Research Article | Open Access
Module Actions on Iterated Duals of Banach Algebras
Let be a Banach algebra and its second dual equipped with the first Arens product. We consider three -bimodule structures on the fourth dual of . This paper discusses the situation that makes these structures coincide.
1. Introduction and Preliminaries
Let be a Banach algebra. It is well known that, on the second dual space of , there are two multiplications, called the first and second Arens products which make into a Banach algebra (see [1, 2]). By definition, the first Arens product on is induced by the left -module structure on . That is, for each , , and , we have Similarly, the second Arens product on is defined by considering as a right -module. The Banach algebra is said to be Arens regular if on the whole of .
For any fixed , the map and are continuous on . Thus, with the -topology, is a right topological semigroup, and is a left topological semigroup. The following sets are called the first and the second topological centres of , respectively. One can verify that is Arens regular if and only if . For example, each -algebra is Arens regular, and for locally compact group , the group algebra is Arens regular if and only if is finite .
For a detailed account of Arens product and topological centres, we refer the reader to Memoire .
Throughout the paper we identify an element of a Banach space with its canonical image in .
2. -Bimodule Structures on
Suppose that is a Banach algebra and is a Banach -bimodule. According to , is a Banach -bimodule where is equipped with the first Arens product. The module actions are defined by where and are nets in and that converge, in -topologies, to and , respectively.
There are two -bimodule structures on which are not always equal . In the first way as the second dual of can be made into -bimodule by module actions defined as above (for ). We denote this module action by the symbol “”. In the second way as the dual space of is -bimodule by the natural module structures.
Now the Banach algebra has three -bimodule structures.(i)If we consider as the second dual of , then can be an -bimodule by module actions defined as above (for ). We denote this module action by the symbol “•”.(ii)If we consider as the dual of in which , then can be an -bimodule by module structures as follows: for all , , and .(iii)We consider as the dual of in which , so can be an -bimodule by the following module actions: where and are the natural module structures on .
It is straightforward to check that three -bimodule structures on are not coincide, in general.
Theorem 2.1. Let be a Banach algebra. Then
Proof. Suppose that , , and are bounded nets in , and , respectively, such that , and in the -topology. Then, we have Therefore, . Similarly, we obtain This shows that , as required.
Let have the first Arens product . Then the first and second Arens products on which we denote by and , respectively, are induced by the left and right -module structure on .
Similarly, we obtain two Arens product on with respect to the second Arens product on , which will denote by and , respectively. We recall that the topological centres of with respect to each of Arens products can be defined analogously.
The proof of the following result is obvious and we omit it.
Proposition 2.2. Let be a Banach algebra. (i)If is commutative, then and .(ii)If is Arens regular, then and .
Theorem 2.3. For each Banach algebra , the following assertions hold. (i) if and only if .(ii) if and only if .
Proof. We prove (i); the proof of (ii) follows from similar argument.
Assume that . Then . Since Arens regularity of implies that of , so we have for each . Therefore by above proposition we obtain It follows that . The converse is similar.
It is well known that every -algebra is Arens regular and is also a -algebra . Therefore itself is Arens regular, but in general case for Arens regular Banach algebra is not Arens regular (see [7, 8]).
Theorem 2.4. Let be an Arens regular Banach algebra. If one of the following conditions hold: (i) and , for each and ,(ii)the operators and are continuous on ,(iii),then is Arens regular.
Proof. This follows from Theorem 2.3 of .
Remark 2.5. Let be a nonunital Banach algebra with a bounded approximate identity(). Then cannot be both WSC and Arens regular . It follows that for every WSC Banach algebra with a BAI, is not Arens regular unless is unital.
Theorem 2.6. Let be an Arens regular Banach algebra. If one of the conditions of Theorem 2.4 hold, then all -bimodule actions on coincide.
Proof. Let and be bounded nets in and that converge, in -topologies, to and , respectively. Then for each , we have Consequently, . By hypothesis is Arens regular and so Thus, for each , Therefore . Now by Theorem 2.1, three -bimodule structure on coincide.
Example 2.7. Let , with pointwise product. Then is an Arens regular Banach algebra which is not reflexive, but . Therefore by above theorem three -bimodule structures on coincide.
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Copyright © 2011 Abbas Sahleh and Abbas Zivari-Kazempour. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.