Approximately Quasi Inner Generalized Dynamics on Modules. { } t t R

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1 Joural of Scieces, Islamic epublic of Ira 23(3): (22) Uiversiy of Tehra, ISSN 6-4 hp://jscieces.u.ac.ir Approximaely Quasi Ier Geeralized Dyamics o Modules M. Mosadeq, M. Hassai, ad A. Nikam Deparme of Mahemaics, Faculy of Scieces, Mashhad Brach, Islamic Azad Uiversiy, Mashhad, Islamic epublic of Ira eceived: February 2 / eved: 9 Jue 22 / Acceped: Ocober 22 Absrac We ivesigae some properies of approximaely quasi ier geeralized dyamics ad quasi approximaely ier geeralized derivaios o modules. I paricular, we prove ha if A a -algebra, δ he geeraor of a geeralized dyamics { } sequeces { }, { } o a A-bimodule M safyig T ad here ex wo b c of self adjoi elemes i A such ha for all x i a core D for δ, δ ( x ) = lim i ( c x xb ), he { } approximaely quasi ier. Keywords: (quasi approximaely ier) Geeralized derivaio; (ier) Geeralized omorphm; (approximaely quasi ier) Geeralized dyamics Iroducio Throughou he paper A ad B are Baach algebras, M ad N are A ad B bimodule, respecively ad B ( A ) he se of all bouded liear operaors o A. A oe parameer group { } ϕ of bouded liear operaors o A a mappig ϕ : B ( A ) safyig ϕ = I ad ϕ s ϕϕ s + =. A oe parameer group { ϕ } called uiformly (srogly) coiuous if ϕ : B ( A ) coiuous wih respec o he orm (srog) operaor opology. We defie he ifiiesimal geeraor d of ϕ as a mappig d : D( d ) A A such ha ϕ ( a) a = lim, d a where ϕ ( a) a D( d ) = a A : lim exs. Also we defie he resolve se ρ ( d ) o be he se of all complex umbers λ for which λ I d iverible, cf. [3]. A auomorphm o algebra A a iverible liear operaor ϕ : A A such ha ( ab ) ( a ) ( b ) ϕ ϕ ϕ = ad ϕ( a ) = ϕ( a). A auomorphm ϕ called ier if here exs a uiary eleme u A such ha ϕ ( a) = u au. I easy o check ha if { } ϕ a oe parameer group of auomorphm o A wih he geeraor d, he d a derivaio ad coversely if d a bouded derivaio o A, he d iduces a uiformly coiuous group of orrespodig auhor, Tel.: +98(5)842952, Fax: +98(5)84242, mosadeq@mshdiau.ac.ir 245

2 Vol. 23 No. 3 Summer 22 Mosadeq e al. J. Sci. I.. Ira d auomorphms { e }. A oe parameer group of auomorphms { ϕ } o A said o be approximaely ier if here exs a sequece { c adjoi elemes i A such ha for each ad a A, ic ic e ae ϕ a uiformly o every compac subse of, cf. [5] A Hilber module over algebra A a algebraic lef A module M wih a A valued ier produc, which A liear i he firs ad cojugae liear i he secod variable such ha M a Baach 2. space wih respec o he orm x = x, x The Hilber Module M called full if he closed liear spa, x, y x, y M M M of all elemes of he form equal o A, cf. [8] We ivesigae some properies of approximaely quasi ier geeralized dyamics ad quasi approximaely ier geeralized derivaios o modules. I paricular, we prove ha if A a algebra, δ he geeraor of a geeralized dyamics { T } o a A bimodule M safyig T ad here ex wo sequeces { b }, { c } of self adjoi elemes i A such ha for all x i a core D for δ, δ x lim i(c x xb ) approximaely =, he { } quasi ier. The reader referred o [4], [6], [8] ad [2] for more deails o algebras ad Hilber module ad o [5] ad [5] for more iformaio o dyamical sysems. Geeralized Isomorphm ad Geeralized Dyamics o Modules I h secio, we defie he geeralized dyamics o modules ad meio he relaio bewee geeralized dyamics ad geeralized derivaios. For h aim, we eed he followig defiiios: Defiiio A liear mappig δ : M M said o be a geeralized derivaio if here exs a derivaio d : A A such ha δ ( ax ) = aδ ( x ) + d ( a ) x ( x M, a A). We call δ a d derivaio. As a example, le c A ad defie δc : M M by δ c ( x ) = cx xb. The δ c a d c -derivaio, where dc ( a) = ca ac. Th dc derivaio called a ier geeralized derivaio. We recall ha a liear mappig T : M N said o be a geeralized module map if here exs a liear homomorphm : A B T ax = ( at ) ( x), ϕ such ha ϕ for all a Ax, M. Th map called a ϕ module map. I he case ha ϕ a auomorphm ad T a bijecive liear mappig, T said o be a ϕ omorphm or geeralized omorphm, cf. [2] Le ϕ : A A be a liear edomorphm. I easy o see ha a liear map T : M M a ϕ module map if ad oly if = ( ) + ( ϕ ) + ( ϕ ( a) a) T ( x ) x T ax ax a T x x a a x.() Example If d a bouded derivaio o A ad δ a bouded d derivaio o M, he e δ d a e omorphm, cf. [2] Defiiio 2 Le A be a algebra. A geeralized omorphm T : M M called quasi ier if here ex uiary elemes uv, A such ha for each x M, T ( x) = uxv. Defiiio 3 Suppose ha { T } a oe parameer group of bouded liear operaors o M such ha for each, T a ϕ omorphm. If moreover { T } uiformly coiuous, he i called ϕ dyamics or geeralized dyamics o M. We defie he ifiiesimal geeraor δ of T as a mappig δ : D( δ) M M such ha δ where ( x ) = T ( x) x lim, T ( x) x D ( δ ) = x M : lim exs. From ow o, A cosidered as a algebra. emark Usig he relaio (), i ca be proved ha dyamics o if { ϕ } a uiformly coiuous A wih he ifiiesimal geeraor d ad { T } a 246

3 Approximaely Quasi Ier Geeralized Dyamics o Modules ϕ dyamics o M wih he ifiiesimal geeraor δ, he δ a everywhere defied d derivaio. oversely, if δ a bouded d derivaio, he i iduces he uiformly coiuous oe parameer group e δ d of e omorphmsy Example. { } Theorem Le bc e wo self adjoi elemes i A ad δcb, M: M be he ier geeralized derivaio δ cb, ( x ) = i ( cx xb ). The here ex a uiformly coiuous oe parameer group { ϕ } of ier auomorphms o A ad a uiformly coiuous oe of quasi ier ϕ parameer group { } omorphms o M such ha d c he geeraor of { ϕ } ad δ cb, he geeraor of { T }. ic ib proof. Take T ( x ) = e xe ad The rivially T ( ax ) ϕ ( a ) T ( x ) T = ad T ( T ( x )) = e ( e xe ) e s = e. e xe. e = e = T ic ib ic ib i( + s) c i( + s) b + s Also akig xe ( x ). u ic = e ad v T x x = u xv x Thus = ( u x xv ) v u x xv u x x + x xv ( u I + I v ) x. ϕ a e ae ib = e, we have T I u I + I v as. ic ic =. = I ad Therefore { T } a uiformly coiuous oe parameer group of quasi ier ϕ omorphms. Moreover, ic ib T x x e xe x lim = lim ic ib ic ib = lim( ice xe ie xbe ) = i ( cx xb ) = δ cb, ( x ). The secod equaliy follows from he L Hopial rule. A similar argume shows ha { ϕ } a uiformly coiuous oe parameer group of ier auomorphms o A wih he geeraor d c. We ed h secio wih he followig useful lemma which ca be foud i [3] ad []. Lemma Le M be a full Hilber A module ad le a A.The a = if ad oly if ax = for all x M. Approximaely Quasi Ier Geeralized Dyamics o Modules Defiiio 4 A geeralized dyamics { T } o A bimodule M called approximaely quasi ier if here exs wo sequeces { c } ad { b adjoi elemes i A such ha for each, ic ib T () = s lim T(), where T ( ) x = e xe which, i ur, meas ha for each ad x ic lim ib T x = e xe. M, Theorem 2 Le { T } be a geeralized dyamics safyig T ad δ be geeraor. If here ex wo sequeces { b } ad { c adjoi elemes i A such ha δ = s δ lim( ), c where δ ( x ) = i ( c x xb ) he { } c,, b approximaely quasi ier. proof. δ c iduces he uiformly coiuous oe ic ib parameer group T () x = e xe y Theorem. Now by assumpio he rage of δ dese i M ad by Troer-Kao approximaio heorem [3] for each T = s lim T., dyamics o he T be a geeralized dyamics o a full Hilber Theorem 3 Le { ϕ } be a algebra A ad { } module M. If { } approximaely quasi ier, he { ϕ } approximaely ier. 247

4 Vol. 23 No. 3 Summer 22 Mosadeq e al. J. Sci. I.. Ira proof. Sice { T } approximaely quasi ier, he here ex wo sequeces { c } ad { b adjoi elemes i A such ha for each, x M, ic ib T () x = lim T () x, where T ( ) x = e xe. Take ic ic () ϕ a e ae = ad le z M. The here exs x M such ha z = T x. Thus for all ad a A we have ϕ ϕ az. az. ϕ = ϕ at. x at. () x. ϕ. = ϕ at x at x ϕ + ϕ at. x at. () x T () ax T () ax + T () x T () x. Thus lim ϕ ( a ) = ϕ () a(by Lemma ). emark 2 (i) I he sese of Theorem 2, a geeralized dyamics { T } o A bimodule M safyig T wih he geeraor δ approximaely quasi ier if here ex wo sequeces { b } ad { c adjoi elemes i A such ha δ = s δ lim( c, ). b (ii) Followig he mehod as saed i he proof of Theorem 3, i ca be show ha if δ a quasi approximaely ier geeralized d derivaio ad D ( δ ) a full Hilber Dd module, he d approximaely ier. Defiiio 5 A geeralized derivaio δ called: (i) quasi approximaely ier if here ex wo sequeces { b } ad { c adjoi elemes i A such ha for each x D( δ), δ ( x ) = lim δc ( x ). (ii) approximaely bouded if here ex a sequece δ of bouded geeralized derivaios o M such { } ha { δ } coverges srogly o δ o D ( δ ). Because of boudedess of ier geeralized derivaio, each quasi approximaely ier geeralized derivaio also approximaely bouded. O he oher had every geeralized derivaio o a uial commuaive semi simple Baach algebra, a uial simple algebra or a Vo-Neuma algebra geeralized ier, cf. [2]. Therefore each approximaely bouded geeralized derivaio o he meioed spaces approximaely ier. Theorem 4 Le δ be he geeraor of geeralized dyamics { T } o A bimodule M safyig T. If δ a quasi approximaely ier geeralized d derivaio ad ( δ) D( ( δ)) dese i M, he { T } approximaely quasi ier. proof. Sice δ a quasi approximaely ier geeralized d derivaio, so here ex wo sequeces { b } ad { c adjoi elemes i A such ha δ ( x ) = lim δc,. b x Also δ c iduces he uiformly coiuous oe parameer group ic ib T x = e xe y Theorem. By emark 2, i eough o show ha h aim we have δ = s lim( δ ). For c ( δc ) ( δ)( x ) ( δ) ( δ)( x ) ( δc ) ( δ)( x ) ( δc ) ( δc )( x ) = ) ( δc ) ( δ)( x ) ( δc )( x ) ( δ)( x ) ( δ )( x ) Sice []). δ c c (By Hille-Yosida heorem Now he desiy of ( )( D ) ha ( δ ) ( δ) c δ δ i M implies (srogly). Defiiio 6 A subse D of domai DS of a closed liear operaor S o a Baach space X called a core for S, if S he closure of resricio o D. Theorem 5 Le δ be he geeraor of geeralized dyamics { T } o A bimodule M Safyig T. If here ex wo sequeces { b } ad { c } of self adjoi elemes i A such ha for all x i a core D for δ, δ ( x ) = lim δc ( x ), he { T } approximaely quasi ier. 248

5 Approximaely Quasi Ier Geeralized Dyamics o Modules proof. Firs oe ha δ c iduces he uiformly ic coiuous oe parameer group T x e xe ib =. Also by Hille-Yosida Theorem λ = ρ ( δ ) ρ ( δ, ) ( δ ) c. Furher ad he rage ( ) M. δ ad δ of δ We are goig o show ha ( δ ) ( δ) c (srogly). For h aim, le M : = {( δ )( y) : y D } ( ) = δ. Firs we show ha M dese D i M. Le x M. Sice ( δ ) = M, so here exs z D δ such ha x = z δ ( z ). Bu D a core for δ. Thus here exs a sequece { y } i D such y z ad y δ ( y ) z δ( z ) = x. ha Hece M dese i M. Now we show ha δ c o M o ( δ ). For, le z B. coverges srogly There exs y D such ha z y δ ( y ) by assumpio δ δ. Therefore y y, ( δ, ) ( δ) z ( z) ( δ, ) δ, δ ( δ) z = ( ) ( δ δ) δ c z ( δ δ) = c y = ad Fially, give x M ad >. Sice M dese i M, so here ex z B ad N such ha z x < ad for each N 3 ( δ ) ( z ) <. Therefore 3 ( δ, ) ( δ) x ( x) ( δc ) ( z ) ( δc ) ( x), ( δ ) ( z ), δ ( z ) ( δ) ( x ) + ( δc ) ( z ) ( δ) ( z ) + ( δc, ) ( ) b z x + δ z x + 2 < + =. 3 3 osequely, ( δ ) ( δ) c srogly o M. Le { ϕ } be a esuls ad Dcussio dyamics o a 3 algebra A wih he geeraor d, { T } be a geeralized dyamics o a full Hilber module M such ha. I has T ad le δ be he geeraor of { } bee proved ha each of he followig implies ha { ϕ } approximaely ier ad { } approximaely quasi ier: (i) There ex wo sequeces { b } ad { c adjoi elemes i A such ha ( δ ) = s lim( δ ), where δ ( x ) = i ( c x xb ) c c,. b (ii) δ a quasi approximaely ier geeralized d derivaio ad ( δ) ( D ( δ)) dese i M. c are (iii) There ex wo sequeces { b } ad { } wo sequeces of self adjoi elemes i A such ha δ ( x ) = lim δ ( x ), for all x i a core D for δ. c Ackowledgemes The auhors would like o hak he referee for heir useful commes ad suggesios. Also we should appreciae Prof. Moslehia for h precious revig. efereces. Abbaspour, Gh., Moslehia, M.S., ad Nikam, A., Dyamical sysems o Hilber -modules. Bull. Iraia Mah. Soc., 3: 25-35(25). 2. Abbaspour, Gh., Moslehia, M.S., ad Nikam, A., Geeralized Derivaios o modules. Bull. Iraia Mah. Soc., 32: 2-3(26). 3. Amyari, M., ad Nikam, A., A Noe o Fler modules, 249

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