ϕ -MEANS OF SOME BANACH SUBSPACES ON A BANACH ALGEBRA

Transcription

1 THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY, Seres A, OF THE ROMANIAN ACADEMY Volue 3, Nuber 4/202, pp ϕ -MEANS OF SOME BANACH SUBSPACES ON A BANACH ALGEBRA Al GHAFFARI, Saaneh JAVADI Senan Unversty, Departent o Matheatcs, Senan, Iran E-al: aghaar@senanacr, sjavad62@galco In ths paper, aong the other thngs, we study the concept o ϕ -aenablty o a Banach algebra A, where ϕ s a nonzero ultplcatve lnear unctonal on A We present a ew results n the theory o ϕ -aenable Banach algebras, and we obtan necessary and sucent condtons or A to have a let nvarant ϕ -ean Let Wap( A ) be the Banach space o all wealy alost perodc unctonals on A Our second purpose n ths paper s to present several characterzatons o the exstence o a let (rght) nvarant ϕ -ean on Wap( A ) Other results n ths drecton are also obtaned Key words: Banach algebra, ϕ -aenablty, ϕ -eans, wea alost perodc, wea topology INTRODUCTION The concept o aenablty or Banach algebras was rst ntroduced by Johnson n [4] Accordng to Johnson' s denton, a Banach algebra A s aenable every dervaton ro A nto the dual A -bodule E s nner or all Banach A -bodules E Ths concept o aenablty has occuped an portant place n the research o Banach algebras, operator algebras and haronc analyss In [7], Lau ntroduced and nvestgated a large class o Banach algebras whch he called F -algebras Later, F -algebras were tered Lau algebras They are Banach algebras A such that the dual A s a von Neuann algebra and the dentty o A s a ultplcatve lnear unctonal on A The concept o let aenablty or a Lau algebra has been extensvely extended or an arbtrary Banach algebra by ntroducng the noton o ϕ -aenablty Let A be an arbtrary Banach algebra and ϕ be a character o A, that s a hooorphs ro A onto C A s called ϕ -aenable there exsts a bounded lnear unctonal on A satsyng, ϕ = and, a =ϕ ( a), or all a A and A Ths concept consderably generalzes the noton o let aenablty or Lau algebras For ore detals on ϕ -aenablty o a Banach algebra the nterested reader s reerred to [3], [6] and [8] Recently the noton o -aenable hypergroups was ntroduced and studed n [2], [3] and [8] The an purpose o ths paper s to nvestgate the ϕ -aenablty or certan Banach subspaces o dual Banach algebras We contnue our wor [0] n the study o aenablty o a Banach algebra A dened wth respect to a character ϕ o A Varous necessary and sucent condtons are ound or a Banach algebra to possess a let nvarant ϕ -ean We prove that Wap( A ) has a let (rght) nvarant ϕ -ean and only a certan copact setopologcal segroup S (to be dened) assocated wth { λ ; a P a ( A, ϕ )} contans a let (rght) zero Other results n ths drecton are also obtaned We obtan sucent condtons and soe necessary condtons about A to have a let nvarant ϕ -ean 2 NOTATION AND PRELIMINARY RESULTS In ths paper, the second dual A o a Banach algebra A wll always be equpped wth the rst Arens product whch s dened as ollows For ab, A, A and n, A, the eleents a and o A and n A are dened by

2 2 ϕ-eans o soe Banach subspaces on a Banach algebra 303 ab, =, ab, n, a = n, a, n, =, n, respectvely Wth ths ultplcaton, A s a Banach algebra and A s a subalgebra o A [5] We wrte A A or the closed lnear span n A o { a; A, a A} When A has a bounded rght approxate dentty the Cohen-Hewtt actorzaton theore (Theore 3222 o [2]) shows that n act A A= { a; A, a A} A unctonal A or whch { a; a } s relatvely copact n the wea (nor) topology o A s sad to be wealy alost perodc (alost perodc) The set o wealy alost perodc (alost perodc) unctonals on A s denoted by Wap( A ) ( Ap( A )) (see [7] and []) I A s a Banach algebra, we shall denote by B( Wap( A )) the usual Banach algebra o bounded lnear operators on Wap( A ) B( Wap( A )) s a setopologcal segroup under operator ultplcaton and the wea operator topology We shall spea o any subsegroup o B( Wap( A )) as a segroup o operators Fnally, we say that an eleent a o A s ϕ -axal t satses a =ϕ ( a) = Let P( A, ϕ ) denote the collecton o all ϕ -axal eleents o A [5] When A s an Lau algebra and ϕ s the dentty o the von Neuann algebra A, the ϕ -axal eleents are precsely the postve lnear unctonals o nor on A and hence span A Let X( A, ϕ ) denote the closed lnear span o P( A, ϕ ) I A and a A, we also consder λ a ( ) = a Throughout the paper, ( A) wll denote the set o all hooorphss ro A onto C 3 MAIN RESULTS Let A be a Banach algebra and let X be a closed subspace o a X whenever X and a A A We say that X s nvarant Denton 3 Let A be a Banach algebra and let X be a closed subspace o A wth ϕ X that s nvarant A contnuous unctonal on X s called a let nvarant ϕ -ean on X the ollowng propertes holds:, ϕ =,, a =ϕ ( a), ( X, a A) LEMMA 32 Let A be a Banach algebra and let ϕ ( A) Suppose that A has a bounded approxate dentty Then A has a let nvarant ϕ -ean and only ( A A ) has a let nvarant ϕ -ean Proo Observe rst that nvarance propertes o ϕ -eans are transtted by heredty onto subspaces So ( A A ) has a let nvarant ϕ -ean To prove the converse, assue that n s a let nvarant ϕ -ean on A A Let A such that extends n and = n Choose a A wth ϕ ( a) = Let { e } I be a bounded approxate dentty n A For A and b A, we have a, b =, ba = n, ba = l n, e ba =ϕ ( ba)l n, e =ϕ ( b)l n, e a =ϕ ( b) n, a =ϕ ( b) a, Ths shows that a s a let nvarant ϕ -ean on A Let G be a locally copact group and let LUC( G ) denote the Banach space o all bounded let unorly contnuous unctons on G For L ( G) and φ L ( G) we now that φ=φ (where φ ( x) = ( x ) φ( x ), here beng the odular uncton on G ) Thus

3 304 Al Ghaar, Saanech Javad 3 LUC( G) = L ( G) L ( G) = L ( G) L ( G) [2] Lea 32 shows that LUC( G) ϕ -ean and only L ( G) has a let nvarant ϕ -ean has a let nvarant Rear 33 Let A be a Banach algebra wth a bounded approxate dentty and ϕ ( A) Suppose that A adts a let nvarant ϕ -ean By Theore 4 n [6], there exsts a bounded net { a } I n A such that aa ϕ( a) a 0 or all a A and ϕ ( ) = or all I Thereore baa ϕ( a) ba 0 or a every a, b A Conversely, suppose that there exsts a bounded net { a } I n A such that ϕ ( a ) = or all I and baa ϕ( a) ba 0 or all a, b A Then every wea cluster pont o { a } I n A clearly satses, ϕ = and, a =ϕ ( a), or all A A and a A By Lea 32, A has a let nvarant ϕ -ean Suppose that or each A there exsts n A such that n n, = ϕ = and n, ab = ϕ ( ab) n, or all a, b A There exsts a net { a } I n A such that a n n the wea 2 topology Let = nn Then, ϕ = n, ϕ = and = Moreover, ϕ ( b), a =ϕ ( b) n n, a = l ϕ ( b) n, aa = l ϕ( b) ϕ ( aa ) n, = l ϕ ( a) n, ba = ϕ ( a) nn, b =ϕ ( a), b or every ab, A By Theore 2 n [0], A adts a let nvarant ϕ -ean o nor Thus we have shown that a Banach algebra A has a let nvarant ϕ -ean o nor and only or each A there exsts A such that n = n, ϕ = and n, ab =ϕ ( ab) n, or all ab, A n Proposton 34 Let A be a Banach algebra and ϕ ( A) Then the ollowng stateents are equvalent: () There exsts a let nvarant ϕ -ean wth = ; () ε (0, ),,, A and,, n n A, then =,, sup{ Re ϕ ( ( ) n ϕ ( n) ) p, ; p A, p, ϕ =, p < +ε} 0; () There exsts a net { } I n A such that l, ϕ = and = or all and or every a A, a ϕ( a) 0 n the wea topology o A Proo () ples ( ) Suppose that ε (0,),,,, A and bounded nets a, b A, I, such that can assue that a, b Obvousely, A adts a let nvarant ϕ -ean o nor, say Let,,, Snce A s wea dense n a, b n n the wea topology o n n A A [5], there are A (Notce that we have the sae drected set I ; otherwse tae the product o ther drected sets) ( ( ) n ( n ) ), l ( ( a ) b ( b ) a ), = ϕ ϕ = = ϕ ϕ l ( ), ( ), = ϕ a b ϕ b a = = l ϕ( a ) ϕ ( b )(,, ) 0, = = we have () ples () Conversely, suppose that () holds It s sucent to show that

4 4 ϕ-eans o soe Banach subspaces on a Banach algebra 305 sup{ Re ( ϕ( a ) b ϕ ( b ) a ) c, ; c A, ε<ϕ ( c) < +ε, c < +ε} 0, = where N, ε (0,),,, A and a, a, b, b A (see Theore 2 n [0] and ts proo) Let us assue on the contrary that there exst ε (0,),,, A and a, a, b, b A such that sup{ Re ϕ ( ( a ) b ϕ ( b ) a ) c, ; ε<ϕ ( c) < +ε, c < +ε } <β< 0 = By assupton, we can choose p A such that p, ϕ =, p < +ε and Re ( ( ) ( ) ), β ϕa b ϕ b a p = 2 By the Goldstne' s theore [2], there exsts a net { c } J n A such that c p n the wea topology and c p < +ε or all J Put c =ϕ ( c ) c For soe J, we have c < +ε, ε<ϕ ( c ) < +ε, ϕ( c ) p, ϕ <ε and β ϕ ( ( a) b ϕ ( b) a) p, ϕ ( ( a) b ϕ ( b) a) c, <, 2 or all Ths shows that β β Re ( ϕ( a ) b ϕ ( b ) a ) p, < Re ( ϕ( a ) b ϕ ( b ) a ) c, + <, 2 2 = = whch s a contradcton So () ples () Clearly, ( ) ples ( ) Suppose that ( ) holds I such a net { } I exsts, then every wea cluster pont o t n A clearly satses, ϕ =, = and, a =ϕ ( a), or all A and a A Ths copletes the proo I S s a segroup o operators on Wap( A ), the orbt O( ) o an eleent o Wap( A ) s dened to be { T ( ); T S} S wll be called wealy alost perodc each orbt has copact closure n the wea topology o Wap (A) In the ollowng theore, we obtan necessary and sucent condtons or Wap( A ) to have a let nvarant ϕ -ean Varous necessary and sucent condtons ound or wealy alost perodc unctons on a locally copact topologcal segroup to possess a let nvarant ean (see [9], [20], [23] and [24]) THEOREM 35 Let A be a Banach algebra and ϕ ( A) The closure S o S = { λa ; a P ( A, ϕ )} n the wea operator topology s a copact convex setopologcal segroup n the sae topology Moreover, aong the ollowng two propertes, the plcaton () () hold I X ( A, ϕ ) = A, then ( ) ( ) () Wap( A ) has a let nvarant ϕ -ean P (, A ϕ )w ; () The segroup S has a let zero, that s, there exsts soe T S S S such that SoT = S or any Proo For Wap(A), { a; a P ( A, ϕ )} has copact closure n the wea topology o A It ollows that { λa ( ); a P ( A, ϕ )} also has copact closure n the sae topology or { λ ; a P a ( A, ϕ )} s wealy alost perodc It s nown that S s wealy alost perodc, then the wea operator closure, S o S n B ( Wap( A)) s a copact setopologcal segroup wth wea operator topology, see Theore 3 n [6]

5 306 Al Ghaar, Saanech Javad 5 Next assue () holds and let P (, A ϕ )w be a let nvarant ϕ -ean on ( ) { a } I n P (, ) A ϕ wth the property that a and aa a 0 n the wea topology o Wap A Choose a net Wap( A ) or any a P( A, ϕ ) By copactness o S, we can assue that λ a converges to soe L S n the wea operator topology, passng to a subnet necessary We cla that LoT = L or all T S Let a P( A, ϕ ), Wap( A) and p Є Wap( A ) Choose a net { b β } β J n A wth the property that b β p n the wea topology o Wap( A ) and bβ p or all β J [5] Hence bβ p n the wea topology o Wap( A ) Snce s n Wap( A ), { a ; a p } s relatvely wealy copact [7] Now b, p all belong to the wealy copact set { a ; a p } on whch the wea topology and the wea topology concde Hence p Wap( A) Snce p Wap( A), we have p, Loλ L( ) = l p, λ oλ λ ( ) = l p, ( aa a) a a a a = l p, aa a = 0 We conclude that Lo λ a = L Snce { λ ; a a P ( A, ϕ )} s dense n S n the wea operator topology, we ust have LoT = L or any T S and () ples () Conversely assue () and let S be a let zero o S, then Soλ = S or any a P( A, ) ϕ There s a net { a } I n P (, ) A ϕ such that λ S a n the wea operator topology By the Banach-Alaoglu' s theore [2], wthout loss o generalty we ay assue that a n the wea topology o Wap( A ) We set out to prove that s a let nvarant ϕ -ean on Wap( A ) Clearly Wap( A) and a P( A, ) ϕ, we have β P (, A ϕ )w For every, a, =, ( a), = l, aa, a = = l, λ o λ ( ), λ ( ) = a a a = So, λ ( ) S, ( ) = 0 a Ths shows that s a let nvarant ϕ -ean on Wap( A ) We can replace the wea topology and the wea operator topology n the above Theore by the strong topology and the strong operator topology, respectvely, we have THEOREM 36 Let A be a Banach algebra and ϕ ( A) The closure S o S = { λa ; a P ( A, ϕ )} n the strong operator topology s a copact convex topologcal segroup n the sae topology Moreover, aong the ollowng two propertes, the plcaton () () hold I X ( A, ϕ ) = A, then () () () Ap (A) has a let nvarant ϕ -ean P (, A ϕ )w ; () The segroup S has a let zero We ntroduce soe addtonal concepts A lnear unctonal Wap( A) s called a rght nvarant ϕ - ean on Wap( A ), ϕ = and a, =ϕ ( a), whenever Wap( A) and a A A let nvarant and rght nvarant ϕ -ean on Wap( A ) s called nvarant ϕ -ean Kanuth, Lau and Py n [5] showed that an eleent o Wap( A ) s an nvarant ϕ -ean or Wap( A ) and only there exsts a bounded net { a } I n A such that ϕ ( a ) = or all and or each a A, aa ϕ( a) a 0, aa ϕ( aa ) 0

6 6 ϕ-eans o soe Banach subspaces on a Banach algebra 307 THEOREM 37: Let A be a Banach algebra and ϕ ( A) I Wap( A ) has a let nvarant ϕ -ean P (, A ϕ )w and a rght nvarant ϕ -ean n P (, A ϕ )w, then the copact setopologcal segroup S contans a let zero and a rght zero Moreover, = n and t s the unque nvarant ϕ -ean on Wap( A ) Proo Assue that Wap( A ) has a rght nvarant ϕ -ean n P (, A ϕ )w There s a net { b β } β J n P( A, ϕ ) such that or all a P( A, ϕ ), ba b 0 β β and also b β n n the wea topology o Wap( A ) An arguent slar to the proo o Theore 35 shows that the segroup S has a rght zero, say R n Snce L s a let zero o S and R n s a rght zero o S, we have L = LoRn = Rn Let Φ: Wap( A) B( Wap( A)) be dened by Φ ( p)( ) = p, ϕ Clearly Φ s a bounded lnear operator and an soetry I Φ( n) S, then Theore 34 n [2] ples that there s a p n Wap( A ), an n Wap (A), an c n R, and ε > 0 such that Re p, b c < c+ ε Re p, Φ ( n)( ) = Re n, p, ϕ β or all β J We conclude that Re np, c < c +ε Re n, p, ϕ o By Goldstn' s theore [2], there exsts a net { c } L n A such that c p n the wea topology Wap( A ) Snce Wap( A ) s Arens regular [7], we have Re n, p, ϕ = lre n, ϕ ( c ) = lre n, c = Re np, Ths s a contradcton We conclude that For each a A, Wap( A) and b A, < Re n, p, ϕ Φ n) ( S Clearly Φ (n) s a rght zero and so Φ ( n ) = Rn na, b = n, a ( b) =ϕ( a) n, b =ϕ ( a) n, b Thereore na =ϕ( a) n Ths shows that n, a =, n ( a ) =ϕ ( a), n =ϕ ( a) n,, and so n s a rght nvarant ϕ -ean n P (, A ϕ )w Consequently L ( ) =Φ ( n)( ) =Φ ( n)( ) = n, ϕ or every Wap( A) Hence = n= n s the unque nvarant ϕ -ean on Wap( A ) As a sple applcaton o the precedng theores, let G be a locally copact group Consder the Banach space W( G ) o all wealy alost perodc unctons on G wth supreu nor [4] It s nown that WG ( ) = WapLG ( ( )) [22] By the Ryll-Nardzews theore, W( G ) has a let and rght nvarant ean [9] By Theore 37, there exsts exactly one nvarant ean on WG ( ) Rear 38 () Let A be a Banach algebra and ϕ ( A) I Wap( A ) has a rght nvarant ϕ -ean n P (, A ϕ )w, then S has a rght zero, say R n Let Cb ( S ) be the space o bounded contnuous unctons on S wth usual sup nor It s easly checed that the ap M, where M ( ) = ( R n ), s a let nvarant ean on Cb ( S ) [4] The reader s reerred to [4] or ore noraton on aenablty o segroups Conversely, let Cb ( S ) have a let nvarant ean Then the ap ( S, T) SoT o S S S s an A -representaton

7 308 Al Ghaar, Saanech Javad 7 (see [] or detals) By theore n [], there exsts soe T S such that SoT = S or every S S Thus S has a rght zero and so Wap( A ) has a rght nvarant ϕ -ean () Let A be a Banach algebra and ϕ ( A) Under the wea topology, P( A, ϕ ) s a setopologcal segroup I Cb ( P ( A, ϕ )) has a let nvarant ean, then Cb ( S ) has a let nvarant ean (see Lea 2 n [6]) Thereore Wap( A ) has a rght nvarant ϕ -ean COROLLARY 39 Let A be a relexve Banach algebra and ϕ ( A) ϕ -ean P (, A ϕ )w and a rght nvarant ϕ -ean nvarant ϕ -ean on A I A has a let nvarant n P (, A ϕ )w, then = n and t s the unque Proo Assue that A s a relexve Banach algebra For A, { a; a } has copcat closure n the wea topology o A As A s relexve, { a; a } has copact closure n the wea topology [2] Thereore A = Wap( A) By Theore 37, A has a unque nvarant ϕ -ean THEOREM 30 Let A be a Banach algebra and ϕ ( A) () Let M be the set o all let nvarant ϕ -eans on Ap( A ) Then M s a copact convex topologcal segroup under Arens product and the wea topology o Ap( A ) ; () Let M be the set o all let nvarant ϕ -eans on Wap( A ) Then M s a copact convex setopologcal segroup under Arens product and the wea topology o Wap( A ) Proo () M s clearly a convex segroup under Arens product and s wea copact n we need s to show that the ap ( n, ) ns contnuous Let (, n ) ( n, ) n M M Let be any eleent n Ap( A ) Snce n wea and hence nor Thereore Ap( A ) All be a net n M M convergng to n wea, we conclude that n n n, n,, n, n +, n, n n n +, n, n 0 () A slar arguent as above ples ths part REFERENCES L N ARGABRIGHT, Invarant eans and xed ponts: A sequel to Mtchell's paper, Trans Aer Math Soc, 30, (968) 2 A AZIMIFARD, -aenable hypergroups, Math Z, 265, (200) 3 A AZIMIFARD, On the -aenablty o hypergroups, Monatsh Math, 5, 3 (2008) 4 J F BERGLUND, H D JUNGHENN, P MILNES, Analyss on segroups, uncton spaces, copactcatons, representons, New Yor, H G DALES, Banach algebra and autoatc contnuty, London Math Soc Monogr Ser, Clarendon Press, K deleeuw, I GLICKSBERG, Applcatons o alost perodc copactcatons, Acta Math, 05, (96) 7 DUNCAN, S A R HOSSEINIUN, The second dual o a Banach algebra, Proc Roy Soc Ednburgh Sect A, 84, (979) 8 F FILBIR, R LASSER, R SZWARC, Reter' s condton P and approxate denttes or hypergroups, Monatsh Math, 43, (2004) 9 B FORREST, Invarant eans, rght deals and the structure o setopologcal segroups, Segroup Foru, 40, (990) 0 A GHAFFARI, A ALINEJAD, ϕ-aenablty o Banach algebras, Subtted A GHAFFARI, Strongly and wealy alost perodc lnear aps on segroup algebras, Segroup Foru, 76, 95 06(2008) 2 E HEWITT, K A ROSS, Abstract Haronc analyss, Vol I, II, Sprnger Verlage, Berln, 963, Z HU, M S MONFARED, T TRAYNOR, On character aenable Banach algebras, Studa Math, 93, (2009) 4 B E JOHNSON, Cohoology n Banach algebras, Me Aer Math Soc, 27 (972)

8 8 ϕ-eans o soe Banach subspaces on a Banach algebra E KANIUTH, A T LAU, J PYM, On character aenablty o Banach algebras, J Math Anal Appl, 344, (2008) 6 E KANIUTH, A T LAU, J PYM, On ϕ -aenablty o Banach algebras, Math Proc Cabrdge Phlos Soc, 44, (2008) 7 A T LAU, Analyss on a class o Banach algebras wth applcatons to haronc analyss on locally copact groups and segroups, Fund Math, 8, 6 75 (983) 8 M S MONFARED, Character aenablty o Banach algebras, Math Proc Cab Phl Soc, 44, (2008) 9 A L T PATERSON, Aenablty, Aer Math Soc Math Survey and Monogrraphs 29, Provdence, Rhode Island, J P PIER, Aenable locally copact groups, John Wley And Sons, New Yor, W RUDIN, Functonal analyss, McGraw Hll, New Yor, A ULGER, Contnuty o wealy alost perodc unctonals on L ( G ), Quart J Math Oxord Ser 2, 37, (986) 23 Jaes C S WONG, On the segroup o probablty easures o a locally copact segroup II, Canad Math Bull, 30, (987) 24 Jaes C S WONG, Topologcal segroups and representatons I, Trans Aer Math Soc, 200, (974) Receved Noveber 4, 20