# φ-Multipliers on Banach Algebras and Topological Modules

Authors: Marjan Adib
Journal: Abstract and Applied Analysis
 (2015)

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## Abstract

We prove some results concerning Arens regularity and amenability of the Banach algebraMφA of all φ-multipliers on a given Banach algebra A. We also consider φ-multipliers in the general topological module setting and investigate some of their properties. We discuss the φ-strict and φ-uniform topologies on MφA. A characterization of φ-multipliers on L1G-module LpG, where G is a compact group, is given.


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## Body

## 1. Introduction

The concept of a multiplier was introduced by Helgason [1] as follows. Let A be a commutative and semisimple Banach algebra and let Δ(A) be its maximal ideal space. Let A^ denote the Gelfand representation of A as a subalgebra of the algebra of continuous functions on Δ(A). A bounded continuous function g on Δ(A) is a multiplier on A if gA^⊆A^. The general theory of multipliers on faithful Banach algebras was developed by Wang [2] and Birtel [3].Recall that a mappingT:A→A is called a left (resp., right) multiplier on A if
(1)T(xy)=Txy(resp.,T(xy)=xT(y))
for all x,y∈A. We say T is a multiplier on A if it is both a left multiplier and a right multiplier on A.We denote withM(A) the algebra of all multipliers on A.A Banach algebraA is called left (resp., right) faithful if, for all x∈A, xA={0} (resp., Ax={0}) implies that x=0; A is called faithful if it is both left and right faithful.In [4] we generalized the concept of multipliers on faithful Banach algebras to φ-multipliers as follows. Let A be a Banach algebra and let φ:A→A be an algebra homomorphism. A linear continuous mapping T:A→A is called a left (resp., right) φ-multiplier on A if
(2)T(xy)=T(x)φ(y)(resp.,T(xy)=φ(x)T(y))
for all x,y∈A. We say T is a φ-multiplier on A if it is both a left φ-multiplier and a right φ-multiplier on A. We denote by  Mφ(A) (resp., Mφl(A),Mφr(A)) the collection of all φ-multipliers (resp., left φ-multipliers, right φ-multipliers) on A.It turns out that this concept is considerably more general than the concept of multipliers on Banach algebras. Also by using some well-known homomorphisms like Jordan homomorphism, spectrum preserving homomorphism, and idempotent preserving homomorphism, we can transfer these useful properties from homomorphismφ to the algebra of φ-multipliers.In [4], we studied various properties of φ-multipliers, for instance, the faithfulness of the Banach algebra Mφ(A) and the existence of a bounded approximate identity in the range of a φ-multiplier. Finally, as an example, we have characterized φ-multipliers on L1(G).In Section2 we are concerned by Arens regularity and amenability of the Banach algebra Mφ(A) under some suitable conditions. We introduce the notion of Jordan φ-multiplier and prove that every Jordan φ-multiplier is a φ-multiplier whenever the range of φ is dense in the algebra.In Section3 we extend the notion of φ-multipliers on Banach algebras to topological modules and investigate some of their properties. We discuss the φ-strict and φ-uniform topologies on Mφ(A) and apply our results to L1(G)-module Lp(G).LetX be a topological vector space and let A be a topological algebra, both over the same field K(=RorC). Then X is called a topological left A-module if it is a left A-module and the module multiplication a,x→a·x from A×X into X is separately continuous. If b(A) denotes the collection of all bounded sets in A, then module multiplication (a,x)→a·x is called b(A)-hypocontinuous [5] if, given any neighborhood G of 0 in X and any D∈b(A), there exists a neighborhood H of 0 in X such that D·H⊆G. Clearly, joint continuity ⇒ hypocontinuity ⇒ separate continuity. A mapping ψ from a left A-module X into another left A-module Y is called an A-module homomorphism if ψ(a·x)=a·ψ(x) for all a∈A and x∈X.


## 2. Some Properties ofφ-Multipliers on Banach Algebras

Let us start with the following result proved in [4].Theorem 1 (see [4, Theorem 2.2]).
LetA be a faithful commutative Banach algebra and let φ be an idempotent homomorphism on A. Then Mφ(A) is a Banach algebra. Moreover, if A2=A and φ∘T=T∘φ for all T∈Mφ(A), then Mφ(A) is a faithful commutative Banach algebra.Definition 2.
LetA be a Banach algebra and let φ be a homomorphism from A to A. The mapping T:A→A is called a left (resp., right) Jordan φ-multiplier on A if for all x∈A(3)T(x2)=T(x)φ(x)(resp.,T(x2)=φ(x)T(x)).Tis called a Jordan φ-multiplier on A if it is both a left Jordan φ-multiplier and a right Jordan φ-multiplier on A.Theorem 3.
LetA be a faithful commutative Banach algebra and let φ be a homomorphism from A to A with dense range. Then T is a φ-multiplier if and only if T is a Jordan φ-multiplier.Proof.
It is clear that everyφ-multiplier is a Jordan φ-multiplier. Conversely, suppose T is a Jordan φ-multiplier. Then
(4)Tx+y2=φ(x+y)T(x+y)=φ(x)T(x)+φ(x)T(y)+φ(y)T(x)+φ(y)T(y)
for all x,y∈A.
On the other hand, we have(5)Tx+y2=T(x2+2xy+y2)=φ(x)T(x)+2T(xy)+φ(y)T(y).
Comparing (4), (5) we obtain
(6)2T(xy)=φ(x)T(y)+φ(y)T(x).
From (6) and using commutativity of A, for each sequence {zn}n=1∞⊂A we have
(7)2Txyzn=φ(y)T(xzn)+φ(xzn)T(y)=φ(y)[φ(x)T(zn)+φ(zn)T(x)]2+φxznTy,
so we have
(8)2Txyzn=φ(y)φ(x)T(zn)+φ(y)φ(zn)T(x)hh+2φxφznTy·2-1;
similarly by using (6) we have
(9)2Txyzn=φ(x)φ(y)T(zn)+φ(x)φ(zn)T(y)hhh+2φyφznTx·2-1;
comparing (8), (9) we obtain
(10)limn→∞φ(x)φ(zn)T(y)=limn→∞φ(y)φ(zn)T(x)
for all x,y,zn∈A. Since φ has dense range and A is a faithful commutative Banach algebra, we have
(11)φxTy=Txφy;
hence T is a φ-multiplier.We mention that Theorem3 holds for certain noncommutative cases, but not in general. For instance, Zalar has proved in [6] that any left (right) Jordan multiplier on a 2-torsion free semiprime ring is a left (right) multiplier. Vukman [7] has shown that an additive map φ:R→R, where R is a 2-torsion free semiprime ring, with the property that 2φ(a2)=aφ(a)+φ(a)a for all a∈A, is a multiplier.The following example shows that, in general, the above theorem need not hold for noncommutative Banach algebras.Example 4.
Consider the subalgebra(12)A=abc0ad00a∣a,b,c,d∈C
of the algebra of all 3×3 matrices. It is obvious that A is a Banach algebra with respect to the norm given by
(13)abc0ad00a=a+b+c+d.
Let
(14)X=010000000
and define a continuous linear map φ:A→A by φ(A)=AX+XA. By a straightforward calculation one can prove that
(15)BAX+XAB=BXA+AXB,A,B∈A.
If ∘′′ denotes the Jordan product A∘B=AB+BA, then we have φ(A∘B)=A∘φ(B) for each A,B∈A and hence φ is a right Jordan multiplier. Also A∘φ(B)=φ(1) for all A,B∈A with A∘B=1. If we consider
(16)A=000001000
then φ(A)≠0 and Aφ(1)=0, where 0,1 are the zero matrix and the identity matrix, respectively. Thus φ is not a right multiplier.Lemma 5 (see [8]).
LetA be an amenable Banach algebra and let ψ be a continuous homomorphism of A onto a dense subalgebra of a Banach algebra B. Then B is amenable.Theorem 6.
(a) LetA be a unital commutative Banach algebra and let φ be an idempotent homomorphism on A such that φ commutes with each S∈Mφ(A). If A is Arens regular then Mφ(A) is Arens regular.
(b) LetA be a commutative Banach algebra and let φ be as in part (a). If A is amenable then Mφ(A) is amenable.Proof.
(a) Defineμ:A→Mφ(A) by μ(a)=φa, where φ(b)a=φ(ab). The homomorphism μ is onto. Namely, if S∈Mφ(A), then S=φS(e). Of course, μ**:A**→(Mφ(A))** has the same property, as well. Let F′,G′∈(Mφ(A))**. Then there exist F,G∈A** such that μ**(F)=F′, μ**(G)=G′. Let ∘ and ∘′ be two Arens multiplications on the second dual A**. Thus,
(17)F′∘G′=μ**(F)∘μ**(G)=μ**(F∘G)=μ**(F∘′G)=μ**F∘′μ**G=F′∘′G′.
(b) LetT∈Mφ(A)   and {eα:α∈I} be a bounded approximate identity in A (see [9]). A simple computation shows that T=limαφT(eα), which means μ(A)¯=Mφ(A). So by Lemma 5 we conclude that Mφ(A) is amenable.Theorem 7.
LetA be a unital Banach algebra and let φ:A→A be a spectrum preserving homomorphism with dense range. Then each T∈Mφ(A) is spectrum preserving.Proof.
Leta∈A and λ∉σ(T(a)). Since φ has dense range, there exists a sequence {cn}n∈A such that (T(a)-λ)limnφ(cn)=1. Thus
(18)φa-λlimnTcn=limn(φ(a)-λ)T(cn)=limnφ(a-λ)T(cn)=limnT(a-λ)φ(cn)=(T(a)-λ)limnφ(cn)=1.
Similarly limnT(cn)(φ(a)-λ)=1. Thus λ∉σ(φ(a)). Since σ(φ(a))=σ(a), we have λ∉σ(a).
Now, letλ∉σ(a). Then there exists b∈A such that (a-λ)b=1. Thus
(19)(T(a)-λ)φ(b)=T(a-λ)φ(b)=T((a-λ)b)=1.
Similarly φ(b)(T(a)-λ)=1. Hence λ∉σ(T(a)), which means σ(T(a))=σ(a).


## 3.φ-Multipliers on Topological Modules and Their Properties

Now, we considerφ-multipliers in the general topological module setting and investigate some of their properties.Definition 8.
LetA be a topological algebra and let X,Y be two topological A-bimodules and let φ be a nonzero and continuous idempotent A-module homomorphism on X. A linear and bounded mapping T:X→Y is called a left (resp., right) φ-multiplier if Ta·x=Tφx·a (resp., T(x·a)=a·T(φ(x))) for all a∈A,x∈X. We say T is a φ-multiplier if it is both a left φ-multiplier and a right φ-multiplier.
We denote byMφ(X,Y) (resp., Mφl(X,Y),Mφr(X,Y)) the collection of all φ-multipliers (resp., left φ-multipliers, right φ-multipliers).
It is easy to check thatφ∈Mφ(X,X). So Mφ(X,X)≠{0}.Example 9.
LetA be a topological algebra, X an A-bimodule, and φ an idempotent A-module homomorphism on X. For each a∈A the mapping φa:X→X defined by φ(x)a=a·φ(x) is a left φ-multiplier on X.Proof.
Leta,b∈A and x∈X,
(20)φb·xa=a·φ(b·x)=a·b·φ(x)=b·a·φ(φ(x))=b·φ(φ(x)),aφx·ba=a·φ(x·b)=a·φ(x)·b=a·φ(φ(x))·b=φ(φ(x))·b.a
Hence φa∈Mφ(X,X).In the sequel,A denotes a topological algebra and X,Y are two topological A-bimodules. In general, φ is an A-module homomorphism on X such that it is also idempotent, linear, and continuous. Sometimes φ is on A; it will be mentioned when this happens.Lemma 10.
M
φ
l
(
X
,
Y
) is a left A-module.Proof.
M
φ
l
(
X
,
Y
) denotes the vector space of all left φ-multipliers from X to Y. Let T∈Mφl(X,Y) and a∈A be arbitrary. Define a*T as (a*T)(x):=T(x·a) where x∈X is arbitrary. Since the equalities
(21)a*Tb·x=T(b·x·a)=b·T(φ(x·a))=b·T(φ(x)·a)=b·(a*T)(φ(x))
hold for all b∈A and x∈X, we conclude that a*T is a left φ-multiplier. Then, since X is an A-bimodule and T is linear, Mφl(X,Y) is a left A-module.Definition 11.
AnA-bimodule X is said to be commutative if a·x=x·a holds for all a∈A and x∈X.Definition 12 (see [10, 11]).
LetX be a left (resp., right) A-module. A is said to be left (resp., right) faithful in X if, for any x∈A, a·x=0 (resp., x·a=0) for all a∈A implies that x=0. If X is an A-bimodule then A is said to be faithful in X if it is both left and right faithful in X.The following definition generalizes Definition12.Definition 13.
LetX be a left (resp., right) A-module. A is said to be left (resp., right) φ-faithful in X if, for any x∈A, φ(a)·x=0 (resp., x·φ(a)=0) for all a∈A implies that x=0. If X is an A-bimodule then A is said to be φ-faithful in X if it is both left and right φ-faithful in X.Definition 14.
LetA be a topological algebra, X a commutative A-bimodule, and φ an idempotent A-module homomorphism on A. For any x∈X, define
(22)φx:A⟶Xbyφx(a)=x·φ(a),a∈A.
It is easy to see that φx∈Mφl(A,X). Now, we define ψ:X→Mφl(A,X), by ψ(x)=φx.Lemma 15.
LetA be a topological algebra with an approximate identity {eλ:λ∈I} and let X be a topological A-bimodule. If A is φ-faithful in X and ψ:X→Mφl(A,X) is onto, then A is left faithful in Mφl(A,X).Proof.
LetT∈Mφl(A,X). In view of Lemma 10, it is enough to show that T=0 if a*T=0 for all a∈A. Since ψ is onto, there exist x∈X such that T=φx. Therefore for any b∈A and λ∈I,
(23)x·φ(b·eλ)=φ(b·eλ)x=(eλ*φx)(b)=0.
The continuity of φ implies that x·φ(b)=0. Now, since A is φ-faithful in X, we conclude that x=0. Hence T=φx=0.Definition 16.
LetA be a Hausdorff topological algebra and (X,τ) a Hausdorff topological A-bimodule. Let φ be an A-module homomorphism on A. The φ-uniform operator topology uφ (resp., φ-strong operator topology sφ) on Mφl(A,X) is defined as the linear topology which has a base of neighborhoods of 0 consisting of all sets of the form
(24)N(φ(B),V)={T∈Mφl(A,X):T(φ(B))⊆V},
where B is a bounded (resp., finite) subset of A and V is a neighborhood of 0 in X. Clearly sφ≤uφ.Theorem 17.
Letφ be an A-module homomorphism on A and let (X,τ) be a topological A-bimodule with b(A)-hypocontinuous module multiplication.
Then(Mφl(A,X),sφ) and (Mφl(A,X),uφ) are topological left A-modules.Proof.
By Lemma10, Mφl(A,X) is a left A-module. Now, let us prove that the module multiplication (a,T)→a*T from A×Mφl(A,X) into Mφl(A,X) is separately continuous in uφ-topology. Let T∈Mφl(A,X) and {aα:α∈I} be a net in A with aα→a∈A and let D be a bounded subset of A and let V be a neighborhood of 0 in X. By b(A)-hypocontinuity, there exists a balanced neighborhood H of 0 in X such that φ(D)·H⊂V. Since T and φ are continuous, there exist α0∈I such that
(25)aα*Tφb-a*Tφb=T(φ(b)·aα)-T(φ(b)·a)=φ(b)·[T(φ(aα))-T(φ(a))]∈φ(D)·H⊂V
for all b∈D and α≥α0. Hence aα*T→uφa*T.
Next, leta∈A and {Tα:α∈I} be a net in Mφl(A,X) such that Tα→uφT∈Mφl(A,X) and let D be a bounded subset of A and let V be a neighborhood of 0 in X. Since the mappings φ and Ra(x)=xa are continuous, it follows that φ(D)·a is a bounded subset in A. So there exist α0∈I such that
(26)a*Tα-a*Tφb=Tα(φ(b)·a)-T(φ(b)·a)=(Tα-T)(φ(D)·a)⊆V
for all α≥α0 and b∈D. Hence a*Tα→uφa*T. That means (Mφl(A,X),uφ) is a left topological module. A similar computation shows that (Mφl(A,X),sφ) is a left topological module.Lemma 18.
LetA be a topological algebra with an approximate identity {eλ:λ∈I} and let (X,τ) be a commutative A-bimodule and φ an idempotent A-module homomorphism on A. Then ψ(X)sφ¯=Mφl(A,X).Proof.
LetT∈Mφl(A,X), and let B be a finite subset of A and let V be a neighborhood of 0 in X. For each a∈A we have eλφ(a)→φ(a). Then, since T is continuous and B is finite, there exist λ0∈I such that T(eλφ(a))-T(φ(a))∈V, for all a∈B and λ≥λ0. Then, for any a∈B and λ≥λ0(27)φφaT(φ(eλ))λ-Tφa=Tφeλ·φ(φ(a))-T(φ(a))=T(eλφ(a))-T(φ(a))∈V.
Therefore φ{T(φ(eλ))}λ→sφT.Theorem 19.
LetA be a topological algebra with an approximate identity {eλ:λ∈I} and let X be a commutative topological A-bimodule such that (X,τ) is complete. Suppose φ is an idempotent A-module homomorphism on A and A is φ-faithful in X. Then (X,τ) is isomorphic to (Mφl(A,X),sφ).Proof.
Letψ be as in Definition 14. It is obvious that ψ is a continuous module homomorphism. We first show that ψ is onto. In view of Lemma 18, it is enough to prove that ψ(X) is sφ-closed. Let T∈ψ(X)¯sφ. Then there exists a net {xα}α⊆X such that φxα→sφT. It follows that the net {xα·φ(a)}α={φxα(φ(a))}α is τ-Cauchy in X for every a∈A. Now, since A is φ-faithful in X, the net {xα} is τ-Cauchy in X. By completeness of (X,τ), there exist x∈X such that xα→x. Hence φxα→sφφx. By uniqueness of limit in Hausdorff space T=φx. Therefore ψ(X) is sφ-closed.
To show thatψ is one-to-one, let x,y∈X such that φx=φy. Then for any a∈A, x-y·φ(a)=0. Since A is φ-faithful in X, this implies that x=y. Thus ψ is one-to-one.Definition 20.
LetA be a topological algebra and let X be a topological A-bimodule. The uniform topology γφ (strict topology βφ) on Mφl(A,X) is defined as the linear topology which has a base of neighborhoods of 0 consisting of all sets
(28)N′(φ(D),G)={T∈Mφl(A,X):φ(D)*T⊂G},
where D is a bounded (finite) subset of A and G is a neighborhood of 0 in (Mφl(A,X),uφ)[(Mφl(A,X),sφ)].Lemma 21.
LetA be a topological algebra with a bounded approximate identity {eλ:λ∈I} and let X be a topological A-bimodule. Then uφ=γφ and sφ=βφ.Proof.
Let{Tα}α be a net in Mφl(A,X) with Tα→uφT. Let G=N(φ(C),V) be a neighborhood of 0 in uφ-topology. Since φ is continuous, φ(C)φ(D) is a bounded subset of A for each bounded subset D of A. Then there exist α0 such that
(29)(φ(D)*(Tα-T))(φ(C))=(Tα-T)(φ(C)φ(D))∈V
for all α≥α0. That means φ(D)*(Tα-T)∈G. Hence Tα→γφT.
Conversely, let{Tα}α be a net in Mφl(A,X) with Tα→γφT. Let D be a bounded subset of A and let V be a closed neighborhood of 0 in X. Choose C={eλ}λ. Then there exist α0 such that
(30)Tα-TφD=limλ(Tα-T)(φ(C)φ(D))=limλ(φ(D)*(Tα-T))(φ(C))∈V
for all α≥α0. That means Tα-T∈N(φ(D),V). Hence Tα→uφT.At the end we characterize theφ-multipliers on Lp(G), where G is a compact Abelian group. Of course, Lp(G) is a Banach algebra and several authors studied its multipliers. For instance, Larsen [5] showed that a linear transformation T:Lp(G)→Lp(G), where G is a locally compact Abelian group, is a multiplier if and only if there exists a unique φ∈L∞(G^) such that Tf^=φf^ for each f∈Lp(G).However, we now considerLp(G) as a left Banach module over the group algebra L1(G). Namely, the algebra L1(G) acts on Lp(G) through the convolution L1(G)*Lp(G)=Lp(G).Example 22.
LetG be a compact Abelian group and let φ:L1(G)→Lp(G) be an idempotent L1(G)-module homomorphism with dense range. If T:Lp(G)→Lp(G) is a φ-multiplier then there exists a unique function HT∈Lp(G) such that
(31)T(φ(f))=φ(f)*HT,(f∈L1(G)).Proof.
Let{eβ}β be a bounded approximate identity in L1(G). Then {T(φ(eβ))}β⊆Ball(Lq(G))*. By the Alaoglu theorem, there exists a function HT∈Lp(G) such that T(φ(eβ))→weak*HT. Then for each f∈L1(G)(32)φ(f)*T(φ(eβ))⟶weak*φ(f)*HT.
On the other hand, since T is a φ-multiplier,
(33)φ(f)*T(φ(eβ))=T(φ(f*eβ))⟶T(φ(f))
for each f∈L1(G). By uniqueness of limit, T(φ(f))=φ(f)*HT.
To show thatHT is unique, let ψ be a second function in Lp(G) such that T(φ(f))=φ(f)*ψ for each f∈L1(G). Since φ has dense range, T(φ(f))=f*ψ for each f∈L1(G). Therefore
(34)f^(γ)ψ-HT^(γ)=0
for each f∈L1(G) and γ∈G^. By compactness of G, for each γ∈G^ there exist f∈L1(G) such that f^(γ)≠0. Hence the semisimplicity of Lp(G) implies that HT=ψ.



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