# The Relationship between Two Involutive SemigroupsS and S

Authors: S. M. Mohammadi; J. Laali
Journal: Journal of Function Spaces
 (2014)

---

## Abstract

LetS 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.


---

## Body

## 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 [3] 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 [4].The purpose of this paper is to continue the subject; that is, we investigate some properties betweenA
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 termsemigroup 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 beHausdorff 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 operatorT is amultiplieron 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 adouble 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 semigroupS 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 [6].For a Banach algebraA, theleft (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 isleft (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 aleft (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 thatS 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 semigroupS 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 algebraA, we add the following definition:(iv)
(
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
)
*.DefineT
Δ
(
a
)
=
T
(
a
*
)
*. Then T is *-mapping if and only if  T
Δ
=
T. We say an element s
∈
S ispositive 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, letS
1 be also *-semigroup. Then, the map T
:
S
→
S
1 is apositive map if T
(
S
+
)
⊆
S
1
+.A complex value functionφ on S is said to bepositive definiteif ∑
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 ofA is denoted by Φ
A and it is the set of all multiplicative linear functionals on A.Define the set of positive elements ofA by
(9)
A
+
=
{
∑
i
=
1
n
‍
a
i
*
a
i
:
a
i
∈
A
,
n
∈
N
}
.A linear functional onA 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 apositive trace if
(10)
f
(
a
a
*
)
=
f
(
a
*
a
)
(
a
∈
A
)
.
The *-algebra A ishermitian if σ
(
a
)
⊆
R for all a
∈
A, a
=
a
*, where σ
(
a
) is the spectrum of a.LetA 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 [5] and the articles [7–9].


## 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 [3] 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 [4].The purpose of this paper is to continue the subject; that is, we investigate some properties betweenA
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 termsemigroup 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 beHausdorff 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 operatorT is amultiplieron 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 adouble 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 semigroupS 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 [6].For a Banach algebraA, theleft (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 isleft (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 aleft (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 thatS 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 semigroupS 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 algebraA, we add the following definition:(iv)
(
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
)
*.DefineT
Δ
(
a
)
=
T
(
a
*
)
*. Then T is *-mapping if and only if  T
Δ
=
T. We say an element s
∈
S ispositive 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, letS
1 be also *-semigroup. Then, the map T
:
S
→
S
1 is apositive map if T
(
S
+
)
⊆
S
1
+.A complex value functionφ on S is said to bepositive definiteif ∑
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 ofA is denoted by Φ
A and it is the set of all multiplicative linear functionals on A.Define the set of positive elements ofA by
(9)
A
+
=
{
∑
i
=
1
n
‍
a
i
*
a
i
:
a
i
∈
A
,
n
∈
N
}
.A linear functional onA 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 apositive trace if
(10)
f
(
a
a
*
)
=
f
(
a
*
a
)
(
a
∈
A
)
.
The *-algebra A ishermitian if σ
(
a
)
⊆
R for all a
∈
A, a
=
a
*, where σ
(
a
) is the spectrum of a.LetA 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 [5] and the articles [7–9].


## 2. Topological SemigroupS
T with an Involution

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.
LetS 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.
LetT 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.
LetS be a *-semigroup. Then(i)
S
T is *-semigroup if and only if (
T
,
T
Δ
) is a double centralizer;
(ii)
ifT is a *-mapping then S
T is *-semigroup. The converse holds if T is a multiplier and S is faithful.Proof.
(i) Suppose thatS
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 thata
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 thatT 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, letT 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 semigroupS 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 thatZ
(
S
) is a commutative subsemigroup of S. Now, we have the following result.Corollary 4.
LetS 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 situationL
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 allx
,
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, leta
=
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.
LetS 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.
LetS 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.
LetS 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 thatT 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: (i)
f
T
(
s
i
*
∘
s
j
)
=
f
T
(
s
j
*
∘
s
i
)
¯;
(ii)
f
T
(
s
i
*
∘
s
i
)
≥
0;
(iii)
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
.


## 3.A
T as a C
*-Algebra

We know that a BanachC
*-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 theC
*-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 thatT is a continuous left multiplier on S.Theorem 7.
LetA be a Banach algebra. Then consider the following.(i)
The operatorT
:
A
T
→
A is homomorphism and norm decreasing.
(ii)
IfA is a *-algebra and T is a *-mapping then T is a positive and P
A
T
⊆
P
A
T.
(iii)
LetA be hermitian. Then every element of Φ
A
T is a positive trace if T is a *-mapping.Proof.
(i) Leta
,
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 thatT 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 thatA 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.
LetA
T be a C
*-algebra. Then consider the following.(i)
The left multiplierT is bounded below.
(ii)
The mapT
:
A
T
→
A is an isometry.
(iii)
T
(
A
) is a closed set.Proof.
(i) Suppose thatA
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 Theorem7, (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 multiplierT 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 onT under which A
T is C
*-algebra.Theorem 9.
LetA 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.
LetA
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 Theorem2, (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.
LetA 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.
LetA
T be a C
*-algebra. Then, by Theorem 8, (ii), T
:
A
T
→
A is an isometry.
For the converse, by Lemma1, 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 ofA to second dual A
*
* that was studied by many authors in [7–9]. Now, we consider it for A
T.Theorem 11.
LetA be a Banach algebra. Consider the following. (i)
If(
T
,
R
) is a double centralizer then so is (
T
*
*
,
R
*
*
), with respect the first (or second) Arens product on A
*
*.
(ii)
LetA 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.
(iii)
IfT is *-mapping then A
T
*
*
*
* admits an involution.Proof.
Suppose thatT
∈
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 thatA 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 thatT is *-mapping. Then (
T
,
T
Δ
) is double centralizer so, by (ii), A
T
*
*
*
* admits an involution.Theorem 12.
LetS be a Hausdorff topological semigroup with an involution and let A be *-algebra. Consider the following.(i)
Let(
T
,
T
Δ
) be a double centralizer. Then A is a *-algebra if and only if so is A
T.
(ii)
LetT 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.
(iii)
IfT is *-mapping then A
T
*
*
*
* admits an involution.Proof.
(i) Suppose thatS is *-algebra. Then by Theorem 2, S
T is also *-algebra.
Now, we consider the converse implication. LetS
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) IfT 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, ifA 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.



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*Source: 851237-2014-08-17.xml*