Internat. J. Math. & Math. Sci.
VOL. 14 NO. 2 (1991) 227-232
227
AN EXTENSION OF HELSON-EDWARDS THEOREM TO BANACH MODULES
SIN-El TAKAHASI
Yamagat a University
Department of Basic Technology
Faculty of Engineering
Yonezawa 992
JAPAN
(Received September 23, 1988)
An extension of the Helson-Edwards theorem for the group algebras to Banach
ABSTRACT.
modules over commutative Banach algebras is given.
This extension can be viewed as a
generalization of Liu-Rooij-Wang’s result for Banach modules over the group algebras.
KEY WORDS AND PHRASES.
Multiplier, Banach modules, bounded approximate identity,
compact abelian group, completely regular.
I.
INTRODUCTION.
Let A be a commutative complex Banach algebra with a bounded approximate identity
{u}
of norm
and denote by
field of complex numbers.
A the
The space
class of all nonzero homomorphisms of A into the
A’
with the Gelfand topology, is called the
carrier space of A. Let X be a Banach left A-module.
of A into X is called a multiplier of X.
A continuous module homomorphism
We introduce a family
{X:
e
A
of Banach
A-modules such that any multiplier T of X can be represented as a function T on
with
T()
e
X
for each
e
A"
A
In this setting we give an extension of the Helson-
Edwards theorem for the group algebras to Banach modules.
We also observe that this
extension can be viewed as a generalization of Liu-RooiJ-Wang’s result for Banach
modules over the group algebras.
We further consider a local property of multipliers
when A is completely regular.
2.
REPRESENTATION THEOREM OF MULTIPLIERS.
For each
A’
let
and define
x
M denote the
maximal modular ideal of A corresponding to
p{M,X +
e,lX},
(,
where sp denotes the closed linear span and
e
Note that X # does not depend on the choise of
is an element of A with
e.
(e)
I.
S. TAKAHASI
228
Throughout the remainder of this note we will assume
(
p(AX)
(2.1)
{0}.
A, the condition (2.1)
In the case of X
The space
A s(M)
is equivalent to the semlslmpliclty of A.
is called the essential part of X and is denoted by X
e.
a bounded approximate identity, it follows that X
factorization theorem (see Doran-Wlchman [I]).
X
A"
for all
exist
X
In fact
a l,...,a
M
s-(M
e
let #
and
x
Since A has
AX from the Cohen-Hewitt
e
We also have
(2.2)
X)
x
n
X#
x
%A’
y
X
and e
e
>
O.
X#
Since x
there
X such that
jx- I -(- >yJj <
i=l
/-
Therefore for each I, we have
n
l]UxX
Letting e
+ 0,
[ uxalx i
c
sp(M#X)
uxe)y)jJ
(u%
ulx sp(M#X)
we obtain that
Consequently, we have that x
+
for all I.
and hence
<
e.
Since x
XVOXeC sp(MX).
Xe,
llm
ulx
x.
The reverse
inclusion is immediate.
We denote by M(A, X), or simply M(X), the class of all multipliers of X.
_
Then
MkX) also becomes a Banach A-module under the module multiplication defined by
[aT)b
a(Tb).
For each x
multiplier of X, so that
(a)
of A into X defined by
X, the mapping
becomes a module homomorphism of X into M(X).
ax is a
Also it can
be easily observed that
for all T
M(X) and
Now, for each
.
.
Xe and TM MX
A’ where the bar denotes the norm
#A’ let X# X/X be the quotient of
TA
(2.3)
closure.
So
X by X
Banach A-module under the natural module structure and the quotient norm.
x
X, let x()
x + X
function o defined on
o()E
X for each
A"
Denote by
bx
A vector field on
Of course,
(x
A is
X) is a
the class of all vector fields on
where a denotes the Gelfand transform of a
Then
A"
HX
A-module under the module multiplication
vector field on
becomes an
A
be the natural image of x in X
with
X#Forbecomes
each
A.
A
beomes a Banach A-module under the norm
defined by
(ao)()
Define
j[
11. and X {:
x
()o(),
a
a
EXTENSION OF HELSON-EDWARDS THEOREM TO BANACH MODULES
_
229
Wih the above notations, we have the following representation theorem of
multipliers.
M(X), then there
THEOREM 2.1. (i) If T
that Ta
aT for all a
bx.
A
such
T is a continuous module
(li) The mapping T
A.
isomorphism of MiX) into
exists a unique vector field T on
A" Since eauk ()euk 6 M for all k, it
follows from (2.3) that
T(eauk) -a()T(euk) MX for all k. Hence, after taking
the limit with respect to
we obtain
T(ea) a()TeG_ MX. Note also that
Let T M(X), a
PROOF.
A and
,
Ta ( X
from (2.3).
e
Then there exist c
A and y
X such that Ta
cy, so that
T(ea) Ta -eTa (c -ec)y 6_ MX. We therefore have
ra a()Te (Ta
Tea) + (Tea -()re ) MX + MX (MX) X
Setting T()
Te(), we obtain that () a()T() (aT)(). In other words,
Ta-
A.
aT for all a
re()
,Ti$)
X such allthat Ta
If o
=()o()
o()
TA
AT
ao for all a6 A, then
A’
so that T
To show that this mapping is inJectlve, let T
[0} from (i), so TA
Also TA
A
k2.2) and our assumption (2.1),
TA
X"
Xe(X
(
A
We thus obtain T
This proves (i).
o.
It
T is a continuous module homomorphism of M(X) into
is immediate from (i) that T
qbx
for
Ta
O, and (li)
A
C-
-(MX)
M(X) with T
Xe
from (2.3).
O.
Then
Therefore, by
{0}.
is proved.
A Banach left A-module X is said to be order-free if for
X with x
every x
A with
0 there exists a
ax 0.
COROLLARY 2.2.
Let x e X.
PROOF.
e
or X is order-free, then x
0
Note first that
2()x
(a
This implies (2.4).
A,
ax, a
ax
In fact, for each
ax
X
If either x
0.
implies x
e
(2.4)
x6_ X.
A’
()e)x
Now let
a()(!
x
X with x
x
e
0.
MX
By the
(l-e) XC X #
above theorem and (2.4), we
have
x () e,(*lx(*) (e,x)(*) xe, (*) e,x(,) -,(,)x(,)
for all
A’
either x e X
3.
e
so that
or X
x
x.
The.
s order-free,
x
then x
0 and he.ce
{0}.
Accordingly,
0.
EXTENSION OF HELSON-EDWARDS THEOREM.
We give a characterization of multipliers of an order-free Banach A-module
is similar to [2, Theorem 1.2.4] and Liu, van RooiJ, and Wang [3, Lemma 1.3].
which
230
S. TAKAHASI
_
Let X be order-free and T a mapping of A into X.
COROLLARY 3.1.
fo[lowlng conditions are equivalent.
(i)
T6. M(X).
(ii)
T is linear and continuous;
(lii) T(ab)
.
aTb for all a,b
PROOF.
a,b
T(abul)
to
==>
(i)
,
TMC
abul -’a()bulr-. M
Since
X for
-()T(bul),-TM
T(ab)
()rb X
we obtain
T(ab)
aTb by Corollary 2.2.
==>
(iii)
..
A"
(ii) follows immediately from (2.3).
A"
A and
A.
for every
X
,
for all
Then the
==>
(ii)
(iii).
Let
it follows from (ii) that
Hence, after taking the limit with respect
aTb, so that
Then, by (2.4), T(ab)=
all
a
To show that T is linear, let a,b e. A and e,B scalars.
(i).
cT(aa + Bb)
T(aac
(aa + b)Tc
Bbc)
ecTa + BcTb
Then
aaTc + bTc
c(aTa + BTb)
eTa + BTb.
Since X is order-free, T(0m + Bb)
A and lira Ta
To show the continuity of T, let lira a
a
n
n
for all c A.
.
for all b
Let M(A)
bTa
So Ta
aTb
lim
anTb
x
X.
Then
bx
lim bTa n
x and hence T is continuous by the closed graph theorem.
{T: T M(X)}.
-
The following result is an extension of the Helson-
Edwards theorem for the group algebra of a locally compact Abelian group (see Rudin
[4, Theorem 3.8.
THEOREM 3.2.
PROOF.
,,,
X
Note first that
If T(-" M(X) with T
2.2.
a
).
Let
A.
x for all
Suppose conversely that AoC M(X).
bco
element of
T(ab
3.1.
T
Xe,
aba
bS
x
Sb e X
e
Let a
bc for some b, c
from (2.3).
a.
M(X).
X as observed in the proof of Corollary
a, then, by Theorem 2.1, aa
theorem, a can be written as a
Then, ao
,
Then, AaC M(X) if and only if
A.
aT
Ta
Ta(M(X)
for all
By the Cohen-Hewitt factorization
A
S.
M(X) with co
Choose S
A.
Hence, by Corollary 2.2, there
is a unique
If a,b are arbitrary elements of A, then
say Ta, such that ao
aTb by Corollary
X
a(ba)
Tb by (2.4). Since TA
e’ T(ab)
aT
Note that Xe is an order-free Banach A-odule. Then, by Corollary 3.1,
T 6 M(X) and the theorem is proved.
M(X).
Consequently,
M(A, X e )C M(A, X)
We will observe that Theorem 3.2 can be viewed as a generalization of Liu-RooiJWang’s result [3, Theorem 2.3].
Let G be a compact Abelian group and X a Banach/ (G)-module. Let x7 7x for each 7
the dual group of G. Also denote by
the class of all mappings p of into X such that p(7)e
for
G.
Set
every
7
7(f) ](7) (7 ,f L(G)), where ] is the Fourier transform of ]. Note
x7
7= (1- 7)x and x 7 is isometrically module-isomorphic to
that for each 7 e
Also since
"s-’ L(G), it follows that
x
,x
x.
EXTENSION OF HELSON-EDWARDS THEOREM TO BANACH MODULES
For each x e X
and hence X satisfies (2 I).
{:
and set
K<
then
fo
fOe X for every f
o(y)
set
.
O(y)()
can be extended to a multiplier of X if and
EX
.
(G)).
*f ( G, f
Clearly f()
M(X)
T
r
LI(G).
X for every f (
PROOF.
to G
the restriction of
denote by
X}.
COROLLARY (Liu-Roo[j-Wang). 0
only if fp
231
X for every f
LI(G).
Then
or
LI(G)
T G for some
0
and
Suppose conversely that
LI(G),
each f
choose
X with f
xfE
f.
We then have
for each y G.
f(y))(#y)
(fo)(,7)
So if
(xf(y))
f(,) xf(, T)
y:-
for all
G and f
some T e M(X) from
so that o(T)
o(T)
4.
TT
TT
TX
6
< LI(G).
eorem
e X
for all
for all f e
T)X for all y ( G.
(1
LI(G)
and hence
T for
Therefore,
3.2.
<
xf 6 X
Thus fo
T7
t
X
and so
TIC.
Consequently, p
G.
But O(T),
LOCAL PROPERTIES OF LTIPLIERS.
We wilt consider local properties of multipliers.
To do this, we introduce the
following notation which is exactly similar to one given in Rfckart [5, 2.7.13].
near a
Then g is said to belong to
Let g<
and
(or at infinity) provided there exists a neighborhood V of # (or
DEFINITION.
A
point
ECEX#.
EX#
point of
A
.
If
belongs to near every
Z sach that glV
g Is said to belong locally to
’lV.
infinity) and an element g’
and at infinity, then
proof of one.
Assume A to be completely regular and let g be a submodule of
THEOREM 4.1.
If
g
s
#
PROOF.
,
belongs locally to
Since g belongs to
compact complement K and
gO <
then
g (.
at infinity, there exists an open set U 0 of
Also since g belongs to
with
g]U O.
golU0
every point of K, there exists a finite open covering
gilUf glUi
{gl,...,gn}
of Z with
of the identity (of. [5, Theorem 2.7.12]).
e
{U1,...,U n}
#A
with
near
of K and a finite
Note that A admits a partion
el,...,e n A such that
U i)
ker(
K
ker
and
A
dulo
+
ts an identity for
e +
ei6
A
1,..., n), where ker K denotes the kernel of K. Set
subset
(i
.
[5, 2.7.16] and we refer to the
The following result is similar to one given tn
en
(i
,n).
Then there eists
S. TAKAHASI
232
kl -e)o 0 +
o’
Then
o’
is obv[ously in Z.
elo
+
enOn
o’
o.
+
We further assert
In fact,
[f
U0,
then we
have
O’)
I -e0))o #) +
e(,) +
l
0 u.
Z
e
i
ei#)o
(#)
(0)
i
and {i:
K, then e()
If 0
o’()
l
+
n, # 4 U.}
i
U
#
O,
so that
ei(#)oi(#) e7.U ei()o(
i
e()o())
Consequently, o’
o and the theorem is proved.
Because M(X) is a submodule of NX
0,
we obtain the following local property of
multipliers from the preceding theorem.
COROLLARY 4.2.
M(X), then o
Assume A to be completely regular.
If
% M(X).
t
X belongs
Let A contain local identities (cf. [5, 3.6.11]) and T 6 M(X).
locally to
The closure
T(0) 0} is called the support of T and is denoted by supp T. If supp T
In fact, by [5, Theorem
X with T
is compact, then there exists a unique x
x
e
3.6.13], A has an identity for A modulo ker(supp T), say e. Set x Te. So the
of {0 6
A:
desired result follows from Theorem 2.1 and Corollary 2.2.
Similarly, we obtain that for each compact set K of
A’
there exists x e
Xe with
REFERENCES
I.
DORAN, R.S. and WICHMANN, J., A_p_proximate Identities and Factorlzation in Banach
Modules, Lecture Notes in Math. 768, Sprlnger-Verlag, New York-Heidelberg,
1979.
2.
LARSEN, R., An Introduction to the Theorey of Multipliers, Springer-Verlag, New
York-Heidelberg, 1971.
3.
LIU, T.S., VAN ROOIJ, A.C.M. and WANG, J.K., A Generalized Fourier Transformation
for Ll(G)-modules, J. Austral. Math. Soc. (Series A) 36 (1984), 365-377.
4.
RUDIN, W., Fourier Analysis on Groups, New York, N.Y.: Interscience Publishers,
Inc., 1962.
5.
RICKART, C.E., General Theorey of Banach Alebras, Van Nostrand, Princeton,
N.J.. 1960.