#### Abstract

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.

#### Acknowledgments

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.