Grothendiecks inequalities revisited

y

Antonio M Peralta and Angel Ro drguez Palacios

Abstract

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 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 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 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 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 Let M respectively M

p

and For every couple V W of real respectively complex

JBWtriples and every separately weakcontinuous bilinear form U on V

W there exist normone functionals V and W satisfying

   

 

     

jU x y j M kU k kxk kxk ky k ky k

 

for al l x y V W

In the case of complex JBtriples the interval of variation of the constant

p

M can b e enlarged with M see PR Remark Preciselywe

have the following theorem

p

Theorem Let M and Then for every couple E F of

complex JBtriples and every bounded bilinear form U on EF there exist

normone functionals E and F satisfying

   

 

     

jU x y j M kU k kxk kxk ky k ky k

 

for al l x y E F

As in the Little Grothendiecks inequality we do not know if the value

is allowed in the previous Theorem However we can take

whenever we change normone functionals with states Indeed when in the

pro of of Theorem Theorem and PR Corollary replace Corollary

we obtain the following theorem

Theorem Let E F becomplex respectively real JBtriples M

p p p



respectively M and let U beaboundedbilin

ear form on EF Then thereare D BLE I and D BLF I

E F

such that

jU x y j M kU kkjxkj kjy kj

 

for al l x y E F

Big Grothendiecks inequality

p

Another interesting question is whether the interval M is

valid in the complex case of Theorem The rest of the pap er deals with

the armative answer of this question The following prop osition gives a

rst answer in the particular case of biduals of JBtriples We recall that

if E and F are complex JBtriples then every b ounded bilinear form U on

e

EF has a unique separately weakcontinuous extension denoted by U

to E F see PR Lemma

p

Prop osition Let M and Then for every couple E F

of complex JBtriples and every bounded bilinear form U on E F there

exist normone functionals E and F satisfying

   

 

     

e

jU j M kU k k k k k k k k k

 

for al l E F

Proof By Theorem there are normone functionals E and

 

F satisfying

 

 

     

e

jU x y j M kU k kxk kxk ky k ky k

 

for all x y E F

Since the rst assertion of Theorem assures that E and F are strong

dense in E and F resp ectively for every E F wehavenets

x E and y F converging to and resp ectively in the strong

top ology hence they converge also in the weak top ology of E and F

resp ectively Let now x E since for i f g the seminorm kk is

i

e

strongcontinuous by and the separately weakcontinuity of U we

have

 

     

e

jU x j M kU k kxk kxk k k k k

 

for all x EF Using the same argument but xing F instead

of x Ewe nish the pro of

By BDH Prop osition every JBWtriple is isometrically isomorphic

to a weakclosed ideal of its bidual Indeed given a JBWtriple V then

there exists a weakclosed ideal P of V such that J is a triple

P V

isomorphism and hence weakcontinuous from V onto P where de

P

notes the natural pro jection from V onto P and J denotes the natural

V



It is also known that J j emb edding of V onto V

P

V

Big Grothendiecks inequality

We can know state the complex case of Theorem with constant M

p

p

Theorem Let M and For every couple V W of

complex JBWtriples and every separately weakcontinuous bilinear form

U on VW there exist normone functionals V and W

   

satisfying

 

     

ky k kxk ky k jU x y j M kU k kxk

 

for al l x y V W

e

Proof Let U the unique separately weakcontinuous extension of U to

V W By Prop osition there exist normone functionals V

 

and W satisfying

 

 

     

e

jU j M kU k k k k k k k k k

 

for all V W

By the previous comments there are weakclosed ideals P and QofV

and W resp ectively such that

J V P

V P V

and

J W Q

W Q W

b

are triple isomorphisms Let us now dene another bilinear form U on

b b

Then U is separately weak J V W by U U J

W V

b e

contin uous and extends U to V W so U U In particular

e

U x y U J x J y

P V Q W

for all x y V W

 

It is well known that V P P and W Q Q so the norm

   

one functionals given in decomp ose and

i i

i i i i

i f g where

   

P P k k k k

i i i i

REFERENCES

and

   

Q Q k k k k

i i i i

 

for i f g Now taking x V and norm one elements e P and e P

i i

j j j

such that e k i j f g applying the orthogonality of P and

i i i

P we get

    

x xe x xe k k x

V V V V V

i i i i

i

   

x x e kxk

V

i i e

V

i



V

i

 

where e V if if we can take as e any

i V V i

i i



k k

V

i

e

other normone functional in V Similarly we get normone functionals

i

in W such that

 

k y k ky k

W

e

i

i

for all y W i f g

Finally applying we get





     

kxk kxk ky k jU x y j M kU k ky k

e e

e e





for all x y V W

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A M Peralta and A Ro drguez Palacios

Departamento de Analisis Matematico Facultad de Ciencias

Universidad de Granada

Granada Spain

ap eraltagoliatugres and apalaciogoliatugres