Abstract

We prove some results concerning Arens regularity and amenability of the Banach algebra of all -multipliers on a given Banach algebra . We also consider -multipliers in the general topological module setting and investigate some of their properties. We discuss the -strict and -uniform topologies on . A characterization of -multipliers on -module , where is a compact group, is given.

1. Introduction

The concept of a multiplier was introduced by Helgason [1] as follows. Let be a commutative and semisimple Banach algebra and let be its maximal ideal space. Let denote the Gelfand representation of as a subalgebra of the algebra of continuous functions on . A bounded continuous function on is a multiplier on if . The general theory of multipliers on faithful Banach algebras was developed by Wang [2] and Birtel [3].

Recall that a mapping is called a left (resp., right) multiplier on if for all . We say is a multiplier on if it is both a left multiplier and a right multiplier on .

We denote with the algebra of all multipliers on .

A Banach algebra is called left (resp., right) faithful if, for all , (resp., ) implies that ; is called faithful if it is both left and right faithful.

In [4] we generalized the concept of multipliers on faithful Banach algebras to -multipliers as follows. Let be a Banach algebra and let be an algebra homomorphism. A linear continuous mapping is called a left (resp., right) -multiplier on if for all . We say is a -multiplier on if it is both a left -multiplier and a right -multiplier on . We denote by   (resp., ) the collection of all -multipliers (resp., left -multipliers, right -multipliers) on .

It turns out that this concept is considerably more general than the concept of multipliers on Banach algebras. Also by using some well-known homomorphisms like Jordan homomorphism, spectrum preserving homomorphism, and idempotent preserving homomorphism, we can transfer these useful properties from homomorphism to the algebra of -multipliers.

In [4], we studied various properties of -multipliers, for instance, the faithfulness of the Banach algebra and the existence of a bounded approximate identity in the range of a -multiplier. Finally, as an example, we have characterized -multipliers on .

In Section 2 we are concerned by Arens regularity and amenability of the Banach algebra under some suitable conditions. We introduce the notion of Jordan -multiplier and prove that every Jordan -multiplier is a -multiplier whenever the range of is dense in the algebra.

In Section 3 we extend the notion of -multipliers on Banach algebras to topological modules and investigate some of their properties. We discuss the -strict and -uniform topologies on and apply our results to -module .

Let be a topological vector space and let be a topological algebra, both over the same field . Then is called a topological left -module if it is a left -module and the module multiplication from into is separately continuous. If denotes the collection of all bounded sets in , then module multiplication is called -hypocontinuous [5] if, given any neighborhood of in and any , there exists a neighborhood of in such that . Clearly, joint continuity hypocontinuity separate continuity. A mapping from a left -module into another left -module is called an -module homomorphism if for all and

2. Some Properties of -Multipliers on Banach Algebras

Let us start with the following result proved in [4].

Theorem 1 (see [4, Theorem ]). Let be a faithful commutative Banach algebra and let be an idempotent homomorphism on . Then is a Banach algebra. Moreover, if and for all , then is a faithful commutative Banach algebra.

Definition 2. Let be a Banach algebra and let be a homomorphism from to . The mapping is called a left (resp., right) Jordan -multiplier on if for all is called a Jordan -multiplier on if it is both a left Jordan -multiplier and a right Jordan -multiplier on .

Theorem 3. Let be a faithful commutative Banach algebra and let be a homomorphism from to with dense range. Then is a -multiplier if and only if is a Jordan -multiplier.

Proof. It is clear that every -multiplier is a Jordan -multiplier. Conversely, suppose is a Jordan -multiplier. Then for all .
On the other hand, we have Comparing (4), (5) we obtain From (6) and using commutativity of , for each sequence we have so we have similarly by using (6) we have comparing (8), (9) we obtain for all . Since has dense range and is a faithful commutative Banach algebra, we have hence is a -multiplier.

We mention that Theorem 3 holds for certain noncommutative cases, but not in general. For instance, Zalar has proved in [6] that any left (right) Jordan multiplier on a 2-torsion free semiprime ring is a left (right) multiplier. Vukman [7] has shown that an additive map , where is a 2-torsion free semiprime ring, with the property that for all , is a multiplier.

The following example shows that, in general, the above theorem need not hold for noncommutative Banach algebras.

Example 4. Consider the subalgebra of the algebra of all matrices. It is obvious that is a Banach algebra with respect to the norm given by Let and define a continuous linear map by . By a straightforward calculation one can prove that If denotes the Jordan product , then we have for each and hence is a right Jordan multiplier. Also for all with . If we consider then and , where are the zero matrix and the identity matrix, respectively. Thus is not a right multiplier.

Lemma 5 (see [8]). Let be an amenable Banach algebra and let be a continuous homomorphism of onto a dense subalgebra of a Banach algebra . Then is amenable.

Theorem 6. (a) Let be a unital commutative Banach algebra and let be an idempotent homomorphism on such that commutes with each . If is Arens regular then is Arens regular.
(b) Let be a commutative Banach algebra and let be as in part (a). If is amenable then is amenable.

Proof. (a) Define by , where . The homomorphism is onto. Namely, if , then . Of course, has the same property, as well. Let . Then there exist such that , . Let and be two Arens multiplications on the second dual . Thus,
(b) Let    and be a bounded approximate identity in (see [9]). A simple computation shows that , which means . So by Lemma 5 we conclude that is amenable.

Theorem 7. Let be a unital Banach algebra and let be a spectrum preserving homomorphism with dense range. Then each is spectrum preserving.

Proof. Let and . Since has dense range, there exists a sequence such that . Thus Similarly . Thus . Since , we have .
Now, let . Then there exists such that . Thus Similarly . Hence , which means .

3. -Multipliers on Topological Modules and Their Properties

Now, we consider -multipliers in the general topological module setting and investigate some of their properties.

Definition 8. Let be a topological algebra and let be two topological -bimodules and let be a nonzero and continuous idempotent -module homomorphism on . A linear and bounded mapping is called a left (resp., right) -multiplier if (resp., ) for all . We say is a -multiplier if it is both a left -multiplier and a right -multiplier.
We denote by (resp., ) the collection of all -multipliers (resp., left -multipliers, right -multipliers).
It is easy to check that . So .

Example 9. Let be a topological algebra, an -bimodule, and an idempotent -module homomorphism on . For each the mapping defined by is a left -multiplier on .

Proof. Let and , Hence .

In the sequel, denotes a topological algebra and are two topological -bimodules. In general, is an -module homomorphism on such that it is also idempotent, linear, and continuous. Sometimes is on ; it will be mentioned when this happens.

Lemma 10. is a left -module.

Proof. denotes the vector space of all left -multipliers from to . Let and be arbitrary. Define as where is arbitrary. Since the equalities hold for all and , we conclude that is a left -multiplier. Then, since is an -bimodule and is linear, is a left -module.

Definition 11. An -bimodule is said to be commutative if holds for all and .

Definition 12 (see [10, 11]). Let be a left (resp., right) -module. is said to be left (resp., right) faithful in if, for any , (resp., ) for all implies that . If is an -bimodule then is said to be faithful in if it is both left and right faithful in .

The following definition generalizes Definition 12.

Definition 13. Let be a left (resp., right) -module. is said to be left (resp., right) -faithful in if, for any , (resp., ) for all implies that . If is an -bimodule then is said to be -faithful in if it is both left and right -faithful in .

Definition 14. Let be a topological algebra, a commutative -bimodule, and an idempotent -module homomorphism on . For any , define It is easy to see that . Now, we define , by .

Lemma 15. Let be a topological algebra with an approximate identity and let be a topological -bimodule. If is -faithful in and is onto, then is left faithful in .

Proof. Let . In view of Lemma 10, it is enough to show that if for all . Since is onto, there exist such that . Therefore for any and , The continuity of implies that . Now, since is -faithful in , we conclude that . Hence .

Definition 16. Let be a Hausdorff topological algebra and a Hausdorff topological -bimodule. Let be an -module homomorphism on . The -uniform operator topology (resp., -strong operator topology ) on is defined as the linear topology which has a base of neighborhoods of consisting of all sets of the form where is a bounded (resp., finite) subset of and is a neighborhood of in . Clearly .

Theorem 17. Let be an -module homomorphism on and let be a topological -bimodule with -hypocontinuous module multiplication.
Then and are topological left -modules.

Proof. By Lemma 10, is a left -module. Now, let us prove that the module multiplication from into is separately continuous in -topology. Let and be a net in with and let be a bounded subset of and let be a neighborhood of in . By -hypocontinuity, there exists a balanced neighborhood of in such that . Since and are continuous, there exist such that for all and . Hence .
Next, let and be a net in such that and let be a bounded subset of and let be a neighborhood of in . Since the mappings and are continuous, it follows that is a bounded subset in . So there exist such that for all and . Hence . That means is a left topological module. A similar computation shows that is a left topological module.

Lemma 18. Let be a topological algebra with an approximate identity and let be a commutative -bimodule and an idempotent -module homomorphism on . Then .

Proof. Let , and let be a finite subset of and let be a neighborhood of in . For each we have . Then, since is continuous and is finite, there exist such that , for all and . Then, for any and Therefore .

Theorem 19. Let be a topological algebra with an approximate identity and let be a commutative topological -bimodule such that is complete. Suppose is an idempotent -module homomorphism on and is -faithful in . Then is isomorphic to .

Proof. Let be as in Definition 14. It is obvious that is a continuous module homomorphism. We first show that is onto. In view of Lemma 18, it is enough to prove that is -closed. Let . Then there exists a net such that . It follows that the net is -Cauchy in for every . Now, since is -faithful in , the net is -Cauchy in . By completeness of , there exist such that . Hence . By uniqueness of limit in Hausdorff space . Therefore is -closed.
To show that is one-to-one, let such that . Then for any , . Since is -faithful in , this implies that . Thus is one-to-one.

Definition 20. Let be a topological algebra and let be a topological -bimodule. The uniform topology (strict topology ) on is defined as the linear topology which has a base of neighborhoods of consisting of all sets where is a bounded (finite) subset of and is a neighborhood of in .

Lemma 21. Let be a topological algebra with a bounded approximate identity and let be a topological -bimodule. Then and .

Proof. Let be a net in with . Let be a neighborhood of in -topology. Since is continuous, is a bounded subset of for each bounded subset of . Then there exist such that for all . That means . Hence .
Conversely, let be a net in with . Let be a bounded subset of and let be a closed neighborhood of in . Choose . Then there exist such that for all . That means . Hence .