Razborov flag algebras as algebras of measurable functions

0.12 MB·English

by  Tim D. Austin

Razborov flag algebras as algebras of measurable
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Abstract

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ThesearesomebriefnotesonthetranslationfromRazborov’srecently-developed
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notion of flag algebra ([9]) into the lexicon of functions and measures on certain
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h abstractCantorspaces(totally disconnected compactmetricspaces).
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1 The objects of interest

1
v
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Consider a universal first-order theory T with equality in a language L that contains only
3
5 predicate symbols; assume T has infinite models. Examples include the theories of undi-
1
rected and directed graphsand hypergraphs,possiblywithloops.
.
1
0 In [9], Razborov develops a formalism for handling the ‘leading order’ statistics of large
8 finite models of such theories. The central objects of his theory are the positive R-
0
: homomorphismsof ‘flag algebras’. Here we shall relate these to measures on a subset of
v
Xi theCantorspaceQi≤kKiNi (whereeach Ki isitselfsomeCantorspace)whichiscompact
andsuchthatbothsetandmeasureareinvariantunderthecanonicalcoordinate-permuting
r
a action of Sym (N). Examples of such Cantor spaces include the spaces of models over
0
N of certain kinds of theory, and our identification will begin here. In particular we will
identify (under some restrictions) a flag algebra with an algebra of measurable functions
on the underlying Cantor space of models, and this will lead to an identification of the
positive homomorphisms of the flag algebra with certain measures by virtue of classical
resultsoffunctionalanalysis.
This will take us through the first three sections of [9]. Section 4, 5 and 6 of [9] relate
to a more precise variational analysis of certain examples of these homomorphisms, and
wewillnotdiscussthishere. Forcertain specialexamplesofthetheoriesappearinginthe
studyofflag algebras (particularlythetheoriesofhypergraphsand directedhypergraphs),
1
the associated probability measures on KNi fall into the classical study of ‘partial
Qi≤k i
exchangeability’ in probability theory, and quite complete structure theorems describing
all such measures are available following work of Hoover [5, 6], Aldous [1, 2, 3] and
Kallenberg [7]. However, these general results do not encompass the variational analysis
undertaken in Section 5 of [9] and in [10], and it is not clear that adopting the older
probabilistic formalism makes much difference to this. We direct the reader to [4] for a
surveyofthisoldertheoryand itsrelationswithcombinatorics.
Although it is not assumed in [9], let us here assume for simplicity that the predicates
of T have bounded maximum arity, say by k. The general case can be recovered as a
suitable‘inverselimit’in whicheverpicture is chosen for the description that follows; the
necessary modifications are routine. We will also assume that L is countable, although
thisassumptioncanalsobedroppedwithonlyacosmeticincreaseincomplexity,andwill
writeL forthesetofpredicatesinLofarityifori ≤ k. Otherwiseweadoptthenotation
i
and definitions of [9], and have tried to avoid conflict with such further notation as we
introduceourselves.
Given this arity bound, a general countably infinite model of such a theory T can be
identified with a non-uniform coloured looped directed hypergraph ofmaximal rank k, in
which for each rank i the ‘colour’ of an i-tuple a ∈ Ni is a truth-assignment for ℓ(a) for
eachpredicateℓ ∈ Lofarityi,andsomaybeinterpretedasapointofK := {0,1}Li. Let
i
us work henceforth only with models whose set of variables (or ‘vertices’) is a subset of
N. ThusweshallviewoutmodelsofT aspointsoftheCantorspace KNi satisfying
Qi≤k i
all additional constraints imposed by the sentences of T; we shall denote the subset of
these points by Ω. In general, given a model M of T with vertex set S and S ⊆ S we
1
shall write M| for the submodel of M with vertex set S . If T is free then all points of
S1 1
thisspacearepossible;otherwisetheconstraintsimposedbyT carveoutsomesubspaceof
KNi,whichisanintersectionofafamilyoffinite-dimensionalsubsetscorresponding
Qi≤k i
to individualconstraints(since any individualinterpretation of a sentence in T oversome
particular finite set of vertices in N simply carves out some clopen subset of KNi
Qi≤k i
dependingonlyonthoseverticesascoordinates). Thissetofmodelsisthereforeacompact
subspaceofanabstractCantorspace,andsoitselfanabstractCantorspace. Thissubspace
isalsoclearlyinvariantunderpermutationsofthevariablesetN;letuswriteSym forthe
0
group of finitely-supported such permutations (it will prove convenient to have imposed
thefinite-supportconditionwhenwecometoworkwiththisgrouplater). Forconvenience,
if a theory T is clear from the context, we shall refer to a cylinder subset of Ω to mean
theintersectionofΩwitha cylindersubsetofthebigproductspace KNi.
Qi≤k i
Infact,ouranalyticformalismwillapplytoanyclosedSym -invariantsubspaceof KNi
0 Qi≤k i
for compact metricspaces K , K , ..., K , and thereader is free to view Ω in thisgener-
1 2 k
2
ality.
The relevant data for what follows are the nonempty Sym -invariant compact subset Ω
0
together with the canonical action of the finitely-supported permutation group Sym on
0
the vertex set N, and with its usual topological structure and Borel σ-algebra Σ. By mild
notational abuse we will identify an element g ∈ Sym directly with the corresponding
0
autohomeomorphismofΩ.
We need to allow a few more constructions. Henceforth all finite models of T will be
assumedtohavevertexsetsequaltoinitialsegmentsoftheintegers. Foragivensuchfinite
model, say σ on the vertex set [k], we let Ω be thesubset of thoseω ∈ Ω with ω| = σ,
σ [k]
and let Σ be its Borel σ-algebra (which is just theideal in Σ of thesubsets ofΩ ⊆ Ω in
σ σ
the usual way). This σ is called a ‘type’ in [9]. We write 0 for the empty type. We now
consider isomorphism classes of those finite T-models that contain a distinguished copy
of σ as a submodel; that is, of pairs F = (M,θ) for M a finite model and θ a particular
embeddingσ ֒→ M. Theseextensionsarecalled‘σ-flags’in[9],andFσ iswrittenforthe
collection of them, Fσ for the subcollection of those with vertex set of size ℓ (so Fσ = ∅
ℓ ℓ
unlessℓ ≥ k),andwecannowspecifytheobviousnotionsofembeddingandisomorphism
forflags.
Restricting our attention to σ-flags F = (M,σ) with M a model on some [ℓ] and σ itself
serving as its distinguished copy in M (so we have implicitly ordered the vertices of M
so that σ = M| ), we can identify F with a finite-dimensional cylinder set A in Ω
[k] F
containedinΩ :
σ
A := {ω ∈ Ω : ω| = M}
F [ℓ]
(of course, this is also just Ω in our earlier notation associating Ω to σ; our choice of
M σ
different letters reflects the different roˆles of M and σ here). In fact, we could have made
anyassignmentF 7→ A asabovebutcorrespondingtoanyotherchoiceoflocationsinN
F
for the vertices of M that are not vertices of σ, and it would not matter; the above choice
isconvenientlyconcrete.
Occasionallyweshall need torelate flags overdifferent typesσ and theassociated spaces
Ω . Ingeneral,theappearanceofσasasubmodelofσ′doesnotguaranteethatspecifically
σ
σ = σ′|[k] for some k; but we can always identify Ωσ′ as a subspaces of Ωσ by suitably
reorderingtheverticesofσ′. In[9]thistechnicalmatterismore-or-lessavoidedaltogether
owing to the early decision to work entirely with isomorphism classes of models; in the
pictureofthespacesΩ ,however,wepaythismodestpriceforthesakeofretainingamore
σ
concretepictureofthepointsofthisspaceand(eventually)measuresonthem,whichhave
moreclassical and easily-analyzedstructuresin certain otherrespects.
Finally, we will want to consider coordinate-permutations that preserve Ω ; that is, that
σ
3
permute only coordinates in N \ [k]. Let us call the group of these Sym ≤ Sym (even
σ 0
thoughit actuallydependsonlyon k).
2 Some analysis and measure theory
Having laid out the objects of study in the previous section, we will now recall those
additional functional-analytic ideas in terms of which we’ll later give an account of flag
algebras. The functional analysis needed does not extend beyond a graduate-level intro-
duction to probability theory and the contents of any good second course on functional
analysis;Yosida[12], forexample,coversourneeds.
Let us write C(Ω ) and M(Ω ) for the usual Banach spaces of real-valued continuous
σ σ
functionsand signed Radon measures on Ω respectively;the Riesz-Kakutani representa-
σ
tion identifies M(Ω ) isometrically with C(Ω )∗. Moreover, we write Mσ for the sub-
σ σ
spaceofmeasuressupportedonΩ andinvariantunderfinitely-supportedpermutationsof
σ
the coordinates in N \ [k]; we will call these σ-exchangeable, and sometimes abbreviate
this to just ‘exchangeable’ when σ is clear from the context. We write (Mσ)⊥ for the
annihilatorofthisconsidered asa subspaceofC(Ω )∗,
σ
(Mσ)⊥ = {f ∈ C(Ω ) : hf,µi = 0∀µ ∈ Mσ};
σ
as usual, the dual-of-the-quotient Banach space C(Ω )/(Mσ)⊥ ∗ can be isometrically
(cid:0) σ (cid:1)
identifiedwithMσ. Let q bethequotientmap C(Ω ) → C(Ω )/(Mσ)⊥.
σ σ σ
Now,givenanyf ∈ C(Ω ),it‘looksthesame’asanyf◦gforg ∈ Sym toallmeasuresin
σ σ
Mσ. Write T for the tail σ-subalgebra Σ of Σ , and (with a slight
σ Tm≥k+1 [k]∪{m,m+1,...} σ
abuse of notation) L∞(T ) for the space of bounded T -measurable functions that are
σ σ
definedµ-a.e. foreveryµ ∈ Mσ andundertheequivalencerelationof“equalityµ-a.e. for
every µ ∈ Mσ”. Clearly these are invariant under the action of Sym . By the pointwise
σ
ergodictheoremfortheamenablegroupSym (oranymoreelementaryargumentforthis
σ
veryspecializedexampleofaSym -system),wemaytaketheaverageofthecompositions
σ
f ◦ g over different g to obtain some T -measurable function f¯on Ω which is defined
σ σ
µ-almost everywhere for every µ ∈ Mσ and is invariant under Sym , and hence actually
σ
specifies a member of L∞(T ). Observe that f¯ = h¯ for f,h ∈ C(Ω ) if and only if
σ σ
f − h ∈ (Mσ)⊥, and so our map f 7→ f¯embeds C(Ω )/(Mσ)⊥ as a subspace Vσ of
σ
L∞(T ); general nonsense now shows also that this is an isometric embedding, so Vσ is
σ
closed.
Furthermore, Vσ is actually a subalgebra of L∞(T ). To show that it is closed under
σ
multiplication,wesupposef,h ∈ C(Ω ),andnowconsidertheproductsf·(h◦g)forany
σ
4
sequenceofpermutationsg thatpushesh‘furtherandfurtherout’,inthefollowingsense:
for any m ≥ 1, there are finite subsets A,B ⊂ N \ [k] such that f and h are uniformly
(1/m)-close to functions depending only on vertices in A and B (respectively), and now
wechooseg that movesall pointsofB into N\A. Lettingm → ∞ thisgivesa sequence
g for which, in terms of their dependence on coordinates, f and h ◦ g are closer and
m m
closertoindependent.
Now the point is that the quotients q (f ·h◦g) converge in C(Ω )/(Mσ)⊥ to a member
σ σ
that depends only on q (f) and q (h) — this follows from an elementary step-function
σ σ
approximation argument and use of all the permutation invariance. Let us call this the
asymptoticproduct ofq (f) and q (g). It isnow aroutineexerciseto check thisis actu-
σ σ
ally aC∗-algebra product on C(Ω )/(Mσ)⊥ and corresponds exactlyto the usualproduct
σ
offunctionsin Vσ.
(Alternatively,onecanfindanactualcontinuousfunctiononΩ whoseimageinVσ repre-
σ
sentsthisproduct;letusillustrateonecheapwaytodothisincaseσ = 0. Letψ : 2N → N
1
and ψ : 2N+1 → N be bijections, and let us use the same letters for the corresponding
2
φ1 φ2
adjoint maps Ω → Ω(2N) ∼= Ω and Ω → Ω(2N+1) ∼= Ω, where we temporarily write Ω(S)
forthespaceofmodelsofT withvertexset S. Then clearly thefunctionsf ◦φ ◦ψ and
1 1
g◦φ ◦ψ dependondisjointsetsofcoordinates;theirproductisthesoughtrepresentative
2 2
forq (f)q (g).)
σ σ
Thus we have identified C(Ω )/(Mσ)⊥ with a closed subalgebra Vσ of L∞(T ) (with
σ σ
a newly-defined product). Let us call functions in Vσ simple if they are the images of
simple(equivalently,finite-dimensional)functions in C(Ω ). We can describe the simple
σ
functions naturally as follows: to any fixed nonempty cylinder set A ⊆ Ω depending
σ
on coordinates in J ⊂ N \ [k] corresponds a collection of finite models of the theory T
on the vertex set J (with some multiplicities), and now the averaged-function 1 (ω) for
A
ω ∈ Ω is just the sum of the densities with which each of those finite models appears
σ
isomorphically as a submodel of ω (now summing over the multiplicities). Referring to
such a function 1 for A corresponding to a single model on J (so that our general A is
A
a disjoint union of such) as a statistics function, the simple functions in Vσ are now just
linearcombinationsofstatisticsfunctions. WewriteVσ forthedensesubspaceofthese.
0
3 Description of flag algebras
We will now identify a flag algebra and its homomorphisms with a family of measurable
functionsandanassociatedsetofprobabilitymeasures,andshowhowvariousresultsper-
5
tainingtotheformertranslateintofactsaboutthelatter. Ingeneralwewillrefertothetwo
resulting pictures as the ‘flag algebra picture’ and the ‘measure theoretic picture’ respec-
tively. WewillfollowthesectionalstructureofSections2and3of[9]. Wewillnotbroach
themorespecificoptimizationproblemstudiedinSections4and5(or,forthatmatter,the
more complete result of [10]); while they too can presumably be translated into the mea-
sure theoretic picture, aside from rendering the underlying objects in a better-established
analyticlightthisdoesn’tseem greatlyto changetheargumentsthat areinvolved.
Informally,thebackgroundpurposeof[9]istosetupafamilyof‘proxies’forthestatistics
oflargemodelsofT thatenjoysomeadditional‘analytic’structuremakingthemeasierto
handle for the study of certain kinds of question; two such questions on extremal statis-
tics are then analyzed in these terms in Section 5 of [9] and in the follow-up paper [10].
In [9] these proxies are certain R-valued homomorphismsof flag algebras, and the actual
elements of the flag algebras are of secondary importance (although they do continue to
appear in a supporting role occasionally later in the paper). The manipulation of flag al-
gebras is mostly for the purpose of setting up these homomorphisms. However, in the
measure theoretic picture we can say at once what these homomorphisms correspond to
— they are the ergodic Sym -invariant probability measures on Ω — and so the effort
σ σ
we expend on setting up the correspondence between flag algebras and certain spaces of
measurablefunctionswillultimatelyberequiredonlytoshowthatthehomomorphismsof
theformerreally are identifiedwithapriori-knownobjectsrelated to thelatter.
Section 2
In[9],RFσ denotesthefreeR-vectorspaceonFσ. WehavealreadyselectedforeachF ∈
Fσ some associated clopen subset A ⊆ Ω , and we will now extend this by associating
F σ
(withsomecareful normalization)toeach member λ F ∈ RFσ a linearcombination
Pj k j
oftheindicatorfunctions1 oftheseA ,eachofwhichisacontinuousfunctionsinceA
AF F F
is clopen. This will define a linear operator Φ : RFσ → C(Ω ) with image some funny-
σ
shaped subspace contained within the space of simple functions in C(Ω ). It turns out
σ
that in order to make contact between the calculus of [9] and addition and multiplication
of functions on Ω we need to introduce some nontrivial normalizing constants: our final
σ
identificationis
1
F 7→ 1 ;
|F|! AF
λ
k
XλkFj 7→ X |F |!1AFj.
j
j j
6
The arbitrariness in our choice of subset A corresponding to F is reflected in a similar
F
arbitrariness in this linear map Φ into C(Ω ); however, this disappears at the next step,
σ
whenwe defineaflag algebraas a quotientofRFσ.
LetKσ denotethesubspaceofRFσ generatedbythelinearcombinationsF˜− p(F˜,F)F
PF∈Fσ
ℓ
fordifferentℓ ≥ |V(F˜)|,andforcertainrealnumbersp(F˜,F)giveninDefinition1of[9]
(we will suppress their exact form here). From that definition one can check at once that,
given our chosen normalization in the definition of Φ above, the values p(F˜,F) are such
thatΦ(Kσ) ispreciselythefollowingsubspaceofΦ(RFσ):
Φ(a) ∈ Φ(Kσ) if and only if, as a simple function on Ω , Φ(a) may be
σ
chopped up furtherinto alinear combinationof simplefunctionscorrespond-
ing to cylinder sets over some common large finite subset of N, say Φ(a) =
b 1 , such that we can cluster this sum according to some partition
Pt∈J t Bt
J = J ,
Si∈I i
Φ(a) = Xbt1Bt = XXbt1Bt,
t i∈I t∈Ji
sothatfor each i ∈ I:
• all theB fort ∈ J are isomorphictosomefixedA , F ∈ Fσ;
t i Fi i
• and b = 0.
Pt∈Ji t
Alternatively,we can describe this by saying that somere-arrangement of the terms b 1
t Bt
bydifferentpermutationsofN\[k]sumstozero: thatis,thereareg ,...,g ∈ Sym with
1 t σ
b 1 exactlycancelingto zero.
Pt t gt(Bt)
It now follows from a little compactness argument that Kσ is also precisely the set of
those a ∈ RFσ for which Φ(a) ∈ C(Ω ) is in the annihilator (Mσ)⊥ (indeed, that
σ
Φ(Kσ) ⊆ (Mσ)⊥ is immediate; for the opposite inclusion we can argue that if a 6∈ Kσ
thenbywitnessingourinabilitytofindadecompositionofΦ(a)asaboveforcylindersets
over larger and larger finite subset of N, we can extract some sequence of members of
M(Ω ) that are invariant for the vertex-permutations in some corresponding exhausting
σ
sequence of finite subsets of Sym 1, and that converge to some member of Mσ that does
σ
notannihilateΦ(a)).
WenowconsiderthequotientspaceAσ := RFσ/Kσ. Bytheabove,q ◦Φfactorsthrough
σ
this quotient to give an injective map Ψ : Aσ → Vσ. Moreover, since any finite dimen-
sional cylinder set contained in Ω is equivalent under Mσ to some A upon a suitable
σ F
1Noticethathereisanappealtoourrestrictiontofinitely-supportedpermutations.
7
permutation of coordinates in N \ [k], the image of Ψ is actually the subspace Vσ of all
0
simplefunctionsinVσ;as such, itisdense.
The next step is to consider products. The definition of a product for a,b ∈ RFσ/Aσ
given in [9] now just translates into the product of Ψ(a) and Ψ(b) as L∞-functions on
Ω ; and the proof in [9] that this product is well-defined mostly becomes a proof in the
σ
measuretheoreticpicturethatthisproductremainsintheimageofΨ(thatis,inVσ). This
0
completes our identification of the flag algebra Aσ with the dense subalgebra Vσ of Vσ,
0
which is itself a norm-closed Banach subalgebra of L∞(T ). The basic properties of the
σ
productcontainedin Lemma2.4,forexample,are nowimmediate.
InSubsection2.2of[9]isintroducedthe‘downwardoperator’. Thisapplieswhenwehave
asubmodelσ′ ofσ,saywithvertexset[k′]forsomek′ ≤ k. Byre-labelingtheverticesof
σ′ ifnecessary,wecanidentifyΩσ withaclopensubsetofΩσ′ (withA(σ′,σ),infact);since
this subset is clopen, the extension operator J : C(Ωσ) ֒→ C(Ωσ′) obtained by extending
acontinuousfunctiononΩ tobeidenticallyzeroelsewhereiswell-defined(inparticular,
σ
its output is still a continuous function). One checks at once that J factorizes through the
quotientsC(Ωσ) → Vσ ⊆ L∞(Tσ) and C(Ωσ′) → Vσ′ ⊆ L∞(Tσ′). This factorization is
themeasuretheoreticpictureofthedownward operator. Onceagain, a normalizingfactor
appears in[9]to makethesumscomeoutright.
Subsection 2.3 turns to the ‘upwards operator’. The definition of this and the properties
it enjoys depend more heavily on the precise shape of the theory T than most of the fore-
going, and we shall not translate the results of this subsection in detail. Instead, let us
examineonlyaleadingspecial caseofthisoperator.
Given again some type σ on [k] extending σ′ on [k′], k′ ≤ k, we now wish to make a
passage from Vσ′ to Vσ in the following way. Any point of Ωσ defines a point of Ωσ′
simplyby ignoringthevertices in {k′ +1,k′ +2,...,k} (and so sendingN\[k] to (N−
(k−k′))\[k] = N\[k′]);thisgivesacontinuousmapfromΩσ toΩσ′,sothatcomposition
gives a homomorphismC(Ωσ′) → C(Ωσ). This may descend to a map Vσ′ → Vσ which
is then necessarily also a homomorphism: this requires that any member of (Mσ′)⊥ be
sent to a member of (Mσ)⊥ by this composition map, which in some sense tells us that
the space of models Ω is still ‘large enough’ to support a sufficiently large collection of
σ
Symσ-invariant measures compared with Ωσ′. A formal version of this property appears
intheflag algebrapictureasaconditionthatacertain memberofAσ isnotazero-divisor,
and under this assumption the existence of a suitable homomorphism (which translates
into the abovementioned factorizability) is proved directly for flag algebras as Theorem
2.6 in [9]; this property is also related to the more immediate property of a theory that it
havethe‘amalgamationproperty’.
8
ThisisanexamplethatcanbeextendedtorelatethealgebrasAσ (orVσ)arisingfromtwo
different theories T , T given an interpretation of onein terms of the other: this interpre-
1 2
tation again defines a continuous map from the Cantor space corresponding to one to that
corresponding to the other, and again we then face the question of whether the resulting
homomorphismofalgebras ofcontinuousfunctionsdescendsunderourquotientingoper-
ation. The considerations to this purpose in [9] apply in this general setting, but we will
notexaminethedetailsfurtherhere.
When a homomorphism can be obtained, some of the other results that follow on the
properties of this map are now translations of certain basic facts for concrete spaces of
measurable functions (to which they already appear structurally very similar): Theorem
2.8(a)of[9],forexample,assertsinourpicturethatE [fh|Ξ] = fE [h|Ξ]iff isalready
µ µ
Ξ-measurable,and 2.8(b)istheruleofiteratedconditionalexpectations.
Section 3
The overall approach of the first three sections of [9] is to define a flag algebra first, and
thentoobtainacollectionof‘proxies’or‘limitobjects’forthestatisticsoflargemodelsof
atheoryintermsofthem. Theroˆleoftheseistobeplayed bythemultiplicativefunctions
φ : Aσ → Rthatarenon-negativeontheimageofanysingleflagF ∈ Fσ;thesetofthese
iswrittenHom+(Aσ,R).
Having identified the flag algebra Aσ as thedense subalgebra Vσ of thecommutativeC∗-
0
algebra Vσ so that the images of single flags correspond to the single statistics functions,
wecaneasilycheckthatnon-negativityontheseimpliesnon-negativityonanymemberof
Vσ that is itself a non-negative-valued function; this follows easily since a non-negative
0
simplefunction can always be written as a linear combination of indicator functions with
non-negativecoefficients.
We now observe that given the non-negativity of such a φ it can be extended to a mul-
tiplicative linear functional on the whole of Vσ. We can now, if we wish, apply certain
standard representation theorems to this space: either by further exploiting its vector lat-
tice structure following the results of Yosida and Kakutani, as presented in Section XII.5
of[12],orbycomplexifyingtheconstructionsofarandusingthe(arguablymorepopular)
Gelfand-Naimark Theorem. At any rate, this identifies φ with a point of the spectrum of
Vσ (for one or other interpretation of ‘spectrum’). In fact, this identification is more-or-
less implicitin Remark 4 of Subsection3.2 of[9], althoughthere we stillrequire someof
thebasicstructureofHom+(Aσ,R)to havebeen idenfitied.
However, given our identification with the measure-theoretic picture, we have an alterna-
9
tivetotheabove. SinceVσ ∼= C(Ω )/(Mσ)⊥,asalinearfunctionalonVσ wecanidentify
σ
φ with a member of Mσ (uniquely, since (C(Ω )/(Mσ)⊥)∗ ∼= Mσ): a Sym -invariant
σ σ
measure on the space Ω that we started with, rather than a point of some abstractly-
σ
produced new space SpecVσ. This has the advantage that we retain the theory itself T,
codedas itisintothe‘shape’ofthespaceΩ . It isnoweasy tocheck thatthosemeasures
σ
in Mσ that are multiplicativeon Vσ are precisely the ergodic Sym -invariant probability
σ
measuresonΩ (thesearemultiplicativeonVσ sinceanymemberofVσ isµ-a.s. constant
σ
if µ ∈ Mσ is ergodic). This establishes the identification of the flag algebra homomor-
phismswithexchangeablemeasures2.
As remarked in Remark 3 of Section 3 of[9], workingwith arbitrary multiplicativelinear
functionals on Aσ is problematic: the point is that without non-negativity these need not
be extendable to the whole of Vσ at all (equivalently, they may not be continuous for the
normtopologyofVσ). Indeed, thefact, mentionedinRemark 3, thatAσ isanon-finitely-
0
generated free commutativealgebra overR, is precisely what wouldallow us toconstruct
adiscontinuoussuch functionalusingtheaxiomofchoice.
Now the order defined on Aσ in Definition 5 of Section 3 of [9] (in terms of the above
notion of ‘positivity’ for a homomorphism, which must be introduced first) is precisely
the usual pointwise order on Vσ as a set of real-valued functions; in the setting of ab-
0
stract flag algebras, where we are unable to define anything ‘pointwise’, the functionals
of Hom+(Aσ,R) are needed as a replacement to formulate this definition. Given this,
Theorem 3.1 requires only that composition with a homeomorphism and conditional ex-
pectationarenon-negativeoperators betweenfunctionspaces. Also,the‘probability-like’
convergence results of Subsection 3.1 (somewhat based on [8]) now really are about the
classicalvaguetopologyonaset ofprobabilitymeasures.
In Subsection 3.2 of [9] averages of homomorphisms are taken with respect to actual
probability measures; in the measure-theoretic picture these become classical ergodic de-
compositions. Specifically,givensomeextensionσ ofσ ,weknowthatuponorderingthe
0
vertices of σ so that σ = σ| , then Ω = A becomes a clopen subset of Ω ; and
0 [k0] σ (σ0,σ) σ0
now if µ is an ergodic Sym -invariant probability measure on Ω for which µ(Ω ) > 0
σ0 σ0 σ
we may consider the conditioned measure µ(·|Ω ) := µ(· ∩ Ω )/µ(Ω ) (defined free
σ σ σ
from any measure-theoretic ambiguity, since µ(Ω ) > 0). Since Sym ≤ Sym fixes
σ σ σ0
Ω , this defines nowa Sym -invariantprobabilitymeasureon Ω ; however,itmay not be
σ σ σ
ergodic under the action of this subgroup Sym , and the resulting ‘ensemble’ of homo-
σ
morphismsobtainedin thissubsectionis simplyitsergodicdecomposition.
2Althoughwe noteinpassingthatobtainingthemfroma priorconstructionof thefunctionalgebrasis
reminiscentofthealternativerouteintothetheoryofintegrationandmeasureestablishedbySegalin[11];
anapproachthathashadlastingconsequencesfortheformulationof‘non-commutativeintegration’inthe
settingofgeneralvonNeumannalgebras.
10