Discontinuity of the product in multiplier algebras

Publicacions Ma emá iques, Vol 34 (1990), 397-401 . Abs ac DISCONTINUITY OF TI-IE PRODUCT IN MULTIPLIER ALGEBRAS M . OUDADESS En i e unc ions ope a e in comple e locally A-con ex algeb as bu no con inuously . Ac ually squa ing is no always con inuous . The coun e - example, we gi e, is a mul iplie algeb a . 1 . In oduc ion W . Zelazko cons uc s ([13]) an example o a non m-con ex algeb a on which all en i e unc ions ope a e . Doing so he sol es a p oblem s a ed in [12] . The example u ns ou o be a uni o mly A-con ex algeb a . The p oblem in ques ion had also been sol ed in [10] . We ob ain ha en i e unc ions do no ope a e con inuously . Ac ually we show by a coun e -example, which is a mul iplie algeb a, ha he p oduc is no (globally) con inuous in gene al . On he o he hand we ob ain he e ha i is always sequen ially con inuous in any uni al and comple e locally A-con ex algeb a . In he Uni o mly A-con ex case i is hypocon inuous . 2 . Sequen ial con inui y and hypocon inui y o he p oduc Le E be a locally con ex algeb a wi h a opology de ined by he amily o semi-no ms (ha)aEA . Then E is said o be locally A-con ex ([4]) i o e e y xE E and e e y AE A he e exis M(A, x) > 0, N(A, x) > 0 such ha Pa(xy) < M(A,x)px(y)  (y E E) . pa(yx) G N(A, x)Pa(y)  (y E E) . A lócally A-con ex algeb a is said o be Uni o mly A-con ex algeb a i M(A, x) and N(A,x) can be chosen independen ly o A ([5]) . Obse e ha he so-called locally m-con ex algeb as ([9]) a e pa icula cases o he abo e de ini ion . Some examples o all hese classes o algeb as and ela ionships among hem can be seen in ([4]) and ([5]) . 39 8  M . OUDADESS Le E be a commu a i e Banach algeb a wi hou o de i .e i xy = 0 (y E E) hen x = 0 . The mul iplie algeb a M(E) o E is he space o linea ope a o s T e i ying T(x .y) = xT(y)(x, y E E) . Endowed wi h he s ong opology gi en by he amily o semi-no ms (Px)xEE, whe e p,,(T) = JITxIj(x E E), M(E) is a uni al, comple e, uni o mly locally A-con ex algeb a which is no always m- con ex since i is a gene aliza ion (see [1, p . 139]) o he algeb a Cb(R), o [4], ha is no m-con ex . We can endow any uni al locally A-con ex algeb a wi h a locally m-con ex opology M(T) ine han by pu ing q, (x) = sup{pa(x .y) : pa(y) < 1} . This opology is comple e when T is . Mo eo e and M( ) ha e he same bounded se s ([2]) ; indeed i su ices o show ha bounded se s o a e bounded o M(T), bu his ollows om he ac ha any ba el in a comple e locally con ex space is bo ni o ous . P oposi ion 2 .1 . En i e unc ions ope a e on any uni al and compe e locally A-con ex algeb a . P oo - (E, M( )) is a uni al comple e locally m-con ex algeb a and en i e unc ions ope a e on such algeb as ([9]) ; he esul ollows since is coa se han M( ) . P oposi ion 2 .2 . In any uni al and comple e locally A-con ex algeb a (E, T), he p oduc is always sequen ially con inuous . P oo - Le (xn)n and (yn)n be wo sequences con e ging o ze o in (E,' ) . Since T and M(T) ha e he same bounded se s, (ga(x n )) n is bounded o e e y A . Then he conclusion ollows om he ela ion pa(x .y) < ga(x) .pa(y), o e e y x and y . E We can endow any uni al and comple e uni o mly A-con ex algeb a (E, T) wi h a Banach algeb a no m 11 - 11 ine han T, by pu ing lixil = sup{q (x) A E A} ; i is Coch an's no m ([5]) . And, as be o e, we can show ha and 11 - 11 ha e he same bounded se s . This has, as a consequence, he p oposi ion 2 o [13] . Now ecall ha , in a locally con ex algeb a (E, T) .  The mul iplica ion is said o be le ( igh ) hypocon inuous i o each neighbo hood U o O and any bounded se B he e exis s a neighbo hood V o o such ha B .V C U(VB C U) . The mul iplica ion, in E, is called hypocon inuous i i is le as well as igh hypocon inuous . P oposi ion 2 .3 . In a uni al and comple e uni o mly A-con ex algeb a (E, T) he p oduc is always hypocon inuous . P oo . By ([6, p oposi ion 9, p . 155]) i is su icien o p ó e ha i we conside he map u : x ---> L x , whe eLx(y) = xy (x, y E E) hen ; ' o e e y DISCONTINUITY OF THE PRODUCT IN MULTIPLIER ALGEBRAS  399 bounded se B in E, he se u(B) is equicon inuous . Bu his ollows om he ela ion pa(xy) < lix¡l . pa(y)(x, yE E, A EA) and he ac ha T and ha e he same bounded se s . 3 . Discon inui y o he p oduc Ou coun e -example is he mul iplie algeb a o an H*-algeb a . I goes along he lines o ([7, p oblem 111]) . An H*-algeb a is a Banach algeb a E, wi h in olu ion *, which is a Hilbe space unde a scala p oduc ( ., .) such ha . a) lixl¡2 = (x, x), o e e y x in E b) IIx*11 = lixil, o e e y x in E c) x* .x :~ 0, o e e y x in E {0} . d) (x . y, z) = (y, x* .z) = (x, z .y*), o e e y x, y, z in E . In such commu a i e algeb as he e always exis s a comple e o hogonal sys- em o idempo en and sel -adjoin elemen s (we can e en suppose mo e bu his is su icien o ou needs ; c [3]) . Coun e -example 5 .1 . Le (E,  be a commu a i e in ini e dimen- sional H*-algeb a . Wi hou loss o gene ali y suppose E sepa able . Le hen (e l , e 2 ,... , ek . . . ) be a comple e o hogonal sys em o sel -adjoin and indem po en elemen s .  The se { .ek : k = 1,2 . . . . } is unbounded since llek l1 >_ (k = 1, 2,  . . ) and i con ains he null ec o in i s weak closu e (c ., [7, p ob- lem 28]) . Hence he e exis s a ne (k¡)¡ o posi i e in ege s such ha  k ; .ek j con e ges weakly o ze o (i canno be a sequence) . Fo each k, conside he mul iplie Ak de ined by Ak(x) = k' .ek .x whe e k' is he ou h oo o k . We ha e 11Ak ;(x)J12 = (  k ; .ek i , x .x*), hence 11Ak ;(x)jj 0 . Bu Ak ; (x) = k ; .ek ; .x, and . i he e o e we ake in pa icula x = (1, . . .,n~ 1 , ... ), hen A4, (x) = ek i. So JIA4 ¡ (x)jj > 1 . Whence squa ing is no con inuous . Rema ks . 1 . In many in e es ing si ua ions he p oduc is con inuous . This is he case o he example Cb(R) o [4] . I is also so o any mul iplie algeb a M(E) whe e E admi s ac o iza ion, Le o e e y z E E, he e exis x and y in E such ha z = x .y . 2 . In connec ion wi h he p e ious ema k, we can no ice ha he e canno exis a Banach algeb a E admi ing a bounded app oxima e iden i y and such ha he se N = {x : x 2 = 0} is O-dense in E . Indeed he p oduc , and in pa icula squa ing, is con inuous, hence {TeM(E) : T 2 = 0} in ,0-closed . I is also P-dense since i con ains N and E is /i-dense in M(E) o i admi s a bounded app oxima e iden i y . The e o e any elemen o M(E) should be nilpo en . Bu his con adic s he ac ha M(E) is always uni al . 3 . Inciden ally we ge ha an absolu ely con e gen se ies in a comple e locally con ex space does no necessa ily de ine a con inuous mapping . Indeed i (E, T) is a locally uni o mly A-con ex algeb a whose opology T is gi en by a 40 0 M . OUDADESS amily o semino ms (Paja, he e exis s a Banach algeb a no m  and a > 0 such ha Pa(x) < a .11xII, o e e y Aand e e y x . Now i (z) =  a n .zn is an en i e unc ion, hen, o e e y A, Pa(E an .2 n ) < a . 1 : janl .11Xlln .  Hence he se ies is absolu ely con e gen ; bu he p e ious coun e -example shows ha he map x --> (x) is no always con inuous . Acknowledg uen s . 1'm indeb ed o he e e ee o ha ing de ec ed an e o in he i s e sion and ha ing sugges ed many imp o emen s o he w i ing o he second one . Re e ences [1[ M . AKKAR, "E ude Spec ale e s uc u es d'algéb es opologiques e bo - nologiques complé es," Thése d'é a , Uni . d e Bo deaux 1, 1976 . [2] M . AKKAR, L . OUBBI AND M . OUDADESS, ALgéb es A-con exes e p obléme de Michaél, P oc . A .M .S . (á pa ai e) . [3] F .F . BONSALL AND J . DUNCAN, "Comple e no med algeb as," E gebnisse de Ma hema ik, Band 80, Sp inge -Ve lag, 1973 . [4] A .C . COCHRAN . R . KEOW N AND C .R . WILLIAMS, On a class o opo- logical algeb as, Paci ic J . Ma h . 34 (1970), 17-25 . [5] A . C . COCHRAN, Rep esen a ion o A-con ex algeb as, P oc . Ame . Ma h . Soc . 4 1 (1973), 473-479 . [6J A . GROTHENDIECK, Espaces ec o iels opologiques, Pub . Soc . Ma h ., S . Paulo (1964) . [7] P .R . HALMOS, "A Hilbe space p oblem book," Second Ed ., Sp inge -Ve - lag, 1974 . [8] R . LARSEN, "The mul iplie p oblem," Lec . No es Ma h . 105, Sp inge - Ve lag, 1969 . [9] E .A . MICHABL, Locally mul iplica i ely con ex opological algeb as, Memoi s Ame . Ma h . Soc . 11, P o iden e (1952) . [10] M . OUDADESS, Théo émes de s uc u es e p op ié és ondamen ales des algéb es uni o mémen A-con exes, C . R . A ad . Sci . Pa i s 296, Sé ie 1(1983),851-853 . [11] J.K . WANG, Mul iplie s o commu a i e Banach algeb as, Paci ic J . Ma h . (1961), 1131-1149 . DISCONTINUITY OF THE PRODUCT IN MULTIPLIER ALGEBRAS  401 [12] W . ZELAZKO, Selec ed opics in opological algeb as, Lec . No es sé ies 31 (1971), Ma ema isk Ins i u , Aa hus Uni e si e . [13] W . ZELAZKO, A non m-con ex algeb a on which ope a e all en i e unc- ions, Ann . Polinici Ma h . 46 (1985), 389-394 . Ecole No male Supé ieu e Takaddoum B .P . 5118 Raba MAROC Rebu el 26 d'Ab il de 1990