EXTREME INTEGRAL POLYNOMIALS ON A COMPLEX BANACH SPACE

EXTREME INTEGRAL POLYNOMIALS ON A COMPLEX BANACH SPACE

SEÁN DINEEN^∗

Abstract

We obtain upper and lower set-theoretic inclusion estimates for the set of extreme points of the unit balls ofPI(ⁿE)andPN(ⁿE), the spaces ofn-homogeneous integral and nuclear polynomials, respectively, on a complex Banach spaceE. For certain collections of Banach spaces we fully characterise these extreme points. Our results show a difference between the real and complex space cases.

1. Introduction

Geometric properties of spaces of polynomials, e.g. smoothness, extreme points, exposed points, norm attaining polynomials, etc. have been invest- igated by a number of authors in recent years. We refer to [1], [3], [7], [8], [13], [12], [18], [20]. In particular, Ryan-Turett [18] and Boyd-Ryan [3] in their investigations examined the extreme points of the unit ball of the space of integral polynomials defined on arealBanach space. In this paper we study the extreme points of the unit ball of the space of integral polynomials defined on acomplex Banach space. Throughout this paperEwill, unless otherwise stated, denote a Banach space over the complex numbersC. We letBE orB denote the open unit ball ofE. We refer to [11] for basic facts on polynomials on Banach spaces and to [9], [10] for the geometry of Banach spaces.

2. Extreme Points

IfEis a Banach space overCwe let

Ext_R(E)= {x∈E:x =1,x+λy ≤1 for−1≤λ≤1 impliesy=0} and

Ext_C(E)= {x∈E:x =1,x+λy ≤1 for|λ| ≤1 impliesy=0}.

∗The author wishes to thank Bogdan Grecu and Manolo Maestre for helpful remarks.

Received September 20, 2000.

Thus, Ext_R(E)is the set of all (real) extreme points of the unit ball ofE and Ext_C(E) is the set of complex extreme points of the same set. Clearly Ext_R(E) ⊂ Ext_C(E). The inclusion may be proper since Ext_R(L¹(0,1))is empty while Ext_C(L¹(0,1))consists of all unit vectors.

A Banach space is strictly convex (or rotund) (respectively strictlyc-convex) if

Ext_R(E)= {x ∈E:x =1}

(respectively Ext_C(E) = {x ∈ E : x = 1}). As noted above the Banach spaceL¹(0,1)is strictlyc-convex but is not strictly convex.

The role of extreme points in functional analysis, convexity theory, linear programming and optimisation theory is well documented. Complex extreme points were introduced by Thorpe and Whitley [21] in order to prove a strong maximum modulus principle for Banach-valued holomorphic functions. Since their introduction they have proved useful in the study of the Shilov boundary, complex geodesics, invariant metrics, bounded symmetric domains and JB^∗- triple systems. We refer to [2], [4], [5], [14], [15], [22], [23] for details.

3. Integral and Nuclear Polynomials

IfEis a complex Banach spaceP(ⁿE) denotes the space of continuousn- homogeneous polynomials onEendowed with the norm·:= ·B of uni- form convergence over the unit ball B of E. A polynomial P ∈ P(ⁿE) is said to beintegralif there exists a regular Borel measureµon the unit ball of E, BE endowed with the weak^∗topology, such that

(1) P (x)=

BE

φ(x)ⁿdµ(φ)

for allxinE. We letPI(ⁿE)denote the space of alln-homogeneous integral polynomials onEand we endow this space with the norm ·I := infµ where the infimum is taken over allµsatisfying (1).

Ann-homogeneous polynomialP onEisnuclearif there exists a bounded sequence(φj)j ⊂Eand(λj)j ∈l1such that

(2) P (x)=

∞ j=1

λjφj(x)ⁿ

for allx inE. The space of all nuclear n-homogeneous polynomials on E, PN(ⁿE), is a Banach space when PN is defined to be the infimum of _∞

j=1|λj| φjⁿtaken over all representations ofP satisfying (2).

Proposition3.1. LetEdenote a complex Banach space.

(a) The spacePI(ⁿE)is isometrically isomorphic to

n,s,E . (b) PN(ⁿE)⊂PI(ⁿE)⊂P(ⁿE)and· ≤ ·I ≤ ·N. (c) Ifφ∈Ethenφⁿ∈PN(ⁿE)and

φⁿ = φⁿI = φⁿN

(d) IfEhas the approximation property thenPN(ⁿE)is isometrically iso- morphic to

n,s,πEandPN(ⁿE)=P(ⁿE)isometrically.

(e) If1 →

n,s,E (in particular ifE has the Radon-Nikodým Property) thenPI(ⁿE)andPN(ⁿE)are isometrically isomorphic.

(f) Ext_R(PI(ⁿE))⊂ {φⁿ :φ∈E,φ =1}. (g) Ext_R(PI(ⁿE))⊂Ext_R(PN(ⁿE)).

Proof. Parts (a), (b), (c) and (d) are well known. Parts (e) and (f) are due independently to C. Boyd-R. A. Ryan [3] and D. Carando-V. Dimant [6] while (g) can be deduced from the proof of [3, Theorem 2].

Remarks. (i) By (b) and (g) we have

PN(ⁿE)⊂PI(ⁿE) and Ext_R(PI(ⁿE))⊂Ext_R(PN(ⁿE)).

This contrasting information is quite useful since Ext_R(PI(ⁿE))has good ab- stract properties while calculations are easier with elements of Ext_R(PN(ⁿE)). (ii) In general we do not know ifPN(ⁿE)is a dual Banach space but, by (a) and (g), Ext_R(PN(ⁿE))is non-empty.

ForrealBanach space we also have the following result which is in contrast to the results we obtain in the next section forcomplexBanach spaces.

Proposition3.2 ([3], [18]). IfEis a reflexive Banach space overRand n >1then

Ext_R(PI(ⁿE))= { ±φⁿ :φ∈E,φ =1}.

4. Extreme Polynomials

Proposition4.1. IfEis a complex Banach space andn≥1then {φ ∈E :φⁿ∈Ext_R(PN(ⁿE))} ⊂Ext_C(E).

Proof. Supposeφ = 1 andφ ∈Ext_C(E). Then there existsψ ∈ E, ψ =0, such thatφ+λψ ≤1 for allλ∈C,|λ| ≤1. Ifω=e^n+12πi then

n j=0

(φ+ω^jψ)ⁿ=ⁿ

k=0

n

k ψ^kφ^n−k n

j=0

ω^kj . Ifk=0 then_n

j=0ω^kj =0. Hence 1

n+1 n j=0

(φ+ω^jψ)ⁿ =φⁿ.

Sinceφ = 1 and φ +ω^jψ ≤ 1 for all j, Proposition 3.1(c) implies φⁿN =1 and,(φ+ω^jψ)ⁿN ≤1 for allj. Henceφⁿ∈/Ext_R(PN(ⁿE)). This completes the proof.

IfEis a complex Banach space we say thatxandyinEare equivalent if there existsα∈C,|α| =1, such thatx =αy. IfA⊂Ewe denote by[A] the cardinality of the set of equivalence classes inA. For example it is well known that[Ext_R(ⁿ1)]= n. For complex extreme points we obtain a different type of result.

Corollary 4.2. IfE is a finite dimensional complex Banach space of dimension≥2then

[Ext_C(E)]= ∞.

Proof. By Proposition 4.1 and the Krein-Milman Theorem dim(PI(ⁿE))≤ [Ext_C(E)]

for any positive integer n. Since the monomials z^jw^n−j, j = 1,2, . . . , n are linearly independent, dim(PI(ⁿE))= dim(P(ⁿE))is at leastn. Hence [Ext_C(E)]≥nfor alln. This completes the proof.

IfEis a Banach space we say thatA ⊂ E isE-transitiveif for allθ,φ inAthere exists an isometry ofEonto itself,T, such thatφ◦T =θ, that is

tT (φ)=θwhere^tT denotes the transpose ofT.

Corollary4.3. If the set of complex extreme points of the unit ball ofE isE-transitive then

Ext_R(PI(ⁿE))= {φⁿ:φ∈E, φ∈Ext_C(E)} and, in particular,

Ext_R(E)=Ext_C(E).

Proof. By Proposition 3.1(a), (f) and (g) and Proposition 4.1, Ext_R(PI(ⁿE)) contains an element φⁿ where φ ∈ Ext_C(E). Let ψ ∈ Ext_C(E). By E- transitivity of the set of complex extreme points there exists an isometryT of E such that ψ ◦T = φ. If P ∈ PI(ⁿE) and ψⁿ ± PI ≤ 1 then ψⁿ◦T±P◦TI ≤1. Sinceψⁿ◦T =(ψ◦T )ⁿ=φⁿandφⁿ∈Ext_R(PI(ⁿE)) this impliesP◦T =0. HenceP =0 andψⁿ ∈Ext_R(PI(ⁿE)). An application of Proposition 4.1 completes the proof.

Example 4.4. LetA denote a W^∗-algebra , i.e. aC^∗-algebra which is also a dual Banach space. We can suppose, without loss of generality, thatA is aC^∗-subalgebra ofB(H), the space of bounded linear operators on the Hilbert spaceH. By [15] the real and complex extreme points of the unit ball of aC^∗-algebra coincide.

Let

A^⊥= {w∈N (H): trace(vw)=0 for allv∈A }.

whereN (H )is the space of trace class operators onH. By [17, Theorem 4.2.9],A∗:=N (H)/A^⊥is the unique isometric predual ofA. The(A,A∗) duality is given by

(u, w+A^⊥)=trace(uw)=

x∈E

uw(x), x

whereu∈ A, w+A^⊥ ∈A∗andEis an orthonormal basis forH. LetU denote the set of unitary elements inA, i.e.

U = {u∈A :uu^∗=u^∗u=1_H }.

Clearlyuw∈A^⊥foruunitary inA andwinA^⊥. Hence the mapping ut :A∗→A∗, ut(v+A^⊥):=uv+A^⊥

is well defined and easily seen to be an isometry ofA∗. Ifv¹andv²are unitary elements inA thenv2^∗v¹is also unitary and for allw+A^⊥∈A∗we have

(v2◦(v2^∗v1)t, w+A^⊥)=(v2, v2^∗v1w+A^⊥)=

x∈E

v2v^∗2v1w(x), x

=

x∈E

v1w(x), x =(v1, w+A^⊥).

Hence

v1=v2◦(v^∗2v1)t

andU isA∗-transitive. Results in [16] (see also Example 4.6) imply that Ext_C(A)=Ext_R(A)=U

i.e. the real (and complex) extreme points coincide with the set of unitary elements whenA is one of the following :

(a) a commutativeW^∗-algebra, (b)B(H),H finite dimensional, (c) a type II1factor.

Hence, by Corollary 4.3, ifA is any of the aboveW^∗-algebras then Ext_R(PI(ⁿA∗))= {φⁿ :φ∈Ext_C(A)}.

If dim(A)≥2 then Ext_C(A)is a proper subset of{x :x∈A,x =1}. (See for instance [5, Theorem 1.9]). For example ifA =^m_∞then its predual is^m1 and Ext_R(PI(^nm1))consists of allφⁿ whereφlies in the distinguished boundary of the unit polydisc inC^m.

The above method does not extend to allW^∗-algebras. By [16, Corollary 2]

Ext_C(B(H))= {u∈B(H):uu^∗ =IH oru^∗u=IH }.

IfH is infinite dimensional then the forward shift,S, is a non-unitary extreme point of the unit ball ofB(H). By [16] isometries ofB(H )map extreme points to extreme points and unitaries to unitaries. Hence no isometry ofB(H) mapsStoIH and Ext_C(B(H))=Ext_R(B(H))is notB(H)_∗-transitive.

Proposition4.5. IfEis a finite dimensional complex Banach space then {φⁿ:φ∈Ext_R(E)} ⊂Ext_R(PI(ⁿE))

for alln.

Proof. Letφ∈Ext_R(E). Supposeφⁿ∈/Ext_R(PI(ⁿE)). SinceEis finite dimensional Proposition 3.1(e) implies thatφⁿ∈/Ext_R(PN(ⁿE)).

Hence there existP andQinPN(ⁿE),P =Q, andλ∈ R, 0< λ < 1, such that

φⁿ=λP +(1−λ)Q

and P = Q =1.

SinceE is finite dimensional so also is PN(ⁿE). Hence the unit ball of PN(ⁿE)is the convex hull of its extreme points and thus, by Proposition 3.1(f), there exists (φi)^k+l_i=1 ⊂ E, φi = 0, such that λP = _k

i=1φⁿ_i, λPN = _k

i=1φⁿ_i,(1−λ)Q=_k+l

i=k+1φⁿ_i and(1−λ)QN =_k+l

i=k+1φ_iⁿ.

Hence

φⁿN = φⁿ =1=^k+l

i=1

φⁿ_i.

Choosex⁰ ∈ E,x⁰ = 1, such thatφ(x⁰) = 1. On differentiatingφⁿ = _k+l

i=1φⁿ_i atx0we obtain

φ =φⁿ⁻¹(x0)φ=^k+l

i=1

φ_iⁿ⁻¹(x0)φi. Hence

1= φ ≤^k+l

i=1

|φ_iⁿ⁻¹(x0)|φi ≤^k+l

i=1

φiⁿ =1.

and we can choose for eachi,βi ∈C, with|βi| =1 such that φ=

k+l i=1

|φ_iⁿ⁻¹(x0)| · φi · βiφi

φi.

Sinceφ ∈Ext_R(E)this implies that for eachi,φi =αiφfor someαi ∈C.

HenceP = φⁿ = Q. This contradicts our hypothesis and shows thatφⁿ ∈ Ext_R(PN(ⁿE))=Ext_R(PI(ⁿE))and completes the proof.

Example 4.6. IfJ is a JB^∗-algebra then, by [4, Lemma 4.1] and [15, Theorem 11],

Ext_R(J)=Ext_C(J).

Hence Propositions 4.1 and 4.5 imply

{φⁿ :φ∈Ext_R(J)} =Ext_R(PI(ⁿJ))

for any finite dimensional JB^∗-algebra J. This result gives an independent proof of the result in Example 4.4 for finite dimensionalC^∗-algebras.

Example4.7. IfE is a strictly convex finite dimensional Banach space then Ext_R(PI(ⁿE))= {φⁿ :φ ∈E,φ =1}.

This is similar to the result for real Banach spaces quoted above (Proposi- tion 3.2).

To extend Proposition 4.5 to infinite dimensional spaces we requireweak^∗- exposedpoints. For convenience we introduce these in complex form.

Definition4.8. LetEdenote a Banach overC. A linear functionalφ∈E is aweak^∗-exposed point of the unit ball ofE ifφ = 1 and there exists

x ∈E,x = 1, such that(φ(x)) = 1 and(ψ(x)) < 1 for allψ ∈ E, ψ ≤ 1 and ψ = φ. We say thatx weak^∗-exposesφ. We let Exp_ω∗(E) denote the set of all weak^∗-exposed points of the unit ball ofE.

Ifφ:E−→Cis complex linear thenφ˜ := (φ)is a real linear mapping.

Moreover,φ = ˜φandφ(x)= ˜φ(x)−iφ(ix)˜ for allxinE. Hence ifφ1

andφ2are complex linear thenφ1=φ2if and only if(φ1)= (φ2). Ifxweak^∗-exposesφandφ = ^φ1+φ₂ ²withφ1 = φ2 =1 then,|φ1(x)| ≤ 1 and |φ2(x)| ≤ 1 imply φ1(x) = φ2(x) = 1. Hence φ1 = φ2 = φ and Exp_ω∗(E)⊂Ext_R(E).

For our next result, which was motivated by [3, Proposition 5], we require the following Lemma [3, Lemma 4].

Lemma4.9. LetEbe a normed space and letφbe a unit vector inE. Sup- pose that for each finite dimensional subspaceF ofEthere exists a subspace GofE, F ⊂G, such thatφ|Gis an extreme point of the unit ball ofG. Then φis an extreme point of the unit ball ofE.

Proposition4.10. IfEis a complex Banach space then {φⁿ:φ ∈Exp_ω∗(E)} ⊂Ext_R(PI(ⁿE)).

Proof. Letφ∈Exp_ω∗(E)and supposexweak^∗-exposesφ. Given a finite dimensional subspace X of

n,s,E choose a finite dimensional subspace F of E such that x ∈ F and X ⊂

n,s,F. Since x ∈ F, φ|F = 1. Let ψ ∈F,ψ =1 andψ =φ|F. By the Hahn-Banach Theorem there exists ψ˜ ∈ E such thatψ|˜ F = ψ and ˜ψ = ψ = 1. Sinceψ˜ = φ we have (ψ(x))˜ = (ψ(x)) <1 andφ|F ∈Exp_ω∗(F)⊂Ext_R(F).

By Proposition 4.5

(φ|F)ⁿ =φⁿ|F ∈Ext_R(PI(ⁿF ))=Ext_R

n,s,

F .

By Proposition 3.1(a) and Lemma 4.9, φⁿ∈Ext_R

n,s,

E

=Ext_R(PI(ⁿE)).

This completes the proof.

Example4.11. IfEis a complex Banach space let

D(E)={x∈E:x =1 andf (·):= ·is real Gâteaux differentiable atx}.

Then, see [19] and [10],

Exp_ω∗(E)= {φ∈E:φ=f(x)for somex∈D(E)}.

For many classical Banach spaces, e.g.p, 1< p <∞, this result can be used to show that

{φ :φ∈E,φ =1} =Exp_ω∗(E) and hence, by Proposition 4.10, that

Ext_R(PI(ⁿE))= {φⁿ :φ ∈E,φ =1}.

IfE = ¹thenx =(xn)n ∈D(¹)if and only ifxn = 0 for alln. Hence φ := (yn)n ∈ Exp_ω∗(∞)if and only if|yn| =1 for allnand Exp_ω∗(∞) = Ext_R(∞). Since Ext_R(∞)=Ext_C(∞)this implies

{φⁿ:φ∈Ext_R(∞)} =Ext_R(PI(ⁿ1))

and we recover a special case of Example 4.4 since∞ is aW^∗-algebra.

Example4.12. Let^m_∞denoteC^mendowed with the supremum norm. By Proposition 3.1(f)

Ext_R(PI(^nm_∞))⊂ {φⁿ:φ ∈^m1,φ =1}

forn ≥ 2. Ifφ := (w1, . . . , wm) ∈ ^m1,θ := (θ1, . . . , θm) ∈ R^m andσ is a permutation of{1, . . . , m}let

φθ,σ :=(e^iθ1wσ (1), . . . , e^iθmwσ(m)).

It is easily seen thatφ_θ,σⁿ ∈Ext_R(PI(^nm_∞))if and only ifφⁿ ∈Ext_R(PI(^nm_∞)). Hence to show

(3) Ext_R(PI(^nm_∞))= {φⁿ:φ∈^m1,φ =1}.

it suffices to show thatαⁿ ∈ Ext_R(PI(^nm_∞))whereα := (α1, . . . , αm)sat- isfies 0≤ α1 ≤ α2 ≤ · · · ≤ αmand_m

i=1αi = 1. Letl denote the smallest positive integer such thatαl > 0. Ifl = mthenα ∈ Ext_R(^m1)and Proposi- tion 4.5 implies thatαⁿ ∈Ext_R(PI(^nm_∞)). We may thus suppose, from now on, thatl < m.

By Proposition 3.1(d), (PI(^nm_∞)) = P(^nm1) with duality φⁿ, P = P (φ)forP ∈P(^nm1)andφ ∈(^m_∞)= ^m1. By the Krein-Milman Theorem

|P|,P ∈P(^nm1), achieves its maximum over the unit ball of^m1 at a pointφ whereφⁿ∈Ext_R(PI(^nm_∞)).

Now consider the 2-homogeneous polynomial on^m1

Pα(z¹, . . . , zm):=

l≤i<j≤m

zizj+

k≥l

αk−αl

2αk z_k². Since

|Pα(z1, . . . , zm)| = |Pα(0, . . . ,0, zl, . . . , zm)|

≤

l≤i<j≤m

|zi||zj| +

k≥l

αk−αl

2αk |zk|²

=Pα(0, . . . ,0,|zl|, . . . ,|zm|)

≤ Pα ·

k≥l

|zk|

2

it follows that|Pα|achieves its maximum,Pα:= PαBm1, at some point which has the forme^iθ(0, . . . ,0, xl, . . . , xm)wherexi ≥0 fori≥l,_m

i=lxi = 1 andθ ∈R.

Applying the method of Lagrange multipliers to the problem of maximizing

|Pα|on the setT:=

(0, . . . ,0, xl, . . . , xm):xi >0 and _m

i=lxi =1 yields the equations

i>l

xi =λ=1−xl

and m

i=l

xi −xk+ αk−αl

2αk 2xk =λ fork > l. Henceαlxk = αkxl fork ≥ l. Since_m

i=lxi = _m

i=lαi = 1 this impliesαi =xi fori≥l.

Hence

sup{ |Pα(x)|:x ∈T} =

l≤i<j≤m

αiαj+

k≥l

αk−αl

2αk α_k²

= 1 2

m i=l

αi 2

− 1 2αl·

m k=l

αk

= 1

2(1−αl).

Ifl^∗≥l,{l^∗} ⊂S⊂ {l^∗, . . . , m}let S=

x∈R^m:x =(0, . . . ,0, xl^∗, . . . , xm),

xi >0 ifi∈S, xi =0 ifi /∈Sand m

i=l

xi =1

.

An analysis similar to the above shows that sup{ |Pα(x)|:x ∈S} =

i∈Sαi

−αl

2

i∈Sαi = 1 2

1− αl i∈Sαi

Sinceαi>0 wheni≥lthis shows thatPα = |Pα(z1, . . . ,zm)|,_m

i=1|zi|=1, if and only if(z¹, . . . , zm)=e^iθαfor someθ ∈R. Henceα²∈Ext_R(PI(^2m_∞)) and we have established (3) whenn=2.

Ifβ:=(1,1, . . . ,1)∈^m_∞ thenβ =1 andβ, (z1, . . . , zm) =_m

i=1zi

for(z1, . . . , zm)∈^m1. Hence|β|achieves its maximum over the unit ball of^m1

preciselyatallpoints of the forme^iθ(x1, . . . , xm)wherexi ≥0,_m

i=1xi =1 andθ ∈R.

Ifn > 2 thenQ := Pα ·βⁿ⁻² is an n-homogeneous polynomial on ^m1

and |Q| achieves its maximum over the unit ball of ^m1 precisely at those points which have the form e^iθα where θ ∈ R is arbitrary. Hence αⁿ ∈ Ext_R(PI(^nm_∞))and we have established (3) for alln≥2 and allm.

Note that the function

f ((yn)n):=1+Pα

e^iθnyσ (n)_m

n=1

satisfies|f (y)|<|f (x)|for ally =xin the closed unit ball of1. This may be used to show that all finitely supported unit vectors are peak points of different algebras of holomorphic functions (we refer to [2] for details).

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