European Journa of Mathematcs and Computer Scence Vo. No. 1, 2017 ON AUTOMATC CONTNUTY OF DERVATONS FOR BANACH ALGEBRAS WTH NVOLUTON Mohamed BELAM & Youssef T DL MATC Laboratory Hassan Unversty MORO CCO ABSTRACT We treat the probem of the automatc contnuty of the Dervatons n Banach agebras provded wth an nvouton σ. To do ths, we ntroduce and study on a untary agebra provded wth an nvouton a noton whch we ca σ-sem-smpcty. t s based on the study of certan batera deas caed σ-deas. Keywords: -Agebra, -Smpe agebra, -Sem-Smpe Agebra, Automatc Contnuty, Separatng dea. NTRODUCTON n Automatc Contnuty theory we are concerned wth agebrac condtons on a near map between Banach spaces whch make ths map automatcay contnuous. Ths theory has been many deveoped n the context of Banach agebras, and there are exceent accounts on automatc contnuty theory [2, 3, 5] (see aso [6]) n ths assocatve context. n [7] Snger and Wermer proved that the range of a contnuous dervaton on a commutatve Banach agebra s contaned n the Jacobson radca. They conjectured that the assumpton of contnuty s unnecessary. n [] Johnson proved that f A s a sem-smpe Banach agebra, then every dervaton on A s contnuous and hence by the Snger-Wermer theorem t s zero. n ths work, we defne and study on a untary agebra provded wth an nvouton a noton whch caed -sem-smpcty whch generazes the noton of sem-smpcty, t rests on the study of certan batera deas caed -deas. The nterest therefore s to restrct onesef to the eve of a famy of batera deas nstead of consderng a the deas on the eft. Ths noton of -sem-smpcty w aso contrbute to the study of the automatc contnuty of near operators on Banach agebras, n partcuar the contnuty of dervatons. We w show that on a -sem-smpe Banach agebra, every dervaton s contnuous (Theorem 2.2). Premnares n these papers, the agebras consdered are assumed compex, Untary, not necessary commutatve. An nvouton on an agebra A s a mappng: satsfyng the foowng propertes: ( x y) ( x) ( y), ( xy) ( y) ( x), ( ( x)) x, ( x) ( x) K, for a x, y n A. Wth nvouton, A s caed -agebra. An dea of -agebra Progressve Academc Pubshng, UK Page 56 www.dpubcatons.org
European Journa of Mathematcs and Computer Scence Vo. No. 1, 2017 s caed a -dea f ( ) (then ( ) ). Moreover, s sad to be a -mnma (resp. -maxma) dea of A f s mnma (resp. maxma) n the set of nonzero (resp. proper) -deas of A. Observe that f s an dea of A, then + ( ), ( ), ( ) and ( ) are -deas of A. Moreover, f we denoted by the map from A to A defned by ( a ) (, then s a we-defned nvouton on A. Characterzatons of -sem-smpe agebras An agebra A s caed smpe f t has no proper deas. An -agebra A s caed - smpe f t has no proper -deas. We observe that every smpe agebra wth nvouton ( A, ) s a -smpe. The foowng counterexampe shows the converse s not true. Counterexampe 1.1 Let A be a smpe agebra, we denoted by A the opposte agebra A. Consder the agebra B = A A. Provded wth the exchange nvouton defned by: ( x, y) ( y, x), t cear that B s not smpe, snce the deas of B are (0), A, {0}x A and A x{0}. But B s -smpe. ndeed, the ony -deas of B are 0 and B. t s therefore natura to ask under what condtons the converse s true. t s subject to the foowng proposton: Proposton 1.1 Let ( A, ) be -smpe agebra. f the nvouton s ansotropc, then A s smpe. Reca that nvouton s caed ansotropc f : a A, t ( a 0 = 0 a = 0. Let be an dea of A, then () s an -dea. t foows that, () = {0 } or = A. f () = {0 }, then ( x) x 0 x. Snce s ansotropc, then x = 0, a resut that = {0 }. f () = A, then = A Proposton 1.2 Let A s -agebra. Then A s a -smpe f, and ony f, there exst a maxma dea M such that, M ( M ) = {0 }. We assume A s -smpe. Let M be a maxma dea of A. We have M ( M ) s a -dea of A, then M ( M ) = { 0 } or A. f M ( M ) = A, then M = A, whch contradcts the fact that M s a proper dea. Hence, M ( M ) = { 0 }. Assume that, there exsts a maxma dea M such that M ( M ) = { 0 }. Let s a - dea of A. f M, then ( ) = ( M ), where M ( M ) = { 0 }. f M, then M = + A, and we have: ( M ) + = ( ( M ) + ) A =( ( M ) + )( M + ) ( M ) M + =. Whch mpes that ( M ), as a resut, M. Snce M s maxmum dea of A, so t foows that A = Proposton 1.3 [8] Let A an -smpe agebra whch s not smpe. Then, there exsts a sub- agebra smpe unt of A such that A = ( ). Let a proper dea of A. So t foows that ( ) s a -dea, snce A s a - Progressve Academc Pubshng, UK Page 57 www.dpubcatons.org
European Journa of Mathematcs and Computer Scence Vo. No. 1, 2017 smpe agebra, then ( ) = { 0 } or ( ) = A. f ( ) = A then = A, whch s absurd. From where ( ) = { 0 }. There s aso + ( ) s a - dea, then + ( ) = { 0 }, or + ( ) = A. f + ( ) = { 0 }, then = { 0 }, whch contradcts the fact that s proper. Therefore, A = ( ). Let J an dea of A such that J. Accordng to what precedes, A = J ( J ). Let, then there exsts j, j J such that = j + ( j '). However - j = ( j ') ( ) = { 0 }, from where = j, therefore = J. Consequenty, s a mnma dea of A. Let J an dea of, then J s an dea of A. ndeed, et a A and j J, then t exsts, ( ) such that a = + ( ). From where aj = ( + ( ). j =j + ( ) j. However, ( ) j ( ) and ( ) ( ) = { 0 }, consequenty aj = j. Snce s a mnma dea, then J = { 0 } or = J. Thus, a smpe sub-agebra. On other hand, a unta and f 1 ndcates the unt of A, then there exsts e, e such that 1 = e + ( e ). Let x, we are: x = x 1 = x e + x ( e '), but x - x e = x (e ') ( ) = { 0 }, from where x = x e. n the same way, we checked that x = e x. Consequenty, a unta of unt e Proposton 1. Let A be a -agebra and M -maxma dea whch s not maxma. Then there exsts a maxma dea N of A such that M = N ( N ). As M s not maxma, there s a maxma dea N of A such that M N. Snce ( M ) = M ( N ), where M N ( N ). Snce N ( N ) s a -dea of A, t foows therefore that M = N ( N ) Defnton 1.1 Let A be a -agebra. We ca -radca of A, denoted Rad, the ntersecton of a deas - maxma of A. A s caed -sem-smpe f Rad = { 0 }. Proposton 1.5 Let be a -dea of a -agebra A such that Rad. So Rad ( A/ ) = Rad / n partcuar, A / Rad s a -sem smpe. M s a -maxma dea of A. We put A = A / and M = M /. We have: Rad M. So from the foowng canonca somorphsm: A / M A /M whch s -smpe, t foows that A / M s a -smpe agebra. Consequenty, M / s a -dea -maxma of A /. From where: Rad ( A/ ) = { M : M s -maxma dea of A } = M : Ms - maxma dea of A = Rad = Rad / Now, we say that an agebra wth nvouton ( A, ) s -sem-smpe f A s a sum of -mnma deas of A. Progressve Academc Pubshng, UK Page 58 www.dpubcatons.org
European Journa of Mathematcs and Computer Scence Vo. No. 1, 2017 Lemma 1.1 Let A be a -sem-smpe agebra such that A =, where each s a - mnma dea of A. f P s a -mnma dea of A, then there s a subset T of S such that: A P ) ( jt j. Snce are -mnma and a drect sum. ndeed, otherwse P, then there exsts some S such that P s P for a S, whch mpes that P A. Appyng Zorn s emma, there s a subset T of S such that the coecton { : T } { P } s maxma wth respect to ndependence: ( ) P = ( ) P. Settng B = j T j j T j ( ) P, the maxmaty of T mpes that B (0) for a S. Then, j T j the -mnmaty of ye ds that B A. B hence B for a S. Consequenty Coroary1.1 For an agebra wth nvouton ( A, ), the foowng condtons are equvaent: 1) A s a -sem-smpe. 2) A s a drect sum of -mnma deas Exampe.1.2 Let A be the aternatng group on etters. Consder the group agebra [ A ] provded wth ts canonca nvouton σ defned by: 1 ( rg g) rg g ga g A R Progressve Academc Pubshng, UK Page 59 www.dpubcatons.org
European Journa of Mathematcs and Computer Scence Vo. No. 1, 2017 F r o m[ 1 ], the decomposton of the sem-smpe agebra R [ A ] nto a drect sum of R A B B, where each B s nvarant smpe components s as foows: 1 2 B3 under σ. More expcty, B1 R, B2 C,a nd B ( 3 M 3 R). n partcuar, each B s a σ-mnma dea of R [ A ]. Consequenty, R [ A ]. s a σ-sem-smpe agebra. Now, et A be a - A -smpe agebra. Snce A s fntey generated (ndeed, 1 generates A), then A has a fnte ength. Thus A 1, where each s a -mnma dea of A. t s easy to verfy that each s generated by a centra symmetrc dempotent eement e A 2 (.e: e e and ( e ) e ), where 1 e. Moreover, e e 0 1 j for a j. n what foows, we denote by S the set of centra symmetrc orthogona dempotents of A,.e. S e 1... e such that Ae. Let A 1 be a -sem-smpe agebra, we have aready seen that each e such that 1 e 1. Hence, by a centra symmetrc dempotent s generated s a subagebra of A wth unty e. Moreover, s a -smpe agebra for a 1. Consequenty, every - sem-smpe agebra s a drect sum of -smpe agebras. Automatc Contnuty A dervaton D on agebra A s near mappng from A to tsef satsfyng D( xy) D( x) y xd( y) for a x, y A Let D a dervaton of a Banach space X. Then, the separatng dea (D) of X s the subset of X defned by: (D) = { y X / ( x ) X : x 0 and D( x n ) y} Lemma 2.1 [6] Let S be a near operator from a Banach space X nto a Banach space Y. Then; ) (S) s a cosed near space of Y ) S s contnuous f ony f (S) ={0} and ) f T and R are contnuous near operators on X and Y respectvey, and f ST RS, then R ( S) ( S) n n Lemma 2.2 [6] Let S be a near operator from a Banach space X nto a Banach space Y, and et R be a contnuous operator from Y nto a Banach space Z. Then: R ( S) 0. ) RS s contnuous f and f ) R ( S) ( RS ), and ) There s a constant M (ndependent of R and Z ) such that f RS s contnuous then RS M R Proposton 2.1 Let A be a Banach -agebra A, then a -maxma -dea M of A s cosed. n Progressve Academc Pubshng, UK Page 60 www.dpubcatons.org
European Journa of Mathematcs and Computer Scence Vo. No. 1, 2017 f M s a maxma dea of A, then M s cosed,. Otherwse, f M not Maxma, there s a maxma dea N of A such that M = N ( N ) (proposton 1. ). Snce N (resp. ( N )) s cosed, t s deduced that M s cosed n A Proposton 2.2 Let A a Smpe Banach agebra. Then a dérvaton D on A s contnuous. Let ( D ) the separator dea of D n A s smpe, so ( D ) = { 0 } or ( D ) = A. f ( D ) = A, that e A (D), cconsequenty 0 Sp( e ) ([6] theorem 6-16). From where A ( D ) = { 0 }. And by Lemma 2.1, as a resut, D s contnuous Theorem 2.1 Let A a -Smpe Banach -agebra. Then a dérvaton D on A s contnuous. We have A s an agebra smpe, there exsts smpe unta subagebra of A such that : A = ( ) (Proposton 1.3); foowng agebrac somorphsm: A / ( ), one deduces that am a maxma dea of A. From where (resp; ( )) s cosed n A. Consequenty, the agebra A / (resp; A / ( )) s a smpe Banach -agebra. Snce s an dea of A, then so s D( ) ; therefor D( ) / s an dea of A /. As A / s a smpe agebra, so D( ) / = { 0 } or D( ) / = A /. Snce s a maxma dea of A, then D( ) =, so D( ). Consder the functon D ~ on A / defned by: D ~ ( a ) D(. We show that s a dervaton on A /. Note that t s easy to show D ~ s near operator. Moreover, for D ~ ( a )( b ) )= D ~ ( ab ) )= D( ab) = ad( b) D( b. But then, ( a ) D ~ ( b ) + D ~ ( a ) ( b ) = ( a ) ( D( b) ) ( D( )( b ) = ad( b) D( b ad( b) D( b. So D ~ s a dervaton on the smpe Banach a, b A, agebra A /, then by proposton 2.2, D ~ s contnuous. To show that D s contnuous, consder the canonca surjecton : A A/ ; a a whch s contnuous. To show that D s contnuous, we observe frst that o D = D ~ o because for every a A, we have o D ( = ( D ( ) = D( and D ~ ( a )= D ~ ( a ) = D(. Snce D ~ o s contnuous, then; we have ( D ~ o ) = { 0 ~ }, And (D) = ( D ) = { 0 } (Lemma 2.2) and ths mped that ( D). Foowng the same steps, we show that ( D) ( ), then ( D) ( ) 0. Therefore D s contnuous (emma 2.1). Theorem 2.2 Let A a -sem--smpe Banach -agebra. Then a dérvaton D on A s contnuous. Snce A s a -sem-smpe agebra, wrtng (by emma 1.1) A 1 where s a - mnma dea of A and settng L, then 1 L s a -maxma dea of A. f L s a maxma dea, then D exst a maxma dea N such that j j ( L ) L f L s not maxma, then by proposton 1., that * 1, L N N. Consequenty, Progressve Academc Pubshng, UK Page 61 www.dpubcatons.org
European Journa of Mathematcs and Computer Scence Vo. No. 1, 2017 D( L ) D( N ( N) ) D( N ) D( ( N) ) N ( N) L 1. Now, consder the functon D ~ on A / defned by: 1 D ~ ( a L ) D( L. Snce a -maxma dea s cosed (proposton (2.1) and as mentoned n theorem (2.1), we have D ~ s a dervaton on the -smpe Banach agebra A / L. then by theorem 2.1, we have D ~ s contnuous. Consder the canonca surjecton : A A/ L ; a a L whch s contnuous. To show that D s contnuous, we observe frst that o D = D ~ o because for every a A, we have o D ( = ( D ( ) = D( L and D ~ ( a )= D ~ ( a L ) = D( L. Snce D ~ o s contnuous, then; we have ( D ~ o ) = { 0 ~ }, And (D) = ( D ) = { 0 } and ths mped that ( D) L 1. Thus mped that ( D) 1 L 0. Consequenty, D s contnuous REFERENCES [1] M. Bouagouaz; L. Oukhtte. (2000), nvoutons of semsmpe group agebras. Araban Journa for Scence and Engneerng 25 Number 2C, 133 19. [2] H. G. Daes (1978), Automatc contnuty: a survey. Bu. London Math. Soc. 10, 129-183. [3] H. G. Daes (2000), Banach agebras and automatc contnuty. London Mathematca Socety Monographs, Oxford Unversty Press, []. B. E. Johnson (1969), Contnuty of dervatons on commutatve agebras,, Amer. J. Math. 91 1-10. [5] V. Runde (1991), Automatc contnuty of dervatons and epmorphsms. Pacfc J. Math. 17 (1991), 365-37. [6] A. M. Sncar (1976), Automatc contnuty of near operators. Cambrdge Unversty Press. [7]. M. Snger and J. Wermer (1976), Dervatves on commutatve normed agebras, Math. Ann. 129 (1955), 260-26. [8] Y. TDL, L. OUKHTTE, A. TAJMOUAT (2002)., On the Automatc contnuty of the epmorphsms n *-agebras of Banach. JMMS 200: 22 1183-1187 Progressve Academc Pubshng, UK Page 62 www.dpubcatons.org