǫ2 (cid:17)(cid:19)(cid:19) N = Set η := min ( δ, ǫ2 ) and take d , . . . , d k ∈ N such that (cid:18) sup x min (cid:18) δ − . ψ ( x ) , min ≤ i ≤ k (cid:16) ϕ i ( x, d i ) − . ǫ2 (cid:17)(cid:19)(cid:19) N < η ; since the inclusion N ⊆ M U is upward ∀ m + , we have (cid:18) sup x min (cid:18) δ − . ψ ( x ) , min ≤ i ≤ k (cid:16) ϕ i ( x, d i ) − . ǫ2 (cid:17)(cid:19)(cid:19) M U < η. Take a ∈ M U realizing p . Then ψ ( a ) M U = ψ ( x ) p = , whence, since η ≤ δ , wehave min ≤ i ≤ k ( ϕ i ( x, d i )− . ǫ2 ) M U < η ≤ ǫ2 . Choosing i such that ( ϕ i ( a, d i )− . ǫ2 ) M U <η , we get that ϕ i ( x, d i ) p = ϕ i ( a, d i ) M U < ǫ , a contradiction. (cid:3) We will be interested in the following special case of Theorem 2.4:
Corollary 2.5.
Suppose that M is a separable L -structure. Fix a separable substructure N of M U such that the inclusion N ⊆ M U is downward ∃ . Fix also p ∈ S ∀ ( N ) . Thenfor any separable parameter set A with N ⊆ A ⊆ M U , there is q ∈ S ∀ , ∀ ( N, A ) that isan heir of p . Definition 2.6.
Given a fragment ∆ and an L -structure M , we setTh ∆ ( M ) := { σ : σ is a nonnegative L -sentence from ∆ and σ M = } . If N is another L -structure, we write N | = Th ∆ ( M ) if σ N = for all σ ∈ Th ∆ ( M ) .We now prove a result connecting small quantifier-fragments of theories of struc-tures with the existence of embeddings as in the previous theorem. Proposition 2.7.
Suppose that M and N are separable L -structures and m ∈ N . Thenthere is an embedding i : N ֒ → M U that is downwards ∃ m + if and only if M | = Th ∃ m + ( N ) . ROPERTIES EXPRESSIBLE IN SMALL FRAGMENTS OF THE THEORY OF THE HYPERFINITE II FACTOR7
Proof.
First suppose that a downwards ∃ m + -embedding i : N ֒ → M U exists and σ is a nonnegative ∃ m + -sentence such that σ N = . Write σ = inf x ϕ ( x ) with ϕ a ∀ m + -formula. Fix ǫ > 0 and take a ∈ N such that ϕ ( a ) < ǫ . Then ( ϕ ( a ) . − ǫ ) N = , and since this formula is equivalent to a ∀ m + -formula and i is upwards ∀ m + , we have that ( ϕ ( i ( a )) . − ǫ ) M U = . Consequently, ( inf x ( ϕ ( x ) . − ǫ )) M = ;since M is arbitrary, we have that σ M = , as desired.Conversely, suppose that M | = Th ∃ m + ( N ) . Let L N be the language obtainedby adding constants c a for a ∈ N . Set Γ to be the following collection of L N sentences:(1) θ ( c a , . . . , c a n ) , where θ is a nonnegative quantifier-free formula and θ ( a , . . . , a n ) N = ;(2) ǫ . − ϕ ( c a , . . . , c a n ) , where ϕ is a ∃ m + -formula with ϕ ( a , . . . , a n ) N ≥ ǫ If Γ can be shown to be approximately finitely satisfiable in an expansion of M ,then by countable saturation there is an expansion of M U which is a model of Γ ,and this yields the desired embedding. So suppose θ , . . . , θ k are as in (1) and ǫ j . − ϕ j , j =
1, . . . , l , are as in (2). Theninf x (cid:18) max (cid:18) max i = θ i ( x ) , max j = ( ǫ j . − ϕ j ( x ) (cid:19)(cid:19) is equivalent to an ∃ m + -sentence that evaluates to in N , whence, by assump-tion, also evaluates to in M . This completes the proof. (cid:3) Combining Theorem 2.4 and Proposition 2.7, we arrive at:
Corollary 2.8.
Suppose that M is a separable L -structure. Fix a separable substructure N of M U such that M | = Th ∃ m + ( N ) . Fix also p ∈ S M U ∀ m ( N ) . Then for any separableparameter set A with N ⊆ A ⊆ M U and any n < m , there is q ∈ S M U ∀ m , ∀ n ( N, A ) thatis an heir of p . In particular, if M | = Th ∃ ( N ) , then for any p ∈ S M U ∀ ( N ) and anyseparable parameter set A with N ⊆ A ⊆ M U , there is q ∈ S M U ∀ , ∀ ( N, A ) that is an heirof p . Proof of Theorem A
In this section, we apply the abstract results from the previous section to thesetting of II factors. Throughout this section, L is the language of tracial vonNeumann algebras and T is the universal theory of embeddable tracial von Neu-mann algebras. All structures considered in this section will be models of T . ISAAC GOLDBRING AND BRADD HART
Lemma 3.1.
Suppose that M and N are separable with N ⊆ M U . Suppose also that a, b ∈ M U are such that a ∈ Z ( N ′ ∩ M U ) and tp M U ∀ ( a/N ) = tp M U ∀ ( b/N ) . Then b ∈ Z ( N ′ ∩ M U ) .Proof. Since tp M U ∀ ( a/N ) = tp M U ∀ ( b/N ) , we have b ∈ N ′ ∩ M U . Now fix ǫ > 0 .By countable saturation, there are e , . . . , e n ∈ N and δ > 0 such that, for all c ∈ M U , if k [ c, e i ] k < δ for all i =
1, . . . , n , then k [ c, a ] k < ǫ . Consequently,sup x min ( δ − . min i k [ x, e i ] k , k [ x, y ] k − . ǫ ) belongs to tp M U ∀ ( a/N ) , whence it also belongs to tp M U ∀ ( b/N ) . It follows that b ∈ Z ( N ′ ∩ M U ) . So, if c ∈ N ′ ∩ M U , then k [ b, c k ≤ ǫ . Since ǫ was arbitrary, itfollows that [ b, c ] = , and thus b ∈ Z ( N ′ ∩ M U ) , as desired. (cid:3) Corollary 3.2.
Suppose that N ⊆ P ⊆ M U , P ′ ∩ M U is a factor, and every element of S M U ∀ ( N ) admits an heir to S M U ∀ , ∀ ( N, P ) . Then N ′ ∩ M U is a factor.Proof. Take a ∈ Z ( N ′ ∩ M U ) and let p := tp ∀ ( a/N ) . Let q ∈ S ∀ , ∀ ( N, P ) be anheir of p . Let b ∈ M U satisfy q . By the heir property, b ∈ P ′ ∩ M U . If c ∈ P ′ ∩ M U ,then c ∈ N ′ ∩ M U , whence, by the previous lemma, [ b, c ] = . It follows that b ∈ Z ( P ′ ∩ M U ) = C . So b = λ · for some λ ∈ C , so d ( x, λ · ) = belongs to q ,whence it also belongs to p , and thus a = λ · , as desired. (cid:3) Recall the following fact of Nate Brown mentioned in the introduction:
Fact 3.3.
For every separable N ⊆ R U , there is a separable P ⊆ R U with N ⊆ P suchthat P ′ ∩ R U is a factor. We are now able to prove the following more precise version of Theorem A:
Theorem 3.4.
Suppose that N is an embeddable factor such that R | = Th ∃ ( N ) . Then N satisfies the FCEP.Proof. Fix P as in the previous fact, so N ⊆ P ⊆ R U with P ′ ∩ R U a factor. Theproof then follows from Corollary 2.8 and Corollary 3.2. (cid:3) Proof of Theorem B
Let (*) denote the statement: the amalgamated free product of embeddable fac-tors over a property (T) base is once again embeddable.
Lemma 4.1.
Suppose that (*) holds. Then whenever N is a w-spectral gap subfactor ofthe e.c. embeddable factor M , then ( N ′ ∩ M ) ′ ∩ M = N . ROPERTIES EXPRESSIBLE IN SMALL FRAGMENTS OF THE THEORY OF THE HYPERFINITE II FACTOR9
Proof.
In [8], this was proven without a restriction to embeddable factors. Theproof goes through in the embeddable case if one assumes (*) holds. (cid:3)
Recall that if N is a property (T) factor, then N has a Kazhdan set , which is afinite subset F of N that satisfies the following property: there is a K > 0 suchthat for any II factor M containing N as a subfactor, any b ∈ M , and anysufficiently small η > 0 , if k [ a, b ] k < η for all a ∈ F , then there is c ∈ N ′ ∩ M such that k b − c k < Kη . Since k b − E N ′ ∩ M ( b ) k ≤ k b − c k < Kη and E N ′ ∩ M isoperator norm-contractive, it follows that we may assume that c ∈ M as well.(See [4, Proposition 1] for a proof.) Theorem 4.2.
Suppose that (*) holds. Suppose further that N is an embeddable prop-erty (T) II factor, M is an e.c. embeddable factor containing N , and j : M ֒ → R U isdownward Σ . Then j ( N ) ′ ∩ R U is a factor.Proof. Suppose, towards a contradiction, that a ∈ Z ( j ( N ) ′ ∩ R U ) but d ( a, tr ( a ) · ) = ǫ > 0 . Without loss of generality, suppose a is in the unit ball. Let { z , . . . , z n } be a Kazhdan set for N with Kazhdan constant K . Note that R U | = ∀ w (cid:18) max ≤ i ≤ n k [ w, j ( z i )] k = → k [ w, a ] k = (cid:19) , whence, by [2, Proposition 7.14], there is a continuous, nondecreasing function α : R → R satisfying α ( ) = such that R U | = sup w (cid:18) k [ a, w ] k − . α (cid:18) max ≤ i ≤ n k [ w, j ( z i )] k (cid:19)(cid:19) = Set ψ ( x, ~ t ) := sup w ( k [ x, w ] k − . α ( max ≤ i ≤ n k [ w, t i ] k )) , a universal formula suchthat R U | = ψ ( a, j ( ~ z )) = whence R U | = inf x max (cid:18) max ≤ i ≤ n k [ x, j ( z i )] k , ψ ( x, j ( ~ z )) , ǫ − . d ( x, tr ( x ) · ) (cid:19) = Since the latter displayed formula is equivalent to a ∃ -formula, by assumptionwe have M | = inf x max (cid:18) max ≤ i ≤ n k [ x, z i ] k , ψ ( x, ~ z ) , ǫ − . d ( x, tr ( x ) · ) (cid:19) = Fix η > 0 sufficiently small and take b ∈ M such that M | = max (cid:18) max ≤ i ≤ n k [ b, z i ] k , ψ ( b, ~ z ) , ǫ − . d ( b, tr ( b ) · ) (cid:19) < η. If η is sufficiently small, there is b ′ ∈ N ′ ∩ M such that d ( b, b ′ ) < Kη . Forsimplicity, set β := Kη . Now suppose that c ∈ N ′ ∩ M is in the unit ball. Then k [ b, c ] k < η , whence k [ b ′ , c ] k < η + . Since c ∈ N ′ ∩ M was arbitrary, we have d ( b ′ , ( N ′ ∩ M ) ′ ∩ M ) ≤ η + . By Lemma 4.1, since M is e.c. and N hasw-spectral gap in M , we have that ( N ′ ∩ M ) ′ ∩ M = N , so d ( b ′ , N ) ≤ η + ,that is, d ( b ′ , E N ( b ′ )) ≤ η + . However, b ′ ∈ N ′ ∩ M implies E N ( b ′ ) ∈ Z ( N ) = C . It follows that d ( b ′ , tr ( b ′ ) · ) = d ( b, C ) ≤ d ( b, E N ( b ′ )) ≤ η + . Since ǫ − . d ( b, tr ( b ) · ) < η , we have that ǫ − . d ( b ′ , tr ( b ′ ) · ) < η + ( b, b ′ ) < η + ,which is a contradiction as long as +
4β < ǫ . Recalling that β = Kη , we havethat + = ( + ) η , whence choosing η < ǫ2 + , we arrive at the desiredcontradiction. (cid:3) The following is a more precise version of Theorem D; it follows immediatelyfrom Proposition 2.7 and Theorem 4.2.
Corollary 4.3.
Suppose that (*) holds and every embeddable factor N embeds into ane.c. embeddable factor M such that M | = Th ∀ ( R ) . Then every embeddable property(T) factor satisfies the FCEP. The assumption in the previous corollary should be compared to:
Lemma 4.4. If M is an e.c. embeddable factor, then M | = Th ∃ ( R ) .Proof. Since M is a II factor, we may assume that R ⊆ M . Fix an ∃ -sentence σ = inf x sup y inf z ϕ ( x, y, z ) such that σ R = . Fix ǫ > 0 and a ∈ R such that ( inf y sup z ϕ ( a, y, z )) R < ǫ . Fix b ∈ M and an embedding i : M ֒ → R U . Then ( inf z ϕ ( i ( a ) , i ( b ) , z ) R U < ǫ , whence there is c ∈ R U such that ( ϕ ( i ( a ) , i ( b ) , c ) R U <ǫ . Since M is e.c. there is b ′ ∈ M such that ϕ ( a, b, c ′ ) < 2ǫ . Since ǫ is arbitrary,we have that σ M = . (cid:3) Thus, the assumption of Corollary 4.3 comes tantalizingly close to removingany model-theoretic assumption at all, leaving only the operator-algebraic as-sumption (*). 5.
Proof of Theorem C
We begin by explaining exactly what we mean for two structures to be k -elementarilyequivalent. Definition 5.1. If ϕ is a formula and k is a nonnegative integer, we recall whatit means for ϕ to have quantifier depth at most k , written depth ( ϕ ) ≤ k , byinduction on the complexity of ϕ : • If ϕ is atomic, then depth ( ϕ ) ≤ . This follows from the general fact that, for a subfactor P of a II factor Q and a ∈ Q , onehas d ( a, P ′ ∩ Q ) ≤ sup b ∈ P k [ a, b ] k . ROPERTIES EXPRESSIBLE IN SMALL FRAGMENTS OF THE THEORY OF THE HYPERFINITE II FACTOR11 • If ϕ , . . . , ϕ n are formulae, f : R n → R is a continuous function and ϕ = f ( ϕ , . . . , ϕ n ) , then depth ( ϕ ) ≤ max ≤ i ≤ n depth ( ϕ i ) . • If ϕ = sup ~ x ψ or ϕ = inf ~ x ψ , then depth ( ϕ ) ≤ depth ( ψ ) + . Definition 5.2. If M and N are L -structures, we write M ≡ k N if σ M = σ N whenever depth ( σ ) ≤ k . Remark 5.3. If σ is an ∀ m -sentence or a ∃ m -sentence, then clearly depth ( σ ) = m .Consequently, if M ≡ m N , then M | = Th ∀ m ( N ) and N | = Th ∀ m ( M ) .We recall the following Ehrenfeucht-Fraisse game for continuous logic. Definition 5.4.
Let M and N be L -structures and let k ∈ N . G( M, N, k ) denotesthe following game played by two players. First, player I plays either a tuple ~ x ∈ M or a tuple ~ y ∈ N . Player II then responds with a tuple ~ y ∈ N or ~ x ∈ M . The play continues in this way for k rounds. We say that Player II wins G( M, N, k ) if there is an isomorphism between the substructures generated by { ~ x , . . . , ~ x k } and { ~ y , . . . , ~ y k } that maps ~ x i to ~ y i . Definition 5.5. If M and N are L -structures, we write M ≡ EFk N if II has a win-ning strategy for G( M, N, k ) .It is a routine induction to show that M ≡ EFk N implies M ≡ k N . Conversely,one has the following result (see [10, Lemma 2.4]): Fact 5.6.
Suppose that M and N are countably saturated L -structures. Then M ≡ k N if and only if M ≡ EFk N . We are now ready to prove Theorem C. Recall from the introduction that a II factor M has the Brown property if: for every separable subfactor N of M U , thereis a separable subfactor P of M U with N ⊆ P such that P ′ ∩ M U is a II factor. Theorem 5.7.
Suppose that M ≡ R . Then M has the Brown property.Proof. Suppose N is a separable subfactor of M U . It suffices to find a separablesubfactor P of M U containing N such that P ′ ∩ M U is a factor. Indeed, since M ≡ R , M is McDuff, whence P ′ ∩ M U will contain a copy of R U and will thusbe a II factor, as desired.Since M ≡ R and M U and R U are ℵ -saturated, we know that player II has awinning strategy in G( M U , R U , 4 ) . We assume in the following run of the gamethat player II plays according to this strategy. Let player I begin with ~ a , whichis a countable sequence from the unit ball of N which generates N . Let playerII respond with ~ b and let N ∗ denote the separable subfactor of R U generated Here, tuples can be either of finite or countably infinite length. by ~ b . Since R has the Brown property, there is a separable subfactor P ∗ of R U containing N ∗ such that ( N ∗ ) ′ ∩ R U is a factor. Let ~ b be a countable subset ofthe unit ball of P ∗ which, together with ~ b , generates P ∗ . Let player II respondwith ~ a and let P be the separable subfactor of M U generated by ~ a and ~ a . Weclaim that this P is as desired.To see this, suppose that a ∈ Z ( P ′ ∩ M U ) . We wish to show that a ∈ C . Tosee this, let player II respond with b ∈ R U . We claim that b ∈ Z (( P ∗ ) ′ ∩ R U ) ,whence b ∈ C . To see this, suppose that b ∈ ( P ∗ ) ′ ∩ R U . Let player II respondwith a ∈ M U . Since the map ~ a ~ a a a → ~ b ~ b b b extends to an isomorphismbetween the subalgebras they generate, we see that a ∈ P ′ ∩ M U . It follows that a and a commute, whence so do b and b .Now that we have established that b ∈ C , the fact that the strategy is winningalso shows that a ∈ C , as desired. (cid:3) Recall that a McDuff II factor is super McDuff if M ′ ∩ M U is a II factor. In [1,Proposition 4.2.4], it was proven that M has the Brown property if and only ifall N elementarily equivalent to M are super McDuff. Consequently, we arriveat: Corollary 5.8. If M ≡ R , then M is super McDuff. As mentioned in the introduction, if Th ( R ) does not admit quantifier simpli-fication, then these results yield continuum many new examples of separablefactors that are super McDuff and have the Brown property. References [1] S. Atkinson, I. Goldbring, and S. Kunnawalkam Elayavalli,
Factorial commutants and the gen-eralized Jung property for II factors , preprint. arXiv 2004.02293.[2] I. Ben Yaacov, A. Berenstein, C. W. Henson, and A. Usvyatsov, Model theory for metric struc-tures , in Model theory with applications to algebra and analysis. Vol. 2, volume 350 of Lon-don Math. Soc. Lecture Note Ser., 315âĂŞ427. Cambridge Univ. Press, Cambridge, 2008.[3] N. Brown,
Topological dynamical systems associated to II factors , Adv. Math., (2011), 1665-1699. With an appendix by N. Ozawa.[4] A. Connes and V. Jones, Property T for von Neumann algebras , Bulletin of the London Math-ematical Society (1985), 57-62.[5] I. Farah, I. Goldbring, B. Hart, and D. Sherman, Existentially closed II factors , FundamentaMathematicae (2016), 173-196.[6] I. Farah, B. Hart, and D. Sherman, Model theory of operator algebras III: Elementary equivalenceand II factors , Bull. London Math. Soc. (2014), 1-20.[7] I. Goldbring, On Popa’s factorial commutant embedding problem , to appear in the Proceedingsof the AMS.[8] I. Goldbring,
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The universal theory of the hyperfinite II factor is not computable ,preprint. arXiv 2004.02299.[10] I. Goldbring and B. Hart, On the theories of McDuff’s II factors , International MathematicsResearch Notices (2017), 5609-5628. Department of Mathematics, University of California, Irvine, 340 Rowland Hall (Bldg.
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