Compactness properties of a locally compact group and analytic semigroups in the group algebra

Publicacions Ma emá iques, Vol 35 (1991, 131-140 . COMPACTNESS PROPERTIES OF A LOCALLY COMPACT GROUP AND ANALYTIC SEMIGROUPS IN THE GROUP ALGEBRA* JOSÉ E . GALÉ 0 . In oduc ion Le G be a locally compac g oup wi h le Haa measu e M, and le L 1 (G) be he con olu ion Banach algeb a o in eg able unc ions on G wi h espec o p . In his pape we a e conce ned wi h he in es iga ion o he s uc u e o G in e ms o analy ic semig oups in L 1 (G) . In his con ex , a well known esul is he heo em o A . Sinclai which says ha G is me izable i and only i he e is an analy ic semig oup (a z )Rez>0 in L'(G) such ha [al *L 1 (G)] - = [L'(G) * a l ] - =L 1 (G) ([26, p .41]) . O he esul s ela ing he beha io o analy ic semig oups on hal -discs o ce ain linea p ope ies o L l (G) ha e been ob ained in [6] . The ollowing p oblem was aised by J . Es e le in [3, p .460] . Ques ion E . I L'(G) has a non-ze o analy ic semig oup (a z ) R, z>0 which is bounded on he line {Re z = 1} , does i ollow ha G is compac ? Wha abou he con e se ? Also, A . Sinclai asked in [26] o he ela ionships be ween he a e o g ow h on e ical lines o analy ic semig oups in L'(G), and compac ness p ope ies o G . Recall ha G is said o be o polynomial g ow h i o e e y compac neighbo hood K o G he e is a nonnega i e in ege m such ha p(I1 n ) = O(n'n) as n -> oo ([23, p .280]) . I he minimal m o which (*) is sa is ied is he same o all compac s K hen his numbe is called he deg ee o g ow h in G . Such a numbe exis s i , o example, G is connec ed o G is compac (m = 0 in his *This esea ch was s a ed, and a pa o he esul s we e ob ained, du ing he a endence o he au ho a "Semes e on Au oma ic Con inui y and Banach Algeb as" in he Uni e si y o Leeds, June 1987, suppo ed by he U . K . Science and Enginee ing Resea ch Council . The esea ch has also been pa ially suppo ed by he Spanish DGICYT, P oyec o PS87-0059, and he Caja de Aho os de la Inmaculada, Za agoza, Ayuda CB 1188 13 2  J .E . GALÉ las case) . Then, does G ha e polynomial g ow h i and only i he e is an analy ic semig oup (a z ) R, z>0 in L 1 (G) and a nonnega i e in ege N such ha llal+iy11 = O(1 yj') as ly1 --> oo, y E IR? (See [26, p .81]) . Recen ly, T . Py lik has shown ha i an elemen b o a Banach algeb a A is such ha ¡le¡,,¡¡ = O(1 lw), as I i --, oo ( ER) o some nonnega i e in ege N, hen he e exis s an analy ic semig oup (a z )Rez>0 in A such ha ~jal+`y11 = O(1yj'+1) as jy1 -> oo (y E R) ([25]) . J . Dixmie had p o ed in [8, p .17] ha , o a g oup G o polynomial g ow h, such an elemen b exis s in L'(G) and, u he mo e, he numbe N associa ed o b can be chosen as N = m + 1, i m sa is ies condi ion (*) . Thus Sinclai 's ques ion has an a i ma i e answe in one way, bu as a as we know no hing else is known abou he con e se . No ice ha he i s pa o Ques ion E can be iewed as a pa icula case o his con e se . He e, we p esen a i s s age in he s udy o Ques ion E . We p o e ha , i G is a cen al g oup, hen L 1 (G) has a non-ze o analy ic semig oup (a z )Rez>0 such ha {a l+ 'y : y E R} is ela i ely weakly compac i and only i G con ains a compac , open subg oup . I , u he mo e, G is also connec ed hen G mus be compac . We ob ain his heo em as a consequence o he ollowing one, which we p o e in a mo e abs ac se ing : i (a z ) R, z>0 is an analy ic semig oup in a Banach algeb a A such ha {a l+ 'y : yE R} is ela i ely weakly compac , hen he spec um a(al) is a mos coun able . The p oo o his las esul which we gi e he e is based upon elemen a y p ope ies o he weakly almos pe iodic unc ions on he eal line (see [9],[10]) . O he essen ial ing edien s in ou gene al a gumen a e he s uc u al p ope ies o cen al g oups ([17],[18]) . We also show ha o e e y locally compac g oup G con aining a compac and open subg oup, L 1 (G) has an analy ic semig oup which is no m compac on {Re z = 1} . Then, i G is cen al (in his case G is o polynomial g ow h) i s s uc u e allows us o imp o e he numbe N appea ing in Sinclai 's ques ion, wi h espec o he one we would ge om he heo ems gi en by Dixmie and Py lik in [8] and [25], espec i ely . The pape is di ided in o h ee sec ions . In he i s one, we collec de ini ions and basic p ope ies o weakly almos pe iodic unc ions, analy ic semig oups in Banach algeb as, and cen al g oups, ha we shall need in he wo o he sec ions . Sec ion 2 is de o ed o p o e he esul men ioned abo e o analy ic semig oups in Banach algeb as . In sec ion 3 we discuss Ques ion E, and we gi e he e he p ecedings esul s abou g oup algeb as . Acknowledgemen s . We wish o hank B . Aupe i , O . Blasco, J . C . Can- deal, B . Cua e o, J . Es e le, N . G onbwk, T . Rans o d o help ul con e sa ions o commen s abou subjec s o his pape , and A . Hulanicki and T . Py lik o ep in s . 1 . P elimina ies Le IR be he eal line, and deno e by Cb(IR) he Banach algeb a o bounded and con inuous unc ions on R . I E C6^ and E R we de ine E Cb(R) ANALYTIC SEMIGROUPS IN THE GROUP ALGEBRA  13 3 by (s) = (s -}- ) (s E IR) . A unc ion E Cb(R) is said o be weakly almos pe iodic i { : E IR} is ela i ely compac in he weak opology o Cb(IR) . This no ion, which was in oduced by W . Ebe lein in [9], is a gene aliza ion o he well known almos pe iodic unc ions o H . Boh ([4]) . Deno e by W(R) he class o weakly almos pe iodic unc ions on R . E e y con inuous unc ion on R which is null a in ini y, and E e y posi i e de ini e unc ion on R belong o W(R) . The class W(R) enjoys in e es ing p ope ies . We shall only need he ollowing mean e godic heo em ([9, Theo em 15 .2]) Theo em 1 .1 . I E W(R) and A E R, he mean alue M, ( ) = lim ( 1 / ) ( ) e -'a d o exis s . Mo eo e , {A ER : Ma( ) :~ 0} is a mos coun able . Le H be he open igh -hand hal -plane o C, and le A be a Banach algeb a . An analy ic semig oup in A is an analy ic unc ion z --> a', H -> A such ha a x +w = a z a w (z, w E H) . We iden i y an analy ic semig oup wi h i s ange, de- no ed by (a z )Rez>o . Th oughou , whene e we conside analy ic semig oups, we shall w i e a ins ead o a l . We a e in e es ed in a non-ze o analy ic semi- g oup (a z )R,Z>o wi h sup . . R j1a l +' 11 < oo . In his case, i is well known ha i A is commu a i e, and ~¿A is he cha ac e space o A, hen o each cp E q)A he e is a ER such ha W(az) = e"(z E H) (see [26, p .85]) . Also, he Banach algeb a gene a ed by he semig oup equals he Banach algeb a polynomially gene a ed by a ([14, p .379]) . Le G be a locally compac g oup, and le Z(G) be i s cen e, ha is, Z(G) _ { E G : s = s o all s E G} . The g oup G is said o be cen al i he quo ien g oup G/Z(G) is compac . Clea ly, all compac g oups and all locally compac abelian g oups a e cen al . "In ac , many o he ea u es common o he e wo classes appea in hei na u al se ing only when iewed as being cha ac e is ic o cen al g oups . In addi ion he e a e s ong indica ions ha he class o cen al g oups ma ks he u mos deg ee o gene ali y in which all he e ea u es a e s ill p esen " ([18, p .361]) . We shall use he ollowing esul s in sec ion 3 Theo em 1 .2 . Leí G be a cen al g oup . Then he e exisis n >_ 0 such ha G - Rn x Go whe e G o is a locally compac g oup con aining a compac , open, no mal, subg oup K such ha G o /K is abelian . A p oo o his is in [17, p .331] . The class o all connec ed, cen al g oups admi s se e al in e es ing, and equi alen , desc ip ions ([21, p .14,15]) . We shall only need he nex one Theo em 1 .3 . I G is a connec ed cen al g oup hen G = Rn x K, whe e n> 0 and K is a compac (connec ed) g oup . 13 4  J .E . GALÉ The co esponding esul s on he s uc u e o locally compac abelian g oups can be seen in [20 I] . We inish his sec ion wi h a simple obse a ion Lemma 1 .4 . I G is cen al, hen G is o polynomial g ow h . P oo .. The cen e Z(G) is o polynomial g ow h because i is commu a i e ([12, Theo em 7 .8]) . Besides ha , G/Z(G) is compac by de ini ion . Then he esul ollows om [22, p . 167,168] . 2 . Analy ic semig oups which a e ela i ely weakly compac on e ical lines The nex p oposi ion is a c ucial s ep in ou easoning P oposi ion 2 .1 . Le A be a Banach algeb a, and le A* be i s dual Banach space . Suppose ha he e is an analy ic semig oup (a z )Rex>0 in A such ha {al+ 4 y : y E R} is ela i ely weakly compac . Then (i) Fo e e y 9 E A*, he unc ion y --> cp(a2+'y), R -+ C is weakly almos pe iodic . (ii) The weak limii lm (1/ )  a 2+iy e -iñy d y io l  0 exis s in A o each AE R, and so i de ines an elemen ba o A . (iii) The se { . E R : ba :~ 0} is a mos coun able . P o- - (i) Pu (y) = cp(a 2 +'y) (y E R) and T(b)(y) = cp(a1+`yb) (b E A, y E R) . Then T is a con inuous linea unc ion om A o Cb(R), which is also con inuous wi h espec o he co esponding weak opologies, and we ha e ha , = T(al+i ) ( E R) . I ollows ha E W(R) . (ii) Fo each A E 9I, he unc ion y -> a 2 +'ye - 'ay, R --> A is con inuous and hen i is Bochne in eg able o e each in e al [0, ] ( E R), and i s ange Ao is sepa able . Pu b,, = (1/ ) a 2+ ye -iay dy (A E R) 0 By (i), he e exis s ba E A** such ha (ba, cp) = lim _ . cp(b a) . Bu , o each E R, ba belongs o he (weakly) closed con ex hull Ca o he se {a 2 +'ye - ay y E R} CA o . Then Ca is a weakly compac subse o Ao, by he K ein's Theo em ([11, p .553]) . Since ba belongs o CA , we ob ain ha ba E Ao . (iii) Since Ao is sepa able we can choose a sequence (cPn)n>i in A* such ha ¡lb¡¡ = sup e>1 Icp n (b)j o e e y b E Ao . Le X  be a coun able subse o ANALYTIC SEMIGROUPS IN THE GROUP ALGEBRA  13 5 R  such ha cp,,(ba) = 0 i A q X,  Se X = Un>1 X  .  Then, i A 1 X, ¡lba¡¡ = sup a>, 1Wn(ba)j = 0 . The a gumen s which we ha e conside ed o p o e pa s (i), (ii), a ad (iii) o he abo e p oposi ion ha e been aken om [16, p .82,83], [16, p .84], and [1, p .43], espec i ely . They ha e been w i en in i s p esen o m in o de o gi e a mo e elemen a y p oo o such p ope ies, in his con ex . The ollowing heo em is he key poin o his pape . Theo em 2 .2 . Le¡ A be a Banach algeb a which con ains a a analy ic semi- g oup (a z )R, z>o such ha {al+`y : y E R} is ela i ely weakly compac . Then he spec um (a) o a is a mos coun able . P oo .. We can assume ha A is gene a ed as Banach algeb a by he semi- g oup (a')R,=>o . By P oposi ion 2 .1, {A E R : ba :~ 0} is a mos coun able, whe e b>, = weak - li~ (1/ )  ~ . a2+'ye-'ay dy (A E R) , J 0 I cp is a cha ac e o A we ha e ha cp(a = ) = eaZ (z E H), o some a E R . Then, Thus, u(a) is a mos coun able . b lim (1/ ) 0 = li m (1/ ) I e 2a e ¡ay e -iay d y = e a ~ 0 . o A mo e o mal p oo o his ac is also a ailable by using some heo ems o a compac opological semig oups (see [15] o his opic) . Such a p oo elies hea ily o a ideas coming om he basic heo y o egula quasimul iplie s o Banach algeb as, which can be seen in [13] . The p oo , and mo e in o ma ion abou commu a i e Banach algeb as gene a ed by semig oups as in Theo em 2 .2, a e gi en in [27] . Co olla y 2 .3 . Le A be a non-uni al, commu a i e, semisimple, Banach algeb a . Suppose ha i s cha ac e space <DA is connec ed in he Gel and opol- ogy . I (a z )Rez>o is a a analy ic semig oup in A such ha {a 1 +'y : yE R} is ela i ely weakly compac hen a = 0 . P oo . By he hypo hesis o a A, IA is non compac ([24, p .154]) . Also, he unc ion ep -4 cp(a), OPA -+ C is con inuous and null a in ini y, whence 0 E {cp E OPA} - . Mo eo e , o each cp E O ¿A he e is a E 6B wi h W(a) = ea . Thus he spec um u(a) o a is a connec ed subse o 1B . Bu u(a) is also coun able (Theo em 2 .2), and he e o e a(a) = 0 . 13 6  J .E . GALÉ Analogous a gumen s o hose o he p oo o his co olla y will be used o p o e Theo em 3 .3 below . Rema k . The condi ion on he semig oup in he abo e esul s canno be e- placed by he weake assump ion ha sup yeR ¡lal+'yll < oo : le A = Co([0,1]), he Banach algeb a o con inuous unc ions on [0,1] which a e null a = 0 . Se az( ) = ez lo g , whe e log is he b anch o he loga i hm which is eal- alued on he posi i e eal numbe s . Then (a')R, z>0 is an analy ic semig oup in A such ha sup . .R llal+'y11 < oo, bu a(a) = [0,1] . 3 . G oup algeb as We begin by gi ing a esul conce ning he second pa o Ques ion E . Le G be a locally compac g oup, and le Go be an open subg oup o G . I is s aigh o wa d o e i y ha he mapping 0 : -> O( ), L 1 (Go) --> V(G) gi en by O( )( ) = ( ) ( E G o ), and O( )( ) = 0 ( E G - Go) is a con inuous algeb a homomo phism . On he o he hand, i G is a compac g oup hen he e a e sequences (cpn)n>1 o unc ions on G such ha (cpn)n>1 CL 1 (G), 1, W n* CP m = 5 nm CP n (n, m > 1) ([2011, p .14]) . P oposi ion 3 .1 . Le G be a locally compac g oup con aining a compac , open subg oup . Then L l (G) has a non-ze o analy ic semig oup (az)R, z>0 such ha {a 1+¡y : y E IR} is no m compac . P oo .. Le us assume ha G is compac and ake a sequence (IP n ) n > 1 as be o e he p oposi ion . Recall ha H is he open igh hal plane . Pu az( ) = En>1 e - nzW n ( ) ( E G, zE H) . Take a> 0 and z E H such ha Re z >_ a . Then ¡le -nz W n il <_ e-n Re z < e-,n . Hence he unc ion z --> az,H -> Ll(G) is analy ic, and i is also clea ha az * a' ° = a z +' (z, w E H) . Fu he mo e, since a l +' y = a l+i(y+21 ) ( y E R) he con inui y o he mapping y -> a l+4y , R -> L 1 (G) implies he compac ness o {al+'y : y E [0, 27 ]} Conside now he gene al case and le Go be a compac , open, subg oup o G . Le (a z )Re z>0 be he semig oup in L 1 (Go) gi en be o e . Pu V = O(az) (z E H), whe e 0 : L'(G0) -> L1(G) is as be o e . Clea ly, (V)R,,>0 sa is ies he equi ed condi ions . As P oposi ion 3 .1 shows, he e exis non compac locally compac g oups G such ha L l (G) has non-ze o analy ic semig oups which a e no only bounded, bu compac , on {Re z = 1} . We can gi e a pa ial con e se o he abo e p oposi ion . Bu , be o e doing his, le us say a his poin some wo ds abou he ques ion sugges ed by A . Sinclai (see In oduc ion) . As al eady said, i G is a g oup o polynomial g ow h, and i m is a nonnega i e in ege such ha p(K") = O(nm) as n -> oa ANALYTIC SEMIGROUPS IN THE GROUP ALGEBRA  13 7 o some compac neighbo hood K in G, hen he esul s gi en by Dixmie and Py lik in [8] and [25] imply ha he e is a non-ze o analy ic semig oup (a z )Rez>0 in L'(G) such ha j1a l +' yl1 = O(1yj') as ¡y¡ -> oo (y E H), wi h N = m + 2 . I G is cen al, we can imp o e subs an ially he ela ionship be ween m and he g ow h o j1a l +' y11 a he in ini y, by using he s uc u e o G and he same ideas as abo e . P oposi ion 3 .2 . Le¡ G be a cen al g oup and leí m be he minimal non- nega i e in ege o which ¡he condi ion (*) holds . Then he e is a non-ze o analy ic semig oup (a z )Rez>0 in L'(G) such ¡ha¡ j1a l +' Y11 = O((logly1)') as jy1 ->oo(yER),i m>1 . P oo .. By Theo em 1 .2, G - IR" x Go whe e n >_ 0 and G o con ains a compac , open subg oup . The deg ee o g ow h o Rn is n ([12, p .181]), and so n = m . Take in V(R) he Poisson semig oup (P z) R, z>0 , o which 11P1+'YIl = O(log ¡y¡) as ¡y¡ - oo ([26, p .25,29]) . Choose an analy ic semig oup (J)Re z>0 in L l (G o ) as he one gi en in P oposi ion 3 .1 . Then, i a z = Pz® . . .OPzOcz (z E H), he semig oup (a z )Rez>0 is in he enso p oduc Ll(IR') ® L l (G o ) C L 1 (G), and i sa is ies he p ope ies o he s a emen . Nex , we a e going o show ou main esul s conce ning Ques ion E . Le A be a Banach algeb a . I X is a Banach space we deno e by £(X) he space o con inuous endomo phisms on X endowed wi h he uni o m no m . Recall ha a ep esen a ion cp o A on X is a con inuous algeb a homomo phism cp : a --> cp(a),A --~ C(X) . A closed subspace X l o X is said o be in a ianí unde cp i W(a)(X 1 ) C X l o all a E A . I (0) and X a e he only closed subspaces o X ha a e in a ian unde cp, and cp is non null, hen cp is said o be i educible . I X is ini e-dimensional, hen ep is called ini e-dimensional ([5]) . We deno e by F he se o ini e-dimensional i educible ep esen a ions o A . We say ha A is P-semisimple i W(b) = 0 o e e y cp E P and some b E A implies ha b = 0 . Le us conside he con olu ion Banach algeb a L' (R" ; A) o A- alued Boch- ne in eg able unc ions on IRn . Theo em 3 .3 . Leí n >_ 1, and le A be a P-semisimple Banach algeb a . I (a z ) Re z>0 is an analy ic semig oup in L I (R n ; A) wi h {a' 4- 'Y : y ER} ela i ely weakly compac , hen a = 0 . P oo . Le cp E P, wi h espec o A, and de ine ( ®W)( ) = IR n e-i .sw( (s» ds E ,C(X), ( E L 1 (R n ;A), E Rn) . Then ® cp is a ( ini e-dimensional) i educible ep esen a ion o L'(IRn ;A) ([7, p .461]), and he e o e he spec um a( ) o 13 8  J .E . GALÉ ( ® cp)(a) in ,C(X) is con ained in u(a) ([2, p .158]), which is an a mos coun - able subse o R, by Theo em 2 .2 . Now, since ,C(X) is ini e-dimensional, he spec um unc ion is con inuous on ,C(X) ([2, p .8]) which implies ha he spec al adius p is also con inuous on ,C(X) . I ollows ha {p( 0 cp)(a)) = maxaE,,( ) 1 A 1 : ER n} is a connec ed subse o l X 1 : E o(a)} . Mo eo e , lim ,-,,( ® cp)(a) = 0, whence we deduce ha P(( ® cp)(a» = 0 o e e y E R n .  So, o each E R n , he Banach subalgeb a o ,C(X) gene a ed by he semig oup (( ® cp)(az»R Q z>o is adical .  Bu his is no possible unless ( ® cp)(a) = 0 ([26, P oo o Theo em 5 .6, p .80]) . Then we ha e ob ained ha W( .  e - ' .s a(s) ds) = 0 o all E Rn and cp E F . Since A is F-semisimple we ob ain ha ue . e -' .sa(s) ds = 0 o e e y ER, whence, as usually, by composi ion wi h con inuous linea unc ionals on A, we deduce ha a = 0 . The abo e p oposi ion applies o A= L l (G), i G is a (so-called) maaimally almos pe iodic g oup (see [19, p .428]), and his ac gi es ise o he ollowing esul . Theo em 3 .4 . Le G be a cen al g oup such ¡ha¡ V(G) has a non-ze o analy ic semig oup (a z )R, z>o wi h {a l +' y : y E R} ela i ely weakly compac in L l (G) . Then G con ains a compac , open, no mal, subg oup K such ha GIK is abelian . P oo . By Theo em 1 .2, G= Rn x Go wi h n and Go as he e . We ha e o p o e ha n = 0 . Suppose, i possible, ha n > 1 . Since G o is (isomo phic o) a closed subg oup o G, L'(G o ) is semisimple o he se o i s ini e-dimensional i educible ep esen a ions ([21, p .15], [19]) . Also, because o L l (G) = LI(Rn x G o ) = Ll(Rn) ®,, Ll(Go) = Ll(Rn ;Ll(Go) ([2, p .132]), i is enough o ake A= L'(G o ) in Theo em 3 .3 o ob ain ha a = 0, which is a con adic ion . We w i e now he con e se esul announced a e P oposi ion 3 .1 . Co olla y 3 .5 . (i) Le G be a cen al g oup . Then G con ains a com- pac , open, subg oup i and only i V(G) has a non-ze o analy ic semig oup (a')Rez>o such ha {a`] - 'Y : y E R} is no m compac (o ela i ely weakly compac ) in L l (G) . (ii) Le G be a cen al and connec ed g oup . Then G is compac i and only i L 1 (G) has a non-ze o analy ic semig oup (a z )Rez>o such ha {a l +' y : y E R} is no m compac (o ela i ely weakly compac ) in L'(G) . This co olla y is a consequence o P oposi ion 3 .1 and Theo em 3 .4 . Fo pa (ii) we need also Theo em 1 .3 . As we ha e seen, he s udy o Ques ion E o cen al g oups depends on he s udy o he exis ence in L'(R) o analy ic semig oups, bounded on {Re z = 1} . Tliis leads us o pose he ollowing ques ions . Suppose ha (**)  L'(R) con ains a non-ze o analy ic semig oup unc ion on R ? ANALYTIC SEMIGROUPS IN THE GROUP ALGEBRA  13 9 (a z ) R, z>0 such ha sup j1a l+ 'y11 < oo . YER Ques ion 1 . Is cp(a z +iy) (y E IR, ~p E L'(R» a weakly almos pe iodic Ques ion 2 . Does (**) imply ha , o some non-ze o analy ic semig oup (V) Rez>o in L1(R), {b' +s y : y E UF} is ela i ely weakly compac ? Also, o any connec ed, locally compac g oup G, we may ask his o he ques ion Ques ion 3 . Suppose ha he e exis s a non-ze o elemen in L I (G) such ha ¡he spec um a( ) is a mos coun able . Does i ollow ¡ha¡ G mus be compac ? Re e ences [1] L . AMERIOAND G . PROUSE, "Almos -pe iodic unc ions and unc ional equa ions," Van Nos and, New Yo k, 1971 . [2] B . AUPETIT, "P op ié és Spec ales des Algéb es de Banach," Lec u e No es in Ma h . 735, Sp inge Ve lag, Be lin, 1979 . [3] J .M . BACHAR AND OTHERS (ED .), "Radical Banach Algeb as and Au o- ma ic Con inui y," P oc . Long Beach 1981, Lec u e No es in Ma h . 975, Sp inge Ve lag, Be lin, 1983 . [4] H . BOHR, "Almos Pe iodic Func ions," Chelsea Pub . Co ., New Yo k, 1947 . [5] F .F . BONSALL AND J . DUNCAN, "Comple e No med Algeb as," Sp inge - Ve lag, Be lin, 1973 . [6] J .C . CANDEAL AND J .E . CALÉ, On he exis ence o analy ic semig oups bounded on he hal -disc in some Banach algeb as, Bull . London Ma h . Soc . 21 (1989), 573-576 . [7] T .K . CARNE, Rep esen a ion heo y o enso p oduc s o Banach alge- b as, Ma h . P oc . Camb . Phil . Soc . 9 0 (1981), 445-463 . [8] J . DIXMIER, Opé a eu s de ang in¡ dans les ep ésen a ions uni ai es, Pub . Ma h . LH .E .S . 6 (1969), 171-204 . [9] W .E . EBERLEIN, Abs ac e godic heo ems and weak almos -pe iodic unc ions, T ans . Ame . Ma h . Soc . 67 (1949), 217-240 . [10] W .E . EBERLEIN, The spec um o weakly almos -pe iodic unc ions, Mich . J . Ma h . 3 (1958), 137-139 . [11] R .E . EDWARDS, "Func ional Analysis : Theo y and applica ions," Hol , Rineha and Wins on, New Yo k, 1965 .