Let be a semigroup with a left multiplier T on . There exists a new semigroup , which depends on and T, which has the same underlying space as . We study the question of involutions on and a Banach algebra . We find a condition of T under which and the second dual admit an involution. We will show that is -algebra if and only if is an isometry, under mild conditions. Also, is -algebra if and only if so is , under other minor conditions.

1. Introduction and Definitions

1.1. Introduction

Birtel in his paper [1, Theorem 5] has introduced a multiplication on a commutative Banach algebra. In [2, Theorem ], Larsen has defined a new Banach algebra which is related to a certain semisimple commutative Banach algebra and a multiplier of which preserves the regular maximal ideal space and , where is the algebra of all multipliers of . The multiplicative linear functionals of and corresponding to given ideals in are not the same. They do, however, only differ by a multiplicative constant. Yoo in his paper [3] has shown that is a -algebra if is -algebra. The second author studied the implications for properties of and . It is shown that, under certain conditions, the Banach algebras and have the same properties and, in the other case, they have different properties [4].

The purpose of this paper is to continue the subject; that is, we investigate some properties between and , in the category of -semigroup. We establish necessary and sufficient conditions for which and are -algebra.

1.2. Definition

The term semigroup will describe a nonempty set endowed with a associative binary operation mapping on into . If is also a Hausdorff topological space and the binary operation is continuous for the product topology of then is said to be Hausdorff topological semigroup. Let be a semigroup. A map is a left (resp., right) multiplier on if for all . The class of left (resp., right) multiplier on is denoted by . It is easy to check that and are unital semigroup under the operation composition.

An operator is a multiplier on if . The space of all multipliers on is denoted by . A pair is a double centralizer on if For , define , by Then , , , and .

If, in addition, is a Banach algebra then a left (resp., right) multiplier must be a linear map. In this case, we say that a multiplier is on Banach algebra (see [5, Definition ]).

A semigroup is left (resp., right) faithful if (resp., ) for all then . A semigroup is faithful if it is left and right faithful. If is faithful then is multiplier if and only if for all [6].

For a Banach algebra , the left (resp., right) annihilator of is denoted by The set is called the annihilator of . The Banach algebra is left (resp., right) faithful if (resp., ).

A net in a Hausdorff topological semigroup , where is a directed set, is a left (resp., right) approximate identity, or, briefly l.a.i. (resp., r.a.i.) if (resp., ).

A net is an approximate identity (briefly, a.i.) if it is a left and right approximate identity.

A net in Banach algebra is a bounded approximate identity (briefly, b.a.i) if, for all , If a Hausdorff topological semigroup has an a.i., then for all (resp., ),

Now, suppose that and are topological semigroup and Banach algebra, respectively, and let be a left multiplier. We define a new product on and a new product and a norm on by The set with the new product “” (resp., with the new norm ) is denoted by (resp., ). It is easy to see that is semigroup and if is continuous then is topological semigroup. Also, the algebra is a Banach algebra (see [2, Theorem ] and [4, Theorem 2.1]).

An involution on a semigroup is a map such that (i)   , (ii) , (iii) if involution map is continuous, we say that is topological -semigroup, and the involution map is an isomorphism and a homeomorphism of onto .

For an involution in Banach algebra , we add the following definition:(iv).

A Banach algebra with an isometric involution is a Banach -algebra. A -algebra is a Banach -algebra such that Now, suppose that is a -semigroup; then a map is said to be a -mapping if .

Define . Then is -mapping if and only if  . We say an element is positive if for some . We denote the set of all positive elements of by ; that  is, .

Now, let be also -semigroup. Then, the map is a positive map if .

A complex value function on is said to be positive definite if for all and , . We write for all positive definite functions on .

In this paper, the nonzero character space on a Banach algebra of is denoted by and it is the set of all multiplicative linear functionals on .

Define the set of positive elements of by

A linear functional on is positive if . The set of all positive linear functionals on is denoted by . Let . Then is a positive trace if The -algebra is hermitian if for all , , where is the spectrum of .

Let be a Banach algebra and , be its first and second dual space of . For any element , the image of in , under the canonical mapping, will be denoted by . Let . Then, we can follow the two Arens products and equip with the first Arens product and second Arens product , which is defined by using iterated limits as follows: The Banach algebra is said to be Arens regular if , for all .

The details of these constructions may be found in many places, including the book [5] and the articles [79].

2. Topological Semigroup with an Involution

We commence our study of -semigroup by considering the same underlying -semigroup . In the first section, we suppose that is -semigroup. We can find some conditions under which is a -semigroup. Then, under those conditions, we begin to consider the position of and obtain the best situation possible in general.

Lemma 1. Let be -topological semigroup. If is a -mapping then is a multiplier. The converse holds, if is continuous and there exists an approximate identity such that for all .

Proof. Let be a left multiplier. Then for all . This yields that is a right multiplier. On the other hand, is a -mapping. Then for all . It follows that is a multiplier.
Conversely, let be an approximate identity with for all . Then for all . Then is a -mapping.

Theorem 2. Let be a -semigroup. Then(i) is -semigroup if and only if is a double centralizer;(ii)if is a -mapping then is -semigroup. The converse holds if is a multiplier and is faithful.

Proof. (i) Suppose that is a -mapping and is a left multiplier. Then is a right multiplier and It follows that is double centralizer.
For the converse, suppose that , for all in . Then
(ii) Suppose that is a -mapping then and then is a multiplier. This follows that is a double centralizer. By (i), is a -semigroup.
For the converse, let be a multiplier and be a -semigroup. Then, by (i), , for all in . Therefore, for all in . This yields that , since is left faithful.

Definition 3. The center of a semigroup is the subset of which is defined by

It is easy to see that is a commutative subsemigroup of . Now, we have the following result.

Corollary 4. Let be a left (resp., right) faithful -semigroup. Then is a -mapping if and only if and .

Proof. By the symmetry of the situation and , we establish the case of . Suppose that is a -mapping. Then, by Lemma 1, is a multiplier. Hence, .
So, for all in , On the other hand, is left faithful; then , for all and then . Now, by using the equality , for all , we also see that is a -mapping if and only if so is ; that is, if is a -mapping, then, for all in , So, we have , for all in . Then .
For the converse, let and . Then Hence, is a -mapping.

Corollary 5. Let and be a -semigroup and let be left faithful. Then, is a -mapping if and only if is a multiplier.

Proof. Let and be two -semigroups. Then So, by using this equality, is -mapping if and only if is a multiplier.

Theorem 6. Let be a -semigroup, with a -mapping if is a map, then, for each and , we have

Proof. Suppose that is a -mapping. Then, by Theorem 2(ii),    is -semigroup. If and , then we have Hence, . Now, suppose that . Then, we have the following assertions: (i);(ii);(iii) (see [10, Lemma II.3] or [11, Page 68, 69]).
It follows that

3. as a -Algebra

We know that a Banach -algebra is a Banach -algebra which has an isometric involution and the norm on algebra is -norm; that is, for all .

Now, we begin by considering the relation between the -algebra and . In general, there are two subjects; then, we have to consider them. First, we find out some conditions of if is -algebra; second, under such conditions, is -algebra if and only if so is . Now, we have established the following theorem.

Throughout this paper assume that is a continuous left multiplier on .

Theorem 7. Let be a Banach algebra. Then consider the following.(i)The operator is homomorphism and norm decreasing.(ii)If is a -algebra and is a -mapping then is a positive and .(iii)Let be hermitian. Then every element of is a positive trace if is a -mapping.

Proof. (i) Let . Then So, the result follows.
(ii) Suppose that is a -mapping. Then, by Theorem 2, (ii), is a -algebra. Let . Then for some . Hence, for each , and It follows that . If is a positive functional on then It follows that .
(iii) Suppose that is hermitian and . Then for all and is a positive trace [5, Proposition ]. So, we have . On the other hand, Then and It follows that . So is a positive trace.

Theorem 8. Let be a -algebra. Then consider the following.(i)The left multiplier is bounded below.(ii)The map is an isometry.(iii) is a closed set.

Proof. (i) Suppose that is -algebra. Take , we have It follows that . So, is bounded below and an injective.
(ii) By Theorem 7, (i), is homomorphism and norm decreasing. Then . By (i), we then have .
(iii) By (ii), the left multiplier satisfies for all . Now, suppose that . Then there exists a sequence in such that converges to . On the other hand, for all . So, is a Cauchy sequence in Banach algebra of and converges to some . Since is continuous then and then is closed.

In the next theorem, we find some conditions on under which is -algebra.

Theorem 9. Let be a -algebra and with double centralizer . Then, is a -algebra if and only if , for all (or is an isometry).

Proof. Let be -algebra. Then, by Theorem 8, (ii), is an isometry and for all
For the converse, by Theorem 2, (i), is -algebra. It suffices to show that , for all . Let . Then, and then

Theorem 10. Let be a -algebra and be a -mapping. Then is a -algebra if and only if is an isometry.

Proof. Let be a -algebra. Then, by Theorem 8, (ii), is an isometry.
For the converse, by Lemma 1, is a multiplier and, is a double centralizer. So, by Theorem 9, is a -algebra.

We pay attention to the extension of involution of to second dual that was studied by many authors in [79]. Now, we consider it for .

Theorem 11. Let be a Banach algebra. Consider the following. (i)If is a double centralizer then so is , with respect the first (or second) Arens product on .(ii)Let be -algebra and Arens regular, ; then admits an involution if and only if is a double centralizer.(iii)If is -mapping then admits an involution.

Proof. Suppose that , with . Take , . Then, with the first Arens regular, we have Now, with second Arens regular, we have Hence . Similarly, and , . So, is a double centralizer with first and second Arens product.
(ii) Suppose that is -algebra. The involution, “,” has an extension to a continuous anti-isomorphism on , denoted by the symbol “”, such that for all and in . Similarly, , for all and in . So admits an involution if and only if is Arens regular. From (i) and Theorem 2, the proof is complete.
(iii) Suppose that is -mapping. Then is double centralizer so, by (ii), admits an involution.

Theorem 12. Let be a Hausdorff topological semigroup with an involution and let be -algebra. Consider the following.(i)Let be a double centralizer. Then is a -algebra if and only if so is .(ii)Let be a -mapping. Then, if is a -algebra then so is . The converse holds if is an isometry.(iii)If is -mapping then admits an involution.

Proof. (i) Suppose that is -algebra. Then by Theorem 2, is also -algebra.
Now, we consider the converse implication. Let be -algebra and . First, suppose that , for some . So, we have Now, for in , there are two nets and in such that . Since , for all , then
(ii) If is a -mapping then . Therefore is a multiplier and is a double centralizer. If is a -algebra then is a -algebra and is an isometry and is closed set (by (i) and Theorem 8). So, by hypothesis, . Now, we consider -norm on . For , there is , such that . So, we have
For the converse, if is -algebra then by Theorem 2, (ii), is a -algebra and for Then -norm holds on .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.


The authors thank the referees for pointing out that some of the results hold if they take -topological semigroup instead of -algebra. Also, they would like to express their thanks to referees for many constructive suggestions to improve the exposition of the their paper.