# Multipliers of Modules of Continuous Vector-Valued Functions

Authors: Liaqat Ali Khan; Saud M. Alsulami
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

In 1961, Wang showed that ifA is the commutative C
*-algebra C
0
(
X
) with X a locally compact Hausdorff space, then M
(
C
0
(
X
)
)
≅
C
b
(
X
). Later, this type of characterization of multipliers of spaces of continuous scalar-valued functions has also been generalized to algebras and modules of continuous vector-valued functions by several authors. In this paper, we obtain further extension of these results by showing that H
o
m
C
0
(
X
,
A
)
(
C
0
(
X
,
E
)
,
C
0
(
X
,
F
)
)
≃
C
s
,
b
(
X
,
H
o
m
A
(
E
,
F
)
)
, where E and F are p-normed spaces which are also essential isometric left A-modules with A being a certain commutative F-algebra, not necessarily locally convex. Our results unify and extend several known results in the literature.


---

## Body

## 1. Introduction

Characterizations of multipliers on algebras and modules of continuous functions with values in a commutative Banach orC
*-algebra A have been obtained by several authors. In 1961, Wang [1] showed that if A is taken as the commutative C
*-algebra C
0
(
X
) with X being a locally compact Hausdorff space, then M
(
C
0
(
X
)
)
≅
C
b
(
X
). This result has also been generalized to vector-valued functions by several authors (see, e.g., [2–6]). In 1985, Lai [6] showed that if X is a locally compact abelian group and A is a commutative Banach algebra with a bounded approximate identity, then M
(
C
0
(
X
,
A
)
)
≅
C
b
(
X
,
M
(
A
)
u
). In 1992, Candeal Haro and Lai [3] had obtained
(1)
Ho
m
C
0
(
X
,
A
)
(
C
0
(
X
,
E
)
,
C
0
(
X
,
F
)
)
≃
C
s
,
b
(
X
,
Ho
m
A
(
E
,
F
)
)
,
in the case when A is a commutative Banach algebra and E and F are left Banach A-modules.A natural question arises is to investigate the extent to which these characterizations can be made beyond Banach modules. We will focus mainly on the nonlocally convex case by consideringA a commutative complete p-normed algebra, 0
<
p
≤
1, having a minimal approximate identity and E and F being F-spaces which are also left A-modules.We mention that the arguments of earlier authors relied heavily on the fact that, in the case ofA, a Banach algebra, C
0
(
X
,
A
) is isometrically isomorphic to the completed tensor product C
0
(
X
)
⊗
λ
A with respect to the smallest cross norm λ (see [2–5]). We will avoid the use of this technique as it need not work in our case. In fact, when A is not locally convex, ⊗
λ is no longer appropriate; even for A a complete p-normed space, many complications arise (see [7, Section 10.4]; [8, p. 100]).


## 2. Preliminaries

In this section, we include some basic definitions and study various classes of topological algebras considered in this paper.Definition 1 (see [9, 10]).
LetE be a vector space over the field 𝕂
∈
{
ℝ, ℂ
}.(a)
A functionq
:
E
→
ℝ is called an F-seminorm on E if it satisfies the following:
(F
1)
q
(
u
)
≥
0 for all u
∈
E;
(F
2)
q
(
u
)
=
0 if u
=
0;
(F
3)
q
(
α
u
)
≤
q
(
u
)   for all u
∈
E and α
∈
𝕂 with |
α
|
≤
1;
(F
4)
q
(
u
+
v
)
≤
q
(
u
)
+
q
(
v
) for all u
,
v
∈
E;
(F
5
)
ifα
n
→
0 in 𝕂, then q
(
α
n
u
)
→
0 for all u
∈
E.
(b)
AnF-seminorm q on E is called an F
-norm if, for any u
∈
E, q
(
u
)
=
0 implies u
=
0.
(c)
AnF-seminorm (or F-norm) q on E is called a p
-seminorm(resp., p-norm), 0
<
p
≤
1, if it also satisfies
(2)
q
(
α
u
)
=
|
α
|
p
q
(
u
)
∀
u
∈
E
,
α
∈
𝕂
.
(
p
-homogeneous
)
.
(d)
Ifq is an F-norm (resp., a p-norm) on a vector space E, then the pair (
E
,
q
) is called an F-normed(resp., a p
-normed)space.
(e)
AnF-norm (or a p-norm) q on an algebra A is calledsubmultiplicative if
(3)
q
(
a
b
)
≤
q
(
a
)
q
(
b
)
∀
a
,
b
∈
A
.An algebraA with a submultiplicative F-norm (resp., p-norm) q is called an F-normed(resp., p
-normed)algebra.Definition 2.
(
1
) A net {
e
λ
:
λ
∈
I
} in a topological algebra A is called anapproximate identity if
(4)
lim
⁡
λ
e
λ
a
=
lim
⁡
λ
a
e
λ
=
a
∀
a
∈
A
.
(
2
) An approximate identity {
e
λ
:
λ
∈
I
} in an F-normed algebra (
A
,
q
) is said to beminimal if q
(
e
λ
)
≤
1 for all λ
∈
I.IfE and F are topological vector spaces over the field 𝕂
∈
{
ℝ or ℂ
}, then the set of all continuous linear mappings T
:
E
→
F is denoted by C
L
(
E
,
F
). Clearly, C
L
(
E
,
F
) is a vector space over 𝕂 with the usual pointwise operations. Further, if F
=
E, C
L
(
E
)
=
C
L
(
E
,
E
) is an algebra under composition (i.e., (
S
T
)
(
u
)
=
S
(
T
(
u
)
)
,
u
∈
E) and has the identity I
:
E
→
E given by I
(
u
)
=
u
(
u
∈
E
).Definition 3.
Let(
E
,
q
E
) and (
F
,
q
F
) be p-normed spaces. For any linear map T
:
E
→
F, define
(5)
∥
T
∥
q
E
,
q
F
=
sup
⁡
{
q
F
(
T
u
)
:
u
∈
E
,
q
E
(
u
)
≤
1
}
.
Then, by ([10, p. 101-102]), T
∈
C
L
(
E
,
F
) if and only if ∥
T
∥
q
E
,
q
F
<
∞. Further, ∥
·
∥
q
E
,
q
F is an F-norm on C
L
(
E
,
F
) and, for any T
∈
C
L
(
E
,
F
),
(6)
q
F
(
T
u
)
≤
∥
T
∥
q
E
,
q
F
·
q
E
(
u
)
∀
u
∈
E
.
In particular, if T
∈
C
L
(
E
)
=
C
L
(
E
,
E
), we denote
(7)
∥
T
∥
q
E
∶
=
sup
⁡
{
q
E
(
T
(
u
)
)
:
u
∈
E
,
q
E
(
u
)
≤
1
}
.
In this case, for any S
,
T
∈
C
L
(
E
), |
|
S
T
|
|
q
E
≤
|
|
S
|
|
q
E
|
|
T
|
|
q
E; hence (
C
L
(
E
)
,
∥
·
∥
q
E
) is a p-normed algebra.Definition 4.
LetE and F be topological vector spaces. Theuniform operator topology
σ (resp., thestrong operator topology
s) on C
L
(
E
,
F
) is defined as the linear topology which has a base of neighborhoods of 0 consisting of all the sets of the form
(8)
N
(
D
,
W
)
=
{
T
∈
C
L
(
A
)
:
T
(
D
)
⊆
W
}
,
where D is a bounded (resp., finite) subset of E and W is a neighborhood of 0 in F. Clearly, s
≤
σ. In particular, if (
A
,
q
A
) is a p-normed algebra, then the σ-topology on C
L
(
A
) is the one given by the p-norm ∥
·
∥
A
p. In this setting, the strong operator topology s on C
L
(
A
) is given by the family of {
P
a
:
a
∈
A
} of F-seminorms, where
(9)
P
a
(
T
)
=
q
A
(
T
(
a
)
)
,
T
∈
C
L
(
A
)
.Remark 5.
If(
E
,
q
E
) is a general F-algebra, then |
|
T
|
|
q
E need not exist since the set {
u
∈
E
:
q
E
(
u
)
≤
1
} may not be bounded (see ([10, p. 8]; [11, 12]) for counterexamples).Definition 6.
LetX be a Hausdorff topological space and E a Hausdorff topological vector space over the field 𝕂 (=
ℝ or ℂ) with a base 𝒲 of neighborhoods of 0 in E. A function f
:
X
→
E is said tovanish at infinity if, for each neighborhood W of 0 in E, there exists a compact set K
=
K
W
⊆
X such that
(10)
f
(
x
)
∈
W
∀
x
∈
X
∖
K
.
We will denote by C
b
(
X
,
E
) the vector space of all continuous bounded E-valued functions on X and by C
0
(
X
,
E
) the subspace of C
b
(
X
,
E
) consisting of those functions which vanish at infinity. When E
=
𝕂
(
=
ℝ or ℂ
), these spaces will be denoted by C
b
(
X
) and C
0
(
X
). Let C
b
(
X
)
⊗
E denote the vector subspace of C
b
(
X
,
E
) spanned by the set of all functions of the form φ
⊗
u, where φ
∈
C
b
(
X
), u
∈
E, and
(11)
(
φ
⊗
u
)
(
x
)
=
φ
(
x
)
u
,
x
∈
X
.
We mention that, if X is not locally compact, then C
0
(
X
,
E
) may be the trivial vector space {
0
}. For example, if X
=
ℚ, the space of rationals, and E
=
ℝ, then C
0
(
ℚ
,
ℝ
)
=
{
0
}.Remarks 7.
(i) IfE
=
A is an algebra, then C
b
(
X
,
A
) is also an algebra with respect to the pointwise multiplication defined by
(12)
(
f
g
)
(
x
)
=
f
(
x
)
g
(
x
)
,
x
∈
X
.
(ii) IfE
=
A is a commutative algebra, then C
b
(
X
,
A
) is also commutative; in particular, C
b
(
X
) is a commutative algebra.
(iii) IfE is only a vector space, then C
b
(
X
,
E
) is a C
b
(
X
)-bimodule with respect to the module multiplications (
φ
,
f
)
→
φ
·
f and (
f
,
φ
)
→
f
·
φ defined by
(13)
(
φ
·
f
)
(
x
)
=
φ
(
x
)
f
(
x
)
=
(
f
·
φ
)
(
x
)
,
x
∈
X
.
(iv) IfE is a vector space and A is algebra, then C
b
(
X
,
E
) is a left A-module with respect to the module multiplication (
a
,
f
)
→
a
·
f as pointwise action:
(14)
(
a
·
f
)
(
x
)
=
a
f
(
x
)
,
a
∈
A
,
f
∈
C
b
(
X
,
A
)
,
x
∈
X
.
In particular, C
0
(
X
,
E
) is a left A-module.Definition 8.
LetX be a Hausdorff space and E a Hausdorff topological vector space (TVS) over 𝕂
(
=
ℝ or ℂ). Theuniform topology  u on C
b
(
X
,
E
) is the linear topology which has a base of neighborhoods of 0 consisting of all sets of the form
(15)
N
(
X
,
G
)
=
{
f
∈
C
b
(
X
,
E
)
:
f
(
X
)
⊆
W
}
,
where W is a neighborhood of 0 in E. In particular, if E
=
(
E
,
q
E
) is an F-normed space, the u-topology on C
b
(
X
,
E
) is given by the F-norm
(16)
∥
f
∥
q
E
,
∞
=
sup
⁡
x
∈
X
q
E
(
f
(
x
)
)
,
f
∈
C
b
(
X
,
E
)
.


## 3. Main Results

In this section we extend some results of [2–6] from Banach modules to the more general setting of topological modules.Definition 9 (cf. [13, 14]).
Let(
A
,
q
A
) be a commutative p-normed algebra, and let (
E
,
q
E
) be a p-normed space which is also an A-module in the usual algebraic sense. Then E is called anisometric
A
-module if
(17)
q
F
(
a
u
)
≤
q
A
(
a
)
q
F
(
u
)
for
any
a
∈
A
,
u
∈
E
.
If (
A
,
q
A
) has a minimal approximate identity {
e
λ
:
λ
∈
I
}, then E is called anessential
A
-module if lim
⁡
λ
e
λ
u
=
lim
⁡
λ
u
e
λ
=
u for all u
∈
E.Definition 10.
Let(
A
,
q
A
) be a commutative p-normed algebra, and let E
=
(
E
,
q
E
) and F
=
(
F
,
q
F
) be p-normed spaces which are also A-modules. One writes
(18)
Ho
m
A
(
E
,
F
)
=
{
T
(
u
)
for
any
a
∈
A
,
u
∈
E
T
∈
C
L
(
E
,
F
)
:
T
(
a
·
u
)

=

a
·
T
(
u
)
for
any
a
∈
A
,
u
∈
E
}
.
If E is an A-bimodule, then defining a
*
T by
(19)
(
a
*
T
)
(
u
)
=
T
(
u
·
a
)
(
a
∈
A
,
u
∈
E
)
,
Ho
m
A
(
E
,
F
) becomes a left A-module. In fact, for any a
,
b
∈
A, u
∈
E,
(20)
(
a
*
T
)
(
b
·
u
)
=
T
(
(
b
·
u
)
·
a
)
=
T
(
b
·
(
u
·
a
)
)
=
b
·
T
(
u
·
a
)
=
b
·
(
a
*
T
)
(
u
)
.
In particular, Ho
m
A
(
A
,
F
) is a left A-module. If E
=
F
=
A, then Ho
m
A
(
A
,
A
)
=
M
(
A
) is the usual multiplier algebra of A:
(21)
M
(
A
)
=
{
T
∈
C
L
(
A
,
A
)
:
T
(
a
b
)
=
a
T
(
b
)
=
T
(
a
)
b
∀
a
,
b
∈
A
T
∈
C
L
(
A
,
A
)
)
:
T
(
a
b
)
=
a
T
(
b
)
=
T
(
a
)
b
}
,
which is a commutative algebra (without A being commutative) and has the identity I
:
A
→
A,I
(
x
)
=
x
(
x
∈
A
).Lemma 11.
Let(
A
,
q
A
) a commutative p-normed algebra having a minimal approximate identity, and let (
F
,
q
F
) be p-normed space which is an essential isometric A-bimodule. Then, for any v
∈
F,
(22)
∥
L
v
∥
q
F
=
∥
R
v
∥
q
F
=
q
F
(
v
)
,
where L
v
,
R
v
:
A
→
F are the maps given by L
v
(
a
)
=
v
·
a and R
v
(
a
)
=
a
·
v
,
a
∈
A.Proof.
Letv
∈
F. Then
(23)
∥
L
v
∥
q
A
,
q
F
=
sup
⁡
{
q
F
(
L
v
(
a
)
)
:
q
A
(
a
)
≤
1
}
=
sup
⁡
{
q
F
(
v
·
a
)
:
q
A
(
a
)
≤
1
}
≤
sup
⁡
{
q
A
(
a
)
q
F
(
v
)
:
q
A
(
a
)
≤
1
}
=
q
F
(
v
)
.
On the other hand,
(24)
∥
L
v
∥
q
A
,
q
F
=
sup
⁡
{
q
F
(
v
·
a
)
:
q
A
(
a
)
≤
1
}
≥
q
F
(
v
·
e
λ
)
∀
λ
∈
I
,
so
(25)
∥
L
v
∥
q
A
,
q
F
≥
lim
⁡
λ
q
F
(
v
·
e
λ
)
=
q
F
(
lim
⁡
λ
v
·
e
λ
)
=
q
F
(
v
)
.
Hence ∥
L
v
∥
q
A
,
q
F
=
q
F
(
v
). Similarly, |
|
R
v
|
|
q
E
=
q
E
(
v
).Lemma 12.
Let(
A
,
q
A
) a commutative p-normed algebra, and let (
F
,
q
F
) be an essential isometric A-bimodule. If A has an identity e, then Hom
A
(
A
,
F
)
≅
F and M
(
A
)
≅
A.Proof.
We claim that(26)
Ho
m
A
(
A
,
F
)
≅
{
L
T
(
e
)
:
T
∈
Ho
m
A
(
A
,
F
)
}
=
{
L
v
:
v
∈
F
}
≅
F
.
Clearly,
(27)
{
L
T
(
e
)
:
T
∈
Ho
m
A
(
A
,
F
)
}
⊆
{
L
v
:
v
∈
F
}
⊆
Ho
m
A
(
A
,
F
)
.
On the other hand, if T
∈
Ho
m
A
(
A
,
F
), then, for any a
∈
A,
(28)
T
(
a
)
=
T
(
e
a
)
=
T
(
e
)
·
a
=
L
T
(
e
)
(
a
)
.
Hence T
=
L
T
(
e
). Further, by Lemma 11, ∥
L
T
(
e
)
∥
q
A
,
q
F
=
q
F
(
T
(
e
)
). Thus Ho
m
A
(
A
,
F
)
≅
F. In particular, M
(
A
)
≅
A.Density Assumption. In the sequel, we will always assume that, for X a locally compact Hausdorff space and E a topological vector space, C
0
(
X
)
⊗
E
is
u
-dense in
C
0
(
X
,
E
). This assumption is crucial for the proof of our main results. For its justification, we mention that as a consequence of the vector-valued versions of Stone-Weierstrass theorem [8, 12, 15], C
0
(
X
)
⊗
E is u-dense in C
0
(
X
,
E
) in each of the following cases.(a)
Eis locally convex.
(b)
Every compact subset ofX has a finite covering dimension and E is any topological vector space.
(c)
E is an F-space with a basis (e.g., E
=
ℓ
p for p
>
0).
(d)
E has the approximation property.Recall that ifT
∈
M
(
C
0
(
X
,
A
)
), then T
(
a
·
f
)
=
a
·
T
(
f
) for f
∈
C
0
(
X
,
A
) and a
∈
A ([16, Lemma 4.5]). We also mention that if (
A
,
q
A
) is an p-normed algebra having a minimal approximate identity, then, by ([16, Lemma 4.4]), C
0
(
X
,
A
) has an approximate identity and hence it is a faithful topological A-module. Consequently, for any T
∈
M
(
C
0
(
X
,
A
)
), T
(
f
g
)
=
f
T
(
g
)
=
T
(
f
)
g for all f
,
g
∈
C
0
(
X
,
A
); we will write
(29)
∥
T
∥
q
A
∶
=
sup
⁡
{
q
A
(
T
(
f
)
)
:
f
∈
C
0
(
X
,
A
)
,
∥
f
∥
q
A
,
∞
≤
1
}
.
If T
∈
Hom
C
0
(
X
,
A
)
(
C
0
(
X
,
E
)
,
C
0
(
X
,
F
)
), we let
(30)
∥
T
∥
q
E
,
q
F
∶
=
sup
⁡
{
q
F
(
T
(
f
)
)
:
f
∈
C
0
(
X
,
E
)
,
∥
f
∥
q
E
,
∞
≤
1
}
.Definition 13.
Now, letE
=
(
E
,
q
E
) and F
=
(
F
,
q
F
) be F-normed spaces. For any closed subspace U
=
U
s
(
E
,
F
) of C
L
(
E
,
F
) endowed with the strong operator topology s, we define
(31)
C
s
,
b
(
X
,
U
)
=
{
strongly
continu
G
:
X
⟶
U
:
G
is
strongly
continuous
and
bounded
G
:
X
⟶
U
:
}
.
We now define an F-norm on C
s
,
b
(
X
,
U
) by
(32)
∥
G
∥
C
s
,
b
=
sup
⁡
x
∈
X
∥
G
(
x
)
∥
q
E
,
q
F
=
sup
⁡
x
∈
X
sup
⁡
u
∈
E
,
q
E
(
u
)
≤
1
q
F
(
G
(
x
)
(
u
)
)
.
Then C
s
,
b
(
X
,
U
) is a complete p-normed space under the p-norm ∥
·
∥
q
,
∞ defined in (24).Recall that a leftA-module E is called faithful (orwithout order) if, for any u
∈
E, a
·
u
=
0 for all a
∈
A implies that x
=
0 (cf. [13, 14]).Lemma 14.
LetA
=
(
A
,
q
A
) be a commutative complete p-normed algebra, and let E and F be A-modules. Then, for any T
∈
Hom
C
0
(
X
,
A
)
(
C
0
(
X
,
E
)
,
C
0
(
X
,
F
)
),(a)
T
(
a
·
f
)
=
a
·
T
(
f
)   for a
∈
A and f
∈
C
0
(
X
,
E
),
(b)
T
(
φ
·
f
)
=
φ
·
T
(
f
)   for φ
∈
C
0
(
X
) and f
∈
C
0
(
X
,
E
).Proof.
(a) We first note thatC
0
(
X
) is a Banach algebra with a bounded approximate identity, {
ψ
α
} (say). Then, for any a
∈
A
,
u
∈
E, and φ
∈
C
0
(
X
),
(33)
lim
⁡
α
[
(
ψ
α
⊗
a
)
·
(
φ
⊗
u
)
]
=
lim
⁡
α
(
ψ
α
φ
⊗
a
·
u
)
=
φ
⊗
a
·
u
=
a
(
φ
⊗
u
)
.
Since T
∈
Hom
C
0
(
X
,
A
)
(
C
0
(
X
,
E
)
,
C
0
(
X
,
F
)
) and ψ
α
⊗
a
∈
C
0
(
X
,
A
), φ
⊗
u
∈
C
0
(
X
,
E
), we have
(34)
T
(
a
·
(
φ
⊗
u
)
)
=
lim
⁡
α
T
[
(
ψ
α
⊗
a
)
·
(
φ
⊗
u
)
]
=
lim
⁡
α
(
ψ
α
⊗
a
)
·
T
(
φ
⊗
u
)
=
a
·
T
(
φ
⊗
u
)
.
By T being linear and C
0
(
X
)
⊗
E being assumed to be u-dense in C
0
(
X
,
E
), it follows that T
(
a
·
f
)
=
a
·
T
(
f
) holds for all f
∈
C
0
(
X
,
A
) and a
∈
A.
(b) Similar to the above part.We now give the following characterization in the pseudoscaler case by considering bothC
0
(
X
) and C
0
(
X
,
F
) as C
0
(
X
)-modules.Theorem 15.
LetX be a locally compact Hausdorff space and F
=
(
F
,
q
F
) a p-normed space. Then
(35)
Hom
C
0
(
X
)
(
C
0
(
X
)
,
C
0
(
X
,
F
)
)
≅
C
b
(
X
,
F
)
.Proof.
LetT
∈
Hom
C
0
(
X
)
(
C
0
(
X
)
,
C
0
(
X
,
F
)
) and x
∈
X. If φ
,
ψ
∈
C
0
(
X
) with φ
(
x
)
≠
0 and ψ
(
x
)
≠
0, then there is a neighborhood N
(
x
) of x in X such that
(36)
φ
(
t
)
≠
0
,
ψ
(
t
)
≠
0
for
any
t
∈
N
(
x
)
.
Since C
0
(
X
) is commutative and C
0
(
X
,
F
) is a C
0
(
X
)-module, following as in ([1, p. 1135]), we have
(37)
ψ
(
t
)
(
T
φ
)
(
t
)
=
T
(
ψ
·
φ
)
(
t
)
=
T
(
φ
·
ψ
)
(
t
)
=
φ
(
t
)
(
T
ψ
)
(
t
)
and then
(38)
T
(
ψ
)
(
t
)
ψ
(
t
)
=
(
T
φ
)
(
t
)
φ
(
t
)
for
any
t
∈
N
(
x
)
.
Now, for each x
∈
X with φ
(
x
)
≠
0, define g
T
:
X
→
F by
(39)
g
T
(
x
)
=
(
T
φ
)
(
x
)
φ
(
x
)
.
By the above argument, the function g
T
(
x
) defined in this way is independent of the choice of φ
∈
C
0
(
X
); hence g
T is well-defined.
Clearly ifφ
(
x
)
≠
0, then (
T
φ
)
(
x
)
=
g
T
(
x
)
φ
(
x
). The equality also holds when φ
(
x
)
=
0. [To see this, choose ψ
∈
C
0
(
X
) such that ψ
(
x
)
≠
0. Then
(40)
ψ
(
x
)
(
T
φ
)
(
x
)
=
T
(
ψ
φ
)
(
x
)
=
φ
(
x
)
(
T
ψ
)
(
x
)
=
0
,
and so T
φ
(
x
)
=
0.]
Next,g
T
∈
C
b
(
X
,
F
), as follows. For any x
∈
X with φ
(
x
)
≠
0, by Urysohn’s lemma, we can choose a φ
∈
C
0
(
X
) such that ∥
φ
∥
∞
=
|
φ
(
x
)
|. So
(41)
q
F
[
g
T
(
x
)
]
=
q
F
[
T
φ
(
x
)
]
|
φ
(
x
)
|
≤
∥
T
∥
q
F
∥
φ
∥
∞
|
φ
(
x
)
|
=
∥
T
∥
q
F
for all x
∈
X. Hence ∥
g
T
∥
q
,
∞
≤
∥
T
∥
q
F, and so g
T
∈
C
b
(
X
,
F
).
On the other hand, since(42)
q
F
[
(
T
φ
)
(
x
)
]
=
q
F
[
g
T
(
x
)
φ
(
x
)
]
≤
∥
g
T
∥
q
,
∞
∥
φ
∥
∞
,
we have ∥
T
∥
q
F
≤
∥
g
T
∥
q
,
∞. Consequently ∥
g
T
∥
q
F
,
∞
=
∥
T
∥
q
F. This shows that Hom
C
0
(
X
)
(
C
0
(
X
)
,
C
0
(
X
,
F
)
) is isometrically embedded in C
b
(
X
,
F
).
Conversely, for anyg
∈
C
b
(
X
,
F
), we define T
g
:
C
0
(
X
)
→
C
0
(
X
,
F
) by
(43)
T
g
(
φ
)
=
g
·
φ
,
φ
∈
C
0
(
X
)
.
Then one can easily show thatT
g is a multiplier from C
0
(
X
) to C
0
(
X
,
F
) and that ∥
g
∥
q
,
∞
=
∥
T
g
∥
q
F.Now we can establish the main theorem by considering bothC
0
(
X
,
E
) and C
0
(
X
,
F
) as C
0
(
X
,
A
)-modules.Theorem 16.
LetA
=
(
A
,
q
A
) be a commutative complete p-normed algebra, and let E
=
(
E
,
q
E
) and F
=
(
F
,
q
F
) be p-normed spaces which are also essential isometric A-modules. Then
(44)
Hom
C
0
(
X
,
A
)
(
C
0
(
X
,
E
)
,
C
0
(
X
,
F
)
)
≅
C
s
,
b
(
X
,
Hom
A
(
E
,
F
)
)
.
The correspondence between the multiplier T and the function G is given by the following relation:
(45)
hhhlhhhh
(
T
f
)
(
x
)
=
G
(
x
)
·
f
(
x
)
f
o
r
x
∈
X
a
n
d
a
n
y
f
∈
C
0
(
X
,
E
)
.Proof.
LetT
∈
Ho
m
C
0
(
X
,
A
)
(
C
0
(
X
,
E
)
,
C
0
(
X
,
F
)
). Then we can define a map Ψ
T
:
E
→
Ho
m
C
0
(
X
) (
C
0
(
X
)
,
C
0
(
X
,
F
)
) by
(46)
Ψ
T
(
u
)
(
φ
)
=
T
(
φ
⊗
u
)
for
u
∈
E
,
φ
∈
C
0
(
X
)
.
To see that this map is well-defined, first note that Ψ
T
(
u
)
(
φ
)
∈
C
0
(
X
,
F
). For a fixed u
∈
E, the operator Φ
T
(
u
) defines a bounded linear operator from C
0
(
X
) into C
0
(
X
,
F
), since by (46),
(47)
∥
Ψ
T
(
u
)
(
φ
)
∥
q
E
,
∞
=
∥
T
(
φ
⊗
u
)
∥
q
E
,
∞
≤
∥
T
∥
q
E
·
∥
φ
⊗
a
∥
q
E
,
∞
;
further, it is a multiplier since, for any φ
,
ψ
∈
C
0
(
X
),
(48)
Ψ
T
(
u
)
(
φ
ψ
)
=
T
(
φ
ψ
⊗
u
)
=
φ
·
T
(
ψ
⊗
u
)
.
Hence Ψ
T
(
u
)
∈
Hom
C
0
(
X
)
(
C
0
(
X
)
,
C
0
(
X
,
F
)
). By Theorem 15, there exists an element, say g
u, in C
b
(
X
,
F
) such that
(49)
Ψ
T
(
u
)
(
φ
)
=
g
u
·
φ
,
for
u
∈
E
,
φ
∈
C
0
(
X
)
.
Now, we can define a map G
:
X
→
Hom
A
(
E
,
F
) by
(50)
G
(
x
)
(
u
)
=
g
u
(
x
)
for
x
∈
X
,
u
∈
E
.
To see that this map is well-defined, first note that, for a fixed x
∈
X, G
(
x
) is a linear operator from E into F. Moreover, for a
∈
A and φ
∈
C
0
(
X
), we have
(51)
G
(
x
)
(
a
·
u
)
·
φ
(
u
)
=
g
a
u
(
x
)
φ
(
x
)
=
T
(
φ
⊗
a
·
u
)
(
x
)
=
a
·
T
(
φ
⊗
u
)
(
x
)
=
a
·
g
u
(
x
)
φ
(
x
)
=
a
·
G
(
x
)
(
u
)
φ
(
x
)
,
or
(52)
G
(
x
)
(
a
·
u
)
=
a
·
G
(
x
)
(
u
)
.
This implies that G
(
x
)
∈
Hom
A
(
E
,
F
), and hence G
∈
C
s
,
b
(
X
,
Hom
A
(
E
,
F
)
). Next we establish isometry between T and G. For x
∈
X and φ
⊗
u
∈
C
0
(
X
)
⊗
E with ∥
φ
⊗
u
∥
q
E
,
∞
≤
1,
(53)
∥
G
(
x
)
∥
q
E
,
q
F
=
sup
⁡
q
E
(
u
)
≤
1
q
F
[
G
(
x
)
(
u
)
]
=
sup
⁡
q
E
(
u
)
≤
1
q
F
[
g
u
(
x
)
]
≤
sup
⁡
q
E
(
u
)
≤
1
∥
g
u
∥
q
F
,
∞
=
sup
⁡
q
E
(
u
)
≤
1
∥
φ
∥
∞
≤
1
∥
g
u
·
φ
∥
q
F
,
∞
=
sup
⁡
∥
φ
⊗
u
∥
q
E
,
∞
≤
1
∥
T
(
φ
⊗
u
)
∥
q
F
,
∞
=
∥
T
∥
q
E
,
q
F
,
since C
0
(
X
)
⊗
Eis u-dense in C
0
(
X
,
E
). So ∥
G
∥
C
s
,
b
≤
∥
T
∥
q
E
,
q
F. But
(54)
∥
T
(
φ
⊗
u
)
∥
q
F
,
∞
=
∥
g
u
·
φ
∥
q
F
,
∞
≤
∥
g
u
∥
q
F
,
∞
∥
φ
∥
∞
≤
∥
G
∥
C
s
,
b
∥
u
∥
∥
φ
∥
∞
=
∥
G
∥
C
s
,
b
∥
φ
⊗
u
∥
q
E
,
∞
for all φ
⊗
u
∈
C
0
(
X
)
⊗
E. Consequently, ∥
T
∥
q
E
,
q
F
≤
∥
G
∥
C
s
,
b.
Conversely, letG
∈
C
s
,
b
(
X
,
Hom
A
(
E
,
F
)
) and φ
∈
C
0
(
X
). Then G
·
φ is a continuous function on X given by
(55)
(
G
·
φ
)
(
x
)
(
u
)
=
(
G
(
x
)
u
)
φ
(
x
)
,
x
∈
X
,
u
∈
E
.
It is easy to see that G
·
φ vanishes at infinity, and so G
·
φ
∈
C
0
(
X
,
Hom
A
(
E
,
F
)
). For any u
∈
E and φ
∈
C
0
(
X
), G determines a bounded linear operator T from C
0
(
X
,
E
) to C
0
(
X
,
F
) given by
(56)
T
(
φ
⊗
u
)
(
x
)
=
(
G
(
x
)
u
)
φ
(
x
)
.
Again, since C
0
(
X
)
⊗
Eis u
-dense in C
0
(
X
,
E
), it follows that ∥
T
∥
q
E
,
q
F
=
∥
G
∥
C
s
,
b.
SinceE and F are A-modules, for any h
⊗
a
∈
C
0
(
X
)
⊗
A and φ
⊗
u
∈
C
0
(
X
)
⊗
E,
(57)
T
(
(
h
⊗
a
)
·
(
φ
⊗
u
)
)
=
T
(
h
φ
⊗
a
u
)
=
G
(
·
)
(
a
·
u
)
(
h
φ
)
(
·
)
=
a
·
h
(
·
)
G
(
·
)
(
u
)
φ
(
·
)
=
(
h
⊗
a
)
·
T
(
φ
⊗
u
)
.
Hence T is a multiplier on C
0
(
X
,
E
) since C
0
(
X
)
⊗
Eis u-dense in C
0
(
X
,
E
). The isometry between G and T now implies that
(58)
Ho
m
C
0
(
X
,
A
)
(
C
0
(
X
,
E
)
,
C
0
(
X
,
F
)
)
≅
C
s
,
b
(
X
,
Ho
m
A
(
E
,
F
)
)
.


## 4. Applications

As an application of the above results, in particular of Theorem16, we can deduce several known results, as follows.Corollary 17 (see [3]).
LetX be a locally compact Hausdorff space and A
=
(
A
,
|
|
·
|
|
) a commutative Banach algebra, and let E and F be Banach A-modules. Then
(59)
Hom
C
0
(
X
,
A
)
(
C
0
(
X
,
E
)
,
C
0
(
X
,
F
)
)
≅
C
s
,
b
(
X
,
Hom
A
(
E
,
F
)
)
.Corollary 18 (see [3, 5]).
LetX be a locally compact Hausdorff space and A
=
(
A
,
∥
·
∥
) be a commutative Banach algebra with identity of norm 1, and let E be a Banach A-module. Then
(60)
Hom
C
0
(
X
,
A
)
(
C
0
(
X
,
E
)
,
C
0
(
X
,
E
)
)
≅
C
b
(
X
,
E
)
.Corollary 19 (see [16]).
LetX be a locally compact Hausdorff space and A
=
(
A
,
q
) a commutative complete p-normed algebra with a minimal approximate identity. Then
(61)
M
(
C
0
(
X
,
A
)
)
≅
C
s
,
b
(
X
,
M
(
A
)
u
)
.Proof.
This follows from the fact thatHom
A
(
A
,
A
)
=
M
(
A
).Corollary 20 (see [1]).
LetX be a locally compact Hausdorff space. Then
(62)
M
(
C
0
(
X
)
)
≅
C
b
(
X
)
.Proof.
This follows from the fact thatHo
m
C
0
(
X
)
(
C
0
(
X
)
,
C
0
(
X
)
)
≅
C
b
(
X
).Example 21.
LetA
p, 0
<
p
≤
1, denote the algebra of all holomorphic functions in the unit disc D
=
{
z
∈
ℂ
:
|
z
|
≤
1
}:
(63)
φ
(
z
)
=
∑
n
=
0
∞
a
n
z
n
,
z
∈
D
,
for which
(64)
∥
φ
∥
p
=
∑
n
=
0
∞
|
a
n
|
p
<
∞
.
This is a commutative complete p-normed algebra with the pointwise multiplication and has an identity ([7, p. 135]; [17, p. 8]). In this case,
(65)
M
(
C
0
(
X
,
A
p
)
)
≃
C
b
(
X
,
M
(
A
p
)
s
)
≃
C
b
(
X
,
A
p
)
.



---

*Source: 397376-2014-05-12.xml*