Fairly amenable semigroups

173 Pages·2014·0.68 MB·English

University of Tasmania
School of Mathematics and Physics
Fairly Amenable Semigroups
Joshua Thomas Deprez, BComp-BSc (Hons)
May 21, 2014
Supervisor: Dr Des FitzGerald
Submitted in fulfilment of the requirements for the Degree of
Doctor of Philosophy
Abstract
Amenabilitydevelopedalongsidemodernanalysis,asitisacentralpropertylacking
in a group used to show, for example, the Banach-Tarski paradox (Wagon, 1993).
e(cid:277)rstworkingde(cid:277)nitionwasgivenbyvonNeumann(1929),intermsof(cid:277)nitely-
additive measures. A number of useful theorems are capable of being shown using
thisbasicde(cid:277)nition.
e(cid:277)rstmodernde(cid:277)nitionofamenabilitywasgivenbyM.M.Day(1957),whose
concept involved invariant means. For groups this coincides exactly with the von
Neumanncondition: eachinvariantmeancorrespondstoaninvariant(cid:277)nitely-additive
measure,correspondingviaLebesgueintegration. isadvancewassigni(cid:277)cantasit
opened thedoortothe applicationofabstract harmonicanalysis, (cid:277)xed-pointtheo-
rems,andanindustryofconsequences. Amenablegroupssupportalmost-invariant
(cid:277)nite means, and via decomposition this is culminated as the Følner condition, a
statementabout(cid:277)nitesets. Abeliangroupsareamenableasasimpleconsequenceof
the Markov-Kakutani (cid:277)xed-point theorem. A theorem of B. E. Johnson (1972) led
to the development of amenable Banach algebras and C*-algebras, neatly encoding
amenabilityinthemechanicsofcohomologytheory.
While amenability is directly generalisable from groups to semigroups, the two
keyde(cid:277)nitionsdonotcorrespondinthesamewayastheydoforgroups: extracting
a(cid:277)nitely-additivemeasurefromale-invariantmeanyieldswhatmightbecalleda
le preimage-invariant measure, and for groups these merely correspond to the in-
verseelements. AsimplebutsurprisingconsequenceofDay’sde(cid:277)nitionofamenabil-
ity is that semigroups with a zero element are both le and right amenable (Day,
1957). Yet they cannot support a (totally) invariant (cid:277)nitely-additive measure (van
Douwen, 1992, p231). On the other hand, all semigroups with more than one dis-
tinctlezeroarenotleamenable(Paterson,1988),andinparticulartherearemany
non-amenable(cid:277)nitesemigroups,whichisanothercontrasttothegroupcase: all(cid:277)-
nite groups are amenable. is standard de(cid:277)nition of amenability for semigroups
i
ABSTRACT ii
is therefore unintuitive and, perhaps, unsatisfactory. Restricting to better-behaved
classesofsemigroups,suchastheinversesemigroups,doeslittletoimprovethis.
e (cid:277)rst new result of the present work is that there is a weakening of invari-
ancethatcanbeusedinthecontextof(cid:277)nitely-additivemeasurestogeneralisegroup
amenabilitytosemigroupsinadifferentway. ForasemigroupS, a(cid:277)nitely-additive
measure (cid:22) 2 [0;1]P(S) will be called le fairly invariant if, for all s 2 S andA (cid:18) S
such that (cid:21) j is an injection, (cid:22)(sA) = (cid:22)(A). When a semigroup supports such
s A
a (cid:277)nitely-additive measure, then it is le fairly amenable. Fair amenability is a gen-
eralisation of group amenability, and retains some of the useful theorems. Some of
theresultsshownusingthisformulationinclude: asemigroupislefairlyamenable
whenitsatis(cid:277)esaweakenedStrongFølnerCondition,(cid:277)nitesemigroupsareallfairly
amenable,semigroupswithinvolutionareeitherfairlyamenableonboththeleand
the right or not at all, adjoining a zero does not cause a non-fairly amenable semi-
grouptobecomefairlyamenable,directedunionsoffairlyamenablesemigroupsare
fairly amenable, and a variety of examples which are fairly amenable or not fairly
amenable.
ename“amenable”is,asthestorygoes,supposedtobeapun,sinceamenable
groupssupportinvariantmeans. usanimportantquestionforfairamenabilityis,
what condition for a mean is equivalent to the fair invariance of the corresponding
(cid:277)nitely-additivemeasure? Oneapproachisto(cid:280)ipthedualitybetweentheconvolu-
tionactioninℓ1(S)andthedualactioninℓ1(S)upside-down: attemptconvolution
in ℓ1(S) and the dual action in ℓ1(S). In this scenario, the curious will consider
suchill-de(cid:277)nedexpressionsas0(cid:3)(cid:31) . Fortunately,wherevertheconvolutionpartial
S
actionofsonϕ 2 ℓ1(S),i.e. s(cid:3)ϕ,iswell-de(cid:277)nedandbounded,thentheintegral
with respect to a le fairly-invariant measure can be readily computed. It is shown
thatasemigroupSlefairlyamenableif,andonlyif,thereexistsameanmsuchthat
m(ϕ) = m(s(cid:3)ϕ) for all s 2 S and ϕ 2 ℓ1(S) such that s(cid:3)ϕ 2 ℓ1(S). Hence
thenomenclature“fairlyamenable”isjusti(cid:277)edasapunalso.
Some variations on fair amenability and related results are also explored. As a
variation on the (cid:3) partial action, an operator ⊛ is introduced on ℓ1(S), which in-
ducesafullactionofS. Onedrawbackof⊛comparedto(cid:3)isthat,inordertoexpress
fairamenability,anadditionalconditionisrequiredtolimitthescopeofinvariance
⊛
appropriately. Finally,inner invarianceandits“fair”variantarebrie(cid:280)yexplored.
Declarations
Declaration of Originality
is esis contains no material which has been accepted for a degree or diploma
bytheUniversityoranyotherinstitution,exceptbywayofbackgroundinformation
and duly acknowledged in the esis, and to the best of my knowledge and belief
nomaterialpreviouslypublishedorwrittenbyanotherpersonexceptwheredueac-
knowledgement is made in the text of the esis, nor does the esis contain any
materialthatinfringescopyright.
AuthorityofAccessandStatementregardingpub-
lished work contained in the Thesis
Chapter 5 has been sent in a reduced form for potential publication in the journal
SemigroupForum,thoughatthiswritingithasnotyetbeenaccepted. Atsuchtime
thatajournalholdsthecopyrightforcontent,accesstothematerialshouldbesought
fromtheminaccordancewiththeirpolicies.
e remaining non-published content of the esis may be made available for
loanandlimitedcopyingandcommunicationinaccordancewiththeCopyrightAct
1968.
…………………………………… …………………
JoshuaomasDeprez Date
iii
Acknowledgements
Firstlyandmostimportantly,aheartfeltthank-youtomysupervisorDrDesFitzGer-
aldforhiscontinuedpatienceandwisdomduringtheproject,thecoffeesandlunches
attheStaffClub,andhisveryforgivingnature.
I thank the Head of School Professor John Dickey, and Mathematics head Pro-
fessorLarryForbes,forhelpingtogreasethewheelsoftheGraduateResearchOffice.
anksalsotoDrBarryGardnerforhisparticipation.
I thank those responsible for my participation in the Australian Postgraduate
Awardscheme: (cid:277)rstly,theAustralianGovernmentforfundingthescholarship,and
theUniversityScholarshipsOfficeforacceptingmyapplication.
ankstotheGraduateResearchOffice,andalsoapologies: forputtingupwith
mypassive-aggressiveemailsandsomewhatnon-standardpaperwork.
IthanktheSchoolofMathematics&Physicsasawholeforfurnishingmewith
anofficethatIcouldsafelyignoreforlongperiodsoftime. Ashout-outtomymost
excellent office mates: Stephen Walters and Melissa Humphries, and the extremely
occasionalTonyFitzpatrick.
A huge thanks to my friends Paris Butt(cid:277)eld-Addison and Jon Manning, whose
officeintowntendedtobemoreconvenientthanmyuniversityoffice(bothforin-
ternetaccessandproximitytoqualitycoffee).
I thank the Victorian Algebra Group and Australian Mathematical Society for
thricesubsidisingmytravelcoststoandfromtheannualVictorianAlgebraConfer-
ence.
In a way, this work is the culmination of much more than just three-and-a-half
yearsofreasearch,butadditionallymanyyearsofschoolingandundergraduatestudy
aswell. Itisatestamenttothequalityofthepubliceducationsystem. Iwouldliketo
thankallthemotivatingandengagingteachersIhavehadovertheyears. Finally,to
anyonewhosuggestedIcouldhavedonebetterifIhadaprivateschoolbackground
instead: Iwouldn’thavechangedathing.
iv
Contents
Abstract i
Declarations iii
Acknowledgements iv
ListofTables viii
ListofFigures ix
1 Introduction 1
1.1 Whatisamenability? . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Aremarkaboutproofs . . . . . . . . . . . . . . . . . . . . 3
1.2.2 Semigroups,groups,self-actions . . . . . . . . . . . . . . . 3
1.2.3 Functionspaces . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.4 Ultra(cid:277)ltersandultralimits . . . . . . . . . . . . . . . . . . . 7
2 AmenabilityandGroups 11
2.1 Finitely-additivemeasures . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Invariantmeans . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Almost-invariant(cid:277)nitemeans . . . . . . . . . . . . . . . . . . . . . 21
2.4 Følnercriteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4.1 VariationsonFølnercriteria . . . . . . . . . . . . . . . . . 25
2.5 Amenabilityandgrowth . . . . . . . . . . . . . . . . . . . . . . . . 29
2.6 Classicalamenabilityresultsforgroups . . . . . . . . . . . . . . . . 33
2.6.1 ompson’sgroupF . . . . . . . . . . . . . . . . . . . . . . 36
2.7 Topologicalgroupamenability . . . . . . . . . . . . . . . . . . . . . 40
v
CONTENTS vi
2.7.1 TopologicalFølnercriterion . . . . . . . . . . . . . . . . . 41
3 AmenabilityandBanachalgebrasandC*-algebras 43
3.1 Banachalgebraamenability . . . . . . . . . . . . . . . . . . . . . . 43
3.2 C*-algebraicamenability . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3 Banachalgebrasagain . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.4 eweakcontainmentproperty . . . . . . . . . . . . . . . . . . . . 48
4 AmenabilityandSemigroups 50
4.1 Semigroupswith(cid:277)nitely-additivemeasures . . . . . . . . . . . . . . 50
4.2 Semigroupswithmeans . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2.1 Breakdownbetweende(cid:277)nitions . . . . . . . . . . . . . . . . 54
4.3 Classicalamenabilityresultsforsemigroups . . . . . . . . . . . . . 55
4.3.1 FølnerconditionsandatheoremofFrey . . . . . . . . . . . 55
4.3.2 Semigroupsand(cid:277)nitemeans . . . . . . . . . . . . . . . . . 58
4.3.3 Morestandardresults . . . . . . . . . . . . . . . . . . . . . 59
4.3.4 Cancellativeandreversiblesemigroups . . . . . . . . . . . . 61
4.4 Amenableinversesemigroups . . . . . . . . . . . . . . . . . . . . . 63
4.4.1 Weakcontainmentandinversesemigroups . . . . . . . . . 66
5 FairlyAmenableSemigroups 71
5.1 De(cid:277)nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.2 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.2.1 GroupsandFølnercriteria . . . . . . . . . . . . . . . . . . 75
5.2.2 Basicconsequences . . . . . . . . . . . . . . . . . . . . . . 76
5.2.3 Subsemigroups . . . . . . . . . . . . . . . . . . . . . . . . 79
5.2.4 Green’srelations . . . . . . . . . . . . . . . . . . . . . . . . 79
5.2.5 Directproducts . . . . . . . . . . . . . . . . . . . . . . . . 81
5.2.6 Quotientsofsemigroups . . . . . . . . . . . . . . . . . . . 84
5.2.7 Completely0-simplesemigroups . . . . . . . . . . . . . . . 85
5.2.8 Semigroupswithinvolution . . . . . . . . . . . . . . . . . . 94
5.2.9 Cliffordsemigroups . . . . . . . . . . . . . . . . . . . . . . 95
5.2.10 Directedunionsofsemigroups . . . . . . . . . . . . . . . . 97
5.3 Furtherexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.3.1 Graphinversesemigroups . . . . . . . . . . . . . . . . . . . 99
5.3.2 eBaer-Levisemigroup . . . . . . . . . . . . . . . . . . . 104
CONTENTS vii
5.3.3 Freeinversesemigroups . . . . . . . . . . . . . . . . . . . . 106
5.3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.4 Motivationsforfairamenability . . . . . . . . . . . . . . . . . . . . 108
5.4.1 Measureratios . . . . . . . . . . . . . . . . . . . . . . . . . 110
6 FairlyInvariantMeansforSemigroups 112
6.1 econvolutionpartialaction . . . . . . . . . . . . . . . . . . . . . 113
6.2 Integratings(cid:3)f . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.3 Modi(cid:277)edconvolution . . . . . . . . . . . . . . . . . . . . . . . . . 122
⊛
6.4 -invariantmeans . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.5 Generalisedconvolution . . . . . . . . . . . . . . . . . . . . . . . . 134
6.5.1 Generalisedsemigroupconvolution . . . . . . . . . . . . . 135
6.5.2 Restrictedmeansandfunctions . . . . . . . . . . . . . . . . 135
7 MakingOtherConditionsFair 140
7.1 Preimageinvariance . . . . . . . . . . . . . . . . . . . . . . . . . . 140
7.2 Inneramenability . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
⊛ (cid:3)
7.2.1 Inner -and -invariance . . . . . . . . . . . . . . . . . . 145
7.3 Resultsyettobeshown . . . . . . . . . . . . . . . . . . . . . . . . 148
(cid:3)
7.3.1 Bi- -invariantmeans? . . . . . . . . . . . . . . . . . . . . . 149
7.3.2 Almostfairlyinvariant(cid:277)nitemeans? . . . . . . . . . . . . . 150
7.3.3 FairlyamenableAbeliansemigroups? . . . . . . . . . . . . 152
7.3.4 Otheroutstandingquestions . . . . . . . . . . . . . . . . . 153
IndexofSymbols 160
List of Tables
2.1 Othernotationsforthenaturalordualactions. . . . . . . . . . . . . 18
2.2 Summary of the variations on Følner criteria for discrete groups.
ϵ;ϵ > 0,thea :::a ;gareelementsofG,andF;F ;K;Sare(cid:277)nite
0 1 n i
subsetsofG. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3 Examplesofgroupsindifferentgrowthclasses . . . . . . . . . . . . 31
2.4 Importantexamplesof(cid:277)nitely-generatedgroups. . . . . . . . . . . . 39
5.1 Amenabilityversusfairamenabilityondifferentsemigroups. . . . . 108
viii
List of Figures
1.1 Anultra(cid:277)lter((cid:277)lledpoints)onthe(cid:277)nitesetfa;b;cg. . . . . . . . . . 8
2.1 ejourneyofthe(cid:277)rsthalfofthischapter. . . . . . . . . . . . . . . 11
2.2 An overview of the sets mentioned in the functional approach to
topologicalamenability. We(cid:277)ndameaninM(G)asaweak*cluster
pointofasequenceinP(G)^,andcanusevariousanalyticmachin-
eryalongtheway. . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.1 ImplicationsthathavebeenshownbetweentheleFølner-typecon-
ditions and le amenability (A) of a semigroup. Additionally, (FC)
̸) ̸)
(A)and(A) (WFC)ingeneral. (Yang,1987) . . . . . . . . . . 57
5.1 ForeverysetAandelementsthereissomesubsetBsuchthatsA =
sBandsactsinjectivelyonB. . . . . . . . . . . . . . . . . . . . . . 75
5.2 If s acts injectively on A, then it also does so on A \ F , and so
n
jA\F j = jsA\sF j. . . . . . . . . . . . . . . . . . . . . . . . . . 76
n n
5.3 e right Cayley graph for the free semigroup on two generators
fa;bg+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.4 PartoftheleCayleygraphofthebicyclicmonoidB. . . . . . . . . 100
(cid:12) (cid:12)
5.5 Deriving(cid:12)qjpk□ △□ (cid:12)inthebicyclicmonoid. . . . . . . . . . . . 100
n n
5.6 From the le, the 1-rose, 2-rose, and 10-rose. e free categories
overthesearesimplythefreemonoidson1generator,2generators,
and10generators,respectively. . . . . . . . . . . . . . . . . . . . . 103
5.7 earrowsx,y,u,andvformacommutativesquare. Ifforanyother
′ ′
pairofarrowsx andy withcommondomainformacommutative
squarewithuandvthereexistsauniquearrowfromd(x′) ! d(x)
(dashed),thenxandyareapullbackofuandv. . . . . . . . . . . . 103
ix