Explicit sentences distinguishing McDuff's II$_1$ factors

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by  Isaac Goldbring

EXPLICIT SENTENCES DISTINGUISHING MCDUFF’S II FACTORS
1
7 ISAACGOLDBRING,BRADDHART,ANDHENRYTOWSNER
1
0
2 ABSTRACT. Recently,Boutonnet,Chifan,andIoanaprovedthatMcDuff’sex-
amplesofcontinuummanypairwisenon-isomorphicseparableII1factorsarein
n
fact pairwise non-elementarily equivalent. Their proof proceeded by showing
a
thatanyultrapowersofanytwodistinctMcDuffexamplesarenotisomorphic.
J
Inapaperbythefirsttwoauthorsofthispaper,Ehrenfeucht-Fraïssegameswere
7
used to find an upper bound on the quantifier complexity of sentences distin-
2
guishing theMcDuff examples, leavingitasanopenquestiontofindconcrete

] sentencesdistinguishingtheMcDufffactors.Inthispaper,weanswerthisques-
O tionbyprovidingsuchconcretesentences.
L
.
h
t
a
m 1. INTRODUCTION
[ ThefirstexamplesofcontinuummanynonisomorphicseparableII factorswere
1

1 givenbyMcDuffin[4]. Thesesameexampleswereshowntobenon-elementarily
v equivalent (in the sense of continuous logic) by Boutonnet, Chifan, and Ioana in
8
[1]. ThewaytheyprovedthattheMcDufffactorswerenotelementarilyequivalent
2
9 wasbyshowing,foranytwodistinctMcDuffexamples and andanytwoul-
7 trafilters and onN,thattheultrapowers U and MV werenNotisomorphic;by
0 U V M N
standardmodel-theoreticresults,itthenfollowsthat and arenotelementarily
. M N
1 equivalent.
0
In [3], the techniques in [1] were dissected in order to give some information
7
1 aboutthesentences distinguishing theMcDuffexamples. Indeed, ifweenumerate
: the McDuff examples by for α 2ω and k ω is the least digit such that
v Mα ∈ ∈
i α(k) = β(k), then it was shown that there must be a sentence θ with at most
X 6
5k + 3 alternations of quantifiers such that θMα = θMβ. The proof there used
r 6
Ehrenfeucht-Fraïsse games. Thegame-theoretic techniques also hinted atapossi-
a
ble strategy of providing concrete sentences distinguishing the McDuff examples
if concrete sentences distinguishing examples that differed at the first digit could
beobtained. In[3,Section4.1],suchsentences wereobtained, buttheylackedthe
uniformityneededtocarryoutthestrategyoutlined there.
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’sworkwaspartiallysupportedbyNSFCAREERgrantDMS-1349399.
Hart’sworkwaspartiallysupportedbyNSERC.
Towsner’sworkwaspartiallysupportedbyNSFgrantDMS-1600263.
1
2 ISAACGOLDBRING,BRADDHART,ANDHENRYTOWSNER
distinguishing all of the McDuff examples is elucidated. We note that the con-
crete 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].
We list here some conventions used throughout the paper. First, we follow set
theoretic notation and view k ω as the set of natural numbers less than k: k =
∈
0,1...,k 1 . Inparticular, 2k denotesthesetoffunctions 0,1,...,k 1
{ − } { − }→
0,1 . If α 2k, then we set α := α(i) for i = 0,1,...,k 1 and we let
i
{ } ∈ −
α# 2k−1 be such that α is the concatenation of (α )and α#. If α 2ω, then
0
∈ ∈
αk denotestherestriction ofαto 0,1,...,k 1 .
| { − }
Wheneverwewriteatuple~x,itwillbeunderstood thatthelengthofthetupleis
countable (thatis,finiteorcountably infinite).
Wewilluseuppercase letters todenote variables informulae whiletheirlower-
casecounterparts willbeelementsfromalgebras. WewilluseU’sandV’s(some-
timeswithsubscripts)forvariablesrangingoverthesetofunitaries;sinceunitaries
are quantifier-free definable relative to the theory of C∗-algebras, this convention
is harmless. Of course, we will then use u’s and v’s for unitaries from specific
algebras.
Given agroup Γ and a Γ, welet u L(Γ) be the canonical unitary associ-
a
∈ ∈
atedtoa.
FixavonNeumannalgebra . Forx,y ,thecommutatorofxandyisthe
M ∈ M
element[x,y] := xy yx. IfAisasubalgebraof ,thentherelativecommutant
− M
ofAin istheset
M
A′ := x [x,a] = 0foralla A .
∩M { ∈ M| ∈ }
In particular, the center of is Z( ) := ′ . For a tuple ~a from ,
M M M ∩ M M
we write C(~a) to denote A′ , where A is the subalgebra of generated by
∩M M
the coordinates of ~a. Technically, this notation should also mention , but the
M
ambient algebra will always be clear from context, whence we omit any mention
ofitinthenotation.
2. PRELIMINARIES
Inthissection, wegathermostofthebackground materialneeded intherestof
the paper. First, we recall McDuff’s examples. Let Γ be a countable group. For
i 1,letΓ denoteanisomorphiccopyofΓandletΛ denoteanisomorphiccopy
i i
of≥Z. LetΓ˜ := Γ . IfS denotesthegroupofpermutations ofNwithfinite
i≥1 i ∞
support, then thLere is a natural action of S∞ on i≥1Γ (given by permutation of
indices), whence we may consider the semidireLct product Γ˜ ⋊ S . Given these
∞
conventions, wecannowdefinetwonewgroups:
T (Γ) := Γ˜,(Λ ) [Γ ,Λ ] = 0fori j
0 i i≥1 i j
h | ≥ i
and
T (Γ) := Γ˜ ⋊S ,(Λ ) [Γ ,Λ ] = 0fori j .
1 ∞ i i≥1 i j
h | ≥ i
EXPLICITSENTENCESDISTINGUISHINGMCDUFF’SII1 FACTORS 3
Note that if∆ is asubgroup of Γand α 0,1 , then T (∆) isasubgroup of
α
∈ { }
T (Γ). Givenasequence α 2≤ω,wedefineagroupK (Γ)asfollows:
α α
∈
(1) K (Γ):= Γifα = ;
α
∅
(2) K (Γ):= (T T T )(Γ)ifα 2n;
α α0 ◦ α1 ◦··· αn−1 ∈
(3) K istheinductive limitof(K ) ifα 2ω.
α α|n n
∈
Wethenset (Γ) := L(T (Γ)). WhenΓ = F ,wesimplywrite instead
α α 2 α
of (F );thMesearetheMcDuffexamplesreferredtotheintroductioMn.
α 2
M
Given n 1, we let Γ˜ denote the subgroup of T (K (Γ)) given by the
≥ α,n α0 α#
direct sum of the copies of K (Γ) indexed by those i n and we let P :=
α# α,n
≥
L(Γ˜ ). Whenαhaslength1,wesimplyrefertoΓ˜ asΓ˜ andP asP ;if,in
α,n ∅,n n ∅,n n
addition, n =1,thenwesimplyrefertoΓ˜ asΓ˜. Asintroduced in[3],wedefinea
1
generalized McDuff ultraproduct corresponding to α and Γ to be an ultraproduct
ofthe form (Γ)⊗ts,where(t )isasequence ofnatural numbers and is
anonprincipQalUulMtraαfilteronN. s U
Thefollowing definition, implicit in[1]and madeexplicitly in [3], iscentral to
ourworkinthispaper.
Definition 2.1. We say that a pair of unitaries u,v in a II factor are good
1
M
unitaries if,forallζ ,
∈ M
inf ζ η 2 100( [ζ,u] 2 + [ζ,v] 2).
η∈C(u,v)k − k2 ≤ k k2 k k2
Intheterminology of[1],thissaysthatC(u,v)isa(2,100)-residual subalgebra
of .
M
Wewillneedthefollowingkeyfacts,whoseproofsareoutlinedin[3,Facts2.6].
Facts2.2. Suppose thatα 2<ω isnonempty, Γisacountable group, and(t )is
s
∈
asequence ofnaturalnumbers.
(1) Suppose that (m ) and (n ) are two sequences of natural numbers such
s s
that n < m for all s. Further suppose that Γ is an ICC group. Then
s s
( P⊗ts )′ ( P⊗ts ) is a generalized McDuff ultraproduct corre-
U α,ms ∩ U α,ns
spQonding toα# aQndΓ.
(2) Foranysequence(n ),thereisapairofgoodunitaries~ufrom (Γ)⊗ts
s U Mα
suchthat U Pα⊗,tnss = C(~u). Q
(3) Given anyQseparable subalgebra A of U Mα(Γ)⊗ts, there is a sequence
(ns)suchthat U Pα⊗,tnss ⊂ A′∩ U MQα(Γ)⊗ts.
Q Q
3. DISTINGUISHING EXAMPLES AT LEVEL ONE
In this section, we will find sentences that distinguish L(T (Γ)) and L(T (Γ))
0 1
for nonamenable groups Γ. For the purposes of the next section, where the main
theoremofthepaperisproved,wewillactuallyneedtoproveabitmore.
Intherestofthispaper,wesetχ(X,U ,U ) := 100( [X,U ] 2+ [X,U ] 2).
1 2 k 1 k2 k 2 k2
Lemma 3.1. Let Γ be a countable group and α 0,1 . For any t,n N with
∈ { } ∈
t 1,therearea,b T (Γ)suchthat,foranyζ L( T (Γ)),wehave
≥ ∈ t α ∈ t α
kζ −ELL(LtΓfn)(ζ)k22 ≤ χ(ζ,ua,ub)L(LtTLα(Γ)).
4 ISAACGOLDBRING,BRADDHART,ANDHENRYTOWSNER
Proof. Thisfollowsfrom[1,Lemmas2.6-2.10]. (cid:3)
Definition3.2. Wesetψ (V ,V )tobetheformula
m a b
sup((inf max [UX U∗,Y ] ) . 2 max χ(X ,V ,V ))
i j 2 i a b
X~,Y~ U 1≤i,j≤mk k − 1≤i≤mp
andsetτ := inf ψ .
m Va,Vb m
Proposition 3.3. Suppose that Γ is a countable group and that t 1. Then for
≥
anym 1,wehave
≥
τL(LtT1(Γ)) = 0.
m
Proof. Apply Lemma 3.1 with n = 1, obtaining a,b T (Γ). Let V := u
∈ t 1 a a
and Vb := ub. Fix m-tuples ~x,~y ∈ L( tT1(Γ)) andLǫ > 0. For each i =
1,...,m,wehavethat L
kxi −EL(LtΓe)(xi)k22 ≤ χ(xi,ua,ub)L(LtT1(Γ)).
Inparticular, thereisk > 0suchthat
kxi −EL(LtLj≤kΓj)(xi)k2 ≤ qχ(xi,ua,ub)L(LtT1(Γ))+ǫ.
Set x+ := E (x ) and x− := x x+. Let H be the subgroup of
i L(LtLj≤kΓj) i i i − i p
T (Γ) generated by Γ ⋊S and Λ ,...,Λ . For p > 0 sufficiently large,
1 j≤p i p 1 p
settingy+ := E L (y )andy− := y y+,wehave y− ǫ.
i L(LtHp) i i i− i k i k2 ≤
Choose σ S with σ(j) > p for all j m. Let σ := (σ,σ,...,σ)
∞ 1
∈ ≤ ∈
L(T (Γ)). Note that σ ( Γ )σ−1 commutes with L( H ). Let
t 1 1 t j≤m j 1 t p
uL∈ U(L( tT1(Γ))) be theLunLitary corresponding to σ1. It folloLws, for 1 ≤
i,j ≤ m,thLat[ux+i u∗,yj+]= 0,so
ux u∗,y [ux+u∗,y−] + [ux−u∗,y+] + [ux−u∗,y−] .
k i jk2 ≤ k i j k2 k i j k2 k i j k2
Now
[ux+u∗,y−] ux+u∗y− + y−ux+u∗ 2 y− 2ǫ.
k i j k2 ≤ k i j k2 k j i k2 ≤ k j k2 ≤
Hereweusethat conditional expectation isacontractive map, so x+ x
k i k ≤ k ik ≤
1. Since x− 2, one shows that [ux−u∗,y−] 4ǫ in a similar fashion.
k i k ≤ k i j k2 ≤
Finally,wehave
k[ux−i u∗,yj+]k2 ≤ 2kx−i k2 ≤ 2(qχ(xi,ua,ub)L(LtT1(Γ)) +ǫ).
Lettingǫgoto0,wegetthedesiredresult. (cid:3)
Thefollowingisprobably obviousand/orwell-known,butinanyevent:
Lemma 3.4. There is a function υ : R∗ R∗ such that, for every ǫ > 0 and an
→
inclusion N M of II factors, if x N is such that d(x,U(M)) < υ(ǫ), then
1
⊆ ∈
d(x,U(N)) < ǫ.
EXPLICITSENTENCESDISTINGUISHINGMCDUFF’SII1 FACTORS 5
Proof. Letψ(x) := max(d(x∗x,1),d(xx∗,1)). Thenψ isweaklystable, sothere
isη > 0suchthatifN isanyII factorandψ(x)N < η,thend(x,U(N)) < ǫ. Let
1
υ(ǫ) := ∆ (η),where∆ isthemodulusofuniformcontinuityfortheformulaψ.
ψ ψ
Now suppose that N M are II factors and x N is such that d(x,U(M)) <
1
υ(ǫ). Thenψ(x)N = ψ⊆(x)M < η,whenced(x,U∈(N)) < ǫ. (cid:3)
The following result, which is Lemma 4.6 in [1], will be very important to us.
Inwhatfollows,π : Γ Γ˜ isthecanonical embedding withπ (Γ) = Γ .
n n n
→
Fact3.5. SupposethatΓisacountablenon-amenablegroupandQisatracialvon
Neumannalgebra. Thenthereareg ,...,g ΓandaconstantC > 0suchthat,
1 m
foranyn 1,unitaries v ,...,v U(L(Γ˜∈ Q)),andζ L(T (Γ)) Q,
1 m n+1 0
≥ ∈ ⊗ ∈ ⊗
wehavethat
m
ζ C u (g )ζ ζv .
k k2 ≤ k πn k − kk2
Xk=1
Note that in the version of [1] currently available, the lemma only allows for
unitaries inL(Γ˜ )rather thanL(Γ˜ Q). However,theproof readilyadapts
n+1 n+1
⊗
to this more general situation and, indeed, the lemma is used in this more general
formintheproofof[1,Lemma4.4].
Foranonamenable groupΓ,letC(Γ)andm(Γ)beasinFact3.5.
Proposition 3.6. Suppose that Γ is a nonamenable group. Let m = m(Γ), C =
C(Γ),andδ := 1 υ( 1 ). ThenwheneverM isanintermediatesubalge-
200(30)2 2Cm
q
braL(T (Γ)) M L(T (Γ)) Q,itfollowsthatτM δ.
0 0 m
⊆ ⊆ ⊗ ≥
Proof. Suppose,towardsacontradiction,thatv ,v U(M)aresuchthatψ (v ,v )M <
a b m a b
∈
δ. Foreachn,letρ :Γ U(P )begivenbyρ (g) := u . Since (P′ Q)
n → n n πn(g) n n⊗
isdenseinL(T0(Γ)) Q,thereisnsufficiently largesothat S
⊗
max( [ρ (g),v ] , [ρ (g),v ] )< δ
n a 2 n b 2
k k k k
for all g Γ. Fix such an n and set ρ := ρ . It follows that χ(ρ(g),v ,v )M
n a b
∈ ≤
200δ2 forallg Γ.
∈
ByLemma3.1,wemayfinda′,b′ T (Γ)suchthat, forallζ L(T (Γ)), we
0 0
∈ ∈
have
ζ E (ζ) 2 χ(ζ,u ,u )L(T0)(Γ).
k − L(Γ^n+1) k2 ≤ a′ b′
For simplicity, write E instead of E . It then follows that, for all ζ
^
L(Γn+1)⊗Q ∈
L(T (Γ)) Q,wehave
0
⊗
kζ −E(ζ)k22 ≤ χ(ζ,ua′,ub′)L(T0(Γ))⊗Q.
Let g ,...,g Γ be as in Fact 3.5. Since ψ (v ,v )M < δ, we may find
1 m m a b
∈
u U(M)suchthat,forall1 k m,wehave
∈ ≤ ≤
max( [uρ(gk)u∗,ua′] 2, [uρ(gk)u∗,ub′] 2)< 20√2δ+δ 30δ.
k k k k ≤
6 ISAACGOLDBRING,BRADDHART,ANDHENRYTOWSNER
Let v := uρ(g )u∗ U(L(T (Γ)) Q) and let v′ := E(v ). It follows that
k k ∈ 0 ⊗ k k
kvk −vk′k22 ≤ χ(vk,ua′,ub′)L(T0(Γ))⊗Q ≤ 200(30δ)2.Bythechoice ofδ,there is
v′′ U(L(Γ]) Q)suchthat v′ v′′ < 1 . ByFact3.5,wehavethat
k ∈ n+1 ⊗ k k − kk2 2Cm
1
u C ρ(g ) uv′′ C uv uv′′ < ,
k k2 ≤ k k − kk2 ≤ k k − kk2 2
Xk Xk
yieldingthedesiredcontradiction. (cid:3)
4. THE INDUCTIVE CONSTRUCTION
Inthissection,wedescribeaninductiveconstructionofsentencesthatallowsus
tocarryouttheargumenthintedatin[3,Section4.2]. By[3,Section4.2],weknow
thatcentralizers ofgoodunitariesandrelativecommutantsbetweencentralizers of
good unitaries are definable sets, whence wecan quantify over them. Weactually
needtoknowthatwecandothisinauniform mannerthatdoesnotdepend onthe
ambientII factornorthegoodunitariesathand. Suchuniformityisthecontentof
1
thenextlemma. Notethatif isaII factor, u ,u aregood unitaries and
1 1 2
M ∈ M
x ,then:
∈ M
d(x,C(u ,u )) χ(x,u ,u )M
1 2 1 2
• ≤
ifx C(u ,u ),thpenχ(x,u ,u )M = 0.
1 2 1 2
• ∈
Lemma4.1(Quantification Lemma).
(1) Foreveryformulaψ(X,Y~,U~),thereareformulaeψˆ (Y~,U~)andψˆ(Y~,U~)such
s i
that,foranyII factor ,anypairofgoodunitaries~u ,andanytu-
1
M ∈ M
ple~y ,wehave
∈ M
ψˆ (~y,~u)M = sup ψ(x,~y,~u)M : x C(~u)
s
{ ∈ }
and
ψˆ(~y,~u)M = inf ψ(x,~y,~u)M : x C(~u) .
i
{ ∈ }
(2) Forevery formula ρ(X,Y~,U~ ,U~ ),there areformulae ρ (Y~,U~ ,U~ )and
1 2 s 1 2
ρ (Y~,U~ ,U~ ) such that, for any II factor and any two pairs of good
i 1 2 1
M
unitaries ~u ,~u with C(~u ) C(~u ) and any tuple ~y , we
1 2 2 1
∈ M ⊆ ∈ M
have
ρ (~y,~u ,~u )M = sup ρ(x,~y,~u ,~u )M : x C(~u )′ C(~u ))
s 1 2 1 2 2 1
{ ∈ ∩ }
and
ρ (~y,~u ,~u )M = inf ρ(x,~y,~u ,~u )M : x C(~u )′ C(~u )) .
i 1 2 1 2 2 1
{ ∈ ∩ }
Proof. We only prove the infimum statements. We first prove (1). Let α be a
continuous, nondecreasing function suchthatα(0) = 0and
ψ(x,~y,~u) ψ(x′,~y,~u) α(d(x,x′))
| − | ≤
forallx,x′,~y,~u. Weclaimthat
ψˆ(Y~,U ,U ) := inf(ψ(X,Y~,U ,U )+α( χ(X,U ,U )))
i 1 2 1 2 1 2
X
p
EXPLICITSENTENCESDISTINGUISHINGMCDUFF’SII1 FACTORS 7
works. Fix a II factor , a pair of good unitaries u ,u , and a tuple
1 1 2
M ∈ M
~y . Itisclearthat
∈ M
ψˆ(~y,u ,u )M inf ψ(x,~y,u ,u )M : x C(u ,u ) .
i 1 2 1 2 1 2
≤ { ∈ }
Toseetheotherdirection, fixx,x′ andnotethat
∈ M
ψ(x,~y,u ,u )M ψ(x′,~y,u ,u )M +α(d(x,x′)),
1 2 1 2
≤
whence,takingtheinfimumoverx C(u ,u ),wehave
1 2
∈
M
inf ψ(x,~y,u ,u )M : x C(u ,u ) ψ(x′,~y,u ,u )M+α( χ(x′,u ,u ) ),
1 2 1 2 1 2 1 2
{ ∈ } ≤
p
whencethedesiredresultfollowsfromtakingtheinfimumoverx′.
Theproofofpart(2)proceedsinthesamewayoncewefindaformulaζ(X,U~ ,U~ )
1 2
such that, for any II factor , any two pairs of good unitaries ~u ,~u such
1 1 2
M ∈ M
that C(u ) C(u ), and any x , we have that d(x,C(u )′ C(u ))
2 1 2 1
⊆ ∈ M ∩ ≤
ζ(x,~u ,~u )M. Let
1 2
E: C(~u )′ C(~u ), E :M C(~u ),andE : C(~u ) C(~u )′ C(~u )
2 1 1 1 2 1 2 1
M→ ∩ → → ∩
betheusualconditional expecations, soE = E E andd(x,C(~u )′ C(~u )) =
2 1 2 1
x E(x) . Notethat ◦ ∩
2
k − k
x E(x) x E (x) + E (x) E (E (x)) .
2 1 2 1 2 1 2
k − k ≤ k − k k − k
Now x E (x) χ(x,u ,u )M. Asprovedin[3,Section4.2],
1 2 11 12
k − k ≤
p
E (x) E (E (x)) sup [y,E (x)] .
1 2 1 2 1 2
k − k ≤ k k
ry∈C(~u2)
Nownoticethat
[y,E (x)] E (x)y xy + xy yx + yx yE (x) .
1 2 1 2 2 1 2
k k ≤k − k k − k k − k
Letψ(X,Y,U~ ):= 2χ(X,U~ )+ XY YX . Itfollowsthat
2 2 2
k − k
sup [y,E (x)] ψˆ (x,~u )M.
1 2 s 2
k k ≤
y∈C(~u2)
Letting
ζ(X,U~ ,U~ ):= χ(X,U~ )+ ψˆ (X,U~ )
1 2 1 s 2
q q
yieldsthedesired formula. (cid:3)
Repeatedly applying theQuantification Lemmayields:
Theorem4.2(RelativizationTheorem). Foranysentenceθinprenexnormalform,
there is a formula θ˜(U~ ,U~ )such that, for any II factor and any two pairs of
1 2 1
M
goodunitaries~u ,~u withC(~u ) C(~u ),wehave
1 2 2 1
∈ M ⊆
θ˜(~u ,~u )M = θC(~u2)′∩C(~u1).
1 2
Moreover,θ˜isalsoinprenexnormalformandhasthesamenumberofalternations
ofquantifiers asθ.
8 ISAACGOLDBRING,BRADDHART,ANDHENRYTOWSNER
Wenowintroduce theformulae
.
ϕ (U ,U ) := supinfmax(max [Y,U ] ,d(X,Y) χ(X,U ,U )))
good 1 2 i 2 1 2
X Y i=1,2k k −p
and
ϕn(Y~;U~) := sup max [X,Y ] .
≤ X∈C(U~)i=1,...,nk i k2
Inthedefinitionofϕ ,weareabusingnotationandreallymeantheformulaone
≤
obtains from Lemma 4.1. In what follows, we will only need to consider ϕ3 and
≤
denotethisformulasimplybyϕ .
≤
Notethat:
If isan -saturated II factor, thenϕ (u ,u )M = 0ifandonlyif
1 1 good 1 2
• M ℵ
u ,u isapairofgoodunitaries.
1 2
If isany II factor, ~u isa pair of good unitaries, and~y n is
1
• M ∈ M ∈ M
arbitrary, thenϕ (~y,~u)M = 0ifandonlyif~y ~u.
≤
≤
Definition4.3. Givenasentence θ,werecursively defineasequence ofsentences
θ asfollows: Setθ := θ. Supposing thatθ hasbeen defined, wesetθ tobe
n 1 n n+1
thesentence
infmax(ϕ (U~ ),supinfmax(ϕ (U~ ),ϕ (A,U~ ;U~ ),θ˜(U~ ,U~ ))).
good 1 good 2 ≤ 1 2 n 1 2
U~1 A U~2
Whenθ =τ ,wewriteθ for(τ ) . Hereisthemainresultofthispaper:
m m,n m n
Theorem 4.4. For each nonamenable group Γ, there is a sequence (r (Γ)) of
n
positive real numbers such that, for any n,t N with t 1 and any α 2n, we
∈ ≥ ∈
have:
θL(Tα(Γ))⊗t = 0forallm 1 ifα(n 1) = 1;
m,n
≥ −
θL(Tα(Γ))⊗t r (Γ) ifα(n 1) = 0.
m(Γ),n ≥ n −
Proof. We prove the theorem by induction on n. When n = 1, the theorem holds
byPropositions 3.3and3.6.
Inductivelysupposethatthetheoremistrueforn. Fixanon-amenablegroupΓ.
Firstsuppose that α 2n+1 issuch that α(n) = 1. Fixalso m,t 1. Let be
∈ ≥ M
theultrapowerofL(T (Γ))⊗t;byŁos’theorem,itsufficestoshowthatθM =
α m,n+1
0. Fix a pair of good unitaries ~u > 1. Given a , wecan finda pair of good
1
∈ M
unitaries ~u such that ~u > a,~u . We then have that C(~u )′ C(~u )
2 2 1 2 1
∈ M { } ∩
isageneralized McDuffultraproduct corresponding toα# and Γ, whence, bythe
inductivehypothesis,wehavethatθ˜ (~u ,~u )M = θC(~u2)′∩C(~u1) = 0. Itfollows
m,n 1 2
thatθM = 0.
m,n+1
Now suppose, towards acontradiction, that there isno constant r (Γ). Then
n+1
foreachl > 1,thereisα 2n+1andt Nwitht 1suchthatθL(Tαl(Γ))⊗tl <
l ∈ l ∈ l ≥ m(Γ),n+1
1. Without loss of generality, each α = αfor some fixed α 2n+1. Let :=
l l ∈ M
U L(Tα(Γ))⊗tl, a generalized McDuff ultraproduct corresponding to α and Γ.
Q
EXPLICITSENTENCESDISTINGUISHINGMCDUFF’SII1 FACTORS 9
WethenhavethatθM = 0. Let~u beapairofgoodunitarieswitnessingthe
m(Γ),n+1 1
infimum. Takeanya > u~ andthentakeapairofgoodunitaries~u > awitnessing
1 2
theinfimumforthata. WethenhavethatC(~u )′ C(~u )isaMcDuffultraproduct
2 1
corresponding toα# andΓ,whence θ˜ (~u ,∩~u )M = θC(~u2)′∩C(~u1) r (Γ),
m(Γ),n 1 2 n n
≥
contradicting thefactthatθ˜ (~u ,~u )M = 0. (cid:3)
m(Γ),n 1 2
Remark4.5. Note that each τ isequivalent to aformula in prenex normal form
m
that begins with an inf and has three alternations of quantifiers. By the construc-
tion,itiseasytocheck,byinductiononn,thateachθ isequivalenttoaformula
m,n
inprenexnormalformthatbeginswithaninf andhas5n+3alternationsofquan-
tifiers. Thisagreeswiththetheoretical bounds givenin[3].
Corollary 4.6. Suppose that Γ is any countable group and α,β 2ω are such
thatα n 1= β n 1,α(n) = 1,andβ(n) =0. Writeβ = (β n∈+1) β∗. Set
| − | − |
m := m(Tβ∗(Γ))andr := rn+1(Tβ∗(Γ)). ThenθmM,nα+(Γ1) = 0andθmM,nβ+(Γ1)b≥ r.
Remark 4.7. As pointed out in [1], the results there also show, for any count-
able group Γ and any distinct α,β 2ω, that C∗(T (Γ)) and C∗(T (Γ)) are
r α r β
∈
not elementarily equivalent. Our results here do indeed yield concrete sentences
distinguishing these algebras. Asmentioned in[1], the groups T (Γ)are increas-
α
ing unions of Powers groups, whence, by the proof of [2, Proposition 7.2.3], the
unique trace on C∗(Γ) is definable, and uniformly so over all α 2ω. Conse-
r
∈
quently, the θ ’s can be construed as formulae in the language of C∗-algebras
m,n
with imaginary sorts added and, since the completion of C∗(T (Γ)) with respect
r α
totheGNSrepresentation inducedbytheuniquetraceis (Γ),wehavethatthe
α
M
θ ’sdistinguish theC∗(T (Γ))’saswell.
m,n r α
[1]alsoshowthatC∗(T (Γ)) andC∗(T (Γ)) arealsonotelementarily
r α r β
⊗Z ⊗Z
equivalent for distinct α,β 2ω, where is the Jiang-Su algebra. Since the
∈ Z
unique trace in a monotracial exact -stable algebra is definable (and uniformly
Z
so)by[2,Section3.5]andtheclosureofC∗(T (Γ)) initsGNSrepresentation
r α
⊗Z
withrespecttotheuniquetraceisalso (T (Γ)),wealsohaveconcretesentences
α
distinguishingtheC∗(T (Γ)) ’sinMthecasethatΓisexact(e.g. whenΓ = F ).
r α 2
⊗Z
Itwouldbeinteresting toknowiftheuniquetraceonC∗(T (Γ)) isdefinable
r α
⊗Z
ingeneral, thatis,foranarbitrarycountable groupΓ.
REFERENCES
[1] R.Boutonnet,I.Chifan,andA.Ioana,II factorswithnon-isomorphicultrapowers,toappearin
1
DukeMath.J.arXiv1507.06340.
[2] I.Farah,B.Hart,M.Lupini,L.Robert,A.P.Tikuisis,A.Vignati,andW.Winter,Modeltheory
ofC∗-algebras,arXiv1602.08072.
[3] I.GoldbringandB.Hart,OnthetheoriesofMcDuff’sII factors,InternationalMathResearch
1
Notices,toappear.
[4] D.McDuff,UncountablymanyII factors,Ann.ofMath.90(1969)372-377.
1
DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CALIFORNIA, IRVINE, 340 ROWLAND
HALL(BLDG.#400),IRVINE,CA,92697-3875.
E-mailaddress: [email protected]
URL:http://www.math.uci.edu/~isaac
10 ISAACGOLDBRING,BRADDHART,ANDHENRYTOWSNER
DEPARTMENTOF MATHEMATICSAND STATISTICS, MCMASTER UNIVERSITY, 1280 MAIN
ST.,HAMILTONON,CANADAL8S4K1
E-mailaddress: [email protected]
URL:http://ms.mcmaster.ca/~bradd/
DEPARTMENTOFMATHEMATICS,UNIVERSITYOFPENNSYLVANIA,209SOUTH33RDSTREET
PHILADELPHIA,PA19104-6395.
E-mailaddress: [email protected]
URL:http://www.sas.upenn.edu/~htowsner/