Weakly almost periodic Banach algebras on semi-groups

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1 See discussios, stats, ad author profiles for this publicatio at: Weakly alost periodic Baach algebras o sei-groups ARTICLE JANUARY 2015 Source: arxiv DOWNLOADS 19 VIEWS 21 2 AUTHORS, INCLUDING: Ali Rejali Uiversity of Isfaha 55 PUBLICATIONS 73 CITATIONS SEE PROFILE Available fro: Ali Rejali Retrieved o: 05 August 2015

2 WEAKLY ALMOST PERIODIC BANACH ALGEBRAS ON SEMI-GROUPS arxiv: v1 [ath.fa] 26 Ja 2015 BAHRAM KHODSIANI, ALI REJALI Abstract. Let WAP(A) be the space of all weakly alost periodic fuctioals o a Baach algebra A. The Baach algebra A for which the atural ebeddig of A ito WAP(A) is bouded below is called a WAP-algebra. We show that the secod dual of a Baach algebra A is a WAP-algebra, uder each Ares products, if ad oly if A is a dual Baach algebra. This is equivalet to the Ares regularity of A. For a locally copact foudatio seigroup S, we show that the absolutely cotiuous seigroup easure algebra M a(s) is a WAP-algebra if ad oly if the easure algebra M b (S) is so. 1. Itroductio ad Preiaries The dual A of a Baach algebra A ca be tured ito a Baach A odule i a atural way, by settig f a,b = f,ab ad a f,b = f,ba (a,b A,f A ). A dual Baach algebra is a Baach algebra A such that A = (A ), as a Baach space, for soe Baach space A, ad such that A is a closed A subodule of A ; or equivaletly, the ultiplicatio o A is separately weak*-cotiuous. It should be rearked that the predual of a dual Baach algebra eed ot be uique, i geeral (see [6, 9]); so we usually poit to the ivolved predual of a dual Baach algebra. A fuctioal f A is said to be weakly alost periodic if {f a : a 1} is relatively weakly copact i A. We deote by WAP(A) the set of all weakly alost periodic eleets of A. It is easy to verify that, WAP(A) is a (or) closed subspace of A. For ore about weakly alost periodic fuctioals, see [7]. It is kow that the ultiplicatio of a Baach algebra A has two atural but, i geeral, differet extesios (called Ares products) to the secod dual A each turig A ito a Baach algebra. Whe these extesios are equal, A is said to be (Ares) regular. It ca be verified that A is Ares regular if ad oly if WAP(A) = A. Further iforatio for the Ares regularity of Baach algebras ca be foud i [6, 7]. WAP-algebras, as a geeralizatio of the Ares regular algebras, has bee itroduced ad itesively studied i [8]. Ideed, a Baach algebra A for which 2010 Matheatics Subject Classificatio. 43A10, 43A20, 46H15, 46H25. Key words ad phrases. WAP-algebra, dual Baach algebra, Ares regularity, weak alost periodicity. c 2008 Uiversiteti i Prishtiës, Prishtië, Kosovë. Subitted xxxxx xxxxx. Published xxxxx xxxx. Supported by the Ceter of Excellece for Matheatics at the Isfaha uiversity. 1

3 2 B.KHODSIANI, A.REJALI the atural ebeddig of A ito WAP(A) is bouded below, is called a WAPalgebra. It has also kow that A is a WAP-algebra if ad oly if it adits a isoorphic represetatio o a reflexive Baach space. If A is a WAP-algebra, the WAP(A) separate the poits of A ad so it is ω -dese i A. It ca be readily verified that every dual Baach algebra, ad every Ares regular Baach algebra, is a WAP-algebra for copariso see [14]. Moreover group algebras are also always WAP-algebras, however, they are either dual Baach algebras, or Ares regular i the case where the uderlyig group is ot discrete, see [5, Corollary3.7] ad [18, 14]. Aple iforatio about WAP-algebras with further details ca be foud i the ipressive paper [8]. The paper is orgaized as follows. I sectio 2 we itroduce soe algebraic ad topologic properties of WAP-algebras. How to iherit the weakly alost periodicity of Baach algebras to the ideals, quotiet spaces, direct sus ad by hooorphiss are studied. We show that the secod dual of a Baach algebra A is a WAP-algebra, uder each Ares products, if ad oly if A is Ares regular. I sectio 3 we deal with a locally copact foudatio seigroup S, we show that the absolutely cotiuous seigroup easure algebra M a (S) is a WAP-algebra if ad oly if the easure algebra M b (S) is so. 2. Soe algebraic ad topologic properties of WAP-algebras N. J. Youg turs out a Baach algebra A adits a isoetric represetatio o a reflexive Baach space if ad oly if the weakly alost periodic fuctioals of or oe deteries the or o A. The followig propositio provides a property qualificatio of WAP-algebras. Propositio 2.1. Let A be a Baach algebra. The the followig stateets are equivalet. (i) A is a WAP- algebra. (ii) There is a reflexive Baach space E ad a bouded below cotiuous represetatio π : A B(E). (iii) There is a dual Baach algebra (or WAP-algebra) B ad a bouded below cotiuous hooorphis φ : A B. Proof. (i) (ii): The caoical ap ι : A WAP(A) is bouded below ad WAP(A) isdualbaachalgebra. Accordigto[8, Corollary3.8]thereisareflexive Baach space E such that π : WAP(A) B(E) is a isoetric represetatio. So π ι : A B(E) is a bouded below cotiuous represetatio. (ii) (iii) Trivial. (iii) (i): Let ι : A WAP(A) be the caoical ap. Suppose that B is a dual Baach algebra ad φ : A B is a bouded below hooorphis, the φ (WAP(B)) WAP(A). Also: ι(a) = sup{ f(a) : f WAP(A) ad f 1} So ι is bouded below. sup{ φ (g)(a) : g WAP(B) ad φ g 1} 1 = φ φ(a) (sice B WAP(B) is isoetric) Corollary 2.2. Let. 1 ad. 2 be equivalet or o algebra A. The (A,. 1 ) is a WAP-algebra if ad oly if (A,. 2 ) is so.

4 WEAKLY ALMOST PERIODIC... 3 Proof. Sice the idetity ap is bouded below, it is a iediate cosequece of propositio 2.1 (iii). By above corollary, it is coveiet to suppose a WAP-algebra A has a isoetric represetatio o reflexive space E. This ca clearly be achieved by reorig A by a 1 = π(a),(a A) where π is a bouded below represetatio o E. Corollary 2.3. Let A be a WAP-algebra ad B be a closed subalgebra of A. The B is a WAP-algebra. Proof. Sice the iclusio ap is bouded below, it is a iediate cosequece of propositio 2.1 (iii). Propositio 2.4. Let E be a Baach space. The the followig stateets are equivalet: (i) B(E) is a dual Baach algebra. (ii) B(E) is a WAP- algebra. (iii) Eˆ E WAP(B(E)). (iv) E is a reflexive Baach space. Proof. (i) (ii) It is trivial. (ii) (iv)if E {0} is ot a reflexive Baach space, the WAP(F(E)) = {0} (see [18] Theore 3). Thus WAP(B(E)) ca ot separate the poits of F(E) ad so B(E). Thus it s ot a WAP-algebra. w (iv) (i)let (T α ) be a et i B(E) a T,S B(E) such that T α T. The for τ = =1 ξ ξ i Eˆ E, predual of B(E), we have: < ST α,τ > = < ST α (ξ ),ξ > = = =1 < T α (ξ ),S (ξ ) > < T(ξ ),S (ξ ) > =1 =1 < ST(ξ ),ξ >=< ST,τ > =1 Ad siilarly < T α S,τ > < TS,τ >. Hece the ultiplicatio o B(E) is separately w -cotiuous ad B(E) is a dual Baach algebra. (iii) (iv)if E {0} is ot a reflexive Baach space. The WAP(B(E)) = {0}([18, Theore3]) ad Eˆ E = {0}. This is a cotradictio. (iv) (iii) See [18, Theore 1]. The Propositio 2.1(iii)with Corollary 2.3 shows that a WAP-algebra A is isoorphic to a closed subalgebra of a dual Baach algebra, so it is coveiet to suppose that a WAP-algebra is or closed subalgebra of soe dual Baach algebra. This ca be achieved by redefiig suitable or. A large class of WAP-algebras are seisiple cuitative Baaach algebras. We refer to [13] for this kid of algebras. Propositio 2.5. Let A be a seisiple cuitative Baaach algebra, the A is WAP-algebra.

5 4 B.KHODSIANI, A.REJALI Proof. Sice (A) WAP(A) ad (A) seprerates the poits of A ad a 1 = sup{ f(a) : a (A)} is a Baach or o A. We kow that every Baach or o seisiple Baach algebras are equivalet ([13]). Thus A is a WAPalgebra. Let G be a locally copact group ad ω is a Borel-easurable weight fuctio o it. The the Fourier-Stieltjes algebra B(G) ad the weighted easure algebra M b (G,ω) are dual Baach algebras, also the Fourier algebra A(G) ad L 1 (G,ω) are closed ideals i the respectively. Hece all are WAP-algebras. Let (A,A ) be a dual Baach algebra ad I is a ideal i A, we defie the aihilators: I = {f A : f(i) = 0, for all i I}, I = {f A : f(i) = 0, for all i I}. It is easy to see that ( I) is the ω -closure of I i A. Let a WAP(A) = sup{ f(a) : f WAP(A), f 1}. Clearly. WAP(A).. The followig lea show that wheever the seior. WAP(A) could be a or o A. Lea 2.6. Let A be a Baach algebra. The the followig coditios are equivalet. (1) WAP(A) separates the poits of A. (2) (WAP(A)) = {0}. (3) WAP(A) σ(a,a) = A (4). WAP(A) is a or o A. Proof. Clearly (1) ad (4)are equivalet. (1) (2). Let 0 x A. The there is a f WAP(A) such that f(x) 0. Thus x (WAP(A)), so (WAP(A)) = {0}. (2) (3). A = 0 = ( (WAP(A)) = WAP(A) ω. (3) (1). Let WAP(A) ω = A the for 0 x A there is a f A such that f(x) 0. Let f α WAP(A) coverget to f i ω topology the there is a idex α such that f α (x) 0 so WAP(A) separates the poits of A. By usig [18, Theore 1] ad the previous lea, the followig is iediate. Propositio 2.7. The followig stateets are equivalet. (1) A is WAP-algebra. (2) WAP(A) separates the poits of A ad. WAP(A) ad. are equivalet. Theore 2.8. Let A = (A ) be a dual Baach algebra. Let I be a closed ideal i A, the the followig stateets are equivalet. (i) A I is a dual Baach algebra with respect to I. (ii) I is ω - closed ideal i A. Proof. (ii) (i) Let I be ω - closed ideal i A. The A I = A = A I ω ( I) = ( I) For x A ad f A we defie the odule actio of A I over I by (x+i).f = x.f ad f.(x+i) = f.x. It is routie to check that this odule actios are well defied

6 WEAKLY ALMOST PERIODIC... 5 ad A I.( I) ( I) ad ( I). A I ( I), so A I is a dual Baach algebra with respect to I. A (ii) (i). I = ( I) = A = A ( I) thus I ω (A I ) = ( A I ω ). So (I ) = (I ω ) ad thus I = I ω I the ext theore A is a WAP-algebra, so we suppose that A is a closed subalgebra of WAP(A). Theore 2.9. Let A be a WAP- algebra. (i) Let I be a σ(a,wap(a)) closed ideal i A, the A I is a WAP- algebra. (ii) Let B be a Baach algebra ad ϕ : A B be a oto cotiuous hooorphis, the B is a WAP- algebra. Proof. (i)let π be the caoical ap fro A oto A/I. It is well kow that π : (A/I) A is a isoetric isoorphis oto it s rag I. We coclude fro latter isoetry that, WAP(A/I) = WAP(A) I is the predual of I i the dual Baach algebra WAP(A). I is a ω -closed ideal of WAP(A) with respect to both Ares products. So WAP(A) /I = WAP(A/I) Sice A is a WAP-algebra, the A/I is a closed subalgebra of WAP(A) /I ad so it is a WAP-algebra. (ii) Let I = ker(ϕ), the I is closed ideal i A ad A/I = B isoetrically isoorphic. Thus by (i) the Baach algebra B is a WAP-algebra. Let A B deote the direct su of Baach algebras A ad B, ad edow it with coordiate wise operatio. Lea (i) If A ad B are WAP-algebras, the A B is WAPalgebras as well. (ii) If A ad B are dual Baach algebras, the A B is dual Baach algebra as well. (iii) If A ad B are Ares regular Baach algebras, the A B is Ares regular as well. Corollary Let A be the uitizatio of A. The: (i) A is a WAP-algebras if ad oly if A is WAP-algebras as well. (ii) A is a dual Baach algebras if ad oly if A is dual Baach algebra as well. (iii) A is Ares regular Baach algebras if ad oly if A is Ares regular as well. Theore Let A be a Baach algebra ad I is a closed copleeted ideal i A. If I ad A I are dual Baach algebra [respectively, WAP-algebra, Ares regular Baach algebra ], the A is a dual Baach algebra [respectively, WAP-algebra, Ares regular Baach algebra ]. Proof. (i): Let I = E (respectively A I = F ) where E (respectively F) is a I-biodule (respectively A I -biodule). Suppose that A := E F. The A = (E F) = E F = I A I = A. We ca regard A as a A-biodule by a.µ := (x.µ 1,y.µ 2 ) ad µ.a := (µ 1.x,µ 2.y)

7 6 B.KHODSIANI, A.REJALI where a = (x,y) ad µ = (µ 1,µ 2 ),x I,y A I,µ 1 E,µ 2 F. (ii):suppose that I is a closed subalgebra of WAP(I) ad A I is closed subalgebra of WAP(A) /Īω = WAP(A/I) the A = I A I is a closed subalgebra of WAP(I) WAP(A) /Īω. So A is a WAP-algebra. (iii): Suppose that WAP(I) = I ad WAP(A/I) = (A/I). The A = (I A I ) = I ( A I ) = WAP(I) WAP( A I ) = WAP(A). The last equality is a result of iterate it arguet. Let F WAP(I) WAP( A I ) ad (x,y ),(a,b ) are sequeces i uit ball of I A I. The F = (f 1,f 2 ) such that f 1 WAP(I) ad f 2 WAP( A I ). So that F(x,y ).(a,b ) = ( f 1 (x a ), f 2 (y b )) = ( f 1 (x a ), f 2 (y b )) = F(x,y ).(a,b ) ad F WAP(A). Let X ad Y be Baach spaces ad T : X Y be a bouded operator. Recall that the followig coditios are equivalet: (i) T = S, for soe bouded operator S : Y X. (ii) T (Y) X. (iii) T is ω -ω -cotiuous operator. We ow state the ai result of this sectio. Theore Let A be a Baach algebra. The the followig stateets are equivalet: (i) A is a Ares regular Baach algebra. (ii) (A,) is a dual Baach algebra. (iii) (A, ) is a dual Baach algebra. (iv) (A,) is a WAP-algebra. (v) (A, ) is a WAP-algebra. (vi) A is a WAP- Baach algebra ad WAP(A) is a ω -closed set i A. Proof. IfAis Aresregular, the by [1, Propositio1.2]A is dual Baachalgebra so(ii) ad (iii) are equivalet. By [12, Theore 4.10] there is a reflexive Baach space E such that A ad also A are closed subalgebras of B(E). Hece A is a WAP-algebra. Thus (i) (ii) (iii) (iv) were established. We showthat ifa with eitherares producthasarepresetatioo areflexive Baach space the it is a represetatio with aother Ares product. Thus the equality (iv) (v) was established. Let π : (A,) B(E) be a isoetric represetatio o a reflexive Baach space E (See corollary2.2 ad explaatios after it). By [8, Propositio 4.2] π is ω -ω -cotiuous. Let Φ,Ψ A ad (a α ) ad (b β ) be ets i A that ω -covergeigto Φ ad Ψ, respectively. The for each τ Eˆ E < τ,π(φψ) > = α β < τ,π(a α ) π(b β ) > = β α < τ,π(a α ) π(b β ) > (sice τ WAP(B(E))) = < τ,π(φ Ψ) >

8 WEAKLY ALMOST PERIODIC... 7 So π : (A, ) B(E) is a isoetric represetatio. Siilarly for aother Ares product. (v) (i)let (A, ) is a WAP-algebra. The there is a isoetric represetatio π : (A, ) B(E), for soe reflexive Baach space E. Hece by above equalities, π(ψ Φ) = π(ψ) π(φ) = π(ψφ) (Φ,Ψ A ) Sice π is ijective, we coclude A is Arers regular Baach algebra. (vi) (i) Let A be a WAP-algebra ad WAP(A) be a ω -closed set i A. By lea 2.6 WAP(A) σ(a,a) = A = WAP(A). (i) (vi) Let A be a a regular Baach algebra. Sice A is a closed subset of A, A is a WAP-algebra. ad WAP(A) σ(a,a) = A = WAP(A). So WAP(A) is a ω -closed subset of A. Propositio A is a Ares regular Baach algebra if ad oly if A is a Ares regular Baach algebra ad WAP(A ) = WAP(A). Proof. Let A be a Ares regular Baach algebra, the WAP(A ) = A. Also A is a closed subalgebra of A so it is Ares regular Baach algebra. Thus WAP(A) = A ad so WAP(A ) = WAP(A). Coversely, let A be a Ares regular Baach algebra the WAP(A) = A. Thus WAP(A ) = WAP(A) = A, ad so A is Ares regular. There exists a Baach algebra A such that A is Ares regular, but A is ot Ares regular (see [7]). Exaples (1) Let S = (N,i). The l 1 (S) is a WAP-algebra, but the secod dual is t so. (2) Let S = (N,ax). The l 1 (S) is a dual Baach algebra, but the secod dual is t so. (3) Let S be ifiite zero seigroup, the l 1 (S) is a dual Bach algebra but l 1 (S) is t so. 3. absolutely seigroup easure algebras Followig [4], a seitopological seigroup is a seigroup S equipped with a Hausdorff topology uder which the ultiplicatio of S is separately cotiuous. If the ultiplicatio of S is joitly cotiuous the S is said to be a topological seigroup. We write l (S) for the coutative algebras of all bouded coplexvalued fuctios o S. I the case where S is locally copact we also write C(S) ad C 0 (S) for the closed subalgebras of l (S) cosist of cotiuous eleets ad cotiuous eleets which vaish at ifiity, respectively. We also deote the space of all weakly alost periodic fuctios o S by WAP(S) which is defied by WAP(S) = {f C(S) : {R s f : s S} is relatively weakly copact i C(S)}, where R s f(t) = f(ts), (s,t S). The WAP(S) is a closed subalgebra of C(S). May other properties of WAP(S) ad its iclusio relatios aog other fuctio algebras are copletely explored i [4]. LetM b (S)deotesthe BaachspaceofallboudedcoplexregularBoreleasures

9 8 B.KHODSIANI, A.REJALI os withthetotalvariatioor. TheM b (S)withthecovolutioultiplicatio defied by the equatio µ ν,f = f(xy)dµ(x)dν(y) (f C 0 (S)) S S is a covolutio easure algebra as the dual of C 0 (S). Let µ M b (S) ad let L ( µ ) be the Baach space of all bouded Boreleasurable fuctios o S with essetial supreu or, f µ = if{α 0 : {x S : f(x) > α} is µ ull}. Cosider the product liear space Π{L ( µ ) : µ M b (S)}. A eleet f = (f µ ) i this product is called a geeralized fuctio o S if the followig coditios are satisfied: (1) f = sup{ f µ : µ M b (S)} is fiite. (2) if µ,ν M b (S) ad µ ν, the f µ = f ν, µ -a.e. Let GL(S) deote the liear subspace of all geeralized fuctios o S. The M b (S) is isoetrically isoorphic to GL(S), see [15]. We idetify C 0 (S) with its iage by caoical ap ι : C 0 (S) GL(S). The w -topology o GL(S) is σ(gl(s),m b (S)). Let A be a covolutio easure algebra i M b (S), i.e. a orclosed solid subalgebra of M b (S). I other words, for all µ M b (S) ad ν A such that µ ν, we have µ A. The space of all easures µ M b (S) for which the aps x δ x µ ad x µ δ x fro S ito M b (S) are weakly cotiuous is deoted by M a (S), where δ x deotes the Dirac easure at x. The M a (S) is a closed two-sided L-ideal of M b (S) ; see Baker ad Baker [3] or Dziotyiweyi [11]. The locally copact seigroup S is called foudatio if, F a (S), the closure of the set {supp(µ) : µ M a (S)} coicide with S. Let F ad K be oepty subsets of a seigroup S ad s S. We put s 1 F = {t S : st F}, ad Fs 1 = {t S : ts F} ad we also write s 1 t for the set s 1 {t}, FK 1 for {Fs 1 : s K} ad K 1 F for {s 1 F : s K}. The authors ad H.R. Ebrahii- Vishki i [14], for a locally copact topological seigroup S showed that M b (S) is a WAP-algebra if ad oly if the caoical appig R : S S wap is oe to oe, where S wap is the weakly alost periodic copactificatio of S. I this sectio we show that for a foudatio seigroup S, this is equivalet to M a (S) is a WAP-algebra. The followig theore is the ai result of this sectio. Theore 3.1. Let S be a foudatio locally copact topological seigroup with idetity. The M a (S) is a WAP-algebra if ad oly if M b (S) is so. Proof. If M b (S) is a WAP-algebra the M a (S) is a closed subalgebra i it, so M a (S) is a WAP-algebra. As the proof of lea 1.8 [11, p.16] for all x 1,x 2 S with x 1 x 2 there is µ M a (S) such that δ x1 µ δ x2 µ. Let M a (S) is a WAP-algebra, the for all µ M a (S) we have µ = sup{ f,µ : f 1,f WAP(M a (S))} ad by [15, Theore3.7] WAP(S) = WAP(M a (S)),

10 WEAKLY ALMOST PERIODIC... 9 so µ = sup{ f,µ : f 1,f WAP(S)}. Thus there is a f WAP(S) such that µ,r x1 f = δ x1 µ,f δ x2 µ,f = µ,r x2 f Thus R x1 R x2. This eas the caoical appig R : S S wap is oe to oe, where S wap is the weakly alost periodic copactificatio of S. By [14, Theore 3.1] M b (S) is a WAP-algebra. By usig[14, Theore3.1] ad the previous Theore, the followig is iediate. Corollary 3.2. Let S be a foudatio seigroup. The followig stateets are equivalet. (1) M a (S) is WAP-algebra. (2) l 1 (S) is a WAP-algebra. (3) M b (S) is a WAP-algebra. By usig [16, Corollary8] ad Theore2.13, the followig is iediate. Corollary 3.3. Let S be a foudatio seigroup. The followig stateets are equivalet. (1) M a (S) is WAP-algebra. (2) l 1 (S) is a WAP-algebra. (3) M b (S) is a WAP-algebra. The ext exaple shows that i Theore3.1 the coditio S be foudatio seigroup ca ot be oitted. Exaples 3.4. (a) Let S be a locally copact seigroup. IfC 0 (S) WAP(S) the M a (S) is a WAP-algebra. (b) Let S be a copact seigroup, the M a (S) is a WAP-algebra. (c) Let S = R 2 ad (x,y).(α,β) = (αx,αy + β) with the stadard Euclidea topology. The M a (S) = {0} ad so S is ot a foudatio seigroup. I fact, let x = (0,1+ 1 ), x 0 = (0,1), K = [ 1,1] [ 1,1] ad µ Ma r(s). The Kx 1 = ad Kx 1 0 = S. The by cotiuity of µ, we have 0 = µ (Kx 1) µ (Kx 1 0 ) = µ. So M a(s) = {0}. Let f C 0 (S) ad f(0,c) 0 for soe c R. Suppose a = (0,) ad b = (1,γ ) where γ is o the lie y = x+c. The f(a b ) = f(0,c) f(a b ) = 0 So by exaiig the iterate its we certify that f WAP(S). Thus WAP(S) does ot separate the poits of Y ad so M b (S). This show that the ap R : S S wap is ot oe to oe. Therefore M b (S) is t a WAPalgebra ( see [14] also) but clearly M a (S) is a WAP- algebra. Ackowledgets. This research was supported by the Ceter of Excellece for Matheatics at the Isfaha uiversity. Refereces [1] M. Abolghasei, A. Rejali, ad H.R. Ebrahii Vishki, Weighted seigroup algebras as dual Baach algebras, arxiv: [ath.fa]. [2] F. Abtahi, B. Khodsiai, ad A. Rejali, Ares regularity of iverse seigroup algebras, Bull. Iraia Math. Soc. Vol. 40 (2014), No. 6, pp

11 10 B.KHODSIANI, A.REJALI [3] A.C. Baker, J.W. Baker, Algebras of easures o a locally copact seigroups, 1, J.Lodo Math., o.1 (1969), [4] J.F. Berglud, H.D. Jughe ad P. Miles, Aalysis o Seigroups, Wiley, New York, (1989). [5] R.B. Burckel, Weakly Alost Periodic Fuctios o Seigroups, Gordo ad Breach, New york,(1970). [6] H.G. Dales, Baach Algebras ad Autoatic Cotiuity, Claredo Press, Oxford, [7] H.G. Dales, A.T.M. Lau, The Secod Dual of Beurlig Algebras, Me. Aer. Math. Soc. 177 (2005), o [8] M. Daws, Dual Baach algebras: Represetatio ad ijectivity, Studia Math, 178(2007), [9] M. Daws, H.L. Pha ad S. White, Coditios iplyig the uiqeess of the weak topology o certai group algebras, Housto J. Math. 35 (2009), o. 1, [10] H.A.M. Dziotyiweyi, Weakly alost periodic fuctios ad the irregularity of ultiplicatio i seiogroup algebras, Math. Scad.46, (1980), [11] H.A.M. Dziotyiweyi, The Aalogue of the Group Algebra for Topological Seigroups, Reserch otes i atheatics, Pita Publishig,(1984). [12] S. Kaijser, O Baach odules I, Math. Proc. Cabridge Philos. Soc. 90 (1981). [13] E.Kaiuth, A Course i Coutative Baach Algebras Spriger New York [14] H.R. Ebrahii Vishki, B. Khodsiai ad A. Rejali, Weighted seigroup easure algebra as a WAP-algebra, U.P.B. Scietific Bulleti Accepted for publicatio. [15] M. Lashkarizadeh Bai, Fuctio algebras o weighted topological seigroups, Math. Japoica 47, o.2 (1998), [16] A. Rejali, The Ares regularity of weighted seigroup algebras, Sci. Math. Japo, 60, o.1 (2004), [17] A. Ulger, Cotiuity of weakly alost periodic fuctioals o L 1 (G), Quart. J. Math. Oxford(2), 37 (1986), [18] N.J.Youg, Periodicity of fuctioals ad represetatios of ored algebras o reflexive spaces., Proc. Ediburgh Math. Soc.(2)20 (1976/77), Bahra Khodsiai Departet of Matheatics, Isfaha Uiversity, IRAN E-ail address: b khodsiai@sci.ui.ac.ir Ali Rejali Departet of Matheatics, Isfaha Uiversity, IRAN. E-ail address: rejali@sci.ui.ac.ir