7th International Conference onAbstract Harmonic Analysis

Organizing CommitteeJ. AlaminosJ. ExtremeraE. KaniuthA. LauA. ÜlgerA. R. Villena

With the support ofResearch Project MTM2012-31755, Research Project FQM-185 ,Fisymat (Máster/Programa de doctorado),Universidad de Granada,Facultad de Ciencias, andDepartamento de Análisis Matemático

May 20-24, 2013University of Granada

2

Welcome

Itisourgreatpleasuretowelcomeyouto7thInternationalConferenceonAbstractHarmonicAnalysis2013inGranada, Spain. Thisbooklethasbeenput together tohelpyoufindyourwaythroughtheconferenceandmakethemostofyourstayinGranada.

Inadditiontothisprogrambooklet, youcanfindmostoftheinformationonlineat http://www.ugr.es/local/aha2013.

Wehopeyoufindthemuseful–feelfreetoemail aha2013@ugr.es withyourfeedback.

Acknowledgements

Wewouldliketothankthesupportandcooperationofthefollowinginstitutions

• UniversidaddeGranada,• FacultaddeCiencias,• DepartamentodeAnálisisMatemático,• ResearchProjectMTM2012–31755fromtheMinisteriodeEconomíayCompetitividad,• ResearchProjectFQM-185fromJuntadeAndalucia, and• FisyMat(Máster/Programadedoctorado),

JerónimoAlaminosJoséExtremera

EberhardKaniuthAntonyTo-MingLau

AliÜlgerArmandoR.Villena

Theorganizers,http://www.ugr.es/local/aha2013/

aha2013@ugr.es

4

Contents

InstructionsforParticipants 6

Program 9

Participants 15

Abstracts 17

5

InstructionsforParticipants

Forgeneralenquiriesorprojector/computer/internetproblemspleasecontactthelocalorganizers(Jeró-nimoAlaminos, JoséExtremera, ArmandoVillena).

InstructionsforSpeakers

• Thelectureroomwillhaveadigitalprojector, overheadprojector, acomputerandpointers. Thereisalsoablackboardthatcanbeusedsimultaneouslywiththeprojector.

• Speakerscanbringtheirownlaptop.• Pleasearrivesomeminutesbeforethesessionstartstochecktheequipmentworks, and, ifyouwillnotbeusingyourownlaptop, touploadyourslidesfromyourUSB drive. Evenbetter, pleasesendacopyofthetalkbye-mailtotheorganizersinadvance.

• Thetimeallocatedtoeachspeakeris40minutesplus5forquestions.

InstructionsforSessionChairs

• Pleasearrive5minutesbeforethesessionstartstochecktheequipmentworks.• Pleasesticktotheschedule.• Pleasemoderatequestions.• Pleasedonotstartsessionsortalksearly.

Internetaccess

• Thereareseveralwifinetsavailableintheuniversity(cviugr, cviugr-v2, eduroam). Please, selectcviugr. Whenyoupointthebrowsertoawebpage, usetheprovidedusernameandpassword.

• Yourbadgehasaloginname, password.• Theclassroom0.7hasseveralcomputersconnectedtointernetatthedisposaloftheparticipants.• YoucanusetheEduroamfacilitiesforthosewhohaveaworkingaccountontheirownuniversity.

6

Where

• Allthelectureswillbeat “SaladeConferencias”.• LunchandcoffeewillbeservedatthecafeteriaofFacultaddeCiencias. Youcanfindtheticketsforlunchinyourbadge.

• Inroom0.7therewillbecomputersavailabletotheparticipants.

Lecture room

Hotel Granada Center

ComputerRoom

Cafeteria

Figure 1: GroundfloorofFacultaddeCiencias

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ConferencedinnerandvisittotheAlhambra

OntheafternoonofWednesday, 22weplantovisittheAlhambraandcelebratetheconferencedinner.

• Wemeetat16:00attheentrance/hallofthe GranadaCenterHotel.

• At16:15thebuswillleavetothe Alhambra.

• Thevisitwillstartat17:00andwilllastforapproximately3hours.

• Theconferencedinnerwilltakeplacejustafterthevisit. A buswillpickusupattheAlhambraandwillleaveusat MiradordeSanCristóbal. Thereisashortwalkfromthereto RestauranteSanNicolás wherewearegoingtocelebratetheconferencedinner.

• Afterthedinner, thebuswillpickusupagainatMiradordeSanCristóbalandtakeusbacktothehotel.

Dinningout

Thereareplentyofplacestoeatrangingfromsmallbarstorestaurantsalloverthecity. Thehighlightedareasinthefollowingmaparesomeofthemostpopular. Allofthemareatwalkingdistance(10–20minutes)fromthehotel. Youcannotgowrongifyougoaroundthecathedral, Paseodelostristes, PlazaNuevaorthetownhall.

09/05/13 10:24Google Maps

Página 1 de 1http://maps.google.es/?ie=UTF8&ll=37.176338,-3.59843&spn=0.…93,0.024354&t=m&z=16&vpsrc=6&ei=z1uLUe3UFqOPiQbJkoDgBA&pw=2

Figure 2: Granadadowntown

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Program

Monday, May20

08:30–09:10 Registrationandopening

ChairArmandoVillena

09:20–10:00 SpectralsynthesisinFourieralgebrasofdoublecosethypergroupsEberhardKaniuth

10:10–10:50 StronglyinvariantmeansoncommutativehypergroupsRupertLasser

11:00–11:30 CoffeeBreak

ChairGarthDales

11:30–12:10 AmenabilitypropertiesforvariousclassesofpolynomialhypergroupsStefanAlexanderKahler

12:20–13:00 TransferringconvolutionoperatorsfromlocallycompactgroupstoclosedsubgroupAntoineDerighetti

13:00–15:30 Lunch

ChairJoséE.Galé

15:30-16:10 InterpolationsetsandquotientsoffunctionspacesonalocallycompactgroupIMahmoudFilali

16:20–17:00 InterpolationsetsandquotientsoffunctionspacesonalocallycompactgroupIIJorgeGalindo

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Tuesday, May21

ChairAnthonyTo-MingLau

08:30–09:10 Completelybounded Λ setsincompactgroupsParasarMohanty

09:20–10:00 Whenisthequotientalgebra 𝐴/𝐼 isomorphictoasubalgebraof 𝐴∗∗?AliÜlger

10:10–10:50 CompactoperatorsynthesisandspectralsynthesisinharmonicanalysisLyudmilaTurowska

11.00–11:30 CoffeeBreak

ChairNicoSpronk

11:30–12:10 Operatorsynthesis: unionsandproductsIvanTodorov

12:20–13:00 SpectralsynthesisforabsolutelycontinuousfunctionsofhigherorderJoséE.Galé

13:00–15:30 Lunch

ChairJoséExtremera

15:30-16:10 OnweightedOrliczalgebrasSerapÖztop

16:20–17:00 𝑁 -hyperreflexivityofthespaceofbounded 𝑁 -cocyclefromBanachalgebrasEbrahimSamei

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Wednesday, May22

ChairAntoineDerighetti

08:30–09:10 Separable 𝐶∗-algebrasandweak∗-fixedpointpropertyoftheirBanachspacedualGeroFendler

09:20–10:00 Fixedpoint and relatedgeometricpropertieson the Fourier andFourier-StieltjesalgebrasoflocallycompactgroupsAnthonyTo-MingLau

10:10–10:50 KirillovtheorywithoutpolarizationsDetlevPoguntke

11:00–11:30 CoffeeBreak

ChairJorgeGalindo

11:30–12:10 Onthe 𝐿 -FouriertransformnormofsomelocallycompactgroupsAliBaklouti

12:20–13:00 TBAJeanLudwig

13:00–15:30 Lunch

16:00– VisittotheAlhambraandConferenceDinner

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Thursday, May23

ChairLyudmilaTurowska

09:20–10:00 LeftidentitiesformeasurealgebraswithanapplicationtotheConnes-amenabilityofdualBanachalgebrasVokerRunde

10:10–10:50 OnthealgebraoffunctionsgeneratedbypurepositivedefinitefunctionsNicoSpronk

11:00–11:30 CoffeeBreak

ChairJerónimoAlaminos

11:30–12:10 NewresultsonsemigroupsofanalyticfunctionsÓscarBlasco

12:20–13:00 DirectlyfinitealgebrasofpseudomeasuresandpseudofunctionsYemonChoi

13:00–15:30 Lunch

ChairAliÜlger

15:30–16:10 RoughCayleygraphsofcompactlygeneratedgroupsPekkaSalmi

16:20–17:00 ThedualspaceofprecompactgroupsSalvadorHernández

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Friday, May24

ChairÓscarBlasco

08:30–09:10 WaveletcoorbittheoryinhigherdimensionsHartmutFühr

09:20–10:00 ImagesofwavelettransformsKeithTaylor

10:10–10:50 AlgebrasofmultilinearformsonhypergroupsBertSchreiber

11:00–11:30 CoffeeBreak

ChairEberhardKaniuth

11:30–12:10 WeakamenabilityoftheFourieralgebraof 𝑎𝑥 + 𝑏 groupMahyaGhandehari

12:20–13:00 RadicalsofsomesemigroupalgebrasGarthDales

13:00–15:30 Lunch

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Timetable

Monday Tuesday Wednesday Thursday Friday

08:30–09:10 Registrationandopening

Mohanty Fendler Führ

09:20–10:00 Kaniuth Ülger Lau Runde Taylor

10:10–10:50 Lasser Turowska Poguntke Spronk Schreiber

11:00–11:30 CoffeeBreak CoffeeBreak CoffeeBreak CoffeeBreak CoffeeBreak

11:30–12:10 Kahler Todorov Baklouti Blasco Ghandehari

12:20–13:00 Derighetti Galé Ludwig Choi Dales

13:00–15:30 Lunch Lunch Lunch Lunch Lunch

15:30–16:10 Filali Öztop Salmi

16:20–17:00 Galindo Samei Hernández

14

Abstracts

AliBaklouti – Wednesday, 11:30–12:10Onthe 𝐿 -Fouriertransformnormofsomelocallycompactgroups

Let 𝐺 beaseparablelocallycompactunimodulargroupoftypeI,and 𝐺 theunitarydualof 𝐺 endowedwiththeMackeyBorelstructure. WeregardtheFouriertransform ℱ asamappingof 𝐿 (𝐺) toaspaceof 𝜇-measurablefieldofboundedoperatorson 𝐺 definedfor 𝜋 ∈ 𝐺 by

𝐿 (𝐺) ∋ 𝑓 ↦ ℱ𝑓 ∶ ℱ𝑓(𝜋) = 𝜋(𝑓),

where 𝜇 denotesthePlancherelmeasureof 𝐺. For 1 < 𝑝 ≤ 2 and 𝑞 = 𝑝/(𝑝 − 1), wehavetheinequality

‖ℱ𝑓‖ ≤ ‖𝑓‖ , 𝑓 ∈ 𝐿 (𝐺) ∩ 𝐿 (𝐺)

andthemapping 𝑓 ↦ ℱ𝑓 extendstoacontinuousoperator ℱ ∶ 𝐿 (𝐺) → 𝐿 (𝐺). Weareconcernedinthistalkwiththenorm ‖ℱ (𝐺)‖ ofthe 𝐿 -Fouriertransform:

‖ℱ (𝐺)‖ ∶= sup‖ ‖ ≤

‖ℱ 𝑓‖ .

WegiveanestimateofthisnormforsomeclassesofsolvableLiegroupsandwediscussthesharpnessproblem. Forcompactextensionsof ℝ , anextremalfunctionisgivenbyanextensionofagaussianfunc-tion. Besides, asanexampleofnon-compactextension, theuniversalcoveringgroupoftheEuclideanmotiongroupoftheplaneisalsotreatedandanestimateofthenormisobtained.

ÓscarBlasco – Thursday, 11:30–12:10Newresultsonsemigroupsofanalyticfunctions

Westudythemaximalspacesofstrongcontinuityon 𝐵𝑀𝑂𝐴 andtheBlochspace ℬ forsemigroupsofcompositionoperators. Characterizationsaregivenforthecaseswhenthesemaximalspacesare 𝑉𝑀𝑂𝐴orthelittleBloch ℬ . Thesecharacterizationsareintermsoftheweakcompactnessoftheresolventfunctionorintermsofaspeciallychosensymbol 𝑔 ofanintegraloperator 𝑇 .

YemonChoi – Thursday, 12:20–13:00Directlyfinitealgebrasofpseudomeasuresandpseudofunctions

AnalgebrawithidentityissaidtobedirectlyfiniteorDedekindfiniteifallleft-invertibleelementsareinfactinvertible. ByanoldobservationofKaplansky, thisistrueforthe ℓ -groupalgebraofadiscretegroup; theproofgoesthroughpropertiesofthegroupvonNeumannalgebra. I willpresentanextensionofthisresulttothecorrespondingalgebrasof 𝑝-pseudomeasures.

Thereisanaturalwaytodefinedirectfinitenessforalgebraswithoutidentity, anditturnsoutthattheresultabovecontinuestoholdfor 𝐿 (𝐺) when 𝐺 isunimodular; herewehavetoworkinthereducedgroup 𝐶∗-algebraratherthanthevonNeumannalgebra. I willgiveanoutlineoftheproofandgivesomepartialextensionsforalgebrasof 𝑝-pseudofunctions. I willalsopresentanexampleofalocallycompactgroup 𝐺 where 𝐿 (𝐺) isnotdirectlyfinite, andsuggestsomequestionsforfuturework.

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GarthDales – Friday, 12:20–13:00Radicalsofsomesemigroupalgebras

Itiswellknownthatgroupalgebrasaresemisimple. I shallrecallthedefinitionofasemigroupalgebraℓ (𝑆) forasemigroup 𝑆, andthinkwhenthesealgebrasaresemisimple.

InparticularI shalldescribethesemigroupcompactificationofasemigroup, concentratingonthedefinitionofthesemigroup (𝛽ℕ,□). Thedefinitionsinvolvecommutative 𝐶∗-algebrasandthetheoryofultrafilterson ℕ.

ThenI shalltrytoworkoutthe(Jacobson)radicalofthealgebra ℓ (𝛽ℕ,□) andsomerelatedalgebras.Thetheoryinvolvescompact, righttopologicalsemigroups.

ThetalkisbasedonjointworkwithDonaStrauss, YuliyaandYevhenZelenyuk, SemigroupForum,toappear.

AntoineDerighetti – Monday, 12:20–13:00Transferringconvolutionoperatorsfromlocallycompactgroupstoclosedsubgroups

Let 𝐺 bea locallycompactgroupand 𝐻 aclosedsubgroup. Wedefineandstudynaturalmapsof𝐶𝑉 (𝐺) into 𝐶𝑉 (𝐻). Applicationstothediscretizationproblemfor 𝑝-convolutionoperatorsandtotheHerz-Figà-Talamancawillbederived.

GeroFendler – Wednesday, 08:30–09:10Separable 𝐶∗-algebrasandweak∗-fixedpointpropertyoftheirBanachspacedual

Itisshowthatthedual 𝐴 ofaseparable 𝐶∗-algebra 𝐴 isdiscreteifandonlyifitsBanachspacedualhastheweak∗-fixedpointproperty. Wediscussfurthertheequivalenceofpropertiestotheuniformweak∗

Kadec-Kleepropertyof 𝐴∗ andtothecoincidenceoftheweak∗ topologywiththenormtopologyonthepurestatesof 𝐴.

ThelectureisbasedonjointworkwithProf. MichaelLeinert, InstitutfürangewandteMathematik,UniversitätHeidelberg.

MahmoudFilali – Monday, 15:30–16:10InterpolationsetsandquotientsoffunctionspacesonalocallycompactgroupI

Usinginterpolationsets, wedeviseafairlygeneralmethodforestimatingthesizeofquotientsoffunctionspacesonalocallycompactgroup. Thisunifiesapproachesfollowedbymanyotherauthors(includingus)toobtainparticularcases.

ThisisjointworkwithJorgeGalindo, UniversityofJaumeI,Spain.

HartmutFühr – Friday, 08:30–09:10Waveletcoorbittheoryinhigherdimensions

Coorbittheoryprovidesafunctional-analyticframeworkfortheconstructionandstudyofBanachframesarisingfromtheactionofanintegrablerepresentation. Thistalkisconcernedwithexistenceandba-sicpropertiesofcoorbit spacesassociated towavelet transformsarising froman irreducible, square-integrablerepresentationofasemidirectproductofthetype 𝐺 = ℝ ⋊ 𝐻 actingnaturallyon 𝐿 (ℝ ).Here 𝐻 isasuitablychosen, closedmatrixgroup.

Thetalkprovidesaunifiedandrathergeneralapproachtoasettingthatsofarhasonlybeenstudiedforveryspecialchoicesofaffinegroupactions(suchasthesimilitudegroup, ortheshearletgroup). Itestablishesthewell-definednessofascaleofBesov-typecoorbitspaces, andprovidestheexistenceofatomicdecompositionsforthesespacesintermsofsuitablychosenband-limitedSchwartzfunctions.

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Undersuitableassumptionsonthedualactionof 𝐻 I establisheasilyverifiedconcreteconditionsforframeatoms, intermsofvanishingmoments, smoothnessanddecay. Inparticular, theseresultsimplytheexistenceofcompactlysupportedsmoothatoms.

JoséE.Galé – Tuesday, 12:20–13:00Spectralsynthesisforabsolutelycontinuousfunctionsofhigherorder

Let 𝐴𝐶( ) betheconvolutionBanachalgebraon ℝ formedbyfunctions 𝑓 ∈ 𝐶( − )(ℝ) suchthat 𝑓 ( − )

isabsolutelycontinuouson ℝ and 𝑡 𝑓 ( ) ∈ 𝐿 (ℝ), endowedwiththecorrespondingnaturalnorm. Inthetalk, acharacterizationoffunctionsfofspectralsynthesisin 𝐴𝐶( ) isgiven, aswellassomeapplicationstothestudyofstabilityorergodicityoforbitsofoperatorsemigroups.

Thisispartofajointwork, stillinprogress, withMaríaMartínezandPedroJ.Miana.

JorgeGalindo – Monday, 16:20–17:00InterpolationsetsandquotientsoffunctionspacesonalocallycompactgroupII

Wewilldescribehowthemethodproposedintheprevioustalkcanbeappliedtoobtainlinearlyisomet-riccopiesof ℓ (𝜅), with 𝜅 aslargeaspossible, inthefollowingquotientspaces: 𝑊𝐴𝑃(𝐺)/(𝐶 (𝐺) ⊕𝐴𝑃(𝐺)), 𝑊𝐴𝑃(𝐺)/𝐵(𝐺), 𝐶𝐵(𝐺)/𝐿𝑈𝐶(𝐺). Wewillalsoseehowthismethodcanbeadaptedtoshowthat 𝐿 (𝑀 × 𝐻) isextremelynon-Arensregularwhen 𝑀 ismetrizableandnondiscrete. Thisfactulti-matelyleadstotheextremenon-Arensregularityof 𝐿 (𝐺) foranylocallycompactgroup.

ThisisjointworkwithMahmoudFilali, UniversityofOulu, Finland.

MahyaGhandehari – Friday, 11:30–12:10WeakamenabilityoftheFourieralgebraof 𝑎𝑥 + 𝑏 groupInhispaperpublishedin1992, Johnsonconstructedanon-zeroboundedderivationfromtheFourieralgebraoftherotationgroupin3dimensionsintoitsdual. Atthetime, thisresultwassurprisingasitwasnatural, duetothedualitybetweenthe 𝐿 -algebraandtheFourieralgebraofagroup, tosuspectthattheFourieralgebraofanamenablegroupisamenableasaBanachalgebra. Johnson’sderivationshowedthattheFourieralgebraof 𝑆𝑂 (ℝ) isnotevenweaklyamenable.

In this talk, weusepropertiesofsquare-integrablerepresentations toconstructexplicit, non-zeroderivationson theFourieralgebraof the 𝑎𝑥 + 𝑏 group, proving inparticular that this algebra isnotweaklyamenable. ThisresultextendstosemisimpleLiegroupsusingthestructuretheoryofLiegroups.

ThistalkisbasedonajointworkwithY.Choi.

SalvadorHernández – Thursday, 16:20–17:00Thedualspaceofprecompactgroups

Foratopologicalgroup 𝐺 thedualobject 𝐺 isdefinedasthesetofequivalenceclassesofirreducibleunitaryrepresentationsof 𝐺 equippedwiththeFelltopology. Itiswellknownthatif 𝐺 iscompact, 𝐺isdiscrete. Inthispaper, weinvestigatetowhatextentthisremainstrueforprecompactgroups, thatis,densesubgroupsofcompactgroups. Weshowthat: (a)if 𝐺 isanalmost-metrizableprecompactgroup,then 𝐺 isdiscrete; (b)if 𝐺 isacountablenon-metrizableprecompactgroup, then 𝐺 isnotdiscrete; (c)everynon-metrizablecompactgroupcontainsadensesubgroup 𝐺 forwhich 𝐺 isnotdiscrete. Thisextendstothenon-AbeliancasewhatwasknownforAbeliangroups. Wealsoprovethatif 𝐺 isacount-ableAbelianprecompactgroup, then 𝐺 doesnothaveKazhdan’sproperty(T),although 𝐺 isdiscreteifG ismetrizable.

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StefanAlexanderKahler – Monday, 11:30–12:10Amenabilitypropertiesforvariousclassesofpolynomialhypergroups

In contrast to the special case of a locally compact group, the 𝐿 -algebra of a general hypergroupneednotbeweaklyamenable; restrictingtocommutativehypergroups, thesituationdoesnotimprove,whereasthisrestrictionwouldguaranteeevenamenabilityinthegroupcase. Concerningpolynomialhy-pergroups, whichareverydifferentfromgroupsbutpossessarichandhighlydevelopedharmonicanaly-sis, even‘point-amenability’ofthe ℓ -algebra(i.e. thenon-existenceofnon-zeroboundedpointderiva-tionsw.r.t. symmetriccharacters, consideredasaglobalproperty)issurprisinglyrare. At‘AHA2011’,wepresentedasolutiontotheproblemwhetheramenabilityandweakamenabilityofthe ℓ -algebranecessarilycoincideormaydifferforthisclassof(commutativeanddiscrete)hypergroups; tothisend,wefirstdevelopedsomegeneralresultsandthenconsideredtheexampleofultrasphericalpolynomialsasafirstandrathereasyapplication.

Inthistalk, westudythethreeprobablymostimportantone-parametergeneralizationsoftheultras-phericalpolynomials, whicharetheclassesofJacobi, symmetricPollaczekandassociatedultrasphericalpolynomials. Wegivefullcharacterizationsforamenability, weakamenabilityandpoint-amenabilitybypreciselyspecifyingthecorrespondingparameterregions. Since–incontrasttothecommonsubclassofultrasphericalpolynomials–explicitcomputationsaredifficultorevenimpossiblefortheselargerclasses, ourstrategyrequirescombiningmethodsfromharmonicandfunctionalanalysiswithsuitableingredients fromthetheoryoforthogonalpolynomialsandtheirasymptotics, andfromthetheoryofspecialfunctions. Wealsogivesomenewgeneralresultsconcerningpoint-amenability.

EberhardKaniuth – Monday, 09:20–10:00SpectralsynthesisinFourieralgebrasofdoublecosethypergroups

Let 𝐺 bealocallycompactgroup, 𝐾 acompactsubgroupof 𝐺 and 𝐺//𝐾 theassociateddoublecosethypergroup. ThetalkwillreportonrecentspectralsynthesisresultsfortheFourieralgebra 𝐴(𝐺//𝐾) of𝐺//𝐾 and, moregenerally, theFourieralgebraofso-calledultrasphericalhypergroups. Specialemphasisisplacedonthequestionofwhenspectralsynthesisholdsfor 𝐴(𝐺//𝐾), i.e. wheneveryclosedsubsetofthespectrum 𝐺//𝐾 of 𝐴(𝐺//𝐾) isasetofsynthesis.

ThisisjointworkwithSinaDegenfeld-SchonburgandRupertLasser, TechnicalUniversityofMunich

RupertLasser – Monday, 10:10–10:50Stronglyinvariantmeansoncommutativehypergroups

EachcommutativehypergroupK possessesaninvariantmean. Forgroupsmanyconditionsareknown,whichareequivalenttotheexistenceofaninvariantmean. Thisisnotthecaseforhypergroups. WepresentseveralresultsconnectingamenabilityofK withrelatedproperties, e.g. summingsequences,Reitercondition, Følnercondition. Wegivealsoexamplesshowingthattheseconditionsarenotequiv-alentforhypergroups. MoreoverweintroducethenotionofstronglyinvariantmeansandinvestigatetherelationtoastrongReitercondition. Finallywecharacterizethosepolynomialhypergroupshavingastronglyinvariantmean.

AnthonyTo-MingLau – Wednesday, 09:20–10:00FixedpointandrelatedgeometricpropertiesontheFourierandFourier-Stieltjesalgebrasoflocallycom-pactgroups

InthistalkI shalldiscusssomerecentresultsconcerningvariousgeometricpropertiesincludingfixedpointproperties, andRadon-NikodympropertyontheFourierandFourier-Stieltjesalgebrasofalocallycompactgroup.

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JeanLudwig – Wednesday, 12:20–13:00TBA

ParasarMohanty – Tuesday, 08:30–09:10Completelybounded Λ setsincompactgroups

Sidonsetsand Λ -setshasmanyapplicationsinvarioustopicsofmathematics.Inthelastdecadecompletelybounded Λ -sets(Λ -sets), forcompactabeliangroups, arestudiedin

theframeworkofPisier’scanonicaloperatorspacestructureon 𝐿 . Inthistalkwewillseetheexistenceofnon-Sidon Λ -sets, 2 < 𝑝 < ∞, inthedualofanyinfinitecompactabeliangroup 𝐺. Also, wewillextendthedefinitionof Λ tonon-abeliancompactgroupsanddiscusscertainpropertiesanaloguetotheclassicalsettings.

ThistalkisbasedonjointworkwithKathrynE. Hare.

SerapÖztop – Tuesday, 15:30–16:10OnweightedOrliczalgebras

Let 𝐺 bealocallycompactgroupand Φ beaYoungfunction. WestudyweightedOrliczspaces 𝐿 (𝐺),denotedby 𝐿 (𝐺), whichareweighted 𝐿 (𝐺) spacesforaspecialYoungfunction |𝑥| , 1 ≤ 𝑝 < ∞.WeshowthatweightedOrliczspacesareBanachalgebraswithrespecttoconvolutionandnaturalnormundercertainconditionsonweight. Amongotherthings, themaximalidealspacesofthealgebra 𝐿 (𝐺)onanabeliangroup 𝐺 isalsodescribed.

ThistalkisbasedonjointworkwithA.Osancliol.

DetlevPoguntke – Wednesday, 10:10–10:50Kirillovtheorywithoutpolarizations

InhisseminaldoctoraldissertationKirillovconstructedabijectionbetweentheunitarydual ̂𝑁 ofasimplyconnectednilpotentLiegroup 𝑁 andtheorbitspace 𝔫∗/𝑁 (orbitmethod). Later, Bernatextendedthisresulttoexponentialgroups 𝐺.

Traditionallyoneconstructsamap 𝔤∗/𝐺 → ̂𝐺 using(Pukanszky)polarizations. Itrequiressomeworktoseetheindependenceofthechosenpolarizations.

Herewe construct a family of bijections 𝜅 ∶ ̂𝐺 → 𝔤∗/𝐺, 𝐺 an exponential Lie group, in theoppositedirection. Thisconstructiondependsonthechoiceofanabelianideal, whichinawayisamilderarbitrariness. Butstilltheindependencehastobeestablished.

Furthermore, thefamily (𝜅 ) iscanonicalinthesensethatitcanbe(uniquely)characterizedintermsofashortlistofplausibleproperties. Ifonerestrictstonilpotentgroupsonehas, withsomeextrawork,anevennicercharacterizationofthisfamily (𝜅 ).

VolkerRunde – Thursday, 09:20–10:00LeftidentitiesformeasurealgebraswithanapplicationtotheConnes-amenabilityofdualBanachalgebras

A fewyearsago, weclaimedthata[SIN]-group 𝐺 suchthat 𝒲𝒜𝒫(𝐺)∗ hasanormal, virtualdiagonalmustbecompact. Asitturnsout, thereisagapintheproof. Wewillclosethisgapandintheprocesscharacterizethoselocallycompact, semitopologicalsemigroups 𝑆 forwhichthemeasurealgebra 𝑀(𝑆)hasaleftidentity.

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PekkaSalmi – Thursday, 15:30–16:10RoughCayleygraphsofcompactlygeneratedgroups

What shouldbe the analogueofCayley graph in the caseof compactly generated locally compactgroups? ThistalktriestoanswerthequestionbyintroducingthenotionofroughCayleygraph. Givenacompactlygeneratedgroup, thereisanaturalconstructionforagraph–theroughCayleygraph–thatencodesthecoarsestructureofthegroup. ThenaturalactionofagrouponitsCayleygraphhasitsana-logueinthequasi-actionofthecompactlygeneratedgrouponitsroughCayleygraph. Ifthecompactlygeneratedgrouphasacocompactlattice, thentheCayleygraphofthelatticeistheroughCayleygraphoftheambientgroup. Thegeneraldefinitionencompassesalsotherecentnotion, introducedbyKrönandMöller, ofroughCayleygraph(alsocalledrelativeCayleygraph)ofatotallydisconnectedlocallycompactgroup. Timepermitting, weshallalsodiscusssomeapplicationstogrowthandamenability.

EbrahimSamei – Tuesday, 16:20–17:00𝑁 -hyperreflexivityofthespaceofbounded 𝑁 -cocyclefromBanachalgebras

Theconceptofhyperreflexivityhasalreadybeendefined for subspacesof 𝐵(𝑋, 𝑌), where 𝑋 and 𝑌areBanachspaces. Weextendthisconcepttothesubspacesof 𝐵 (𝑋, 𝑌) forany 𝑛 ∈ ℕ, takingintoaccountthe 𝑛-linearstructureof 𝐵 (𝑋, 𝑌). Wecallthenewconcept 𝑛−hyperreflexivity. If 𝐴 isaBanachalgebraand 𝑋 aBanach 𝐴-bimodule, wegivesufficientconditionsunderwhich 𝒵 (𝐴, 𝑋), thespaceofallbounded 𝑛-cocyclesfrom 𝐴 into 𝑋, is 𝑛-hyperreflexive. Todoso, wedefinetwonotionsrelatedtoaBanachalgebra: Thestrongproperty (𝐵) andboundedlocalunits. Thehereditarypropertiesofbothnotionsarestudied. Weapplyourapproachandshowthatthebounded 𝑛-cocyclespacesrelatedtocertainC∗-algebrasandgroupalgebrasare 𝑛-hyperreflexive.

ThisisajointworkwithJafarSoltaniFarsani.

BertSchreiber – Friday, 10:10–10:50Algebrasofmultilinearformsonhypergroups

Forlocallycompacthypergroups 𝐻 , 𝑖 = 1, 2, … , 𝑛, let 𝐶𝐵(𝐻 , … , 𝐻 ) denotetheBanachspaceofcompletelyboundedmultilinearformson 𝐶 (𝐻 ) × ⋯ × 𝐶 (𝐻 ), inthecompletelyboundednorm.𝐶𝐵(𝐻 , … , 𝐻 ) canbegiventhestructureofaBanach ∗-algebraunderamultiplicationandadjointoperationwhichagreewith theconvolutionstructureon themeasurealgebra 𝑀(𝐻 × ⋯ × 𝐻 ). Ifthe 𝐻 areallabelian, 𝐶𝐵(𝐻 , … , 𝐻 ) carriesanaturallydefinedFouriertransformasfunctionsonthespaceofsemicharacterswhichgeneralizestheFouriertransformonhypergroupmeasurealgebras. TheconstructionoftheseBanachalgebraswillbeoutlined, andvariousotheraspectsof 𝐶𝐵(𝐺 , … , 𝐺 ) willbedescribedastimepermits. ThisisjointworkwithRupertLasser.

NicoSpronk – Thursday, 10:10–10:50Onthealgebraoffunctionsgeneratedbypurepositivedefinitefunctions

Let 𝐺 bealocallycompactgroupand 𝐴(𝐺) and 𝐵(𝐺) denoteitsFourierandFourier-Stieltjesalgebras,respectively. Inhis doctoral thesis (completed atU.Alberta), Y.-H.Chengconsidered the subspace𝑎 (𝐺) in 𝐵(𝐺) generatedbypurepositivedefinitefunctionsandthesmallestclosedalgebra 𝑎(𝐺) in𝐵(𝐺) containing 𝑎 (𝐺). If 𝐺 isabelianwithdualgroup ̂𝐺, then 𝑎 (𝐺) = 𝑎(𝐺) ≅ ℓ ( ̂𝐺); whereas𝐴(𝐺) ≅ 𝐿 (𝐺). Thuswethinkof 𝑎(𝐺) asthedualanalogueof ℓ (𝐺). Chenfurtherdemonstratedmanycasesinwhich 𝑎 (𝐺) = 𝑎(𝐺).

Injointwork, recentlypublishedinMonatsh. Math., Chen, B.E.ForrestandI investigatedthestructureof 𝑎(𝐺) forHeisenberggroups ℍ , andfor SL (ℝ). Inbothcases 𝑎 (𝐺) ⊊ 𝑎(𝐺). FortheHeisenberggroups, 𝑎(ℍ ) containstheFourieralgebraofaspecial“compactificationalongthecentre”ofthegroup,andisasubalgebraofthespine, 𝐴∗(ℍ ) (asdefinedbyM.Ilieandthespeaker). InbothcasesI commentontheBanachalgebrastructureof 𝑎(𝐺), namelyoperatoramenabilityproperties.

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KeithTaylor – Friday, 09:20–10:00Imagesofwavelettransforms

Anysquare-integrablerepresentationofalocallycompactgroup 𝐺 admitsanabstractwavelettransform,theimageofwhichisasubspaceoftheFourieralgebra 𝐴(𝐺). Wewillexplorehowthisimagevariesasthegeneratingwaveletvaries.

IvanG.Todorov – Tuesday, 11:30–12:10Operatorsynthesis: unionsandproducts

Thepresenttalk, basedonjointworkwithG.Eleftherakis, willbecentredaroundthenotionofoperatorsynthesis, introducedbyW.B.Arveson in1974. A moregeneral versionof theproblemabout theoperatorsynthesisoftheunionoftwooperatorsyntheticsetswillbeformulated. Severalresultsinthisdirectionwillbediscussedandconsequencesforspectralsynthesisinlocallycompactgroupswillbegiven. Fubiniproductsand J. Kraus’property 𝑆 willbe related to theproblemabout theoperatorsynthesisoftheproductofoperatorsyntheticsets, andageneralresultaboutunionsofproductswillbediscussed. AtthecentreofourapproachlieidempotentSchurmultipliersandintersectionformulasforweak∗ closedspansofspacesofoperators.

LyudmilaTurowska – Tuesday, 10:10–10:50Compactoperatorsynthesisandspectralsynthesisinharmonicanalysis

W.Arvesoninhisfundamentalpaper(Ann. Math. 1974)discoveredaninterplaybetweeninvariantsub-spacesandoperatoralgebrastheoryandspectralsynthesisinharmonicanalysis. Thenotionofoperatorsynthesiswasproposed. Itprovidedapowerfultooltostudydifferentquestionsinharmonicanalysis, op-eratortheory, theoryofmultipliersandsoon. Inthistalkwewilldiscusssetsthatareoperatorsynthetic”modulocompactoperators”or”moduloSchattenideals”. Analogsofsuchsetsinharmonicanalysisaresubsets 𝐸 ⊂ 𝕋 (𝕋 thecirclegroup)suchthatanypseudofunctionsupportedin 𝐸 (anypseudomeasurewhoseFouriertransformisin ℓ andsupportedin 𝐸 annihilatesfunctionsin 𝐴(𝕋) vanishingon 𝐸. Wewillgivedifferentexamplesofsuchsetsanddiscussapplicationstooperatorequations.

AliÜlger – Tuesday, 09:20–10:00Whenisthequotientalgebra 𝐴/𝐼 isomorphictoasubalgebraof 𝐴∗∗?

Let 𝐴 beanarbitraryBanachalgebrawithaboundedapproximateidentity. Weconsider 𝐴∗∗ asaBanachalgebraunderoneoftheArensmultiplications. Themainresultofthistalksaysthat, foranyclosedleftideal 𝐼 of 𝐴 withaboundedrightapproximateidentity, thequotientalgebra 𝐴/𝐼 is isomorphictoaclosedsubalgebraoftheform 𝐴𝑢 of 𝐴∗∗ forsomeidempotent 𝑢 in 𝐴∗∗.

ThisresultisapartofajointworkwithProf. A.To-MingLau.

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