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Grothendieck's Inequalities Revisited Introduction 1 Little Grothendieck's Inequality

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Grothendieck's Inequalities Revisited Introduction 1 Little Grothendieck's Inequality Grothendiecks inequalities revisited y Antonio M Peralta and Angel Ro drguez Palacios Abstract Mathematics Subject Classication C K L L and L Intro duction Let X b e a normed space We denote by S B and X the unit sphere X X the closed unit ball and the dual space resp ectively of X If X is a Banach dual space we write X for a predual of X Little Grothendiecks inequality We recall that a complex JBtriple is a complex Banach space E with a E E which is bilinear and continuous triple pro duct f g E E symmetric in the outer variables and conjugate linear in the middle variable and satises Jordan Identity La bfx y z g fLa bx y z gfx Lb ay zg fx y La bz g for all a b c x y z in E where La bx fa b xg The map La afromE to E is an hermitian op erator with nonnegative sp ectrum for all a in E Supp orted by Programa Nacional FPI Ministry of Education and Science grant DGICYT pro ject no PB and Junta de Andaluca grantFQM y Partially supp orted by Junta de Andaluca grantFQM Little Grothendiecks inequality kfa a agk kak for all a in E Complex JBtriples have been intro duced by W Kaup in order to pro vide an algebraic setting for the study of bounded symmetric domains in complex Banach spaces see K K and U By a complex JBWtriple we mean a complex JBtriple which is a dual Banach space We recall that the triple pro duct of every complex JBW triple is separately weakcontinuous BT and that the bidual E of a com plex JBtriple E is a JBWtriple whose triple pro duct extends the one of E Di Given a complex JBWtriple W and a normone element in the predual W of W we can construct a prehilb ert seminorn kk as follows see BF Prop osition By the HahnBanach theorem there exists z W suchthat z kz k Then x y fx y z g b ecomes a p ositive sesquilinear form on W which do es not dep end on the p oint of supp ort z for The prehilb ert seminorm kk is then dened by kxk fx x z g for all x W If E is a complex JBtriple and is a normone element in E then kk acts on E hence in particular it acts on E Following IKR we dene real JBtriples as normclosed real subtriples of complex JBtriples In IKR it is shown that every real JBtriple E can be regarded as a real form of a complex JBtriple Indeed given a real JBtriple E there exists a unique complex JBtriple structure on the b complexication E E i E and a unique conjugation ie conjugate b b b linear isometry of p erio d on E such that E E fx E x xg The class of real JBtriples includes all JBalgebras HS all real Calgebras G and all JBalgebras Al By a real JBWtriple we mean a real JBtriple whose underlying Banach space is a dual Banach space As in the complex case the triple pro duct of every real JBWtriple is separately weakcontinuous MP and the bidual E of a real JBtriple E is a real JBWtriple whose triple pro duct extends the one of E IKR If U is a real or complex JBtriple and A is a subset of U we denote by A fx U fx A U gg the orthogonal complementof A In PR see also P the authors proved the following appropriated ver sion of the so called Little Grothendiecks inequality for real and complex JBWtriples which avoids the gaps con tained in BF Little Grothendiecks inequality Theorem PR Theorems and p p Let K respectively K and Then for every complex respectively real JBWtriple W every complex respectively real Hilbert space H and every weakcontinuous linear operator T W H there exist normone functionals W such that the inequality kT xkK kT k kxk kxk holds for al l x W Let T be a b ounded linear op erator from a real resp ectively complex JBtriple E to a real resp ectively complex Hilb ert space H since E is a real resp ectively complex JBWtriple and T is a weakcontinuous op erator from E to H then the following result follows from the previous theorem p p Corollary Let K respectively K and Then for every complex respectively real JBtriple E every complex respec tively real Hilbert space H and every bounded linear operator T E H there exist normone functionals E such that the inequality kT xkK kT k kxk kxk holds for al l x E The question is whether in Corollary the value is allowed for some value of the constant K We are going to give an armative answer to this question whenever we replace the prehilb ertian seminorm kk with another prehilb ertian seminorm asso ciated with a state of a real or complex JBtriple Given a BanachspaceX BLX and I will denote the normed algebra X of all b ounded linear op erators on X and the identity op erator on X re sp ectively If u is a normone elemen tinX the set of states of X relativeto u D X u is dened as the non empty convex and weakcompact subset of X given by D X u f B u g X Let E be a complex JBtriple and D BLE I Since for every E x E the map Lx x is an hermitian op erator with nonnegative sp ectrum Little Grothendiecks inequality we can dene the prehilb ertian seminorm kjkj by kjxkj Lx x for all x E such that e We dene and let e S Let S e E E D BLE I by T T e for all T BLE We notice that in E e E and hence in E this case kjkj and kk coincide on e Theorem Let E be a complex respectively real JBtriple H a com plex respectively real Hilbert space and T E H a bounded linear opera tor Then there exists D BLE I such that E p kT xk kT k kjxkj p respectively kT xk kT k kjxkj for al l x E Proof We supp ose that E is a complex JBtriple The pro of for a real JBtriple is the same By Corollary for every n N there are normone n n functionals E such that the inequality p kT xk kT k kxk kxk n n n n p kT k kjxkj kjxkj n n n n e e n n n n n holds for all x E where e S with e i n N E i i Let i f g since D BLE I is weakcompact we can take a weak E n n cluster p oint D BLE I of the sequence i Then the i E e i i y inequalit p kT xk kT kkjxkj holds for all x E From the previous Theorem we can now derive a remarkable result of U Haagerup Corollary H Theorem Let A be a Calgebra H a complex Hilbert space and T A H a bounded linear operator There exist two states and on A such that kT xk kT k x x xx for al l x A Little Grothendiecks inequality Proof By Theorem there exists D BLAI such that A kT xk kT k Lx x for all x A Since for every x A Lx x L L where L and xx x x a R stands for the left and rightmultiplication by a resp ectively we have a R kT xk kT k L x x xx for all x A b Now denoting by b and the p ositive functionals on A given by bx b b L and x R resp ectively we conclude that and x x kbk b are states on A and b k k kT xk kT k x x xx for all x A The concluding section of the pap er PR deals with some applications of the Theorem including certain results on the strongtop ology S WW of a real or complex JBWtriple W We recall that if W is a real or complex JBWtriple then the S WW top ology is dened as the top ology on W generated by the family of seminorms fkk W kk g For every dual Banach space X with a xed predual denoted by X we denote by mX X the Mackey top ology on X relative to its dualitywith X It is worth mentioning that if a JBWalgebra A is regarded as a com plex JBWtriple S A A coincides with the socalled algebrastrong top ology of A namely the top ology on A generated by the family of semi p norms of the form x x x when is any p ositive functional in A R Prop osition As a consequence when a von Neumann algebra M is regarded as a complex JBWtriple S M M coincides with the familiar strongtop ology of M compare S Denition The results of PR allowustoavoid the diculties in R compare PR page and to extend these results to the real case We summarize these results in the following theorem Theorem PR page Corol lary and Theorem see also R Theorem and R Theorem D Little Grothendiecks inequality Let W be a real or complex JBWtriple Then the strongtopology of W is compatible with the duality WW Linear mappings between real or complex JBWtriples are strong continuous if and only if they are weakcontinuous If W is a real or complex JBWtriple and if V is a weakclosed subtriple then the inequality S WW j S V V holds and in V fact S WW j and S V V coincide on bounded subsets of V V Let W be a real or complex JBWtriple Then the triple product of W is jointly S WW continuous on bounded subsets of W and the topologies mWW and S WW coincide on bounded subsets of W Remark In a recent work L J Bunce has obtained an improvement of the third statement Concretely in Bu Corol lary he proves that if W is a real or complex JBWtriple and if V is a weakclosed subtriple then each element of V has a norm preserving extension in W S WW j S V V V From the results related with the strongtop ology we derive a Jarchow typ e characterization of weakly compact op erators from real or complex JBtriples to arbitrary Banach spaces Theorem PR Theorem Let E be a real respectively complex JBtriple X a real respectively complex Banach space and T E X a bounded linear operator The fol lowing assertions are equivalent T is weakly compact There exist a bounded linear operator G from E to a real respectively complex Hilbert space and a function N such that kT xk N kGxk kxk for al l x E and There exist norm one functionals E and a function N such that kT x k N kxk kxk for al l x E and Big Grothendiecks inequality Big Grothendiecks inequality In PR Theorems and we obtained the following result p p Theorem
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