A Characterization of Jacobson Radical in Γ-Banach Algebras

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1 Advaces Pure Matheatcs Publshed Ole Noveber ( A Characterzato of Jacobso Radcal Γ-Baach Algebras Nlash Goswa Departet of Matheatcs Gauhat Uversty Guwahat Ida Eal: la_g3@yahooco Receved August 7 ; revsed Septeber 5 ; accepted October 3 ABSTRACT V V Let V ad V be two -Baach algebras ad R be the rght operator Baach algebra ad L be the left operator Baach algebra of V We gve a characterzato of the Jacobso radcal for the projectve tesor product ters of the Jacobso radcal for R L If ad V are soorphc the we show that ths characterzato ca also be gve ters of the Jacobso radcal for V R L Keywords: Γ-Algebra; Rght Quas Regularty; Tesor Product; Operator Baach Algebra Itroducto I [] usg the rght quas regularty property Kyuo ad Coppage ad Luh gave a characterzato of Jacobso radcal -rgs May terestg results o the teral propertes of Jacobso radcal for -rgs were developed [-5] by dfferet research worers I [6] soe of these results are exteded to -algebras I ths paper we cosder two -Baach algebras V ad V ad cosder ther projectve tesor product V V Let R be the rght operator Baach algebra ad L be the left operator Baach algebra of V We gve a characterzato of Jacobso radcal J V V ters of J R L Before gog to preset our a results we frst gve soe basc terologes (refer to [5-]) whch are eeded our dscusso Defto Let X be a rg havg the ut eleet e A ew ultplcato called the crcle coposto (refer to [5]) o X s defed by: x x xxxx Ths coposto aes sese eve whe X does ot have the ut eleet A eleet x of X s sad to be rght quas regular f t has a rght quas verse wrt ths coposto e there exsts xx such that x x xxxx Defto Let V ad be two lear spaces over a feld F V s sad to be a -algebra over F f for x y z V ; ; a F the followg codtos are satsfed: ) x yv ; x y z x yz ; ) a xy axy xay xay x yz x y x z x y xy xy x y z xz yz V 3) ; 4) The -algebra s deoted by If V ad are ored lear spaces over F the -algebra V s called a -ored algebra f codtos ) to 4) hold ad further 5) xy x y holds A -ored algebra V s called a -Baach algebra f V s a Baach space Ay Baach algebra ca be regarded as a -Baach algebra by sutably choosg Defto 3 A subset I of a -Baach algebra V s sad to be a rght (left) -deal of V f ) I s a subspace of V ( the vector space sese); ) ; x y I y xi xi y V e I V I V I I A rght -deal whch s a left -deal as well s called a two-sded -deal or sply a -deal Defto 4 Let V be a -Baach algebra ad let x V The the appg x defed by y x yxy V s a rght Baach space edoorphs of V The collecto R of all edoorphss geerated by x ; x V s a Baach algebra uder the operatos: x y= x y x x x Copyrght ScRes

2 44 N GOSWAMI where a F ad the or: a x ax a x x y xy x x V Ths Baach algebra s tered as the rght operator Baach algebra of -Baach algebra V We ca slarly defe the left operator Baach algebra L of V as the Baach algebra geerated by the set of all left edoorphss of V the for x where x y xyyv Defto 5 Let V ad V be -Baach algebras over F ad : V V be a appg The s called a -Baach algebra hooorphs f ) ax by ax b y ad ) xy x y for all x y V ; ad ab F Defto 6 Let X ad Y be two ored spaces The projectve tesor or o X Y s defed as: u f x y : u x y where the fu s tae over all (fte) represetatos of u The copleto of X Y s called the projectve tesor product of X ad Y ad s deoted by X Y Let V ad V be -Baach algebras over F ad F soorphc to F The projectve tesor product V V wth the projectve tesor or s a -Baach algebra over F where a ultplcato s defed by the forula: x y xy xx yy where x y V ; x yv; Defto 7 Let V be a -Baach algebra Let A eleet x V s sad to be -rght quas regular wth -rght quas verse y f x yx y x s sad to be a rght quas regular eleet of V f t s -rght quas regular for each Equvaletly a eleet x V s called rght quas regular f for ay there exst v V such that vx v v vx v vv A deal I of V s sad to be rght quas regular f each of ts eleets s rght quas regular We have rght quas regularty s a radcal property a algebra The axal rght quas regular deal s called the Jacobso radcal of V ad t s deoted by J(V) Ma Results I [6] we have the followg Lea regardg rght quas regularty of a -Baach algebra ad ts operator algebra Lea A eleet x of a -Baach algebra V s rght quas s rght quas regular the rght operator Baach algebra R of V Extedg ths result to the projectve tesor product of -Baach algebras we prove Lea regular f ad oly f for all x Let V ad V be two ad -Baach algebras respectvely Let R be the rght operator Baach algebra of V ad L be the left operator Baach algebra of V If x x s rght quas regular V V the x x s rght quas regular R L for ad coversely Proof Sce x x s rght quas regular V V so for ay there exst j j j pj xj xj V V j such that for ay q v v V V j j j j j j q x x q p q x x p v x x v v j j xj x v j j v v x x j j xj x j j v x x v v x x v v x x x xv j j j j j j j j j j () Copyrght ScRes

3 N GOSWAMI 45 Let x x x We tae j j x Now y x j j j x yxy v v x x j xj xj j j x x j xj xj j v v j x x v v j x j xj j v v j x x j x j xj j v v j v x xv v x x v v x x x j j j j j j j j j jx v (by ()) Proof Let x x J V V But v v V V s arbtrary The x x s a rght quas regular eleet of So x + y xy = Thus x e x x s V V By Lea for ay rght quas regular R L x x s a rght quas regular eleet of The coverse follows the sae way I [3] we have defed the followg deal for the R L e projectve tesor product of V ad V xx JR L Lea 3 Let V ad V be two ad -Baach algebras respectvely Let R be the rght operator Baach algebra of So V ad L be the left operator Baach algebra of V Let J x x JR L be a deal of R L We defe: Hece J x x V V : xx J x x J R L where x x j : j ad j x j : x j j The J s a deal of V V Usg the above defed deal ow we gve the characterzato of Jacobso radcal for the projectve tesor product of two -Baach algebras V ters of the Jacobso radcal of the projectve tesor product of correspodg rght ad left operator Baach algebras Theore 4 Let V be a -Baach algebra (over F) wth rght operator Baach algebra R ad left operator Baach algebra L respectvely The the Jacobso radcal of V J R L s gve by: J V V V Thus J V V J R L Coversely let The x x J R L x x J R L So for ay x x R L x x s a rght quas regular eleet of V V s a rght quas regular eleet of By Lea So e x x JV V Copyrght ScRes

4 46 N GOSWAMI L JV V J R Thus J V L V J R Let the -Baach algebras V ad V are soorphc I that case we have the followg result Theore 5 Let V be a -Baach algebra (over F) wth rght operator Baach algebra R ad left operator Baach algebra L respectvely If there exsts a -Baach algebra soorphs f fro V oto V the R L s a hooorphc age of R L Proof Let r lr L where l y r x We defe : R L R L by r l x y xf y where x f x x V Let r R (The dual space of R ) We defe r : R C by r x r x where x f x The r R Slarly for l L we ca defe l L by l y lf y Now let r r l l where r x l y r l r lh h hr L I partcular tag h r l we get r l l r l r r l r r l l r r l l where ad x f x x f x r x l y r x l y y y r x l f r x l f xf y r l xf y r l r lr lr l r l But r ad are arbtrary So R l L r l r l b x f y Thus s well defed Now Let ab F The arl brl arl brl a x y b x y a x y b x y a x f y f x ad where x x f x b x f y a x f y a f( x) f y b f x f y ar l br l Aga r lr l rr l l x x y y () x x y y We have x x V So there exst x x V x f x x f x such that Now x x V ad f x x f x f x x x So the expresso () s equal to Copyrght ScRes

5 N GOSWAMI 47 x x f y y x x f y f y x xf y f y xf y x f y r lr l So : R L R L s a hooorphs Sce f s oto so s also oto Also t ca be show that s oe-oe Thus R L R L Corollary 6 Let the -Baach algebras V ad V as defed Theore 4 are soorphc The we have R L J V V J Rear 7 If the soorphs f fro V oto V s soetrc the we ca show that : R L R L s also a soetry So that case R L J V V J The oto of drect suad for -rgs s dscussed [] by Booth For a -Baach algebra V a deal P s called drect suad f there exsts a -deal Q of V such that every eleet v of V s uquely expressble the for v = p + q p P qq ad V s wrtte as V PQ Clearly f V P Q the for p P qq p q Now we prove: Deducto 8 If P s the drect suad for the -Baach algebra V V the J P s the drect suad for J V V Proof Let V V PQClearly JQ J P ad x = p + q where p P qq Sce x s rght quas regular V V so for ay we have there exsts yv V such that x yx y Let y p q where p P q Q So Let x JV V pq pq pq pq p p p p + qq q q [sce p q But p p p ppad q q q q Q ad P Q So p p p p ad q q q q for ay ad q p ] Thus p s rght quas regular P ad q s rght quas regular Q e p JP ad q JQ Hece J VV = JPJQ I [4] there s a characterzato of Jacobso radcal for -rgs ters of axal regular left deals Lea 9 M Let X be a -rg The J X where the tersecto s over all axal regular left deals M of X Cosderg ths aspect we ca rase the followg proble: Let the structures of axal regular left deals of the operator Baach algebras R ad L are gve Usg ths ca we obta the structure of the Jacobso radcal for V V? I [6] Behres radcal for -Baach algebras s troduced whch cotas the Jacobso radcal Let deote the class of all subdrectly rreducble -Baach algebras V such that the tersecto of all o-zero deals of V cotas a o-zero depotet eleet The upper radcal R B detered by the class s called the Behres radcal for V Lea For a sple -Baach algebra V J V RB V Now aother proble ca be rased: Ca we derve aalogous result as Theore 4 case of the Behres radcal for V V? REFERENCES [] S Kyuo Notes o Jacobso Radcals of Gaa Rgs Matheatca Japoca Vol 7 No 98 pp 7- [] W E Coppage ad J Luh Radcals of Gaa Rgs Joural of the Matheatcal Socety of Japa Vol 3 No 97 pp 4-5 do:969/jsj/34 [3] A C Paul ad A K Azad Jacobso Radcal for Gaa Rgs Rajshah Uversty Studes Part-B Joural of Scece Vol pp 53-6 [4] A C Paul ad Md S Udd O Jacobso Radcal for Gaa Rgs Gat: Joural of Bagladesh Matheatcal Socety Vol 9 9 pp 47-6 [5] K N Raghava The Jacobso Desty Theore ad Applcatos 5 [6] H K Nath A Study of Gaa-Baach Algebras PhD Thess Gauhat Uversty Guwahat [7] W E Bares O the -Rgs of Nobusawa Pacfc Joural of Matheatcs Vol 8 No pp 4- Copyrght ScRes

6 48 N GOSWAMI 4 [8] D K Bhattacharya ad A K Maty Selear Tesor Product of -Baach Algebras Gata Vol 4 No 989 pp 75-8 [9] F F Bosall ad J Duca Coplete Nored Algebras Sprger-Verlag Berl 973 do:7/ [] G L Booth Operator Rgs of a -Rg Math Japoca Vol 3 No 986 pp [] N J Dvsy Rgs ad Radcals George Alle ad Uw Lodo 965 [] N Goswa Soe Results o Operator Baach Algebras of a -Baach Algebra Joural of Assa Acadey of Matheatcs Vol pp 4-48 [3] N Goswa O Levtzl Radcal of Gaa Baach Algebras Global Joural of Appled Matheatcs ad Matheatcal Sceces Press Copyrght ScRes

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