Abstract
In 1961, Wang showed that if is the commutative -algebra with a locally compact Hausdorff space, then . 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 where and are -normed spaces which are also essential isometric left -modules with being a certain commutative -algebra, not necessarily locally convex. Our results unify and extend several known results in the literature.
1. Introduction
Characterizations of multipliers on algebras and modules of continuous functions with values in a commutative Banach or -algebra have been obtained by several authors. In 1961, Wang [1] showed that if is taken as the commutative -algebra with being a locally compact Hausdorff space, then . 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 is a locally compact abelian group and is a commutative Banach algebra with a bounded approximate identity, then . In 1992, Candeal Haro and Lai [3] had obtained in the case when is a commutative Banach algebra and and are left Banach -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 considering a commutative complete -normed algebra, , having a minimal approximate identity and and being -spaces which are also left -modules.
We mention that the arguments of earlier authors relied heavily on the fact that, in the case of , a Banach algebra, is isometrically isomorphic to the completed tensor product with respect to the smallest cross norm (see [2–5]). We will avoid the use of this technique as it need not work in our case. In fact, when is not locally convex, is no longer appropriate; even for a complete -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]). Let be a vector space over the field , .(a)A function is called an -seminorm on if it satisfies the following:() for all ;() if ;() for all and with ;() for all ;(if in , then for all .(b)An -seminorm on is called an -norm if, for any , implies .(c)An -seminorm (or -norm) on is called a -seminorm (resp., -norm), , if it also satisfies (d)If is an -norm (resp., a -norm) on a vector space , then the pair is called an -normed (resp., a -normed) space.(e)An -norm (or a -norm) on an algebra is called submultiplicative if
An algebra with a submultiplicative -norm (resp., -norm) is called an -normed (resp., -normed) algebra.
Definition 2. A net in a topological algebra is called an approximate identity if
An approximate identity in an -normed algebra is said to be minimal if for all .
If and are topological vector spaces over the field or , then the set of all continuous linear mappings is denoted by . Clearly, is a vector space over with the usual pointwise operations. Further, if , is an algebra under composition (i.e., ) and has the identity given by .
Definition 3. Let and be -normed spaces. For any linear map , define Then, by ([10, p. 101-102]), if and only if . Further, is an -norm on and, for any , In particular, if , we denote In this case, for any , ; hence is a -normed algebra.
Definition 4. Let and be topological vector spaces. The uniform operator topology (resp., the strong operator topology ) on is defined as the linear topology which has a base of neighborhoods of consisting of all the sets of the form where is a bounded (resp., finite) subset of and is a neighborhood of in . Clearly, . In particular, if is a -normed algebra, then the -topology on is the one given by the -norm . In this setting, the strong operator topology on is given by the family of of -seminorms, where
Remark 5. If is a general -algebra, then need not exist since the set may not be bounded (see ([10, p. 8]; [11, 12]) for counterexamples).
Definition 6. Let be a Hausdorff topological space and a Hausdorff topological vector space over the field ( or ) with a base of neighborhoods of in . A function is said to vanish at infinity if, for each neighborhood of in , there exists a compact set such that We will denote by the vector space of all continuous bounded -valued functions on and by the subspace of consisting of those functions which vanish at infinity. When or , these spaces will be denoted by and . Let denote the vector subspace of spanned by the set of all functions of the form , where , , and We mention that, if is not locally compact, then may be the trivial vector space . For example, if , the space of rationals, and , then .
Remarks 7. (i) If is an algebra, then is also an algebra with respect to the pointwise multiplication defined by
(ii) If is a commutative algebra, then is also commutative; in particular, is a commutative algebra.
(iii) If is only a vector space, then is a -bimodule with respect to the module multiplications and defined by
(iv) If is a vector space and is algebra, then is a left -module with respect to the module multiplication as pointwise action:
In particular, is a left -module.
Definition 8. Let be a Hausdorff space and a Hausdorff topological vector space (TVS) over or ). The uniform topology on is the linear topology which has a base of neighborhoods of consisting of all sets of the form where is a neighborhood of in . In particular, if is an -normed space, the -topology on is given by the -norm
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 be a commutative -normed algebra, and let be a -normed space which is also an -module in the usual algebraic sense. Then is called an isometric -module if If has a minimal approximate identity , then is called an essential -module if for all .
Definition 10. Let be a commutative -normed algebra, and let and be -normed spaces which are also -modules. One writes If is an -bimodule, then defining by becomes a left -module. In fact, for any , , In particular, is a left -module. If , then is the usual multiplier algebra of : which is a commutative algebra (without being commutative) and has the identity ,.
Lemma 11. Let a commutative -normed algebra having a minimal approximate identity, and let be -normed space which is an essential isometric -bimodule. Then, for any , where are the maps given by and .
Proof. Let . Then On the other hand, so Hence . Similarly, .
Lemma 12. Let a commutative -normed algebra, and let be an essential isometric -bimodule. If has an identity , then and .
Proof. We claim that Clearly, On the other hand, if , then, for any , Hence . Further, by Lemma 11, . Thus . In particular, .
Density Assumption. In the sequel, we will always assume that, for a locally compact Hausdorff space and a topological vector space, is -dense in . 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], is -dense in in each of the following cases.(a)is locally convex.(b)Every compact subset of has a finite covering dimension and is any topological vector space.(c) is an -space with a basis (e.g., for ).(d) has the approximation property.
Recall that if , then for and ([16, Lemma 4.5]). We also mention that if is an -normed algebra having a minimal approximate identity, then, by ([16, Lemma 4.4]), has an approximate identity and hence it is a faithful topological -module. Consequently, for any , for all ; we will write If , we let
Definition 13. Now, let and be -normed spaces. For any closed subspace of endowed with the strong operator topology , we define We now define an -norm on by Then is a complete -normed space under the -norm defined in (24).
Recall that a left -module is called faithful (or without order) if, for any , for all implies that (cf. [13, 14]).
Lemma 14. Let be a commutative complete -normed algebra, and let and be -modules. Then, for any ,(a) for and ,(b) for and .
Proof. (a) We first note that is a Banach algebra with a bounded approximate identity, (say). Then, for any , and ,
Since and , , we have
By being linear and being assumed to be -dense in , it follows that holds for all and .
(b) Similar to the above part.
We now give the following characterization in the pseudoscaler case by considering both and as -modules.
Theorem 15. Let be a locally compact Hausdorff space and a -normed space. Then
Proof. Let and . If with and , then there is a neighborhood of in such that
Since is commutative and is a -module, following as in ([1, p. 1135]), we have
and then
Now, for each with , define by
By the above argument, the function defined in this way is independent of the choice of ; hence is well-defined.
Clearly if , then . The equality also holds when . [To see this, choose such that . Then
and so .]
Next, , as follows. For any with , by Urysohn’s lemma, we can choose a such that . So
for all . Hence , and so .
On the other hand, since
we have . Consequently . This shows that is isometrically embedded in .
Conversely, for any , we define by
Then one can easily show that is a multiplier from to and that .
Now we can establish the main theorem by considering both and as -modules.
Theorem 16. Let be a commutative complete -normed algebra, and let and be -normed spaces which are also essential isometric -modules. Then The correspondence between the multiplier and the function is given by the following relation:
Proof. Let . Then we can define a map by
To see that this map is well-defined, first note that . For a fixed , the operator defines a bounded linear operator from into , since by (46),
further, it is a multiplier since, for any ,
Hence . By Theorem 15, there exists an element, say , in such that
Now, we can define a map by
To see that this map is well-defined, first note that, for a fixed , is a linear operator from into . Moreover, for and , we have
or
This implies that , and hence . Next we establish isometry between and . For and with ,
since is -dense in . So . But
for all . Consequently, .
Conversely, let and . Then is a continuous function on given by
It is easy to see that vanishes at infinity, and so . For any and , determines a bounded linear operator from to given by
Again, since is -dense in , it follows that .
Since and are -modules, for any and ,
Hence is a multiplier on since is -dense in . The isometry between and now implies that
4. Applications
As an application of the above results, in particular of Theorem 16, we can deduce several known results, as follows.
Corollary 17 (see [3]). Let be a locally compact Hausdorff space and a commutative Banach algebra, and let and be Banach -modules. Then
Corollary 18 (see [3, 5]). Let be a locally compact Hausdorff space and be a commutative Banach algebra with identity of norm , and let be a Banach -module. Then
Corollary 19 (see [16]). Let be a locally compact Hausdorff space and a commutative complete -normed algebra with a minimal approximate identity. Then
Proof. This follows from the fact that .
Corollary 20 (see [1]). Let be a locally compact Hausdorff space. Then
Proof. This follows from the fact that .
Example 21. Let , , denote the algebra of all holomorphic functions in the unit disc : for which This is a commutative complete -normed algebra with the pointwise multiplication and has an identity ([7, p. 135]; [17, p. 8]). In this case,
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under the Project no. 3-059/429. The authors therefore acknowledge with thanks DSR technical and financial support. The authors are also grateful to the referee for his many useful suggestions to improve the paper.