Research Article The Arens Algebras of Vector-Valued Functions

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1 Fuctio Spaces, Article ID , 4 pages Research Article The Ares Algebras of Vector-Valued Fuctios I G Gaiev,2 adoiegamberdiev,2 Departmet of Sciece i Egieerig, Faculty of Egieerig, Iteratioal Islamic Uiversity Malaysia, PO Box 0, Kuala Lumpur, Malaysia 2 Departmet of Algebra ad Fuctioal Aalysis, Natioal Uiversity of Uzbekista, Vuzgorodok, 0074 Tashket, Uzbekista Correspodece should be addressed to I G Gaiev; iam@iiumedumy Received 4 September 203; Accepted 5 December 203; Published 23 February 204 Academic Editor: Draga Djordjevic Copyright 204 I G Gaiev ad O I Egamberdiev This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited A class of Ares algebras of vector Baach algebra valued fuctios is cosidered It is show that Ares algebra of vector-valued fuctios is complete locally covex metrizable algebra Itroductio Let (, Σ, μ) be a measure space with fiite measure μ ad let L p =L p (, Σ, μ) be the Baach space of all measurable fuctios f o (, Σ, μ) such that f p =( f p dμ) /p <, ( p<)i[]theset L ω = L p () p is itroduced ad it is show that L ω is locally covex algebra i topology τ geerated by the orms p Suchalgebras are called Ares algebras I [2] itisshowthat(l ω,τ) is GB algebra Properties of algebra L ω are cosidered i [3, 4] Moreover, it is show that dual of L ω is <q L q ad L ω is reflexive I [5] a class of ocommutative aalogue of Ares algebra L ω (M, φ) associated with semifiite vo Neuma algebra ad semifiite trace φ is itroduced ad i [6] itisshowthatl ω (M, φ) is EW algebra I [6 8], properties of ocommutative Ares algebras are cosidered Nocommutative Ares algebras L ω (M, Φ) associated with semifiitevoneumaalgebraadceter-valuedtraceφ are itroduced i [9]adi[0] they are geeralized for the case whe Φ is the Maharam trace Let L p (, X) be of Bocher maps from ito a uital Baach algebra (X, X ), itegrable with degree p, p<,thatis,l p space of vector-valued fuctios, ad L ω (, X) = L p (, X) p I the preset paper we prove that L ω (, X) is complete locally covex metrizable algebra, which we call Ares algebra of vector-valued fuctios It is show that if dual of X has Rado-Nikodym property, the dual space of L ω (, X) is <q L q (, X );ifx is reflexive, the L ω (, X) also is reflexive 2 Prelimiaries Let (, Σ, μ) be a measure space with fiite measure; let L 0 () be the classes of all measurable fuctios o Throughout this paper, X stads for a fixed iitial Baach algebra with the orm X ByL 0 (, X) we deote the classes of all measurable by Bocher mappigs from to X; that is, (2) L 0 (, X) ={f: X: f (ω) X L0 ()} (3) It is kow [] thatl 0 (, X) is algebra with respect to multiplicatio operatio (fg)(ω) = f(ω)g(ω) for ay

2 2 Fuctio Spaces f, g L 0 (, X) ByL p (, X), p<,wedeotethe Baach space of X valued measurable fuctios f o (, Σ, μ) with the orm defied as f p =( f(ω) p /p X dμ) = f(ω) X p ; (4) that is, L p (, X) = {f L 0 (, X) : f(ω) X L p } ad L (, X) = {f L 0 (, X) : λ > 0, f(ω) X λae} = {f L 0 (, X) : f(ω) X L } with the orm defied as f = if {λ > 0 : f (ω) X λae} (5) It is kow [] thatthepair(l (, X), ) is a Baach algebra Thefollowigwellkowpropertiesholdtrue is valid (a) For f, g L p (, X) the Mikowski iequality f+g p f p + g p (6) (b) For h L p ad f L q (, X), whereq:=p/(p ) for p =ad q:=for p=,thehölder iequality is valid hf h p f q (7) Propositio (i) If f L (, X) ad g L p (, X) the fg L p (, X) ad fg p f g p (8) (ii) If p, q, r, /r = (/p) + (/q), f L p (, X), ad g L q (, X),thefg L r (, X) ad fg r f p g q (iii) If f(ω) X ae, f L 0 (, X), adg L p (, X),thef L p (, X) ad f p Proof (i) Let f L (, X) ad g L p (, X) Sice f(ω)g(ω) X f(ω) X f ad L p we have that fg L p (, X) ad fg p =( f (ω) g (ω) p /p X dμ) f ( g (ω) p /p X dμ) = f g p (ii) Let p, q, r, /r = (/p) + (/q), f L p (, X), adg L q (, X) As f(ω) X L p ad L q from iequality f(ω)g(ω) X f(ω) X we get that fg L r (, X) ad fg r = ( f(ω)g(ω) r X dμ)/r ( ( f(ω) X ) r dμ) /r ( f(ω) p X dμ)/p ( g(ω) q X dμ)/q = f p g q (iii) Let f(ω) X ae, f L 0 (, X), adg L p (, X) The f(ω) p X dμ g(ω) p Xdμ < This meas that f L p (, X) ad f p (9) 3 The Ares Algebras L ω (,X) of Vector-Valued Fuctios I this sectio we itroduce otio of the Ares algebras of vector-valued fuctios Put L ω (, X) = L p (, X) ; p (0) that is, L ω (, X) = {f L 0 (, X) : f <, f 2 <,, f p <,}, ad will cosider i L ω (, X) local covex topology τ X is geerated by system of orms { p } p From (ii) Propositio we get that f f 2 f,thatis,thetopologyτ X geerated by coutable system of orms { } = By Theorem III22 [2] it meas that topological vector space (L ω (, X), τ X ) is metrizable space with respect to the metric ρ (f, g) = f g + f g () Theorem 2 (L ω (, X), τ X ) is a complete topological algebra Proof Let {f } be a fudametal sequece i (L ω (, X), τ X ) The {f } is fudametal i L p (, X) for ay p< Sice L p (, X) is complete there exist f p L p (, X) such that f f p p 0as,as f f p q μ()pq/p+q f f, p p (2) for all q<p<,impliesthat f f p q 0This meas that f p =f q if q<p< Therefore f=f p is limit of sequece {f } ad f is a elemet of space L ω (, X) Sice the iequality fg p f 2p g 2p (3) is valid the multiplicatio is cotiuous o L ω (, X)Hece (L ω (, X), τ X ) is a topological algebra Theorem 3 (i) If f L 0 (, X), g L ω (, X), ad f(ω) X ae the f L ω (, X) ad ρ(f, 0) ρ(g, 0) (ii) If g L (, X) ad f L ω (, X) theg f L ω (, X) ad there exist m N ρ (gf, 0) 2 m ρ (f, 0) (4) Proof (i) As g L ω (, X) we have that g L p (, X) for ay p Thefrom f(ω) X ae by Propositio (iii) we get that f L p (, X) for ay p This meas that f L ω (, X) Sice f p for ay p we have ρ(f,0)= f + f g + g =ρ(g,0) (5) (ii) Let g L (, X) ad f L ω (, X); thef L p (, X) for all p ByPropositio (i) g f L p (, X)

3 Fuctio Spaces 3 for all p Therefore gf L ω (, X)Wechosem N that g 2 m TheusigPropositio (i) we get ρ(gf,0)= k= gf + gf g f + g f 2 m f 2 k +2 m f 2 m f 2 k + f =2 m ρ(f,0) k= (6) Theorem 4 L (, X) is dese i L ω (, X) ad L ω (, X) is dese i L p (, X) Proof Let f L ω (, X) There exists sequece of idempotets e from L 0 () such that e,where is uit elemet o L 0 () The f(ω)e (ω) X = f(ω) X e (ω) L (); that is, f(ω)e (ω) L (, X) Letf = fe Thef L (, X) ad f(ω) f (ω) X = f(ω) f(ω)e (ω) X = f(ω)((ω) e (ω)) X = f(ω) X ((ω) e (ω)) 0 As orm p is order cotiuous i L p space for all p we get that f f p = f(ω) X ((ω) e (ω)) p 0 for all p as Therefore ρ(f,f)= f f + f 0, (7) f as The desity L ω (, X) i L p (, X) follows from desity L (, X) i L ω (, X) Let φ,φ 2,,φ L ω ad x,x 2,,x XThewe ca defie fuctio f: Xby settig f (ω) = i= φ i (ω) x i, (8) for almost all ω We will deote L ω X = {f L 0 (, X) : f(ω) = i= φ i(ω)x i,φ i L ω,x i X,i =,, N} L ω Xis subspace of L ω (, X), because f(ω) p X dμ i= φ i(ω) p x i p X dμ = i= x i p X φ i(ω) p dμ < for all p Theorem 5 L ω Xis dese i L ω (, X) Proof Let f L ω (, X)Thef L p (, X) for all p Let s (ω) be the sequece simple X valued fuctios; that is, every s (ω) has form { m i= χ B i (ω)x i :B i Σ,B i B j = Ø,x i X},suchthats (ω) f(ω) ae o The s (ω) X f(ω) X almost everywhere o Let A ={ω: s (ω) X 2 f(ω) X }Ifwesetg (ω) = s (ω)χ A (ω) we have that g (ω) f(ω) ae o ad sup g (ω) f(ω) sup X g (ω) + X f(ω) X (9) 3 f(ω) X By domiated covergece we have that g (ω) f(ω) p Xdμ 0, (20) for all p ad of course g L ω X 4 Dual of Ares Algebras L ω (,X) I this sectio we study dual of Ares algebras of vectorvalued fuctios It is show that if dual of X has Rado-Nikodym property, the dual space of L ω (, X) is <q L q (, X );ifx is reflexive, the L ω (, X) also is reflexive Defiitio 6 (see [3]) Recall that a Baach space Y is said to possess the Rado-Nikodym property, with respect to μ, if for ay μ-cotiuous vector measure m:σ Yof bouded total variatio, there exists a Bocher itegrable fuctio f: Ysuch that m (A) = A f dμ, (2) for all A ΣItiskow[3]thatdualofBaachalgebrac 0, c 0 =l has the Rado-Nikodym property AclassofBaachalgebrasX whose duals X have the Rado-Nikodym property is studied i [4] Theorem 7 If the dual space X of Baach algebra X has Rado-Nikodym property with respect to μ, the the dual of L ω (, X) is the space E= <q L q (, X ), ad for ay l (L ω (, X)) there exists g Esuch that (22) l (f) = gfdμ(f L ω (, X)) (23) Proof Sice L ω (, X) is coutably ormed space by [5]dual of L ω (, X) is the space <q (L p (, X)),where(/p) + (/q) = AsX has the Rado-Nikodym property with respect to μ by Theorem IV [3] (L p (, X)) =L q (, X ), where (/p) + (/q) = Hece,thedualofL ω (, X) is the space E= L q (, X ), (24) <q where (/p) + (/q) = Let l be a cotiuous liear fuctioal o L ω (, X)The by [5] there exists atural umber p( p<)ad C>0 such that l(f) C f p (25) It meas l (L p (, X)) Usig the geeral form of cotiuous liear fuctioals i L p (, X) (see [3], IV) we get that there exists g L q (, X ) Esuch that l (f) = gf dμ (26)

4 4 Fuctio Spaces Theorem 8 If X is a reflexive Baach algebra the L ω (, X) is reflexive Proof Let X be a reflexive Baach algebra Sice every reflexive space has Rado-Nikodym property with respect to μ,by Theorem 7, L ω (, X) =( L q (, X )) <q = L p (, X) =L ω (, X) p = L p (, X ) p (27) [2] L V Katorovich ad G P Akilov, Fuctioal Aalysis,vol4, Pergamo Press, Oxford, UK, 2d editio, 982 [3] J DiesteladJ J UhlJr, Vector Measures, America Mathematical Society, Providece, RI, USA, 977 [4] H S Mustafayev, A class of Baach algebras whose duals have the Rado-Nikodym property, Archiv der Mathematik,vol87, o5,pp ,2006 [5] B Z Vulikh, Itroductio to Fuctioal Aalysis, Pergamo Press, New York, NY, USA, 963 [6] H H Schaefer, Topological Vector Spaces, Spriger, New York, NY, USA, 97 This meas that L ω (, X) is semireflexive space By Theorem 2 L ω (, X) is (F)-space Sice by Corollary II7 [6] every(f)-space is barreled we have that L ω (, X) is barreled Hece by IV5 [6] L ω (, X) is reflexive Coflict of Iterests The authors declare that there is o coflict of iterests regardig the publicatio of this paper Ackowledgmet The first author (I G Gaiev) ackowledges the MOHE Grat FRGS Refereces [] R Ares, The space L ω ad covex topological rigs, Bulleti of the America Mathematical Society,vol52,pp93 935,946 [2] GRAlla, Oaclassoflocallycovexalgebras, Proceedigs of the Lodo Mathematical Society, vol7,o3,pp9 4, 967 [3]SJBhatt, OAres algebral ω, Glasik Matematički, vol 5(35), o 2, pp , 980 [4] SJBhatt, OAres algebral ω II, Glasik Matematički,vol 6(36), o 2, pp , 98 [5] A Ioue, O a class of ubouded operator algebras II, Pacific Mathematics,vol66,o2,pp4 43,976 [6] BSZakirov, NocommutativeAresalgebras, Uzbek Mathematical Joural,o,pp7 24,997(Russia) [7] R Z Abdullaev, Spaces dual to ocommutative Ares algebras, Uzbek Mathematical Joural, o2,pp3 7,997 (Russia) [8] RZAbdullaevadVIChili, Aresalgebras,associatedwith commutative vo Neuma algebras, Aales Mathématiques Blaise Pascal,vol5,o,pp 2,998 [9] A A Alimov, Ares algebras associated with a ceter-valued trace, Uzbek Mathematical Joural, o, pp 2 8, 2000 (Russia) [0] A A Alimov, Ares algebras ofmeasurable operators for Maharam traces, [] V I Chili, I G Gaiev, ad K K Kudaĭbergeov, The Gelfad-Naĭmark theorem for C -algebras-algebras over a rig of measurable fuctios, Russia Mathematics, vol 52, o 2, pp58 66,2008

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