JFS Journal of Function Spaces 2314-8888 2314-8896 Hindawi Publishing Corporation 10.1155/2014/851237 851237 Research Article The Relationship between Two Involutive Semigroups S and S T Is Defined by a Left Multiplier T Mohammadi S. M. 1 Laali J. 2 Djordjevic Dragan 1 Department of Mathematics Science and Research Branch, Islamic Azad University Tehran Iran azad.ac.ir 2 Department of Mathematics Faculty of Mathematical Science and Computer Kharazmi University 50 Taleghani Avenue Tehran 15618 Iran khu.ac.ir 2014 1482014 2014 08 04 2014 12 06 2014 15 07 2014 17 8 2014 2014 Copyright © 2014 S. M. Mohammadi and J. Laali. 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.

Let S be a semigroup with a left multiplier T on S . There exists a new semigroup S T , which depends on S and T, which has the same underlying space as S . We study the question of involutions on S T and a Banach algebra A T . We find a condition of T under which S T and the second dual A T * * * * admit an involution. We will show that A T is C * -algebra if and only if T : A T A is an isometry, under mild conditions. Also, A is C * -algebra if and only if so is A T , 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 1.3 . 1 ], Larsen has defined a new Banach algebra A T which is related to a certain semisimple commutative Banach algebra A and a multiplier T of A which preserves the regular maximal ideal space and Mul ( A ) = Mul ( A T ) , where Mul ( A ) is the algebra of all multipliers of A . The multiplicative linear functionals of A and A T corresponding to given ideals in Δ ( A ) = Δ ( A T ) are not the same. They do, however, only differ by a multiplicative constant. Yoo in his paper  has shown that Mul ( A ) is a C * -algebra if A is C * -algebra. The second author studied the implications for properties of A and A T . It is shown that, under certain conditions, the Banach algebras A and A T have the same properties and, in the other case, they have different properties .

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

1.2. Definition

The term semigroup will describe a nonempty set S endowed with a associative binary operation mapping on S × S into S . If S is also a Hausdorff topological space and the binary operation is continuous for the product topology of S × S then S is said to be Hausdorff topological semigroup. Let S be a semigroup. A map T : S S is a left (resp., right) multiplier on S if (1) T ( a b ) = T ( a ) b ( T ( a b ) = a T ( b ) ) for all a , b S . The class of left (resp., right) multiplier on S is denoted by Mu l l ( S ) ( resp . , Mu l r ( S ) ) . It is easy to check that Mu l l ( S ) and Mu l r ( S ) are unital semigroup under the operation composition.

An operator T is a multiplier on S if T Mu l l ( S ) Mu l r ( S ) . The space of all multipliers on S is denoted by Mul ( S ) . A pair ( T , R ) is a double centralizer on S if (2) T Mu l l ( S ) , R Mu l r ( S ) , a T ( b ) = R ( a ) b ( a , b S ) . For s S , define L s : S S , R s : S S by (3) L s ( t ) = s t , R s ( t ) = t s ( t S ) . Then L s Mu l l ( S ) , R s Mu l r ( S ) , L s L t = L s t , and R s R t = R t s .

If, in addition, S 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 1.4 . 25 ]).

A semigroup S is left (resp., right) faithful if s x = t x (resp., x s = x t ) for all x S then s = t . A semigroup S is faithful if it is left and right faithful. If S is faithful then T : S S is multiplier if and only if s T t = ( T s ) t for all s , t S .

For a Banach algebra A , the left (resp., right) annihilator of A is denoted by (4) Ann l ( A ) = { a A : a A = 0 } ( Ann r ( A ) = { a A : A a = 0 } ) . The set Ann l ( A ) Ann r ( A ) is called the annihilator of A . The Banach algebra A is left (resp., right) faithful if Ann l ( A ) = { 0 } (resp., Ann r ( A ) = { 0 } ).

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

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

A net ( e α ) in Banach algebra A is a bounded approximate identity (briefly, b.a.i) if, for all a A , (5) lim α e α a = a = lim α a e α , sup α I e α < . If a Hausdorff topological semigroup S has an a.i., then for all T Mu l l ( S ) (resp., R Mu l r ( S ) ), (6) T ( a ) = lim α L T ( e α ) ( a ) ( resp . R ( a ) = lim α R T ( e α ) ( a ) ) , w w w w w w w w w w w w i ( a S ) .

Now, suppose that S and A are topological semigroup and Banach algebra, respectively, and let T be a left multiplier. We define a new product on S and a new product and a norm on A by (7) a b = a T ( b ) ( a , b S , A ) c T = c T ( c A ) . The set S ( resp . , A ) with the new product “ o ” (resp., ( A . o ) with the new norm · T ) is denoted by S T (resp., A T ). It is easy to see that S T is semigroup and if T is continuous then S T is topological semigroup. Also, the algebra A T is a Banach algebra (see [2, Theorem 1.3 . 1 ] and [4, Theorem 2.1]).

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

For an involution in Banach algebra A , we add the following definition:

( α s + β t ) * = α ¯ s * + β ¯ t *       ( α , β C , s , t A ) .

A Banach algebra with an isometric involution is a Banach * -algebra. A C * -algebra is a Banach * -algebra A such that (8) a * a = a 2 ( a A ) . Now, suppose that S is a * -semigroup; then a map T : S S is said to be a * -mapping if T ( a * ) = T ( a ) * .

Define T Δ ( a ) = T ( a * ) * . Then T is * -mapping if and only if   T Δ = T . We say an element s S is positive if s = x * x for some x S . We denote the set of all positive elements of S by S + ; that  is, S + = { x * x : x S } .

Now, let S 1 be also * -semigroup. Then, the map T : S S 1 is a positive map if T ( S + ) S 1 + .

A complex value function φ on S is said to be positive definite if i j = 1 n c i c ¯ j φ ( x j * x i ) 0 for all c 1 , c 2 , , c n C and x 1 , x 2 , , x n S , n N . We write P ( S ) ( resp . , P ( S T ) for all positive definite functions on S ( resp . , S T ) .

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

Define the set of positive elements of A by (9) A + = { i = 1 n a i * a i : a i A , n N } .

A linear functional on A is positive if f ( A + ) R + . The set of all positive linear functionals on A is denoted by P A . Let f P A . Then f is a positive trace if (10) f ( a a * ) = f ( a * a ) ( a A ) . The * -algebra A is hermitian if σ ( a ) R for all a A , a = a * , where σ ( a ) is the spectrum of a .

Let A be a Banach algebra and A * , A * * be its first and second dual space of A . For any element a A , the image of a in A * * , under the canonical mapping, will be denoted by a ^ . Let F , G A * * . Then, we can follow the two Arens products and equip A * * with the first Arens product and second Arens product , which is defined by using iterated limits as follows: (11) F G = w * - lim a F w * - lim b G ( a b ) , F G = w * - lim b G w * - lim a F ( a b ) . The Banach algebra A is said to be Arens regular if F G = F G , for all F , G A * * .

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

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

Lemma 1.

Let S be * -topological semigroup. If T is a * -mapping then T is a multiplier. The converse holds, if T is continuous and there exists an approximate identity ( e α ) such that T ( e α ) * = T ( e α ) for all α .

Proof.

Let T be a left multiplier. Then (12) T Δ ( a b ) = T ( ( a b ) * ) * = ( T ( b * ) a * ) * = a T Δ ( b ) for all a , b S . This yields that T Δ is a right multiplier. On the other hand, T is a * -mapping. Then (13) T ( a ) = T ( a * * ) = T ( a * ) * = T Δ ( a ) , for all a S . It follows that T is a multiplier.

Conversely, let ( e α ) be an approximate identity with T ( e α ) * = T ( e α ) for all α . Then (14) T ( a * ) = lim α T ( e α a * ) = lim α T ( e α ) a * = lim α ( a T ( e α ) ) * = lim α ( T ( a e α ) ) * = T ( a ) * , for all a S . Then T is a * -mapping.

Theorem 2.

Let S be a * -semigroup. Then

S T is * -semigroup if and only if ( T , T Δ ) is a double centralizer;

if T is a * -mapping then S T is * -semigroup. The converse holds if T is a multiplier and S is faithful.

Proof.

(i) Suppose that S T is a * -mapping and T is a left multiplier. Then T Δ is a right multiplier and (15) a T ( b ) = a b = ( b * a * ) * = T ( a * ) * b = T Δ ( a ) b ( a , b S ) . It follows that ( T , T Δ ) is double centralizer.

For the converse, suppose that a T ( b ) = T Δ ( a ) b , for all a , b in S . Then (16) ( a b ) * = ( a T ( b ) ) * = ( T Δ ( a ) b ) * = b * T ( a * ) = b * a * .

(ii) Suppose that T is a * -mapping then T = T Δ and then T is a multiplier. This follows that ( T , T Δ ) is a double centralizer. By (i), S T is a * -semigroup.

For the converse, let T be a multiplier and S T be a * -semigroup. Then, by (i), a T ( b ) = T Δ ( a ) b , for all a , b in S . Therefore, (17) T ( a * ) * b = T Δ ( a ) b = a T ( b ) = T ( a ) b for all a , b in S . This yields that T ( a * ) = T ( a ) * , since S is left faithful.

Definition 3.

The center of a semigroup S is the subset Z ( S ) of S which is defined by (18) Z ( S ) = { a S : a b = b a b S } .

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

Corollary 4.

Let S be a left (resp., right) faithful * -semigroup. Then L a ( resp . , R a ) is a * -mapping if and only if a Z ( S ) and a * = a .

Proof.

By the symmetry of the situation L a and R a , we establish the case of L a . Suppose that L a is a * -mapping. Then, by Lemma 1, L a is a multiplier. Hence, L a ( x ) y = x L a ( y ) .

So, for all x , y in S , (19) a x y = x a y . On the other hand, S is left faithful; then a x = x a , for all x and then a Z ( S ) . Now, by using the equality a x = x a , for all x S , we also see that L a is a * -mapping if and only if so is L a * ; that is, if L a * is a * -mapping, then, for all x in S , (20) L a * ( x ) = a * x = L a * ( x * * ) = L a * ( x * ) * = ( a * x * ) * = x a = a x = L a ( x ) . So, we have a * x = a x , for all x in S . Then a = a * .

For the converse, let a = a * and a Z ( S ) . Then (21) L a ( x * ) = L a * ( x * ) = ( x a ) * = ( a x ) * = L a ( x ) * . Hence, L a is a * -mapping.

Corollary 5.

Let S and S T be a * -semigroup and let S be left faithful. Then, T is a * -mapping if and only if T is a multiplier.

Proof.

Let S and S T be two * -semigroups. Then (22) a T ( b ) = a b = ( b * a * ) * = T ( a * ) * b             ( a , b S ) . So, by using this equality, T is a * -mapping if and only if T is a multiplier.

Theorem 6.

Let S be a * -semigroup, T M u l l ( S ) with a * -mapping if T : S T S is a map, then, for each f P ( S ) and { s 1 , , s n } S , we have (23) 0 i , j = 1 n ( f T ( s i * s j ) f T ( s j * s i ) ) [ i = 1 n f T ( s i * s i ) ] 2 .

Proof.

Suppose that T is a * -mapping. Then, by Theorem 2(ii),    S T is * -semigroup. If f P ( S ) and { t 1 , , t m } S , { c 1 , , c m } C then we have (24) i , j = 1 m c i c ¯ j f T ( t i * t j ) = i , j = 1 m c i c ¯ j f ( T ( t i * ) T ( t j ) ) = i , j = 1 m c i c ¯ j f ( T ( t i ) * T ( t j ) ) 0 . Hence, f T P ( S T ) . Now, suppose that { s 1 , , s n } S . Then, we have the following assertions:

f T ( s i * s j ) = f T ( s j * s i ) ¯ ;

f T ( s i * s i ) 0 ;

f T ( s i * s j ) f T ( s j * s i ) = | f T ( s i * s j ) | 2 f T ( s i * s i ) f T ( s j * s j ) (see [10, Lemma II.3] or [11, Page 68, 69]).

It follows that (25) 0 i , j = 1 n | f T ( s i * s j ) | 2 = i , j = 1 n f T ( s i * s j ) f T ( s j * s i ) i , j = 1 n f T ( s i * s i ) f T ( s j * s j ) = [ i = 1 n f T ( s i * s i ) ] 2 .

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We know that a Banach C * -algebra is a Banach * -algebra which has an isometric involution and the norm on algebra is C * -norm; that is, a * a = a 2 for all a A .

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

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

Theorem 7.

Let A be a Banach algebra. Then consider the following.

The operator T : A T A is homomorphism and norm decreasing.

If A is a * -algebra and T is a * -mapping then T is a positive and P A T P A T .

Let A be hermitian. Then every element of Φ A T is a positive trace if T is a * -mapping.

Proof.

(i) Let a , b A . Then (26) T ( a b ) = T ( a T ( b ) ) = T ( a ) T ( b ) , T ( a ) a T . So, the result follows.

(ii) Suppose that T is a * -mapping. Then, by Theorem 2, (ii), A T is a * -algebra. Let a A T + . Then a = i = 1 n a i * a i for some a i , a i * A ( 1 i n ) . Hence, T ( a i * ) = T ( a i ) * for each i , and (27) T ( a ) = i = 1 n T ( a i * T ( a i ) ) = i = 1 n T ( a i ) * T ( a i ) . It follows that T ( A T + ) A + . If f is a positive functional on A then (28) f T ( A T + ) f ( A + ) R + . It follows that f T P A .

(iii) Suppose that A is hermitian and φ Φ A . Then φ ( a * ) = φ ( a ) ¯ for all a A and φ is a positive trace [5, Proposition 1.10.22 ]. So, we have φ P A . On the other hand, (29) φ T ( a * a ) = φ ( T ( a * ) T ( a ) ) = φ ( T ( a ) ) ¯ φ ( T ( a ) ) 0 . Then φ T P A T and (30) φ ( T ( a * o a ) ) = φ ( T ( a ) ) φ ( T ( a ) * )    = φ ( T ( a ) T ( a * ) ) = φ T ( a T ( a * ) ) = φ T ( a a * ) . It follows that φ T ( a * a ) = φ T ( a a * ) . So φ T is a positive trace.

Theorem 8.

Let A T be a C * -algebra. Then consider the following.

The left multiplier T is bounded below.

The map T : A T A is an isometry.

T ( A ) is a closed set.

Proof.

(i) Suppose that A T is C * -algebra. Take a A , we have (31) a T 2 = a * a T = a * T ( a ) T T ( a ) a T = T ( a ) a T . It follows that a T T ( a ) . So, T is bounded below and an injective.

(ii) By Theorem 7, (i), T is homomorphism and norm decreasing. Then T ( a ) a T . By (i), we then have T ( a ) = a T .

(iii) By (ii), the left multiplier T satisfies (32) T ( a ) = T a , for all a A . Now, suppose that y T ( A ) ¯ . Then there exists a sequence ( x n ) in A such that ( T ( x n ) ) converges to y A . On the other hand, (33) T ( x n ) - T ( x m ) = T x n - x m for all n , m . So, ( x n ) is a Cauchy sequence in Banach algebra of A and converges to some x A . Since T is continuous then y = T ( x ) T ( A ) and then T ( A ) is closed.

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

Theorem 9.

Let A be a C * -algebra and T M u l ( A ) with double centralizer ( T , T Δ ) . Then, A T is a C * -algebra if and only if T ( a ) = a T , for all a A + (or T : A T + A is an isometry).

Proof.

Let A T be C * -algebra. Then, by Theorem 8, (ii), T : A T A is an isometry and for all a A + (34) T ( a ) = a T .

For the converse, by Theorem 2, (i), A T is * -algebra. It suffices to show that a * a T = a T 2 , for all a A . Let a A . Then, a * a A + and then (35) a * a T = a * T ( a ) T = T ( a * a ) T = a * a T T = a 2 T 2 = a T 2 .

Theorem 10.

Let A be a C * -algebra and T be a * -mapping. Then A T is a C * -algebra if and only if T : A T A is an isometry.

Proof.

Let A T be a C * -algebra. Then, by Theorem 8, (ii), T : A T A is an isometry.

For the converse, by Lemma 1, T is a multiplier and, ( T , T Δ ) is a double centralizer. So, by Theorem 9, A T is a C * -algebra.

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

Theorem 11.

Let A be a Banach algebra. Consider the following.

If ( T , R ) is a double centralizer then so is ( T * * , R * * ) , with respect the first (or second) Arens product on A * * .

Let A be * -algebra and Arens regular, T M u l l ( A ) ; then A T * * * * admits an involution if and only if ( T , T Δ ) is a double centralizer.

If T is * -mapping then A T * * * * admits an involution.

Proof.

Suppose that T Mu l l ( A ) , R Mu l r ( A ) with a T ( b ) = R ( a ) b . Take F A * * , G A * * . Then, with the first Arens regular, we have (36) T * * ( F G ) = w * - lim a F w * - lim b G T ( a b ) = w * - lim a F w * - lim b G T ( a ) b = w * - lim a F T ( a ) G = T * * ( a ) G . Now, with second Arens regular, we have (37) T * * ( F G ) = w * - lim b G w * - lim a F T ( a b ) = w * - lim b G w * - lim a F T ( a ) b = w * - lim b G T * * ( F ) b = T * * ( F ) G . Hence T * * Mu l l ( A * * ) . Similarly, R * * Mu l r ( A * * ) and F T * * ( G ) = R * * ( F ) G , F T * * ( G ) = R * * ( F ) G . So, ( T * * , R * * ) is a double centralizer with first and second Arens product.

(ii) Suppose that A is * -algebra. The involution, “ * ,” has an extension to a continuous anti-isomorphism on A * * , denoted by the symbol “ ~ ”, such that (38) ( F G ) ~ = w * - lim a F w * - lim b G [ ( a b ) ] * = w * - lim a F w * - lim b G [ ( b * ) ( a * ) ] = w * - lim a F w * - lim b G [ ( b ) * ( a ) * ] = G ~ F ~ , for all F and G in A * * . Similarly, ( F G ) ~ = G ~ F ~ , for all F and G in A * * . So A * * admits an involution if and only if A is Arens regular. From (i) and Theorem 2, the proof is complete.

(iii) Suppose that T is * -mapping. Then ( T , T Δ ) is double centralizer so, by (ii), A T * * * * admits an involution.

Theorem 12.

Let S be a Hausdorff topological semigroup with an involution and let A be * -algebra. Consider the following.

Let ( T , T Δ ) be a double centralizer. Then A is a * -algebra if and only if so is A T .

Let T be a * -mapping. Then, if A T is a C * -algebra then so is A . The converse holds if T : A T A is an isometry.

If T is * -mapping then A T * * * * admits an involution.

Proof.

(i) Suppose that S is * -algebra. Then by Theorem 2, S T is also * -algebra.

Now, we consider the converse implication. Let S T be * -algebra and a , b S . First, suppose that a = T ( a 1 ) , b = T ( b 1 ) , for some a 1 , b 1 S . So, we have (39) ( a b ) * = ( T ( a 1 ) T ( b 1 ) ) * = ( T ( a 1 b 1 ) ) * = T Δ ( ( a 1 b 1 ) * ) = T Δ ( b 1 * a 1 * ) = T Δ ( b 1 * T ( a 1 * ) ) = T Δ ( T Δ ( b 1 * ) a 1 * ) = T Δ ( b 1 * ) T Δ ( a 1 * ) = ( T ( b 1 ) ) * ( T ( a 1 ) ) * = b * a * . Now, for a , b in S = T ( S ) ¯ , there are two nets ( a α ) and ( b β ) in S such that a = lim α T ( a α ) , b = lim β T ( b β ) . Since ( T ( a α ) T ( b β ) ) * = T ( b β ) * T ( a α ) * , for all α , β , then (40) ( a b ) * = lim α lim β ( T ( a α ) T ( b β ) ) * = lim α lim β T ( b β ) * T ( a α ) * = ( lim β T ( b β ) ) * ( lim α T ( a α ) ) * = b * a * .

(ii) If T is a * -mapping then T = T Δ . Therefore T is a multiplier and ( T , T Δ ) is a double centralizer. If A T is a C * -algebra then A T is a * -algebra and T is an isometry and T ( A ) is closed set (by (i) and Theorem 8). So, by hypothesis, T ( A ) = A . Now, we consider C * -norm on A . For a A , there is a 1 A , such that a = T ( a 1 ) . So, we have (41) a a * = T ( a 1 ) T ( a 1 ) * = T ( a 1 ) T ( a 1 * ) = T ( a 1 a 1 * ) = a 1 a 1 * T = a 1 T 2 = T ( a 1 ) 2 = a 2 .

For the converse, if A is C * -algebra then by Theorem 2, (ii), A T is a * -algebra and for a A (42) a a * T = a T ( a * ) T = T ( a a * ) T = a a * T T = a T 2 . Then C * -norm holds on A T .

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.

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