Arens Regularity of Certain Class of Banach Algebras
We study Arens regularity of the left and right module actions of on , where is the nth dual space of a Banach algebra , and then investigate (quotient) Arens regularity of as a module extension of Banach algebras.
1. Introduction and Preliminaries
In 1951, Arens showed that every bounded bilinear map on normed spaces has two natural but different extensions and from to . The first extension of is constructed by forming in turn the following bilinear maps: The bilinear map is the unique extension of which is -separately continuous on . The second extension of can be made in the same way if we start by transpose map instead of , which is defined by . Similarly, it is the unique extension of that is -separately continuous on . It is easy to check that where and are nets in and that converge, in -topologies, to and , respectively. According to , is said to be Arens regular if .
For the product map of a Banach algebra , we denote and by the symbols and , respectively. These are called the first and second Arens products on . The Banach algebra is said to be Arens regular if on the whole of . The higher extensions and of and Arens products on can be defined similarly. For any fixed , the maps 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, the group algebra for locally compact group is Arens regular if and only if is finite . The reader is referred to [3, 4] for more information on Arens products and topological centres.
Throughout the paper we identify an element of a Banach space with its canonical image in . Also for closed linear subspace of we write .
In , Eshaghi Gordji and Filali obtained significant results related to the topological centres of Banach module actions and regularity of bilinear maps. They showed that if enjoys a bounded approximate identity, then the left (right) module action of on is regular if and only if is reflexive; see also .
In this paper, under certain conditions we prove that the left and right module actions of on are regular, where has not bounded approximate identity. Then we apply this fact to determine Arens regularity and quotient Arens regularity of certain class of Banach algebras.
2. Arens Regularity of Module Extension Banach Algebras
Suppose that is a Banach -bimodule with the left and right module actions and , respectively. 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.
Now suppose that . Then with norm and product is a Banach algebra which is known as a module extension Banach algebra. The second dual of is identified with , as a Banach space. Also the first Arens product on is specified by It is straightforward to check that if and only if (a),(b) is - continuous,(c) is - continuous, (see [5, 8]).
If has the second Arens product , then is an -bimodule in the same way. We denote this module action by the symbol “•”. The second Arens product on and second topological centre of can be defined analogously. Thus, the Banach algebra is Arens regular if and only if is Arens regular and
We consider as a Banach -bimodule equipped with its own multiplication. Then can be made into a Banach -bimodule in a natural fashion . Clearly, regularity of implies that of but the converse is not true in general. For example, let be a nonreflexive Banach space and let be a nonzero element of such that . Then the product turns into a Banach algebra , such that is Arens regular for all .
Now we consider the bilinear mappings One can verify that is Arens regular for all but is not regular for each . This shows that is not regular. However, is Arens regular.
We commence with the next result which studies Arens regularity of the left and right module actions of on .
Theorem 2.1. Let be a Banach algebra and .(i)If , then the right module action of on is Arens regular.(ii)If , then the left module action of on is Arens regular.
Proof. We prove (i) that the assertion (ii) can be proved similarly.
Since , as a direct sum of -bimodules, it is enough to show that the result is valid for , and it can be deduced for , analogously. To this end let and let be bounded net in that is -convergent to . Since , for each there exist and such that . It follows that; for each in the weak topology. So we have that Therefore the right module action of on is regular, as required.
Corollary 2.2. Let be the left module action of a Banach algebra on . If is onto and , then is Arens regular.
The following theorem, which is the main one in the paper, characterizes Arens regularity of .
Theorem 2.3. Let be an Arens regular Banach algebra. If , then, for all , is Arens regular and
Proof. Since is Arens regular, is a dual Banach algebra with predual space . Let and . Then the inclusion shows that is -continuous linear functional on and so it must be in . It follows that for all , and hence for each . Similarly, we obtain . Then by Proposition 2.16 of  is Arens regular and One may verify that and, since , that we have . Thus the result is established for . An easy induction argument now finishes the proof.
Corollary 2.4. Let be an Arens regular Banach algebra. If , then the following assertions hold for all .(i) is Arens regular.(ii) is an -submodule of .
Let , with pointwise product. Then is an Arens regular Banach algebra which is not reflexive but satisfies . Therefore by the preceding corollary is Arens regular.
It is easy to verify that regularity of the left and right module actions of on are equivalent for each Arens regular Banach algebra which is commutative.
Remark 2.5. It is well known that each -algebra is Arens regular and is also a -algebra , and therefore itself are Arens regular. This shows that for each , and hence is Arens regular. But in general, is not Arens regular. Indeed, it is Arens regular if and only if is reflexive .
3. Quotient Arens Regularity of Module Extension Banach Algebras
Let be a Banach algebra with a bounded approximate identity and let , the subspace of consisting of the functionals of the form , for all and . By Cohen’s factorization theorem , is a closed -submodule of . It is also left introverted in ; that is, for each and . Then is a Banach algebra by the following (first Arens type) product:
As in , the Banach algebra is said to be left quotient Arens regular if , where
Similarly, is an -module and is right introverted in . As mentioned above, the second Arens product on induces naturally a Banach algebra product on , which is denoted by . The topological centre and right quotient Arens regularity can be defined analogously. Obviously, every Arens regular Banach algebra is quotient Arens regular but the converse does not hold; see example 38 of . Also a direct proof shows that, if is an ideal in , then is quotient Arens regular.
Proposition 3.1. Suppose that the Banach algebra is a left ideal in . Then is a left ideal in for all .
Proof. We first show that, if is a left ideal in , then it is also a left ideal in for each . So let and . Then, for all , there exist and such that . By assumption , and therefore . This shows that is -continuous linear functional on and so . Since , for some and . Then we have that It follows that , and thus is a left ideal in . An easy induction argument now finishes our claim. Therefore by definition is a left ideal in for each .
In general, the above result is not valid if we replace with . For example, let be the group algebra of an infinite compact group . Then is an ideal in , as is well known, but is not ideal in . By additional hypothesis we have the next result.
Theorem 3.2. If the Banach algebra is a left ideal in and the right module action of on is regular, then is a left ideal in .
Proof. The result is straightforward for the case . So we give the proof for . Let and . Then a similar argument to what has been used in the proof of the preceding proposition shows that . On the other hand, regularity of the right module action of on implies that is -continuous linear functional on and so it must be in . Thus, . Therefore by definition we have that , and hence is a left ideal in . A similar discussion reveals that the result will be established for .
Corollary 3.3. Let be a Banach algebra such that is an ideal in . Suppose that the left and right module actions of on are regular. Then is quotient Arens regular.
Corollary 3.4. Let be a -algebra and . If is an ideal in , then is quotient Arens regular.
Example 3.5. Let , with pointwise product and . Then is a commutative -algebra which is an ideal in . Therefore by the above corollary is quotient Arens regular for all . Note that, by Remark 2.5, is not Arens regular for the odd case .
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