INVERSE CLOSED ULTRADIFFERENTIAL SUBALGEBRAS ANDREASKLOTZ ABSTRACT. Inpreviousworkwehaveshownthatclassicalapproximationtheorypro- videsmethodsforthesystematicconstructionofinverse-closedsmoothsubalgebras.Now weextendthisworktotreatinverse-closedsubalgebrasofultradifferentiableelements.In particular,CarlemanclassesandDales-Daviealgebrasaretreated. Asanapplicationthe resultofDemko,SmithandMossandJaffardontheinverseofamatrixwithexponential 2 decayisobtainedwithintheframeworkofageneraltheoryofsmoothness. 1 0 2 n a J 1. INTRODUCTION 3 We describe new methods to generate a smooth inverse-closed subalgebra of a given 1 BanachalgebraAandtocharacterizethissubalgebrabyapproximationpropertiesandby weights. RecallthatasubalgebraBofAisinverse-closedinA,if ] A everyb∈BthatisinvertibleinAisactuallyinvertibleinB. F Aprototypeofaninverse-closedsubalgebraistheWieneralgebraofabsolutelyconvergent . h Fourierseries,whichisinverse-closedinthealgebraofcontinuousfunctionsonthetorus. at AnotherexampleisthealgebraC1(T)ofcontinuouslydifferentiablefunctionsonthetorus; m theproofthatC1(T)isinverse-closedinC(T)isessentiallythequotientruleofclassical [ analysis. Manymethodsfortheconstructionofinverse-closedsubalgebrasarebasedongeneral- 1 izationsofthissimplesmoothnessprinciple. InthecontextofBanachalgebras,derivatives v 8 arereplacedbyderivations. TheLeibnizruleforderivationsimpliesthattheirdomainisa 3 Banachalgebra,andbythesymmetryofAthedomainisinverse-closedinA,see[15]. 9 A more refined concept of smoothness can be developed, if A is invariant under the 2 boundedactionofad-dimensionalautomorphismgroup. InthiscasealgebrasofBessel- . 1 Besov type can be defined, and the properties of the group action imply that the spaces 0 definedforminverse-closedsubalgebrasofA,see[20]. 2 Adifferentapproachtosmoothnessisbyapproximationusingapproximationschemes 1 adaptedtothealgebramultiplication.Thislineofresearch,initiatedbyAlmiraandLuther[2, : v 3], yields Banach algebras of approximation spaces that are inverse-closed in A, if A is i X symmetric[15]. Moreover, if A is invariant under the action of the translation group and the approx- r a imation scheme consists of the bandlimited elements of A, we obtain Jackson-Bernstein theoremsthatidentifyapproximationspacesofpolynomialorderwithBesovspaces. Alloftheabovehasbeencarriedoutintwopreviouspublications[15,20]forsmooth- ness spaces of finite order. Now we use the same principles to construct inverse-closed subalgebrasofultradifferentiableelements. Date:January17,2012. 1991MathematicsSubjectClassification. 41A65,42A10,47B47. Keywordsandphrases. Banachalgebra,inverseclosedness,spectralinvariance,Carlemanclasses,Dales- Daviealgebras,matrixalgebra,off-diagonaldecay,automorphismgroup. A.K.wassupportedbyNationalResearchNetworkS106SISEoftheAustrianScienceFoundation(FWF) andtheFWFprojectP22746N13. 1 2 ANDREASKLOTZ Classes of Carleman type are defined by growth conditions on the norms of higher derivationsinthesamewayasforfunctions,andweobtainacharacterizationofinverse- closedCarlemanclassesbyadaptingaproofofSiddiqi[31]. Ifthegrowthofthederiva- tionssatisfiesthecondition(M2’)ofKomatsu,thenanalternativedescriptionoftheCar- lemanclassesasunionofweightedspacesorapproximationspacesisavailable. Whereas Carleman algebras are inductive limits of Banach spaces we can also define BanachalgebrasofultradifferentiableelementsderivedfromagivenBanachalgebra. The constructiongeneralizesanapproachusedbyDalesandDavie[7]forfunctionsdefinedon perfectsubsetsofthecomplexplane,sowecalltheresultingBanachalgebrasDales-Davie algebras. An result of Honary and Abtahi [1] on inverse-closed Dales-Davie algebras of functionscanbeadaptedtothenoncommutativesituation(Theorem32). The general theory has applications to Banach algebras of matrices with off-diagonal decay. The formal commutator δ(A)=[X,A], X =2πiDiag((k)k∈Z), is a derivation on B((cid:96)2),anditsdomaindefinesanalgebraofmatriceswithoff-diagonaldecaythatisinverse- closedinB((cid:96)2)[15,3.4]. ThetranslationgroupactsboundedlyonB((cid:96)2)byconjugation withthemodulationoperatorM =Diag(e2πik·t) , t k∈Zd (1) χ(A)=MAM = ∑ Aˆ(k)e2πik·t fort∈Rd, t t −t k∈Zd whereAˆ(k)isthekthsidediagonalofA, (cid:40) A(l,m), l−m=k, (2) Aˆ(k)(l,m)= 0, otherwise. In [15, 20] the theory of smooth and inverse-closed subalgebras has been applied to de- scribeBanachalgebrasofmatriceswithoff-diagonaldecay. TheapproximationtheoreticcharacterizationofCarlemanCarlemanclassesofGevrey typeonB((cid:96)2)yieldsanewproofofaresultofDemko,SmithandMoss[10]. Theorem1. IfA∈B((cid:96)2)with|A(k,l)|≤Ce−γ|k−l|forconstantsC,γ>0andallk,l,∈Zd, andifA−1∈B((cid:96)2),thenthereexistC(cid:48),γ(cid:48)>0suchthat |A−1|(k,l)≤C(cid:48)e−γ(cid:48)|k−l| forallk,l∈Zd. Insomeinstances,Dales-DaviealgebrasofmatricescanbeidentifiedwithknownBa- nachalgebrasofmatrices,e.g. ifC1 consistsofmatriceswithnorm v0 (cid:107)A(cid:107) = ∑ ∑|A(l,l−k)|, C1 v0 k∈Zdl∈Zd thenD1 (C1 )isaweightedformofthisalgebraforasubmultiplicativeweightv associ- M v0 M atedtoM,seeSection4. Theorganizationofthepaperisasfollows.Firstwerecallsomefactsfromthetheoryof Banachalgebrasandreviewresultsof[15,20]oninverse-closedsubalgebrasofagivenBa- nachalgebradefinedbyderivations,automorphismgroups,andapproximationspaces. In Section3,aftertreatingC∞ classes,ultradifferentiableclassesofCarlemantypeareintro- duced,andnecessaryandsufficientconditionsontheirinverse-closednessaregiven. Car- lemanclassessatisfyingaxiom(M2’)ofKomatsuarecharacterizedbyapproximationand weightconditions.AsanapplicationwegeneralizetheresultofDemko[10]ontheinverses ofmatriceswithexponentialoff-diagonaldecay. Theresultsontheinverse-closednessof Dales-Davie algebras are treated in Section 4. In Section 5 some applications to matrix algebraswithoff-diagonaldecayaregiven. IntheappendixacombinatorialLemmaonthe iteratedquotientruleisproved. Acknowledgment: The author wants to thank Karlheinz Gro¨chenig for many helpful discussions. INVERSECLOSEDULTRADIFFERENTIALSUBALGEBRAS 3 2. PRELIMINARIES 2.1. Notation. The cardinality of a finite set A is |A|. The d-dimensional torus is Td = Rd/Zd. Thesymbol(cid:98)x(cid:99)denotesthegreatestintegersmallerorequaltotherealnumberx. PositiveconstantswillbedenotedbyC,C(cid:48),C ,c,etc.,wherethesamesymbolmightdenote 1 differentconstantsineachequation. We use the standard multi-index notation. Multi-indices are denoted by Greek letters andaread-tuplesofnonnegativeintegers. Thedegreeofxα =x1α1···xdαd is|α|=∑dj=1αj, and Dαf(x)=∂α1···∂αdf(x) is the partial derivative. The inequality β ≤α means that 1 d βj≤αj forallindices j. The p-normonCd isdenotedby|x|p=(cid:0)∑dk=1|x(k)|p(cid:1)1/p A submultiplicative weight on Zd is a positive function v:Zd →R such that v(0)= 1 and v(x+y)≤v(x)v(y) for x,y∈Zd. The standard polynomial weights are v (x)= r (1+|x|)r for r ≥0. The weighted spaces (cid:96)wp(Zd) are defined by the norm (cid:107)x(cid:107)(cid:96)wp(Zd) = (cid:107)xw(cid:107)(cid:96)p(Zd). If w=vr we will simply write (cid:107)x(cid:107)(cid:96)rp(Zd). A weight w on Zd satisfies the Gelfand,Raikov,Shilov(GRS)-conditioniflim w(nx)1/n=1forallx∈Zd. n→∞ ThecontinuousembeddingofthenormedspaceX intothenormedspaceY isdenoted asX (cid:44)→Y. TheoperatornormofaboundedlinearmappingA: X →Y is(cid:107)A(cid:107) . Inthe X→Y specialcaseofoperatorsA: (cid:96)2(Zd)→(cid:96)2(Zd)wewrite(cid:107)A(cid:107) =(cid:107)A(cid:107) or B((cid:96)2(Zd)) (cid:96)2(Zd)→(cid:96)2(Zd) simply(cid:107)A(cid:107) . B((cid:96)2) WewillconsiderBanachspaceswithequivalentnormsasequal. 2.2. InverseclosedBanachalgebras. AllBanachalgebrasareassumedtobeunital. To verifythataBanachspaceAwithnorm(cid:107)(cid:107)A isaBanachalgebraitissufficienttoprove that (cid:107)ab(cid:107)A ≤C(cid:107)a(cid:107)A(cid:107)b(cid:107)A for some constantC. A Banach algebra A is a (Banach) ∗- algebraifithasanisometricinvolution∗,i.e.,(cid:107)a∗(cid:107)A=(cid:107)a(cid:107)A foralla∈A. TheBanach ∗-algebra A is symmetric, if σA(a∗a)⊆[0,∞) for all a∈A, where σA(a) denotes the spectrumofa∈A. Thespectralradiusofa∈AisρA(a)=sup{|λ|:λ ∈σA(a)}. Definition2(Inverse-closedness). IfA⊆BareBanachalgebraswithcommonmultipli- cationandidentity,wecallAinverse-closedinB,if (3) a∈Aanda−1∈B implies a−1∈A. The relation of inverse-closedness is transitive: If A is inverse-closed in B and B is inverse-closedinC,thenAisinverse-closedinC. 2.3. Derivations. A derivation δ on a Banach algebra A with domain D = D(δ) = D(δ,A) a subspace of A is a closed linear mapping δ: D→A that satisfies the Leib- nizrule (4) δ(ab)=aδ(b)+δ(a)b foralla,b∈D. If A is a ∗-algebra, we assume that the derivation and the domain are symmetric, i.e., D=D∗ and δ(a∗)=δ(a)∗ for all a∈D. The domain is normed with the graph norm (cid:107)a(cid:107)D=(cid:107)a(cid:107)A+(cid:107)δ(a)(cid:107)A. Assume that A is a symmetric Banach algebra with a symmetric derivation δ. If 1∈ D(A), then the (symmetric) Banach algebra D(A) is inverse-closed in A. Moreover, δ satisfiesthequotientruleδ(a−1)=−a−1δ(a)a−1,see[15]. Inmoregenerality,let{δ ,···,δ }beasetofcommutingderivationsonA.Thedomain 1 d ofδ δ ...δ ,1≤r ≤disdefinedbyinductionasD(δ δ ...δ )=D(δ ,D(δ ...δ )). r1 r2 rn j r1 r2 rn r1 r2 rn Foreverymulti-indexα theoperatorδα =∏ δαk anditsdomainD(δα)arewellde- 1≤k≤d k fined. InanalogytoCα(Rd)weequipD(δα)withthenorm (cid:107)a(cid:107)D(δα)= ∑(cid:107)δβ(a)(cid:107)A. β≤α 4 ANDREASKLOTZ Sinceδ isassumedtobeaclosedoperatoronA,itfollowsthatδα isaclosedoperatoron j D(δα). IfAissymmetricand1∈D(δ ),1≤k≤d,thenD(δα)isinverse-closedinA.Further- k more,theBanachalgebraA(k)=(cid:84) D(δα)andtheFre´chetalgebraA(∞)=C∞(A)= |α|≤k (cid:84)∞ A(k)areinverse-closedinA[15,3.7]. k=0 2.4. AutomorphismGroups. A(d-parameter)automorphismgroupactingontheBanach algebraAisasetofBanachalgebraautomorphismsΨ={ψ} ofAthatsatisfyψ ψ = t t∈Rd s t ψs+t forall s,t∈Rd andareuniformlybounded,i.e. MΨ=supt∈Rd(cid:107)ψt(cid:107)A→A<∞.If Aisa∗-algebraweassumethatΨconsistsof∗-automorphisms. Wecalla∈Acontinuousandwritea∈C(A),iflim ψ(a)=a. t→0 t Fort∈Rd\{0}thegeneratorδ,definedbyδ(a)=lim ψht(a)−a isaclosedderiva- t t h→0 h tion,andthedomainD(δ,A)ofδ isthesetofalla∈Aforwhichthislimitexists. IfA t t isa∗-algebra,thenδ issymmetric. t TheactionofΨisperiodic,ifψ =ψ forallt∈Rd andall1≤ j≤d. Iftheaction t t+ej ofΨisperiodic,wecandefineFouriercoefficientsofa∈C(A)by (cid:90) aˆ(k)= ψ(A)e−2πik·tdt t Td With the group action Ψ it is possible to define the classical smoothness spaces, see, e.g. [6]. We need the Besov spaces that are defined, using the difference operators ∆k = t (ψ −id)k,bythenorm t (cid:16)(cid:90) dt (cid:17)1/p (cid:107)a(cid:107)Λrp(A)=(cid:107)a(cid:107)A+ Rd(|t|−r(cid:107)∆tka(cid:107)A)p|t|d for 1≤ p≤∞ (standard change for p=∞), r >0 and the integer k >r (every choice of k yields an equivalent norm). Algebra properties of Λp(A) are discussed in [20]. In r particular,Λp(A)isinverse-closedinAforall1≤p≤∞andallr>0,see[20,3.8]. r InasimilarspiritBesselpotentialspacesareintroducedanditcanbeshownthatthey forminverse-closedsubalgebrasofA[20]. 2.5. Approximation Spaces. An approximation scheme on the Banach algebra A is a family(Xn)n∈N0 ofclosedsubspacesofAthatsatisfyX0={0}, Xn⊆Xm forn≤m, and X ·X ⊆X , n,m∈N . If A is a ∗-algebra, we assume that 1∈X and X =X∗ for n m n+m 0 1 n n alln∈N0. Then-thapproximationerrorofa∈AbyXn isEn(a)=infx∈Xn(cid:107)a−x(cid:107)A. For 1≤p<∞andwaweightonN theapproximationspaceEp(A)consistsofalla∈Afor 0 w whichthenorm ∞ (5) (cid:107)a(cid:107)Ep =(cid:0)∑Ek(a)pw(k)p(cid:1)1/p w k=0 isfinite(standardchangeforp=∞).Ifwisastandardpolynomialweight,w=v forsome r r>0, then in order to remain consistent with the existing literature we define Ep(A)= r Ep (A). v r−1/p Algebra properties of approximation spaces are discussed in [3, 15]. In particular, in[15]thefollowingresultisproved. Proposition3. IfAisasymmetricBanachalgebrawithapproximationscheme(Xn)n∈N0 thenEp(A)isinverse-closedinA. r Approximationwithbandlimitedelements. Therelationbetweensmoothnessandapprox- imationisgivenbytheWeierstrasstheoremandJackson-Bernstein-theorems. GivenaBanachalgebrawithautomorphismgroupwesaythata∈Aisσ-bandlimited forσ>0,ifthereisaconstantCsuchthatforeverymulti-indexαtheBernsteininequality (6) (cid:107)δα(a)(cid:107)A≤C(2πσ)|α| INVERSECLOSEDULTRADIFFERENTIALSUBALGEBRAS 5 issatisfied. Anelementisbandlimited,ifitisσ-bandlimitedforsomeσ >0. Inthiscase X ={0}, X ={a∈A: aisn-bandlimited}, n∈N, is an approximation scheme for A 0 n [15,Lemma5.8]. Theorem4(Weierstrassapproximationtheorem). IfAisaBanachalgebrawithautomor- phismgroupΨ,thesetofbandlimitedelementsisdenseinC(A). Theorem5(Jackson-Bernstein-Theorem). LetAbeaBanachalgebrawithautomorphism groupΨ,andassumethatr>0and1≤ p≤∞. If(Xn)n∈N0 istheapproximationscheme ofbandlimitedelements,thenΛp(A)=Ep(A). Inparticular r r (7) a∈Λ∞(A) ifandonlyif E (a)≤Cn−r foralln>0. r n 3. ALGEBRASOFC∞ ANDULTRADIFFERENTIABLEELEMENTS 3.1. C∞ class. As in the scalar case, elements in a Banach algebra with automorphism groupthathavederivationsofallorderscanbecharacterizedbyapproximationproperties. Proposition6. IfAisaBanachalgebrawithautomorphismgroupΨ,and(Xn)n∈N0 isthe approximation scheme that consists of the bandlimited elements of A, then a∈C∞(A) if andonlyifforallr>0lim E (a)kr =0.IftheactionofΨisperiodic,thisisfurther k→∞ k equivalenttolim|k|→∞(cid:107)aˆ(k)(cid:107)A|k|r=0forallr>0. Proof. The proof works as for the scalar case. If a∈C∞(A), then a∈Λ∞ (A) for any r+1 r>0 by the properties of Besov spaces [5, 20]. Using Proposition 5 we conclude that E (a)kr+1 ≤C, and E (a)kr → 0 for k → ∞. For the other inclusion observe that (7) k k impliesa∈Λ∞(A),andfurtherδαa∈Aforallα with|α|=(cid:98)r(cid:99),againbytheinclusion r relationsofBesovspaces[5,20]. IftheactionofΨisperiodic,weusethatforallb∈X |k|∞ (cid:90) (8) aˆ(k)= (ψ(a)−ψ(b))e−2πik·tdt, t t Td andso(cid:107)aˆ(k)(cid:107)A≤C(cid:107)a−b(cid:107)A. Theinfimumofthenormoverallb∈X|k|∞ yields (9) (cid:107)aˆ(k)(cid:107)A≤CE|k|∞(a), and so Ek(a)kr →0 implies (cid:107)aˆ(k)(cid:107)Akr →0. If we assume (cid:107)aˆ(k)(cid:107)Akr →0 for all r>0 then∑k∈Zd(2πi)kaˆ(k)convergesinthenormofAtoδα(a)forallmulti-indicesα,aseach δ isclosedinD(δα),anda∈C∞(A). (cid:3) j 3.2. CarlemanClasses. Definition 7 (cf. [12, 13]). Let A be a Banach algebra with commuting derivations δ1,···,δd, and let M ={Mk}k∈N0 be a sequence of positive numbers with M0 =1. For eachr>0wesaythata∈AisintheBanachspaceC (A),ifthenorm r,M (cid:107)δα(a)(cid:107)A (cid:107)a(cid:107)Cr,M(A)= sup r|α|M α∈Nd |α| 0 isfinite. TheCarlemanClassC (A)istheunionofthespacesC (A), M r,M C (A)= (cid:91)C (A) M r.M r>0 withtheinductivelimittopology. CallMthedefiningsequenceofC (A). M IfA=(cid:84)d kerδ wecallC (A)trivial,otherwiseC (A)isnontrivial. j=1 j M M Example8. IfM =1forallk,thenC (A)consistsofther-bandlimitedelementsof k 2πr,M A. IfM =k!r forr>0thenJ (A)=C (A)istheGevrey-classoforderr. Inparticular, k r M J (A) consists of the analytic elements of A, i.e., the elements a∈A with convergent 1 expansions∑α∈Nd δαα(!a)tα forsomet>0. Thisfollowsasinthescalarcase,see,e.g.[32]. 0 Consequently,ifr≤1thenJ (A)consistsonlyofanalyticelements. r 6 ANDREASKLOTZ EquivalenceofDefiningSequences. WecalltwodefiningsequencesM,Nequivalent,M∼ N,ifC (A)=C (A). IfckN ≤M ≤CkN forallindiceskandsomeconstantsc,Cthen M N k k k M∼N. Foexample,theGevreyclassJ isalsogeneratedbythesequenceN =krk. r k Werecallastandardconstruction. LetMbeadefiningsequence. The functionassoci- atedtoMis uk (10) T (u)=sup foru>0. M M k≥0 k We call T and T equivalent and write T ∼T , if T (cu)≤T (u)≤T (Cu) for all N M N M N M N u>0 and some positive constants c,C. A function associated to the Gevrey class J is r T (u)=exp(ru1/r). M e Thelog-convexregularizationMcofthesequenceM=(Mk)k∈N0 isthelargestlogarith- micallyconvexsequencesmallerthanM. Proposition9([21,26,27]). Thelog-convexregularizationofMsatisfies uk (11) Mc=sup . k T (u) u>0 M Moreover,TMc =TM andMcc=Mc. Wewillalsoneedthefollowingsimplefactsaboutlog-convexsequences. Lemma 10 ([22, 27]). (1) For all k,l ∈N the sequence M satisfies McMc ≤Mc . (2) 0 k l k+l Thesequence(Mc)1/k isincreasing. k Ifδ ,...,δ aregeneratorsofanautomorphismgroupwecangiveaweaktypecharac- 1 d terizationofC (A). r,M Lemma 11. Assume that the automorphism group Ψ acts on A. An element a∈A is in Cr,M(A)ifandonlyifGa(cid:48),a(t)=(cid:104)a(cid:48),ψt(a)(cid:105),isinCr,M(L∞(Rd)))foralla(cid:48)∈A(cid:48), thedual ofA. Inthiscase(cid:107)a(cid:107)Cr,M(A)(cid:16)sup(cid:107)a(cid:48)(cid:107)A(cid:48)≤1(cid:107)Ga(cid:48),a(cid:107)Cr,M(L∞(Rd)). Proof. Therequiredequivalencefollowsimmediatelyfrom (cid:107)δαa(cid:107)A≤ sup (cid:107)Ga(cid:48),δαa(cid:107)L∞(Rd)= sup (cid:107)DαGa(cid:48),a(cid:107)L∞(Rd)≤MΨ(cid:107)δαa(cid:107)A (cid:107)a(cid:48)(cid:107)A(cid:48)≤1 (cid:107)a(cid:48)(cid:107)A(cid:48)≤1 bydividingwithrαM|α|andtakingsupremaoverallα. TheequalityGa(cid:48),δαa=DαGa(cid:48),ais aconsequenceofelementarypropertiesofGa(cid:48),a[15,Lemma3.20]. (cid:3) Proposition 12 ([14, 27]). Assume that the automorphism group Ψ acts on A, and let M be a defining sequence for C (A). If limM1/k = 0, then C (A) is trivial. If 0 < M k M limM1/k<∞,thenC (A)istheclassofbandlimitedelements. Iflim M1/k=∞,and k M k→∞ k (Mc)1/k (cid:16)(Nc)1/k, thenC (A)=C (A). Moreover, the last condition is equivalent to k k M N T ∼T . M N Proof. Asa∈CM(A)ifandonlyifGa(cid:48),a∈CM(Rd)foralla(cid:48)∈A(cid:48),theconditionsfollow from[27,6.5.III]byaweaktypeargument. Thestatementgiventhereisforfunctionson the real line, but it remains true for functions on Rd. In the proof one has to replace the Kolmogorovinequality[26,6.3.III]bytheCartan-Gornyestimates[27,(6.4.5)]. Theycan beverifiedforfunctionsonRd aswell(see[22,IV.E.,Problem7]). Theequivalencebetweencondition(12)andT ∼T followsdirectlyfromthedefini- M N tionofequivalentassociatedfunctions. (cid:3) Corollary13. Inparticular,weobtainthatCM(A)=CMc(A). INVERSECLOSEDULTRADIFFERENTIALSUBALGEBRAS 7 AlgebrapropertiesofCarlemanclasses. InthissectionweverifythatC (A)isaninverse- M closedsubalgebraofA,ifCM(A)=CMc(A).IfAhasanautomorphismgroupthisfollows formProposition12. Proposition14. EachCarlemanclassC (A)isanalgebra. M Proof. TheproofisasinKomatsu[21]. (cid:3) We need the following technical term: A sequence (uk)k∈N0 of positive numbers is almostincreasing,ifu ≤Cu forallk<landaconstantC>0. k l Lemma15. AssumethatthedefiningsequenceMsatisfiesM=Mc.Thesequence(M /k!)1/k k isalmostincreasingifandonlyifthereisaC>0suchthatforalll∈Nandallindices jk,k=1,...,lwith j=∑lk=1 jk l M M (12) ∏ jk ≤Cj j. j ! j! k=1 k Proof. Assumingthat(M /k!)1/k isalmostincreasingweobtain k M (cid:16)M (cid:17)k/j jk ≤Ck j , j ! j! k andthe“if”partfollowsbymultiplyingtheseestimates. Fortheotherimplicationobserve firstthatStirling’sformulaimpliesthat(M /k!)1/k isalmostincreasingifandonlyifthere k isaC(cid:48)>0suchthat 1/k 1/l M M (13) k ≤C(cid:48) l forallk<l. k l Ifl=rkforanintegerrthen(12)implies 1/k 1/rk M M k ≤C(cid:48) rk . k rk 1/k Ifrk<l<(r+1)k,weuseaninterpolationargument. ByLemma10thesequenceM k isincreasingink,so 1/l 1/kr 1/k M M kr kr 1 M l ≥ kr ≥ k l kr l l C(cid:48) k bywhathasbeenjustproved. Butthisimplies 1/k 1/l 1/l M l M M k ≤C(cid:48) l ≤2C l . (cid:3) k kr l l (cid:3) Remark. (a) For the proof of the direct implication we do not need the condition that M=Mc. (b)Equation(13)impliesthatM1/k→∞if(M /k!)1/k isalmostincreasing. k k Theorem16([25,31]). IfCM(A)=CMc(A)andif(Mk/k!)1/k isalmostincreasing,then C (A)isinverse-closedinA. M Weadaptthemethodof[31]tothenoncommutativesituation. Weneedaformofthe iteratedquotientrulethatwillbeprovedintheappendix. Lemma17. LetE={1,...,d}andδ ,...,δ bederivationsthatsatisfythequotientrule 1 d δ (a−1)=−a−1δ (a)a−1 forall j∈E. j j Foreveryk∈NandeverytupleB=(b ,...,b )∈Eksetδ (a)=δ ...δ (a).Definethe 1 k B b1 bk orderedpartitionsofBintomnonemptysubtuplesas P(B,m)={(B ,···,B ): B=(B ,···,B ),B (cid:54)=0/ foralli}. 1 m 1 m i 8 ANDREASKLOTZ Then |B| (cid:16) m (cid:17) (14) δ (a−1)= ∑(−1)m ∑ ∏a−1δ (a) a−1. B Bi m=1 (Bi)1≤i≤m∈P(B,m) j=1 ProofofTheorem16. Assume that |α|=k. With the notation of Lemma 17 there is a k-tuple B with |B| = k such that δα = δ . As a ∈C (A), we know that (cid:107)δ (a)(cid:107) ≤ B M Bi Ar|Bi|M for some constants C,r >0. The number of (nonempty) partitions of B into |Bi| sets(B) ∈P(B,m)ofcardinalityk is(cid:0) k (cid:1),soweobtainthenormestimate i 1≤i≤m i k1,...,km k (cid:18) k (cid:19)(cid:16) m (cid:17) (cid:107)δα(a−1)(cid:107)A≤ ∑(cid:107)a−1(cid:107)mA+1 ∑ k ,...,k ∏CrkjMkj m=1 k1+···km=k 1 m j=1 kj≥1 (15) k (cid:18) k (cid:19)(cid:16) m (cid:17) =rk ∑(cid:107)a−1(cid:107)m+1Cm ∑ ∏M A k ,...,k kj m=1 k1+···km=k 1 m j=1 kj≥1 Using(12)weobtain k (cid:107)δα(a−1)(cid:107)A≤rkCkMk ∑(cid:107)a−1(cid:107)mA+1Am ∑ 1 m=1 k1+···km=k kj≥1 k (cid:18)k−1(cid:19) =rkCkM ∑(cid:107)a−1(cid:107)m+1Am ≤CkM , k A m−1 1 k m=1 andthisiswhatwewantedtoshow. (cid:3) Corollary18. TheGevreyclassesJ (A)areinverse-closedinA,ifr≥1. r 3.3. Description by Weighted and Approximation Spaces. In this section we charac- terizeCarlemanclassesbyunionsofweightedspacesandofapproximationspaces,ifthe actionoftheautomorphismgroupΨontheBanachalgebraAisperiodicandthesequence MsatisfiesKomatsu’scondition(M2’). Definition 19. Let A be a Banach ∗- algebra with periodic automorphism group Ψ. For 1≤p≤∞andaweightvonZd weintroducetheweightedspacesspaces (cid:16) (cid:17)1/p Cvp(A)={a∈A: (cid:107)a(cid:107)Cvp(A)= ∑ (cid:107)aˆ(k)(cid:107)Ap v(k)p <∞} k∈Zd withtheobviousmodificationfor p=∞,whereaˆ(k)aretheFouriercoefficientsofa(see Section2.4). Remark. If (cid:96)p(Zd) is a Banach algebra with respect to convolution, then Cp(A) is an v v inverse-closedsubalgebraofA.Theproofisastraightforwardadaptionoftheproofof[17, Theorem3.2],basedonthetheoremofBochner-Philips. Lemma20. IfMisadefiningsequenceforC (A),r>0,andT (k)=T (2π|k|∞),then M r,M M r C1 (A)⊆C (A)⊆C∞ (A). Tr,M r,M Tr,M Proof. Assume first that a∈C (A). Let j be an index such that |k |=|k| . Then, by r,M j ∞ l-foldpartialintegration (cid:90) 1 (cid:90) aˆ(k)= ψ(a)e−2πik·tdt= ψ(δl a)e−2πik·tdt. Td t (2πikj)l Td t ej Takingnormsweobtain rlM l (cid:107)aˆ(k)(cid:107)A≤C . (2π|k| )l ∞ INVERSECLOSEDULTRADIFFERENTIALSUBALGEBRAS 9 This relation is valid for all l ∈ N , and therefore also for the infimum, which yields 0 (cid:107)aˆ(k)(cid:107)A≤C/Tr,M(k),ora∈C∞Tr,M(A). Fortheconverseinclusionassumethata∈C1Tr,M, i.e., ∑k∈Zd(cid:107)aˆ(k)(cid:107)ATr,M(k)<∞. For α ∈Nd weestimatethenormofδα(a)by 0 (cid:107)δα(a)(cid:107)A≤ ∑ (cid:107)δα(aˆ(k))(cid:107)A≤ ∑ (2π|k|∞)|α|(cid:107)aˆ(k)(cid:107)A k∈Zd k∈Zd (2π|k| )|α| u|α| ≤(cid:107)a(cid:107) sup ∞ ≤(cid:107)a(cid:107) sup C1Tr,M(A)k∈Zd Tr,M(k) C1Tr,M(A)u>0TM(u/r) =(cid:107)a(cid:107) r|α|Mc , C1 (A) |α| Tr,M thelastequalityby(11),andsoa∈Cr,Mc(A)=Cr,M(A). (cid:3) Corollary21. WiththenotationofLemma20, (cid:91)C1 (A)(cid:44)→C (A)(cid:44)→ (cid:91)C∞ (A), Tr,M M Tr,M r>0 r>0 whereallspacesareequippedwiththeirnaturalinductivelimittopologies. InordertoobtainequalityinCorollary21weimposecondition(M2’)ofKomatsu[21]. Lemma22([28],[21,Prop. 3.4]). IfM isadefiningsequence,thefollowingareequiva- lent: (M2’) Thereexistconstantsc>0,h>1suchthatforallk∈N. (1) T (hr)≥CrT (r)forallr>0. M M (2) TM(λr) ≥exp(cid:0)log(r/c)logλ/logh(cid:1)forallr,λ >0. TM(r) Example23. ThedefiningsequencefortheGevrey-classJ ,r>0satisfies(M2’). r Proposition 24. If A is a Banach algebra with periodic automorphism group and if the definingsequencesatisfies(M2’) C (A)= (cid:91)C1 (A)= (cid:91)C∞ (A)= (cid:91)E∞ (A) M Tr,M Tr,M Tr,M r>0 r>0 r>0 withtheinterpretationthatthesealgebrasaretopologicallyisomorphic. Proof. (see,e.g.,[23])Wesplittheproofintoseveralparts. Byknownpropertiesofinduc- tivelimits[11]itissufficienttoprovethefollowinginclusions. (1) C1 (A)(cid:44)→C∞ (A): ThisfollowsfromLemma20. Tr,M Tr,M (2) C∞ (A)(cid:44)→C1 (A)forsomeλ >0: UsingLemma22,(2),weobtaintheestimate Tλr,M Tr,M ∑ (cid:107)aˆ(k)(cid:107)ATr,M(|k|∞)≤ ∑ (cid:107)aˆ(k)(cid:107)ATr,M(λ|k|∞)exp(cid:0)−log(|kc|∞)llooggλh(cid:1) k∈Zd k∈Zd (cid:16)|k| (cid:17)−logλ/logh ≤ sup(cid:107)aˆ(k)(cid:107)ATr,M(λ|k|∞) ∑ c∞ . k∈Zd k∈Zd Ifwechooseλ suchthatlogλ/logh>d,thesumontherighthandsideoftheinequality isconvergent. (3) E∞ (A)(cid:44)→C∞ (A)followsfrom(9). Tr,M Tr,M (4) C∞ (A)(cid:44)→E∞ (A)forsomeκ>0willbeverifiedwithoutlossofgeneralityforr= Tr,M Tκr,M 2π.Theapproximationerrorofa∈C∞T2π,M(A)canbeestimatedbyEl(a)≤∑|k|∞≥l(cid:107)aˆ(k)(cid:107)A≤ (cid:107)a(cid:107)C∞T2π,M∑|k|∞≥lT2−π1,M(|k|).AsT2π,M(u)=TM(u)isincreasing,wecanreplacethesumby 10 ANDREASKLOTZ anintegral. Weassumethatlissolargethat log(l/c) >2d,andobtain logh (cid:90) (cid:90) ∞ 1 ∑ T−1(|k|)≤ T−1(|k|)dk≤C(cid:48) ud−1du M M T (u) |k|∞≥l |k|∞≥l l M =C(cid:48)ld(cid:90) ∞ 1 vd−1dv≤C(cid:48) ld (cid:90) ∞vd−1e−log(llo/gc)hlogvdv T (lv) T (l) 1 M M 1 =C(cid:48) ld (cid:90) ∞vd−1−lolgo(gl/hc)dv=C(cid:48) ld 1 ≤C(cid:48) ld d−1, TM(l) 1 TM(l) log(l/c)−d TM(l) logh where we have used(2) of Lemma 22 in the second line. Applying (1) of Lemma 22 d times we obtain T (l)≥CldT(l/hd) with a constant C independent of l. Substituting M thisinthecurrentestimateweobtainEl(a)≤C(cid:107)a(cid:107)C∞T2π,MTM−1(l/hd),andtheconstantCis independentofl. So(cid:107)a(cid:107)E∞ ≤C(cid:107)a(cid:107)C∞ ,andthatiswhatwewantedtoshow. (cid:3) T2πhd,M T2π,M Foramoregeneraldiscussionofapproximationresultssee[29]. 4. DALES-DAVIEALGEBRAS InthissectionweassumethatΨisaoneparameterautomorphismgroupactingonthe BanachalgebraA. We define Banach algebras that are determined by growth conditions on the sequence ((cid:107)δk(a)(cid:107)A)k∈N0 by adapting a similar construction introduced in [7] for scalar functions inthecomplexplane. Definition 25. Let M=(M ) be an algebra sequence, that is, a sequence of positive k k≥0 numberswithM =1and Mk+l ≥ MkMl forallk,l∈N .TheDales-DaviealgebraD1 (A) 0 (k+l)! k! l! 0 M consistsoftheelementsa∈Awithfinitenorm ∞ (cid:107)a(cid:107)D1(A)= ∑Mk−1(cid:107)δk(a)(cid:107)A. M k=0 ThespaceD1 (A)isindeedaBanachalgebra. ThiswillbeprovedinProposition28. M Example26. RecallthatthenormofaderivationonC1v0(A)(seeDefinition19)is(cid:107)δk(a)(cid:107)C1v0(A)= ∑l∈Z(cid:107)aˆ(l)(cid:107)A(2π|l|)k.ForthenormonD1M(C1v0(A))weobtain ∞ ∞ (2π|l|)k (cid:107)a(cid:107)D1M(A)=k∑=0Mk−1l∑∈Z(cid:107)aˆ(l)(cid:107)A(2π|l|)k=l∑∈Z(cid:107)aˆ(l)(cid:107)Ak∑=0 Mk . Letusdefinetheweightv associatedtoMby M ∞ (2π|l|)k (16) v (l)= ∑ . M M k=0 k Thus we obtain D1 (C1 (A))=C1 (A). For this example we have established a relation M v0 vM betweenthegrowthofderivativesandweights. Werecallsomenotionsfromcomplexanalysis(see, e.g. [24]). Foranentirefunction f letM (r)=sup |f(x)|. Theorder of f isρ =lim loglogM (r)/logr. If f has f |x|≤r f r→∞ f finiteorderρf, thetypeof f isσf =limr→∞logMf(r)r−ρf.Ifσf =0, wesaythat f has minimaltype. Inthefollowinglemmasomebasicpropertiesofv arecollected. M Lemma27. (1) IfMisanalgebrasequence,thenv (|k|)issubmultiplicative. M