ON A CERTAIN PRODUCT OF BANACH ALGEBRAS AND SOME OF ITS PROPERTIES


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1 HE PULISHING HOUSE PROCEEDINGS OF HE ROMANIAN ACADEMY Series A OF HE ROMANIAN ACADEMY Volume 5 Number /04 pp. 9 7 ON A CERAIN PRODUC OF ANACH ALGERAS AND SOME OF IS PROPERIES Hossossein JAVANSHIRI Mehdi NEMAI Yzd University Deprtment of Mthemtics P.O. ox: Yzd Irn Isfhn University of echnology Deprtment of Mthemticl Sciences Isfhn Irn Institute for Reserch in Fundmentl Sciences (IPM School of Mthemtics P. O. ox: ehrn Irn Emil: A we construct nd study product on the Crtesin product spce A nd we denote this lgebr by A. Among other things we chrcterize the set of ll continuous derivtions from A into ( its n th dul spce ( A n nd s n ppliction we study the n wek menbility of A In this pper for two rbitrry nch lgebrs A nd nd homomorphism : nd its reltion with A nd. Moreover we obtin chrcteriztions of (bounded pproximte identities nd study the idel structure of these products. Key words: nch lgebr Arens regulrity wek menbility.. INRODUCION he Lu product of two nch lgebrs tht re preduls of von Neumnn lgebrs nd for which the identity of the dul is multiplictive liner functionl ws introduced nd investigted by Lu []. his pper initited series of subsequent publictions [ ] nd hs hd gret impct. Extension to rbitrry nch lgebrs ws proposed by Monfred [4] with the nottion A θ where θ is nonzero multiplictive liner functionls on. Recently new extension of Lu product hs been studied by htt nd Dbhi []. More precisely for commuttive nch lgebr A nd rbitrry nch lgebr by using the lgebr homomorphism : A they defined Lu product on the Crtesin product spce A perturbing the pointwisedefined product resulting in new nch lgebr s follows A ( b ( b ( + b ( + b ( bb for ll And bb nd they investigted the Arens regulrity nd some notions of menbility of A. Note tht Lu product coincides with Lu product nd θ Lu product of nch lgebrs. However routine observtions show tht the imposed hypothesis on A for the definition of Lu product given by [ p. ] is too strong. As n evidence for this clim recll tht for two rbitrry nch lgebrs A nd nd multiplictive liner functionl θ : C Monfred [4 Corollry.] proved tht the nch lgebr A θ is Arens regulr if nd only if both A nd re Arens regulr but htt nd Dbhi [ heorem.] proved the sme result for A only when A is n Arens regulr commuttive nch lgebr; his is becuse of on the one hnd the commuttivity ssumption is needed in the htt nd Dbhi s definition nd on the other hnd we know tht the second dul of commuttive nch lgebr is commuttive when it is Arens regulr.
2 0 Hossein Jvnshiri Mehdi Nemti his pper continues these investigtions. In Section by chnge in the htt nd Dbhi s definition we show tht Lu products cn be defined for nch lgebrs in firly generl setting; Indeed we remove the commuttivity ssumption imposed on the definition of Lu products given by [] nd we show tht we cn recover most of the results obtined in [] in the generl cse. Moreover we study the idel structure of Lu product of nch lgebrs nd we chrcterize the existence of the (bounded pproximte identity for this product. In the third prt we introduce bimodule ction of ( A on ( A n ( nd we chrcterize the set of ll continuous derivtions from A ( denotes the n th dul spce of A ( into ( A n where ( A n. Finlly we study the n wek menbility of A nd its reltion with A nd nd then we pply our results to some prticulr instnces of nch lgebrs such s quotient spce of nch lgebrs with closed idel nch lgebrs relted to loclly compct group nd unitl nch lgebr.. HE DEFINIION AND SOME ASIC RESULS Let A nd be rbitrry nch lgebrs (not necessry commuttive nd : A be n lgebr homomorphism. If we replced the first coordinte of the ordered pir in eqution ( bove by " + ( b + ( b " then A will be n ssocitive lgebr. So we re led to the following definition where it pves the wy to remove the commuttivity ssumption imposed on A in the definition of A given by [ Pge ]. Definition.. Let A nd be rbitrry nch lgebrs nd : A be n lgebr homomorphism. he Lu product A is defined s the Crtesin product A with the lgebr product ( b ( b :( + b ( + b ( bb nd the norm ( b + b for ll And bb. he reder will remrk tht routine computtions show tht if then A is nch lgebr. In wht follows A nd re rbitrry nch lgebrs nd : A is n lgebr homomorphism with. We now proceed to show tht we cn recover most of the results obtined in [] in the generl cse. First by n rgument similr to the proof of [ heorem. heorem 4.] we cn prove the following improvement of those. Note tht in heorems. nd 4. of [] the commuttivity of A is needed only to conclude tht A is nch lgebr; wheres by Definition. bove this nch spces lwys is nch lgebr without ny further ssumptions on A. Moreover the wek menbility of the nch lgebr A is considered in Corollry.6 below s specil cse of the Propositions.4 nd.5 in section below. HEOREM.. he following ssertions hold. (i Let the Gelfnd spce ( is nonempty nd E {( λλ : λ ( A} F {(0 φ: φ ( }. Set E if ( A is emptyt. hen ( A is equl to the union of its closed disjoint set E nd F. (ii A is menble if nd only if both A nd re menble; (iii A is (pproximtely cyclic menble if nd only if both A nd re (pproximtely cyclic menble. In order to stte our next results we need to set some terminology. In wht follows for nch spce ( n X the nottion X is used to denote the n th dul spce of X nd we lwys consider X s nturlly ( ( embedded into X. For x X nd f X by x f (nd lso f x we denote the nturl dulity
3 On certin product of nch lgebrs nd some of its properties ( ( between X nd X. Now let A be nch lgebr. On A there exists two nturl products extending ( ( the one on A known s the first nd second Arens products of A. For b in A f in A nd Φ Ψ ( ( ( in A the elements f nd Φ f of A nd Ψ Φ of A re defined respectively s follows. f b f b Φf Φ f nd Ψ Φ f Ψ Φ f. Also the second Arens product '' '' is defined using symmetry. Equipped with these products ( A is nch lgebr nd A is sublgebr of it. hese products re in generl not seprtely wek * to wek * ( ( continuous on A. In generl the first nd the second Arens products do not coincide on A nd A is ( ( ( sid to be Arens regulr if these products coincide on A. he first topologicl center Z t ( A of A is ( ( the set of ll Φ A such tht Φ Ψ Φ Ψ for ll Ψ A. In [ heorem.] htt nd Dhbi proved tht if A is commuttive nd Arens regulr nch lgebr then (i ( A ( ( is isometriclly isomorphic to A ( when the lgebrs re equipped with the ( ( ( sme either of its Arens products; (ii Z (( A A Z ( ; [] t [] t (iii A is Arens regulr if nd only if is Arens regulr [] where the second djoint of is continuous homomorphisms in nturl wy. Now by n rgument similr to the proof of [ heorem.] we cn prove the following improvement of it; Indeed we remove the following ssumptions from the httdhbi s result for Arens regulrity of A : he commuttivity ssumption on A he Arens regulrity ssumption on A. ( Note tht the nch lgebr A is commuttive if the underlying nch lgebr is commuttive nd Arens regulr so in heorem. of [] the commuttivity nd Arens regulrity of A re needed only to ( ( conclude tht A nd A [] re nch lgebrs; wheres by Definition. bove these nch spces lwys re nch lgebrs without ny further ssumptions on A. HEOREM.. he following ssertions hold. ( ( ( A is isometriclly isomorphic to the A ; ( (i ( ( ( (ii Z t A Z t A [] Z t (( ( ( ; (iii A is Arens regulr if nd only if both A nd re Arens regulr. We end this section by the following three results which re of interest in its own right. PROPOSIION.4. Let I be left idel of A nd J be left idel of. hen the following ssertions hold. (i If ( J I. hen I J is left idel of A. (ii If I is prime left idel nd I J is left idel of A then ( J I. (iii If A is unitl nch lgebr. hen I J is left idel of A if nd only if ( J I. (iv If I is close left idel of A nd suppose tht A hs left pproximte identity. hen I J is left idel of A if nd only if ( J I. Proof. he proof of (i is routine nd is omitted. For the proof of (ii we need only note tht if I is prime idel of A nd I J is n idel of A then ( J I. o this end ssume tht there exists b J such tht b ( I. hen [] ( 0( b ( + ( b 0 I J for ll I nd A\ I which contrdicts with ssumption.
4 Hossein Jvnshiri Mehdi Nemti 4 (iv Let ( α α be left pproximte identity for A nd suppose tht I nd b J. hen ( 0( b ( + ( b 0 I J α α α for ll α. Since I is closed it follows tht + ( b I. Consequently ( J I. Finlly note tht one cn prove the ssertion (iii by the sme mnner s in the proof of (iv. Next we consider the converse. PROPOSIION.5. Let N be left idel of A nd I { A: ( b N for some b } J { b : ( b N for some A}. hen the following ssertions hold. (i J is left idel in. (ii If is onto then I is left idel of A. (iii If N is closed nd A hs left pproximte identity then necessry condition for the equlity N I J is tht ( J I. Proof. For briefness we only give the proof for (ii nd (iii. Let c I nd A. hen there exists b such tht b (. herefore ( b ( c0 N nd hence c I. Now we give proof for (iii. o this end let N I J nd j J. hen there is i I such tht ( i j N. If now ( α α is left pproximte identity for A then ( α 0( i j N. Hence i + ( j I nd this implies tht ( j I. PROPOSIION.6. he nch lgebr A hs bounded (resp. unbounded left (resp. right or twosided pproximte identity if nd only if A nd hve the sme pproximte identity. Proof. First ssume tht the nets ( α α nd ( b α α re bounded left pproximte identities for A nd respectively. hen the net (( α ( bα bα α is bounded left pproximte identity for A ; Indeed for every ( b A ( ( b b ( b (( ( b + ( b + ( ( b ( b b b. α α α α α α α α α Conversely if (( α bα α is bounded left pproximte identity for A then the nets ( + ( b nd ( b α α re bounded. Also for every A nd b we must hve α α α b b b ( b (0 b (0 b 0 ( + ( b ( b ( 0 ( 0 0. α α α α α α α Since nd b re rbitrry we conclude tht the nets ( α + ( bα α nd ( b α α ( b α α re bounded left pproximte identities for A nd respectively. his completes the proof of this proposition since one cn obtin proof for other cses by similr rgument.. DERIVAIONS INO IERAED DUALS We commence this section with some nottions. In this section A nd re two rbitrry nch ( lgebrs nd : A be n lgebr homomorphism with norm t most. hen ( A the first dul spce of ( ( A cn be identified with A in the nturl wy + ( ( ( ( ( b ( b b b ( ( ( ( ( ( for ll A b A nd b. he dul norm on A is of course the mximum ( ( ( ( norm ( mx { } ( n ( n b b. Now for integer n 0 tke A s the underlying spce of
5 5 On certin product of nch lgebrs nd some of its properties ( A n formulted s follows: nd (. From induction we cn find tht the ( A ( n ( n ( b ( b ( n [ n] ( n ( n ( ( bimodule ctions on ( A n re ( + ( b + ( b b b if n iseven ( n [ n] ( n ( n (( + ( b ( + b b if n isodd ( n ( n ( b ( b ( n [ n] ( n ( n ( ( + ( b + ( b b b if n iseven ( n [ n] ( n ( n ( ( + ( b ( + b b if n is odd ( n ( n ( n ( n [ n] for ll ( b A nd ( b A where denotes the n th djoint of. Let A be nch lgebr nd X be nch A bimodule. hen liner mp D : A X is clled derivtion if Db ( Db ( + D ( b for ll b A. For x X we define d x : A X s follows d x ( x x for ll b A. It is esy to show tht d x is derivtion; such derivtions re clled inner derivtions. A derivtion D : A X is clled inner if there exist x X such tht D d x. We denote the set of ll continuous derivtions from A into X by Z ( AX. he first cohomology group H ( A X is the quotient of the spce of continuous derivtions by the inner derivtions nd in mny situtions trivility of this spce is of considerble importnce. In prticulr A is clled contrctible if H ( A X 0 ( for every nch A bimodule X A is clled menble if H ( A X 0 for every nch A bimodule ( n X for every n 0 A is clled n wekly menble if H ( A A 0 nd wekly menble if A is ( n wekly menble where A is the n th dul module of A when n nd is A itself when n 0. Let us mention tht the concept of wek menbility ws first introduced nd intensively studied by de Curtis nd Dles [] for commuttive nch lgebrs nd then by Johnson [] for generl nch lgebr. Dles Ghhrmni nd Grønbæk [4] initited nd intensively developed the study of n wek menbility of nch lgebrs; see lso [8 0 8]. he following result chrcterize the set of ll continuous derivtions from A into ( A  (n+ bimodule ( A. In the sequel the nottion M is used to denote the Lu product of the nch lgebrs A nd. ( HEOREM.. Let M A nd n 0. hen D Z ( M M n+ if nd only if there exist (n d Z ( A A + (n d Z ( + (n d Z ( A + (n nd bounded liner mp S: A + such tht for ech And b ; (i D(( b ( d( + d ( b S( + d ( b (ii d( b ( d ( b + bd ( ( (iii d( ( b d ( b + d( ( b [n+ ] (iv Sb ( ( ( d ( b + bs ( [n ] (v [n+ ] (vi + Sb ( ( ( d ( b + Sb ( S ( ( d(. ( Proof. Suppose tht D Z ( M M n+ (. hen there exist bounded liner mps D A A + : n ( nd : n D A + such tht D ( D D. Let d( D(( 0 d ( b D((0 b d ( b D((0 b nd S ( D (( 0 for ll A nd b. hen trivilly d d d nd S re liner mps stisfying (i. Moreover for every And bb we hve
6 4 Hossein Jvnshiri Mehdi Nemti 6 nd nd D(( b ( b D(( + b ( + b ( bb ( d + b + b + d bb d bb + S + b + b ( ( ( ( ( ( ( ( D(( b( b + ( b D(( b ( d ( d ( b S( d ( b( b ( b ( d ( d ( b S( d ( b [n+ ] ( d + d b + d + d b b d + d b + S + d b b [n+ ] ( d ( ( d( b b ( ( d + d b d + d b + b S + d b ( ( ( ( ( ( ( (( ( ( ( ( ( ( ( ( ( ( ( ( ( (. ( It follows tht D Z ( M M n+ if nd only if ( nd ( coincide. hus d ( + ( b + ( b + d ( bb d ( + d ( b + d ( ( b + d ( b ( b + d( + d( b + bd ( ( + bd ( ( b [n+ ] ( [n+ ] ( S + ( b + ( b + d ( bb d ( + d ( b + S( b ( + d ( b b + d ( + d ( b + bs( + bd ( b. ( ( ( (4 ( herefore D Z ( M M n+ if nd only if the equtions ( nd (4 re stisfied. Now strightforwrd verifiction shows tht if d d nd d re derivtions nd the equlities (ii (iii (iv (v (vi re stisfied then ( nd (4 re vlid. Applying ( nd (4 for suitble vlues of bb shows tht d d nd d re derivtions nd the equlities (ii (iii (iv (v (vi re lso stisfied s climed. As n immedite corollry we hve the following result which chrcterize the set of ll inner (n+ derivtions from A into ( A bimodule ( A. ( COROLLARY.. Let M A nd n 0. hen D Z ( M M n+ nd if nd only if D b ( d d b S d b b (n D d n+ n+ ( ( b( (( ( + ( ( + ( for ll A nd b where d d (n+ d d + d d ( n + S δ ( n+ [n ] nd δ ( + ( d ( for ll A. b Proof. For the proof we need only note tht if (n+ (n+ then ( d ( S ( D (( 0 d ( (n b(n (n+ (n+ D d for some ( n+ ( n+ A nd + + n n ( ( b( [ ] ( 0 d ( n+ ( d ( + + ( (n+ (n+ for ll A. It follows tht d d (n + nd S δ ( n+. Similrly d d b (n + nd d d ( n+ nd this completes the proof. y replcing (n + by ( n in heorem. nd Corollry. nd using similr rgument one cn obtin the following result. ( HEOREM.. Let M A nd n 0. hen D Z ( M M n if nd only if there exist ( n ( n ( n ( n d Z ( d Z ( A nd bounded liner mps S: A nd R: A A such tht for ech And b ; (i D(( b ( R( + d ( b S( + d ( b [ n] [ n] (ii R( R ( + ( S ( + R ( + ( S (
7 7 On certin product of nch lgebrs nd some of its properties 5 [ n] (iii R( b ( d ( b + ( d ( b + br ( ( [ n] (iv R( ( b d ( b + ( d ( b + R( ( b (v Sb ( ( bs ( (vi Sb ( ( Sb ( (vii S ( 0. Moreover D d if nd only if R d ( n n ( n S 0 d d ( n nd d d ( n. n n ( ( b( + [ ] ( b Now we pply these results to study the n wek menbility of A. Our strting point is the following proposition which gives necessry condition for the (n + wek menbility of A. PROPOSIION.4. Let M A be (n + wekly menble for some n 0. hen A nd re (n + wekly menble. (n Proof. Suppose tht d Z ( A A + (n nd d Z ( +. y heorem. we conclude tht (n+ (n+ the mp D: A A defined by ( + + ( D(( b d ( d ( b S( d ( b ( b A n [ + ] is derivtion where d d nd S d. herefore there exists ( (n+ (n+ (n+ (n+ b A such tht n n ( ( b( D d. Hence d + + d ( + nd d d ( + b n b n by Corollry.. In the following proposition for n 0 we denote by X ( n the closed liner spn of the set ( ( n ( n A A A A ( n in A. he following result with suitble condition gives sufficient condition for the (n + wek menbility of A. ( PROPOSIION.5. Let n be positive integer number such tht X A. If A nd re (n + wekly menble then M is (n + wekly menble. A n ( n A (n Proof. Suppose tht D Z ( M M + (n. hen there exist d Z ( A A + (n d Z ( + (n d Z ( A + (n nd bounded liner mp S: A + such tht D(( b ( d ( + d ( b S( + d ( b (n+ (n+ (n+ (n+ for ll ( b A. y ssumption there exist A nd b such tht d d + nd d d ( n+ [n ]. hus by heorem. we hve S ( + ( d( for ll (n b A. Since A is (n + wek menble it follows tht A A. Hence we conclude tht S d δ [n+ ] (n+ with (n. On the other hnd by heorem. nd Corollry. we hve d ( b ( d ( b + bd ( ( (n+ (n+ for ll A nd b d ( b d ( ( b 0 for ll A nd b. y replcing d ( d d + we conclude tht d ( b 0for ll b. Similrly we cn show tht ( A( n. A. herefore ( (n+ d ( b 0 for ll b. It follows from ssumption tht d d (n+. Hence by heorem.. AA. n D d n+ n+ ( ( b(
8 6 Hossein Jvnshiri Mehdi Nemti 8 We recll from [4 Proposition.] tht if A is wekly menble then.4 nd.5 we hve the following result. A A. hus by Propositions COROLLARY.6. A nd re wekly menble if nd only if M is wek menble. elow we observe tht nwek menbility of A is necessry for n wek menbility of M. PROPOSIION.7. Let M A be n wekly menble for some n 0. hen A is n wekly ( n [ ] menble nd ll derivtions d Z ( with ( ker n d re inner. (n Proof. Suppose tht d Z ( A A + ( n [ ] nd d Z ( such tht ( ker n d. y ( n ( n heorem. we conclude tht the mp D: A A defined by [ n] ( + ( D(( b d ( d ( b ( d ( b ( b A ( n ( n ( n ( n is derivtion. herefore there exists ( b A such tht [ n] D d. It n n ( ( b( follows tht d d ( n + [ n] ( b( n nd d d b ( n. Let I be closed idel in A. hen it is trivil tht the nturl quotient mp q: A A/ I is n lgebr [ n] ( n homomorphism with q. In prticulr ker q I nd ker q I for ll n 0. COROLLARY.8. Let I be closed idel in A nd n 0. Suppose tht A/ I q A is n wekly ( n menble. hen AI / is n wekly menble nd ech derivtion from A into I is inner. As two pplictions of heorems. nd. for the cse tht A is unitl we hve the following ( n chrcteriztions for elements in Z ( A ( A. ( COROLLARY.9. Let A be unitl nd n 0. hen D Z ( M M n+ if nd only if (n where ( + + D(( b d ( d ( b S( d ( b d Z ( A A + (n d Z ( + (n d Z ( A + (n nd S: A + is bounded [n+ ] liner mp such tht d d nd S d. Moreover D is inner if nd only if both d nd d re inner. ( COROLLARY.0. Let A be unitl nd n 0. hen D Z ( M M n if nd only if ( + D(( b d ( d ( b d ( b ( n ( n ( n [ n] where d Z ( A A d Z ( nd d Z ( A such tht d d d. Moreover D is inner if nd only if both d nd d re inner. As n immedite corollry we hve the following result. he detils re omitted. PROPOSIION.. Let A be unitl nd n 0. hen M is n wekly menble if nd only if A nd re n wekly menble. We end this work with the following exmple. EXAMPLE.. (i It is well known tht for ny loclly compct group G the group lgebr L ( G is n wekly menble for ll n N. Now suppose tht G nd G re two loclly compct groups nd let
9 9 On certin product of nch lgebrs nd some of its properties 7 : L ( G L ( G be n lgebr homomorphism with. hen L ( G L ( G is wekly menble by Corollry.6. (ii Let : M( G L ( G be n lgebr homomorphism with. hen [5 heorem.] nd Proposition. implies tht G is discrete if nd only if M ( G L ( G is n wekly menble for ll n N. ACKNOWLEDGEMENS he uthors would like to express their sincere thnks to the referee for invluble comments tht helped us to improve considerbly the presenttion of this pper. he second uthor s reserch ws supported in prt by grnt from IPM (No REFERENCES. HA S. J. DAHI P. A. Arens regulrity nd menbility of Lu product of nch lgebrs defined by nch lgebr morphism ull. Austrl. Mth. Soc. 87 pp ADE W. G. CURIS P. C. DALES H. G. Amenbility nd wek menbility for eurling nd Lipschits lgebrs Proc. Londn Mth. Soc. 55 pp ONSAL F. F.DUNCAN J. Complete normed lgebrs Springer erlin DALES H. G. GHAHRAMANI F. GRØNÆK N. Derivtions into iterted duls of nch lgebrs Studi Mth. 8 pp DALES H. D. GHAHRAMANI F. HELEMSKII A. Y. he menbility of mesure lgebrs J. London Mth. Soc. 66 pp E. VISHKI H. R. KHODAMI A. R. Chrcter inner menbility of certin nch lgebrs Colloq. Mth. pp GHADERI E. NASRISFAHANI R. NEMAI M. Some notions of menbility for certin products of nch lgebrs Colloq. Mth. 0 pp H. AZAR K. RIAZI A. A generliztion of the wek menbility of some nch lgebrs Proc. Rom. Acd. Ser. A. Mth. Phys. ech. Sci. Inf. Sci. pp HU Z. MONFARED M. S. RAYNOR. On chrcter menble nch lgebrs Studi Mth. 9 pp JOHNSON. E. Cohomology in nch Algebrs Mem. Amer. Mth. Soc JOHNSON. E. Wek menbility of group lgebrs ull. Lodon Mth. Soc. pp JOHNSON. E. Derivtions from L ( G into L ( G nd L ( G Lecture Notes in Mth. 59 Springer erlin New York pp LAU A.. Anlysis on clss of nch lgebrs with pplictions to hrmonic nlysis on loclly compct groups nd semigroups Fund. Mth. 8 pp MONFARED M. S. On certin products of nch lgebrs with pplictions to hrmonic nlysis Studi Mth. 78 pp MONFARED M. S Chrcter menbility of nch lgebrs Mth. Proc. Cmbridge Philos. Soc. 44 pp NASRISFAHANI R. NEMAI M. Essentil chrcter menbility of nch lgebrs ull. Aust. Mth. Soc. 84 pp NASRISFAHANI R. NEMAI M. Cohomologicl chrcteriztions of chrcter seudomenble nch lgebrs ull. Aust. Mth. Soc. 84 pp YAZDANPANAH. Wek menbility of tensor product of nch lgebrs Proc. Rom. Acd. Ser. A Mth. Phys. ech. Sci. Inf. Sci. pp Received Februry 4 04
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