A NOTE ON A CLASS OF BANACH ALGEBRA-VALUED POLYNOMIALS

IJMMS 32:3 (2002) 189–192 PII. S0161171202203348 http://ijmms.hindawi.com

© Hindawi Publishing Corp.

A NOTE ON A CLASS OF BANACH ALGEBRA-VALUED POLYNOMIALS

SIN-EI TAKAHASI, OSAMU HATORI, KEIICHI WATANABE, and TAKESHI MIURA

Received 10 March 2002

LetFbe a Banach algebra. We give a necessary and sufficient condition forFto be finite dimensional, in terms of finite typen-homogeneousF-valued polynomials.

2000 Mathematics Subject Classification: 46H99.

1. Introduction and results. LetEandF be complex Banach spaces. We denote by L(ⁿE,F)the Banach space of all continuousn-linear mappingsAfromEⁿintoF en- dowed with the normA =sup{A(x1,...,x_n):x_j ≤1, j=1,...,n}. A mappingP fromEintoFis called a continuousn-homogeneous polynomial ifP(x)=A(x,...,x) (for allx∈E) for someA∈L(ⁿE,F). We denote byP(ⁿE,F)the Banach space of all continuous n-homogeneous polynomials P fromE intoF endowed with the norm P =sup{P(x):x ≤1}. Also a mappingPfromEintoF is called a finite type n-homogeneous polynomial ifP(x)=f1(x)ⁿb1+···+f_k(x)ⁿb_k(for allx∈E), where f1,...,f_k∈E^∗ andb1,...,b_k∈F. We denote byP_f(ⁿE,F)the space of all finite type n-homogeneous polynomialsPfromEintoF. Then we havePf(ⁿE,F)⊆P(ⁿE,F). In- deed, letP∈P_f(ⁿE,F). Then we writeP(x)=f1(x)ⁿb1+ ··· +f_k(x)ⁿb_k(x∈E)for somef1,...,f_k∈E^∗andb1,...,b_k∈F. Set

Ax1,...,x_n

= k i=1

f_ix1

···f_ix_nb_i, x1,...,x_n

∈Eⁿ. (1.1)

Then A is a continuous n-linear mapping from Eⁿ into F and P(x)=A(x,...,x) (x∈E). Hence P∈P(ⁿE,F). We are now interested in the case that F is a Banach algebra. Let

Pf_nE,F

=ϕ₁ⁿ+···+ϕⁿ_k:ϕ_j∈B(E,F) (j=1,...,k), k∈N, (1.2) whereϕⁿ_j(x)=(ϕ_j(x))ⁿ(x∈E). Then we havePf(ⁿE,C)=P_f(ⁿE,C)andPf(ⁿC,F)⊆ Pf(ⁿC,F) (see [1, Section 1]). Also, we have Pf(ⁿE,F) ⊆P(ⁿE,F). Indeed, let P ∈ Pf(ⁿE,F). Then we can writeP =ϕ₁ⁿ+ ··· +ϕⁿ_k for some ϕ1,...,ϕ_k∈B(E,F). Set A(x1,...,xn)=_k

i=1ϕi(x1)···ϕi(xn),(x1,...,xn)∈Eⁿ. ThenAis a continuousn- linear mapping fromEⁿintoF andP(x)=A(x,...,x) (x∈E). HenceP∈P(ⁿE,F).

Now, for eachn∈N, we say that an algebra F has ther_n-property if, given any b∈F, we can find elementsa1,...,ap∈F such thatb=p

i=1aⁿ_i. We also say that an algebraF has ther-property ifFhas thern-property for eachn∈N.

190 SIN-EI TAKAHASI ET AL.

Proposition1.1(see [1]). (1)Every unital complex algebra has ther-property.

(2)LetEbe a Banach space andF be a Banach algebra. ThenPf(ⁿE,F)⊆Pf(ⁿE,F) if and only ifF has ther_n-property.

In [1], it is remarked that, given an arbitrary Banach space(F,+,·), we can always define a product◦and a norm·∗onFin order that(F,+,◦,·∗)is a unital Banach algebra and · ∗ is equivalent to · . By Proposition 1.1 and the above remark, Lourenço-Moraes proved the following proposition.

Proposition1.2(see [1]). LetEbe a Banach space. The following are equivalent:

(a)Eis a finite-dimensional space;

(b)Pf(ⁿE,F)=Pf(ⁿE,F)for everyn∈Nand for every Banach algebraF with the r_n-property;

(c)P_f(ⁿE,F)=Pf(ⁿE,F)for everyn∈Nand for every unital Banach algebraF. Remark1.3. By the proof ofProposition 1.2(see [1]), we see that each of the fol- lowing two statements are also equivalent to one of, hence all of, (a), (b), and (c) in Proposition 1.2:

(b) P_f(¹E,F)=Pf(¹E,F)for every unital Banach algebraF;

(d) Pf(ⁿE,F)=Pf(ⁿE,F)for everyn∈Nand for every Banach spaceF. In this note we show the following result, which is opposite toProposition 1.2.

Proposition1.4. LetF be a Banach algebra. Then the following are equivalent:

(a)F is a finite-dimensional space;

(b)Pf(ⁿE,F)⊆P_f(ⁿE,F)for everyn∈Nand for every Banach spaceE;

(c)Pf(¹E,F)⊆Pf(¹E,F)for every Banach spaceE.

In particular, in the unital case, we have the following proposition.

Proposition1.5. LetF be a unital Banach algebra. Then the following are equiv- alent:

(a)F is a finite-dimensional space;

(b)Pf(ⁿE,F)=P_f(ⁿE,F)for everyn∈Nand for every Banach spaceE;

(c)Pf(¹E,F)=Pf(¹E,F)for every Banach spaceE.

2. Proofs

Lemma2.1. Letnbe any positive integer and letx1,...,x_nben-real variables. Then n

i=1

x_i= 1 2ⁿn!

ε₁,...,εn=±1

ε1···ε_n



ⁿ

k=1

ε_kx_k





n

(2.1)

holds.

Proof. For eachmwith 0≤m≤n, let

P_mx1,...,x_n

=

ε1,...,εn=±1

ε1···ε_n



ⁿ

k=1

ε_kx_k





m

. (2.2)

A NOTE ON A CLASS OF BANACH ALGEBRA-VALUED POLYNOMIALS 191 Then we havePm(0,x2,...,xn)=Pm(x1,0,...,xn)= ··· =Pm(x1,...,x_n−1,0)=0. In- deed since

P_mx1,...,x_n

=

ε2,...,εn=±1

ε2···ε_nx1+ε2x2+···+ε_nx_nm

−

ε2,...,εn=±1

ε2···ε_n

−x1+ε2x2+···+ε_nx_nm, (2.3)

it follows thatP_m(0,x2,...,x_n)=0. Similarly, P_mx1,0,...,x_n

= ··· =P_mx1,...,x_n−1,0

=0. (2.4)

Therefore, we have

Pm

x1,...,xn

=0, (2.5)

for eachm=0,1,2,...,n−1 and Pn

x1,...,xn

=Kn

n i=1

xi, (2.6)

for some constantKn, becausePm(x1,...,xn)ism-homogeneous forx1,...,xn. Hence we only show thatK_n=2ⁿn!. Note that

K_n=P_n(1,...,1)=

ε1,...,εn=±1

ε1···ε_n



ⁿ

k=1

ε_k





n

. (2.7)

ThenK1=2. Now, for eachmwith 0≤m≤n, letαm=

ε₁,...,εn=±1ε1···εn(_n

k=1εk)^m. Then by (2.5) and (2.6), we haveα0=α1= ··· =α_n−1=0 andαn=Kn. Hence,

K_n+1=

ε1,...,ε_n+1=±1

ε1···ε_n+1



ⁿ⁺¹

k=1

ε_k





n+1

=

ε₁,...,εn=±1

ε1···εn



ⁿ

k=1

εk+1





n+1

−

ε₁,...,εn=±1

ε1···εn



ⁿ

k=1

εk−1





n+1

=

n+1

m=0

n+1 m

ε₁,...,εn=±1

ε1···ε_n



ⁿ

k=1

ε_k





m

−

n+1

m=0

n+1 m

ε1,...,εn=±1

ε1···ε_n(−1)^n+1−m



ⁿ

k=1

ε_k





m

=

n+1 m=0

n+1 m

1−(−1)^n+1−m

ε₁,...,εn=±1

ε1···εn



ⁿ

k=1

εk





m

= n m=0

n+1 m

1−(−1)^n+1−m αm

= n+1

n

1−(−1)ⁿ⁺¹⁻ⁿK_n

=2(n+1)Kn,

(2.8)

so that we haveKn=2ⁿn!(n=1,2,...)inductively.

192 SIN-EI TAKAHASI ET AL.

Proof ofProposition1.4. (a)⇒(b). Let{u1,...,uN}be a basis ofFandg1,...,gN

the corresponding coordinate functionals, that is, gi(uj)=δij (i,j=1,...,N). Let P∈Pf(ⁿE,F). Then we can writeP(x)=

i=1(Ti(x))ⁿ (x∈E)for someT1,...,T∈ B(E,F). Let

f_ij(x)=g_j T_i(x)

(x∈E), (2.9)

for eachi=1,...,, j =1,...,N. Then we have Ti(x)=N

j=1fij(x)uj (x ∈E, i= 1,...,), and hence byLemma 2.1,

P(x)= i=1



^N

j=1

f_ij(x)u_j





n

= i=1

N j1=1

···

N jn=1

f_ij1(x)···f_ijn(x)u_j1···u_jn

= i=1

N j1=1

···

N jn=1

1 K_n

ε₁,...,εn=±1

ε1···εn



ⁿ

k=1

εkfij_k(x)





n

uj₁···ujn

= i=1

N j₁=1

···

N jn=1

ε1,...,εn=±1

f_{i,j1,...,jn,ε1,...,εn}(x)nb_{j1,...,jn,ε1,...,εn},

(2.10)

for eachx∈E, wheref_{i,j1,...,jn,ε1,...,εn}=ε1f_ij1+ ··· +ε_nf_ijn∈E^∗andb_{j1,...,jn,ε1,...,εn}= (1/Kn)ε1···εnuj₁···ujn∈F. Therefore we haveP∈P_f(ⁿE,F).

(b)⇒(c). This is trivial.

(c)⇒(a). Suppose that Pf(¹E,F)⊆ P_f(¹E,F) for every Banach space E. Note that P_f(¹F,F)= {T ∈B(F,F): dimT (F) <∞}andPf(¹F,F)=B(F,F). Then by hypothe- sis, the identity map ofF onto itself is finite dimensional and so isF.

Proof ofProposition1.5. This follows immediately from Propositions1.1and 1.4.

References

[1] M. L. Lourenço and L. A. Moraes,A class of polynomials from Banach spaces into Banach algebras, Publ. Res. Inst. Math. Sci.37(2001), no. 4, 521–529.

Sin-Ei Takahasi: Department of Basic Technology, Applied Mathematics and Physics, Yamagata University, Yonezawa992-8510, Japan

E-mail address:[email protected]

Osamu Hatori: Department of Mathematical Sciences, Graduate School of Science and Technology, Niigata University, Niigata950-2181, Japan

E-mail address:[email protected]

Keiichi Watanabe: Department of Mathematics, Faculty of Science, Niigata Univer- sity, Niigata950-2181, Japan

E-mail address:[email protected]

Takeshi Miura: Department of Basic Technology, Applied Mathematics and Physics, Yamagata University, Yonezawa992-8510, Japan

E-mail address:[email protected]