# Quasimultipliers onF-Algebras

Authors: Marjan Adib; Abdolhamid Riazi; Liaqat Ali Khan
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

We investigate the extent to which the study of quasimultipliers
can be made beyond Banach algebras. We will focus mainly on the class ofF-algebras, in particular on complete k-normed algebras, 0<k≤1, not necessarily
locally convex. We include a few counterexamples to demonstrate that some of
our results do not carry over to general F-algebras. The bilinearity and joint continuity of quasimultipliers on an F-algebra A are obtained under the assumption of strong factorability. Further, we establish several properties of the strict and
quasistrict topologies on the algebra QM(A) of quasimultipliers of a complete
k-normed algebra A having a minimal ultra-approximate identity.


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## Body

## 1. Introduction

A quasimultiplier is a generalization of the notion of a left (right, double) multiplier and was first introduced by Akemann and Pedersen in [1, Section 4]. The first systematic account of the general theory of quasimultipliers on a Banach algebra with a bounded approximate identity was given in a paper by McKennon [2] in 1977. Further developments have been made, among others, by Vasudevan and Goel [3], Kassem and Rowlands [4], Lin [5, 6], Dearden [7], Argün and Rowlands [8], Grosser [9], Yılmaz and Rowlands [10], and Kaneda [11, 12].In this paper, we consider the notion of quasimultipliers on certain topological algebras and give an account, how far one can get beyond Banach algebras, using combination of standard methods. In particular, we are able to establish some results of the above authors in the framework ofF-algebras or complete k-normed algebras.


## 2. Preliminaries

Definition 2.1.
LetE be a vector space over the field 𝕂(=ℝ or ℂ). (1)
A functionq:E→ℝ is called an F-seminorm on E if it satisfies
(F1)
q(x)≥0 for all x∈E,(F2)
q(x)=0 if x=0,(F3)
q(αx)≤q(x) for all x∈E and α∈𝕂 with |α|≤1,(F4)
q(x+y)≤q(x)+q(y) for all x,y∈E,(F5)
ifαn→0 in 𝕂, then q(αnx)→0 for all x∈E.(2)
AnF-seminorm q on E is called an F-norm if, for any x∈E, q(x)=0 implies that x=0.(3)
AnF-seminorm (or F-norm) q on E is called k-homogeneous([13, page 160]; [14, pages 90, 95]), where 0<k≤1, if it also satisfies
(F3′)
q(αx)=|α|kq(x)  for all  x∈E and α∈𝕂.(4)
Ak-homogeneous F-seminorm (resp., F-norm) on E is called, in short, a k-seminorm(resp., k-norm).Definition 2.2.
(1) A vector space with anF-norm q is called an F-normed space and is denoted by (E,q); if it is also complete, it is called an F-space. Clearly, any F-normed space (E,q) is a metrizable TVS with metric given by d(x,y)=q(x-y), x,y∈E.
(2) AnF-seminorm (or F-norm) q on an algebra A is called submultiplicative if
(2.1)q(xy)≤q(x)q(y),∀x,y∈A.
An algebra with a submultiplicative F-norm q is called an F-normed algebra; if it is also complete, it is called an F-algebra. An algebra with a submultiplicative k-norm q is called an k-normed algebra.A complete k-normed algebra is also called a k-Banach  algebra in the literature.Theorem 2.3.
(a) If(E,τ) is TVS, then its topology  τ can be defined by a family of  F-seminorms  (see [15, pages 48–51]; [16, pages 2-3]).
(b) If(E,τ) is a metrizable TVS, then  τ may be defined by a single  F-norm  (see [13, 15, 17]).
(c) If(E,τ) is a Hausdorff locally bounded TVS, then  τ may be a single  k-norm for some  k,  0<k≤1 (see [13, 14]).Note that if(E,q) is an F-normed space, then, for any ɛ>0, the set {x∈E:q(x)≤ɛ} is a neighbourhood of 0 in E, but it need not be a bounded set. In case, if {x∈E:q(x)≤ɛ} is bounded for some ɛ>0, then (E,q) becomes a Hausdorff locally bounded TVS and hence, by Theorem 2.3(c), a k-normed space for some k, 0<k≤1.Definition 2.4.
LetA be an algebra over 𝕂(ℝ or ℂ) and τ a topology on A such that (A,τ) is a TVS. Then the pair (A,τ) is called a topological algebra if it has a separately continuous multiplication. A topological algebra A is said to be locally boundedif it has a bounded neighbourhood of 0 (see [18, page 39]). If (A,τ) is a complete Hausdorff locally bounded topological algebra, then its topology can be defined by a submultiplicative k-norm q, 0<k≤1 [18, page 41].
By a famous result of Arens (see [18, page 24]), every Baire metrizable topological algebra has jointly continuous multiplication; in particular, every F-algebra has jointly continuous multiplication.
For the general theory and undefined terms, the reader is referred to [13, 15–17, 19] for topological vector spaces, to [13, 14, 20] for F-normed and k-normed spaces, and to [18, 21, 22] for various classes of topological algebras.
IfE and F are topological vector spaces over the field 𝕂(=ℝ or ℂ), then the set of all continuous linear mappings T:E→F is denoted by CL(E,F). Clearly, CL(E,F) is a vector space over 𝕂 with the usual pointwise operations. Further, if F=E,  CL(E)=CL(E,E) is an algebra under composition (i.e., (ST)(x)=S(T(x)), x∈E) and has the identity I:E→E given by I(x)=x(x∈E).
We now state the following three versions of the uniform boundedness principle for reference purpose.Theorem 2.5 (see [23, page 142, principle 33.1]).
LetX be a complete metric space and  ℋ={fα:α∈J} a family of continuous real-valued functions on  X. If  ℋ is pointwise bounded from above, then on a certain closed ball  B⊆X it is uniformly bounded above, that is, there exists a constant  C>0 such that
(2.2)fα(x)≤C,∀α∈J,x∈B.Theorem 2.6 (see [14, page 39]; [19, page 465]).
LetEbe an  F-space and  F any topological vector space. Let  ℋ⊆CL(E,F) be a collection such that  ℋ is pointwise bounded on  E. Then  ℋ is equicontinuous; hence, for any bounded set  D in  E,  ∪{T(D):T∈ℋ} is a bounded set in  F.The following version is for bilinear mappings.Theorem 2.7.
LetE and F be  F-spaces and  G  any TVS.  (a)
A collectionℋ  of bilinear mappings from  E×F into G is equicontinuous if and only if each  f∈ℋ is separately continuous and  ℋ is pointwise bounded on  E×F.  In particular, every separately continuous bilinear map f:E×F→G is jointly continuous (see [13, page 172]; [19, page 489]).(b)
Letfn:E×F→G be a sequence of separately continuous bilinear mappings such that limn→∞fn(x,y)=f(x,y) exists for each (x,y)∈E×F. Then {fn} is equicontinuous and f is bilinear and jointly continuous (see [19,page 490]; [24, page 328]).Definition 2.8.
(1) A net{eλ:λ∈I} in a topological algebra A is called an approximate identity if
(2.3)lim⁡λeλa=a=lim⁡λaeλ,∀a∈A.
(2)
An approximate identity{eλ:λ∈I} in an F-normed algebra (A,q) is said to be minimal if q(eλ)≤1 for all λ∈I.(3)
An algebraA is said to be left(resp., right)faithful if, for any a∈A, aA={0} (resp., Aa={0}) implies that a=0; A is called faithful if it is both left and right faithful. One mentions that A is faithful in each of the following cases:
(i)
A is a topological algebra with an approximate identity (e.g., A is a locally C*-algebra);(ii)
A is a topological algebra with an orthogonal basis [25].Definition 2.9.
A topological algebraA is called(1)
factorable if, for each a∈A, there exist b,c∈A such that a=bc,(2)
strongly factorable if, for any sequence {an} in A with an→0, there exist a∈A and a sequence {bn} (resp., {cn}) in A with bn→0 (resp., cn→0) such that an=abn (resp., an=cna) for all n≥1.
Clearly, every strongly factorable algebra is factorable. Factorization in Banach and topological algebras plays an important role in the study of multipliers and quasimultipliers. There are several versions of the famous Hewitt-Cohen's factorization theorem in the literature (see, e.g., the book [26] and its references). Using the terminology of [27], we state the following version in the nonlocally convex case.Theorem 2.10 (see [27]).
LetA be a fundamental  F-algebra with a uniformly bounded left approximate identity. Then  A is strongly factorable.Definition 2.11.
Let(A,q) be an F-normed space (in particular, an F-normed algebra). For any T∈CL(A), let
(*)‖T‖q=sup⁡⁡{q(T(x))q(x):x∈A,x≠0}.
It is easy to see that if q is a k-norm, 0<k≤1, (resp., a seminorm) on A, then ∥·∥q is a k-norm (resp., a seminorm) on CL(A); further, in these cases, we have alternate formulas for ∥T∥q as
(2.4)‖T‖q=sup⁡⁡{q(T(x)):x∈A,q(x)=1}=sup⁡⁡{q(T(x)):x∈A,q(x)≤1},
for each T∈CL(A) (see [14, pages 101-102]; [28, pages 3–5]; [19, page 87]).Remark 2.12.
In an earlier version of this paper, the authors had erroneously made the blank assumption that, forq an F-norm on A, ∥T∥q given by * always exists for each T∈CL(A). We are grateful to referee for pointing out that this assumption cannot be justified in view of the following counterexamples.
(1)
First,∥T∥q need not be finite for a general F-norm. For example, let A=ℝ2, q(x1,x2)=|x1|+|x2|1/2, T(x1,x2)=(x2,x1). Then q is an F-norm on A, but ∥T∥q=∞: for any n∈ℕ,
(2.5)‖T‖q≥q[T(n,n2)]q(n,n2)=q(n2,n)q(n,n2)=n2+n1/22n⟶∞.(2)
Even when considering the subspace of thoseT for which ∥T∥q<∞, then ∥·∥q need not always be an F-norm, since (F5) need not hold. For example, for a fixed sequence (pn) with 0<pn≤1, pn→0, consider the F-algebra A of sequences (xn)⊆ℝ with |xn|pn→0 and q((xn))=supn≥1|xn|pn. Then ∥T∥q<∞ for all multipliers of A, but ∥·∥q is not an F-norm, that is, it makes the space CL(A) into an additive topological group but not into a topological vector space (as it would lack the continuity of scalar multiplication in the absence of (F5) (cf. [14, Example 1.2.3, page 8]).
In view of the above remark, we will need to assume that(A,q) is a k-normed space (or a k-normed algebra) whenever ∥T∥q is considered for T∈CL(A) or CL(A×A,A).
Some useful properties of(CL(A),∥·∥q) are summarized as follows.Theorem 2.13 (see [14, pages 101-102]).
Let(A,q) be a  k-normed space (in particular, a  k-normed algebra) with  0<k≤1. Then: (a)
a linear mappingT:A→A is continuous  ⇔∥T∥q<∞;(b)
∥·∥q is a  k-norm on  CL(A);(c)
q(T(x))≤∥T∥q·q(x) for all  x∈A;(d)
for anyS,T∈CL(A), ∥ST∥q≤∥S∥q∥T∥q; hence  (CL(A),∥·∥q) is a  k-normed algebra;(e)
ifA is complete, then  (CL(A),∥·∥q) is a complete  k-normed algebra.Remark 2.14.
The referee has enquired if the present theory can be considered for the class of locally convexF-algebras. It is well known (e.g., [22, page 33]; [21, page 9]) that, for this class of topological algebras, the topology can be generated by an increasing countable family of seminorms {qn}, which need not be submultiplicative but satisfy the weaker condition qn(xy)≤qn+1(x)qn+1(y); however, for locally m-convex F-algebras, the seminorms can be chosen to be submultiplicative. In view of this, we believe that a study of quasimultipliers can possibly be made on locally m-convex algebra F-algebras parallel to the one given by Phillips (see [29, pages 177–180]) for multipliers.


## 3. Multipliers onF-Normed Algebras

In this section, we recall definitions and results on various notions of multipliers on an algebraA (as given in [30–33]) which we shall require later in the study of quasimultipliers (see also [25, 29, 34–40]). In fact, we shall see that the proofs of most of the results on quasimultipliers are based on the properties of left, right, and double multipliers.Definition 3.1 (see [31]).
LetA be an algebra over the field 𝕂(ℝ or ℂ).
(1)
A mappingT:A→A is called a
(i)
multiplieron A if aT(b)=T(a)b for all a,b∈A,(ii)
left  multpilier on A if T(ab)=T(a)b for all a,b∈A,(iii)
right multiplier on A if T(ab)=aT(b) for all a,b∈A.(2)
A pair(S,T) of mappings S,T:A→A is called a double multiplieron A if aS(b)=T(a)b for all a,b∈A.
Some authors use the termcentralizer instead of multiplier (see, e.g., [30, 31]).
LetM(A) (resp., Mℓ(A),Mr(A)) denote the set of all multipliers (resp., left multipliers, right multipliers) on A and Md(A) the set of all double multipliers on an algebra A. For any a∈A, let La,Ra:A→A be given by(3.1)La(x)=ax,Ra(x)=xa,x∈A.
Clearly, La∈Mℓ(A), Ra∈Mr(A), and (La,Ra)∈Md(A).
For convenience, we summarize some basic properties of multipliers in the following theorems for later references.Theorem 3.2 (see [31]).
LetA be an algebra. Then, (a)
Mℓ(A)∩Mr(A)⊆M(A);(b)
ifA is faithful, then  M(A)⊆Mℓ(A)∩Mr(A) and hence  M(A)=Mℓ(A)∩Mr(A);(c)
ifA is commutative and faithful, then  Mℓ(A)=Mr(A)=M(A);(d)
Mℓ(A) and Mr(A) are algebras with composition as multiplication (i.e.,  (T1T2)(x)=T1(T2(x))) and have the identity  I:A→A,   I(x)=x(x∈A);(e)
M(A) is a vector space; if, in addition,  A is faithful, then  M(A) is a commutative algebra (without  A being commutative) with identity I.Theorem 3.3 (see [31]).
LetA be a faithful algebra. Then, (a)
if(S,T)∈Md(A), then  (i)  S and  T are linear and  (ii)  S∈Mℓ(A) and  T∈Mr(A). In particular, every  T∈M(A) is linear;(b)
Md(A) is an algebra with identity  (I,I) under the operations
(3.2)(S1,T1)+(S2,T2)=(S1+S2,T1+T2),λ(S1,T1)=(λS1,λT1)(λ∈K),(S1,T1)(S2,T2)=(S1S2,T2T1);(c)
let(S1,T1),(S2,T2)∈Md(A). If  S1=S2, then  T1=T2; if   T1=T2  then  S1=S2;(d)
ifA is commutative, then  Md(A) is commutative and  Md(A)=M(A); in fact, if (S,T)∈Md(A), then  S=T.Definition 3.4.
One defines mappingsμℓ:A→Mℓ(A), μr:A→Mr(A), and μd:A→Md(A) by
(3.3)μl(a)=La,μr(a)=Ra,μd(a)=(La,Ra),a∈A.Theorem 3.5 (see [31]).
LetA be an algebra, and let  μℓ,  μr, and  μd be the mappings as defined above. Then, (a)
μℓ,μr,  and  μd are linear;(b)
μℓ and  μd are algebra homomorphisms, while  μr is an algebra antihomomorphism;(c)
μℓ (resp.,  μr) is 1-1  ⇔A is left (resp., right)   faithful;  μd is 1-1  ⇔A is faithful;(d)
μℓ (resp.,  μr) is onto  ⇔A has left (resp., right) identity;  μd is onto  ⇔A has an identity.Theorem 3.6 (see [31]).
LetA be an algebra. (a)
For anya∈A and  T∈Mℓ(A),  TLa=LT(a)∈μℓ(A); hence  μℓ(A) is a left ideal in  Mℓ(A).(b)
For anya∈A and  T∈Mr(A),  TRa=RT(a)∈μr(A);  hence  μr(A) is a left ideal in  Mr(A).(c)
Suppose thatA is faithful. Then, for any  a∈A and  (S,T)∈Md(A),
(3.4)(La,Ra)(S,T)=(LT(a),RT(a))∈μd(A),(S,T)(La,Ra)=(LS(a),RS(a))∈μd(A);
hence  μd(A) is a two-sided ideal in  Md(A).Regarding the continuity of multipliers, we state the following.Theorem 3.7.
(a) Suppose thatA is a strongly factorable  F-normed algebra. If  T∈Mℓ(A) (resp., Mr(A)), then  T is linear and continuous  (see [32, 33]).
(b) Suppose thatA is a faithful   F-algebra. If  (S,T)∈Md(A), then  S and  T are linear and continuous; in particular each  T∈M(A)  is linear and continuous  (see [31, 33]).Convention.
In the remaining part of this paper, unless stated otherwise,A is a topological algebra and M(A) (resp., Mℓ(A),Mr(A)) denotes the set of all continuous linear multipliers (resp., left multipliers, right multipliers) on A and Md(A) denotes the set of all double multipliers (S,T) on A with both S and T continuous and linear.Definition 3.8 (see [31, 33]).
LetA be a topological algebra. The uniform operator topologyu (resp., the strong operator topologys) on Md(A) is defined as the linear topology which has a base of neighborhoods of 0 consisting of all the sets of the form
(3.5)N(D,W)={(S,T)∈Md(A):S(D)⊆W,T(D)⊆W},
where D is a bounded (resp., finite) subset of A and W is a neighborhood of 0 in A. Clearly, s≤u. Note that the u and s topologies can also be defined on the multiplier algebras M(A),Mℓ(A) and Mr(A) in an analogous way. (The topology s is also called the strict topology in the literature and denoted by β.) There is an extensive literature on the s and u topologies (see, e.g., [30, 33–35, 37–39, 41–45]).Theorem 3.9 (see [33]).
LetA be a faithful  F-algebra, and let  Mt(A) denote any one of the algebras  Mℓ(A), Mr(A), M(A), and Md(A). Then,  (a)
(Mt(A),u) and  (Mt(A),s) are topological algebras with separately continuous multiplication;(b)
(Mt(A),u) and  (Mt(A),s) are complete;(c)
s and  u have the same bounded sets;(d)
if(Mt(A),s) is metrizable, then  s=u on  Mt(A);(e)
ifA has a two-sided approximate identity, then  A is  s-dense in  Mt(A).Remark 3.10.
Let(A,q) be an F-normed algebra.(1)
If(A,q) is a k-normed algebra, the u-topology on Mℓ(A), Mr(A), and M(A) is given by the k-norm
(3.6)‖T‖q=sup⁡⁡{q(T(x))q(x):x∈A,x≠0};
the u-topology on Md(A) is given by the k-norm
(3.7)‖(S,T)‖q=max⁡⁡{‖S‖q,‖T‖q},(S,T)∈Md(A).(2)
Thes-topology on Mℓ(A), Mr(A), and M(A) is given by the family of {pa:a∈A} of F-seminorms, where
(3.8)pa(T)=q(T(a)),T∈Ml(A),Mr(A)orM(A);
the s-topology on Md(A) is given by the family {ra:a∈A} of F-seminorms, where
(3.9)ra(S,T)=max⁡⁡{q(S(a)),q(T(a))},(S,T)∈Md(A).Theorem 3.11.
Let(A,q) be a  k-normed algebra having a minimal approximate identity  {eλ:λ∈I}. Then,   (a)
for anya∈A,  ∥La∥q=∥Ra∥q=q(a); so each of the maps  μℓ:A→(Mℓ(A),u),  μr:A→(Mr(A),u),  μd:A→(Md(A),u) is an isometry and hence continuous;(b)
for any(S,T)∈Md(A),  ∥S∥q=∥T∥q;(c)
if(A,q) is complete, then  A is a u-closed two-sided ideal in  Md(A), under the identification  μd:a→(La,Ra).Proof.
(a) Leta∈A. Then(3.10)‖La‖q=sup⁡b≠0q(La(b))q(b)=sup⁡b≠0q(ab)q(b)≤sup⁡b≠0q(a)⋅q(b)q(b)=q(a).
On the other hand,
(3.11)‖La‖q=sup⁡b≠0q(ab)q(b)≥q(aeλ)q(eλ)≥q(aeλ),∀λ∈I;
so
(3.12)‖La‖q≥lim⁡λq(aeλ)=q(lim⁡λaeλ)=q(a).
Hence ∥μℓ(a)∥q=∥La∥q=q(a). Similarly, ∥μr(a)∥q=∥Ra∥q=q(a). Thus
(3.13)‖μd(a)‖q=max⁡⁡{‖La‖q,‖Ra‖q}=q(a).
(b) Let(S,T)∈Md(A). Using (a), we have
(3.14)‖S‖q=sup⁡a≠0q(S(a))q(a)=sup⁡a≠0‖RS(a)‖qq(a)=sup⁡a≠0sup⁡b≠01q(a)⋅q[RS(a)(b)]q(b)=sup⁡a≠0sup⁡b≠0q[bS(a)]q(a)⋅q(b)=sup⁡a≠0sup⁡b≠0q[T(b)a]q(a)⋅q(b)≤sup⁡a≠0sup⁡b≠0q(T(b))⋅q(a)q(a)⋅q(b)=sup⁡b≠0q(T(b))q(b)=‖T‖q.
Similarly,
(3.15)‖T‖q=sup⁡a≠0‖LT(a)‖qq(a)=sup⁡a≠0sup⁡b≠0q[T(a)b]q(a)⋅q(b)=sup⁡a≠0sup⁡b≠0q[a⋅S(b)]q(a)⋅q(b)≤‖S‖q.
Thus, ∥S∥q=∥T∥q.
(c) In view of Theorem3.6(c), we only need to show that μd(A) is u-closed in Md(A). Let (S,T)∈Md(A) with (S,T)∈u-cl(μd(A)). Choose {aα:α∈J}⊆A such that {Laα,Raα}→u(S,T). By part (a), μd is an isometry. Hence {aα:α∈J} is a Cauchy net in A. Then
(3.16)(S,T)=u-lim⁡αμd(aα)∈μd(A).
Thus μd(A) is u-closed in Md(A).


## 4. Quasimultipliers onF-Algebras and k-Normed Algebras

In this section, we consider the notion of quasimultipliers onF-algebras and complete k-normed algebra and extend several basic results of McKennon [2], Kassem and Rowlands [4], Argün and Rowlands [8], and Yılmaz and Rowlands [10] from Banach algebras to these classes of topological algebras.Definition 4.1 (see [2, 4]).
LetA be an algebra. A mapping m:A×A→A is said to be a quasimultiplier on A if(4.1)m(ab,c)=am(b,c),m(a,bc)=m(a,b)c,
for all a,b,c∈A.
The following Lemma shows in particular that every left multiplier, right multiplier, multiplier, and double multiplier on an algebraA can be viewed as quasimultiplier on A.Lemma 4.2.
LetA be a faithful algebra. (a)
For anyc∈A, define  m=mc:A×A→A by
(4.2)mc(a,b)=acb,∀(a,b)∈A×A.(b)
For anyT∈Mℓ(A), define an associated map  m=mT:A×A→A by
(4.3)mT(a,b)=aT(b),∀(a,b)∈A×A.(c)
For anyT∈Mr(A), define an associated map  m=mT:A×A→A by
(4.4)mT(a,b)=T(a)b,∀(a,b)∈A×A.(d)
For anyT∈M(A), define an associated map  m=mT:A×A→A by
(4.5)mT(a,b)=aT(b),∀(a,b)∈A×A.(e)
For any(S,T)∈Md(A), define an associated map  m=m(S,T):A×A→A by
(4.6)m(S,T)(a,b)=aS(b),∀(a,b)∈A×A.Then each of the maps  m:A×A→A defined above is a quasimultiplier on  A.Proof.
We only prove (e). Let(S,T)∈Md(A), for any a,b,c,d∈A,
(4.7)m(ab,c)=m(S,T)(ab,c)=(ab)S(c)=a[bS(c)]=a[m(S,T)(b,c)]=am(b,c).
In a similar way, m(a,bc)=m(a,b)c.Theorem 4.3.
Suppose that(A,τ) is a strongly factorable  F-algebra. Then, (a)
A mapm:A×A→A is a quasimultiplier  on A  if and only if
(4.8)m(ab,cd)=am(b,c)d,∀a,b,c,d∈A;(b)
every quasimultiplierm on  A is bilinear;(c)
every quasimultiplierm  on  A is jointly continuous.Proof.
(a) Ifm is a quasimultiplier on A, then clearly, for any a,b,c,d∈A,
(4.9)m(ab,cd)=am(a,cd)=am(b,c)d.
Conversely, leta,b,c∈A. Since A  is a strongly factorable and an={b,c,0,0,…}→0, there exist y,z,w∈A such that b=wy, c=wz. Then, using (4.8),
(4.10)m(ab,c)=m(awy,wz)=(aw)[m(y,w)]z=am(wy,wz)=am(b,c).
Similarly, we obtain m(a,bc)=m(a,b)c.
(b) Leta,b,c∈A and α∈𝕂. Choose, as above, x,y,w∈A such that a=wx, b=wy. Then,
(4.11)m(a+b,c)=m(wx+wy,c)=(x+y)m(w,c)=xm(w,c)+ym(w,c)=m(wx,c)+m(wy,c)=m(a,c)+m(b,c).
Similarly, m(c,a+b)=m(c,a)+m(c,b). Next,
(4.12)m(αa,c)=m(αwx,c)=(αw)m(x,bc)=αm(wx,c)=αm(a,c).
First, we show that m is separately continuous. Let a∈A and {xn}⊆A with limit x. Then {xn-x} converges to 0. By strong factorability, there exist a sequence {zn} and an element z of A such that zn→0, xn-x=zzn. Thus,
(4.13)m(a,xn)-m(a,x)=m(a,xn-x)=m(a,zzn)=m(a,z)zn⟶0.
Now, the joint continuity ofm follows directly from Theorem 2.7(a).Theorem 4.4.
LetA be a commutative algebra. Then, (a)
am(b,c)=m(b,a)c for any quasimultiplier on A and a,b,c∈A;(b)
ifA is also faithful, then a bilinear map  m:A×A→A is a quasimultiplier on  A if and only if(4.14)m(a2,b)=am(a,b),m(a,b2)=m(a,b)b∀a,b∈A.Proof.
(a) By hypothesis,(4.15)am(b,c)=m(b,c)a=m(b,ca)=m(b,ac)=m(b,a)c.
(b)
(⇒) This is obvious.
(⇐) For all a,b,c∈A, using (4.14), (4.16)m((a+b)2,c)=(a+b)m(a+b,c)=am(a,c)+am(b,c)+bm(a,c)+bm(b,c).
On the other hand, we have
(4.17)m((a+b)2,c)=m(a2+b2+2ab,c)=m(a2,c)+m(b2,c)+2m(ab,c)=am(a,c)+bm(b,c)+2m(ab,c).
Comparing (4.16) and (4.17), we obtain
(4.18)2m(ab,c)=am(b,c)+bm(a,c).
Now, for alla,b,c,d∈A, using (4.18) twice,(4.19)2m(abd,c)=abm(d,c)+dm(ab,c)=abm(d,c)+12d[2m(ab,c)]=abm(d,c)+12dam(b,c)+12dbm(a,c).
By commutativity of A and using (4.18) twice as above,
(4.20)2m(abd,c)=2m(adb,c)=adm(b,c)+bm(ad,c)=adm(b,c)+12[bam(d,c)+bdm(a,c)]=adm(b,c)+12abm(d,c)+12dbm(a,c).
Comparing (4.19) and (4.20), abm(d,c)=adm(b,c). Since this holds for all a∈A and A is faithful, bm(d,c)=dm(b,c). Hence, for all d∈A,
(4.21)dm(ab,c)=abm(d,c)=a[bm(d,c)]=adm(b,c)=dam(b,c).
Since this holds for all d∈A and A is faithful, m(ab,c)=am(b,c).
A similar computation shows thatm(a,bc)=m(a,b)c. Hence m is a quasimultiplier.Definition 4.5.
LetQM(A) denote the set of all bilinear jointly continuous quasimultipliers on a topological algebra A. Clearly, QM(A) is a vector space under the usual pointwise operations. Further, QM(A) becomes an A-bimodule as follows. For any m∈QM(A) and a∈A, we can define the products a∘m and m∘a as mappings from A×A into A given by
(4.22)(a∘m)(x,y)=m(xa,y),(m∘a)(x,y)=m(x,ay),x,y∈A.
Then a∘m,m∘a∈QM(A), so that QM(A) is an A-bimodule.Definition 4.6.
Let(A,q) be an F-normed algebra. Following [2, 4, 8], we can define mappings
(4.23)ϕA:A⟶QM(A),ϕl:Ml(A)⟶QM(A),ϕr:Mr(A)⟶QM(A),ϕd:Md(A)⟶QM(A),
by
(4.24)(ϕA(a))(x,y)=xay,a∈A,(ϕl(T))(x,y)=xT(y),T∈Ml(A),(ϕr(T))(x,y)=T(x)y,T∈Mr(A),(ϕd(S,T))(x,y)=xS(y),(S,T)∈Md(A),
for all (x,y)∈A×A. By Lemma 4.2, these mappings are well defined.Definition 4.7.
(1) A bounded approximate identity{eλ:λ∈I} in a topological algebra A is said to be ultra-approximateif, for all m∈QM(A) and a∈A, the nets {m(a,eλ):λ∈I} and {m(eλ,a):λ∈I} are Cauchy in A  (see [2]).
(2) A topological algebraA is called m-symmetric if, for each S∈Mℓ(A)∪Mr(A), there is a T∈Mℓ(A)∪Mr(A) such that either (S,T)∈Md(A) or (T,S)∈Md(A)(see[4]).Theorem 4.8.
Let(A,q) be an  F-algebra with a bounded approximate identity  {eλ:λ∈I}. Consider the following conditions. (a)
{eλ:λ∈I} is ultra-approximate.(b)
For anya∈A,  S∈Mℓ(A), and  T∈Mr(A), the nets  {aS(eλ)} and   {T(eλ)a} are Cauchy in  A.(c)
A is  m-symmetric.Then  (a) ⇒ (b) ⇔ (c). If  A is factorable, then  (c) ⇒ (a); hence  (a), (b),  and  (c) are equivalent.Proof.
(a)⇒ (b) Suppose that {eλ:λ∈I} in A is ultra-approximate. Let S∈Mℓ(A) and T∈Mr(A). Put m1=ϕℓ(S) and m2=ϕr(T). Then m1,m2∈QM(A) and, for any a∈A,  {m1(a,eλ)}={aS(eλ)} and {m1(eλ,a)}={T(eλ)a} which are Cauchy in A by hypothesis.
(b)⇒ (c) Suppose that (b) holds, let S∈Mℓ(A)∪Mr(A), and suppose that S∈Mℓ(A). Since A is complete, the map T:A→A given by
(4.25)T(a)=lim⁡λaS(eλ),a∈A,
is well-defined. Since S is continuous, for any a,b∈A,
(4.26)aS(b)=alim⁡λS(eλb)=alim⁡λS(eλ)b=[lim⁡λaS(eλ)]b=T(a)b.
Since A is a faithful F-algebra, by Theorem 3.7(b), (S,T)∈Md(A). Hence A is m-symmetric.
(c)⇒(b) Suppose that (c) holds. Let S∈Mℓ(A) and T∈Mr(A). By (c), there exist T1∈Mr(A),  S1∈Mℓ(A) such that (S,T1),(S1,T)∈Md(A). Then, for any a∈A,
(4.27)aS(eλ)=T1(a)eλ⟶T1(a),T(eλ)a=eλS1(a)⟶S1(a).
Thus, both {aS(eλ)} and  {T(eλ)a}, being convergent, are Cauchy in A.
Suppose thatA is factorable. Then (c) ⇒ (a), as follows. Let m∈QM(A) and a∈A. By factorability, a=bc for some b,c∈A. Define the mappings S,T:A→A by
(4.28)S(x)=m(c,x),T(x)=m(x,b),x∈A.
Then, for any x,y∈A,
(4.29)S(xy)=m(c,xy)=m(c,x)y=S(x)y,T(xy)=m(xy,b)=xm(y,b)=xT(y),
and so S∈Mℓ(A) and T∈Mr(A). By (c), there exist T1∈Mr(A), S1∈Mℓ(A) such that (S,T1),(S1,T)∈Md(A). Then
(4.30)m(a,eλ)=m(bc,eλ)=bm(c,eλ)=bS(eλ)=T1(b)eλ⟶T1(b),m(eλ,a)=m(eλ,bc)=m(eλ,b)c=T(eλ)c=eλS1(c)⟶S1(c).
Hence {m(a,eλ):λ∈I} and {m(eλ,a):λ∈I} are Cauchy in A. So {eλ:λ∈I} is ultra-approximate.Theorem 4.9.
Let(A,q) be an  F-algebra having an ultra-approximate identity  {eλ:λ∈I}. Then, (a)
each of the mapsϕA,  ϕℓ,  ϕr, and  ϕd is a bijection;(b)
ϕd∣M(A)=ϕℓ∣M(A)=ϕr∣M(A);(c)
ϕd∣μd(A)=ϕA.Proof.
(a) We give the proof only forϕd:Md(A)→QM(A). To show that ϕd is onto, let m∈QM(A). Since {eλ} is ultra-approximate, for each x∈A, the nets {m(x,eλ):λ∈I} and {m(eλ,x):λ∈I} are convergent. For each x∈A, define S,T:A→A by
(4.31)S(x)=lim⁡λm(eλ,x),T(x)=lim⁡λm(x,eλ).
Then (S,T)∈Md(A) since, for any x,y∈A,
(4.32)xS(y)=xlim⁡λm(eλ,y)=lim⁡λm(xeλ,y)=m(x,y)=lim⁡λm(x,eλy)=lim⁡λm(x,eλ)y=T(x)y.
Further, we have for any (a,b)∈A×A,
(4.33)[ϕd(S,T)](a,b)=aS(b)=alim⁡λm(eλ,b)=lim⁡λm(aeλ,b)=m(a,b);
that is, ϕd(S,T)=m.
To show that,ϕd is one to one, let (S1,T1),(S2,T2)∈Md(A) with ϕd(S1,T1)=ϕd(S2,T2). Then, for any a,b∈A,
(4.34)ϕd(S1,T1)(a,b)=ϕd(S2,T2)(a,b),oraS1(b)=aS2(b).
Since A is faithful, S1(b)=S2(b),b∈A. So S1=S2. Consequently, by Theorem 3.3(c), also T1=T2. Thus (S1,T1)=(S2,T2).
(b) LetT∈M(A) and x,y∈A. Then, since T∈Mℓ(A)∩Mr(A) and (T,T)∈Md(A),
(4.35)ϕd(T)(x,y)=ϕd(T,T)(x,y)=xT(y)=T(xy).
Also
(4.36)ϕl(T)(x,y)=xT(y)=T(xy);ϕr(T)(x,y)=T(x)y=T(xy).
(c) For anya∈A and (x,y)∈A×A,
(4.37)ϕd(μd(a))(x,y)=ϕd(La,Ra)(x,y)=xLa(y)=xay=ϕA(a)(x,y).
Thus ϕd∣μd(A)=ϕA.We obtain the following lemma for later use.Lemma 4.10.
(a) If(A,q) is a factorable  F-normed algebra having an approximate identity  {eλ:λ∈I}, then
(4.38)lim⁡λq(eλaeλ-a)=0,∀a∈A.
(b) If(A,q) is an  F-algebra having a minimal ultra-approximate identity  {eλ:λ∈I}, then, for any  m∈QM(A),
(4.39)lim⁡λeλm(eλ,x)=lim⁡λm(eλ,x)(exists),∀x∈A.
Compare with [10, page 124].Proof.
(a) Leta∈A. Since A is factorable, there exist x,y∈A such that a=xy. Then
(4.40)lim⁡λ--q(eλaeλ-a)=lim⁡λ--q[(eλxyeλ-eλxy)+(eλxy-xy)]≤lim⁡λ--q(eλx)⋅q(yeλ-y)+lim⁡λ--q(eλxy-xy)=q(x)⋅q(0)+q(0)=0.
Since A is complete and {eλ} is ultra-approximate, for any x∈A, limλm(eλ,x)=y (say) exists. Since q(eλ)≤1,
(4.41)q[y-eλm(eλ,x)]≤q(y-eλy)+q[eλy-eλm(eλ,x)]≤q(y-eλy)+q(eλ)q(y-m(eλ,x))≤q(y-eλy)+q(y-m(eλ,x))⟶0.Definition 4.11.
Let(A,q) be a k-normed algebra. For any m∈QM(A), we define
(4.42)‖m‖q=sup⁡⁡{q[m(x,y)]q(x)q(y):x,y∈A,x,y≠0}.
Clearly,
(4.43)q[m(x,y)]≤‖m‖qq(x)q(y),∀x,y∈A.Theorem 4.12.
Let(A,q) be a  k-normed algebra. Then,  (a)
∥·∥q is a  k-norm on  QM(A);(b)
if(A,q) is complete, then so is  (QM(A),∥·∥q).Proof.
(a) For anym∈QM(A),
(4.44)‖m‖q=0⟺sup⁡x≠0,y≠0q[m(x,y)]q(x)q(y)=0⟺q[m(x,y)]=0,∀x,y∈A,x≠0,y≠0⟺q[m(x,y)]=0,∀x,y∈A⟺m=0.
Also, for any α∈𝕂,q(αx)=|α|kq(x) and so it follows that ∥αm∥q=|α|k∥m∥q.
Next, letm,u∈QM(A), and let ɛ>0. Choose x,y∈A such that
(4.45)‖m+u‖q≤q[(m+u)(x,y)]q(x)q(y)-ɛ.
Then,
(4.46)‖m+u‖q≤q[m(x,y)+u(x,y)]q(x)q(y)-ɛ≤q[m(x,y)]q(x)q(y)+q[u(x,y)]q(x)q(y)-ɛ≤‖m‖q+‖u‖q-ɛ.
Since ɛ>0 is arbitrary, ∥m+u∥q≤∥m∥q+∥u∥q. Thus ∥·∥q is a k-norm on QM(A).
(b) Let{mi:i∈ℕ} be a ∥·∥q-Cauchy sequence in QM(A). Then, for any x,y∈A,
(4.47)lim⁡i,j---q[mi(x,y)-mj(x,y)]=lim⁡i,j---q[(mi-mj)(x,y)]≤lim⁡i,j---‖mi-mj‖q⋅q(x)q(y)=0.
Therefore, for any x,y in A,{mi(x,y)} is a Cauchy sequence in A. Since A is complete, the map m:A×A→A given by
(4.48)m(x,y)=lim⁡imi(x,y),x,y∈A,
is welldefined. Clearly, m is bilinear and, by Theorem 2.7(b), m is jointly continuous. Further, for any a,b,x,y∈A,
(4.49)m(ax,yb)=lim⁡imi(ax,yb)=lim⁡i[ami(x,y)b]=a[lim⁡imi(x,y)]b=am(x,y)b.
Hence, m∈QM(A). Next, ∥mi-m∥q→0 as follows. Let ɛ>0. Since {mi} is ∥·∥q-Cauchy, there exists an integer N≥1 such that
(4.50)‖mi-mj‖q<ɛ2,∀pairsi,j≥N,
that is,
(4.51)q[(mi-mj)(x,y)]q(x)q(y)<ɛ2,∀pairsi,j≥N,x,y∈A,x,y≠0.
Let x,y∈A, x,y≠0. Fix any io≥N in (4.51); since mj(x,y)→m(x,y) in A, letting j→∞ in (4.51),
(4.52)q[mio(x,y)-m(x,y)]q(x)q(y)≤ɛ2.
Then, for any i≥N, using (4.51) and (4.52),
(4.53)q[mi(x,y)-m(x,y)]q(x)q(y)≤q[mi(x,y)-mio(x,y)]q(x)q(y)+q[mio(x,y)-m(x,y)]q(x)q(y)<ɛ2+ɛ2=ɛ.
Thus mi→∥·∥qm.Theorem 4.13.
Let(A,q) be a factorable  k-normed algebra having a minimal approximate identity  {eλ:λ∈I}. Then (a)
each of the mapsϕA,  ϕℓ,  ϕr, and  ϕd defined above is a linear isometry;(b)
for anya,b∈A and  m∈QM(A),  ∥a∘m∘b∥q=q(m(a,b)).Proof.
(a) We give the proof only forϕd:Md(A)→QM(A). Clearly, ϕd is linear. Let (S,T)∈Md(A). Then
(4.54)‖ϕd(S,T)‖q=sup⁡x≠0,y≠0q[ϕd(S,T)(x,y)]q(x)q(y)=sup⁡x≠0,y≠0q[xS(y)]q(x)q(y)≤sup⁡x≠0,y≠0q(x)q(S(y))q(x)q(y)=sup⁡y≠0q(S(y))q(y)=‖S‖q.
To prove the reverse inequality, let ɛ>0. There exists (y≠0)∈A such that ∥S∥q<q(S(y))/q(y)+ɛ. For any λ∈I, since 0<q(yeλ)≤q(y)q(eλ)≤q(y),
(4.55)‖ϕd(S,T)‖q≥q[ϕd(S,T)(eλ,yeλ)]q(eλ)q(yeλ)≥q[eλS(yeλ)]q(yeλ)≥q[eλS(y)eλ]q(y);
hence, in view of factorability, using Lemma 4.10(a),
(4.56)‖ϕd(S,T)‖q≥lim⁡λq[eλS(y)eλ]q(y)=q(S(y))q(y)>‖S‖q-ɛ.
Since ɛ>0 is arbitrary, we obtain ∥ϕd(S,T)∥q=∥S∥q=∥(S,T)∥q, and so ϕd is an isometry.
(b) By (a),ϕA is an isometry and so
(4.57)‖a∘m∘b‖q=sup⁡x≠0,y≠0q[(a∘m∘b)(x,y)]q(x)q(y)=sup⁡x≠0,y≠0q[m(xa,by)]q(x)q(y)=sup⁡x≠0,y≠0q[xm(a,b)y]q(x)q(y)=sup⁡x≠0,y≠0q[ϕA(m(a,b))(x,y)]q(x)q(y)=‖ϕA(m(a,b))‖q=q(m(a,b)).We next consider multiplication inQM(A) in various equivalent ways.Definition 4.14 (see [2, 4]).
LetA be an F-algebra with an ultra-approximate identity {eλ:λ∈I} and m1,m2∈QM(A). Since ϕd is onto, there exist (S1,T1),(S2,T2)∈Md(A) such that
(4.58)ϕd(S1,T1)=m1,ϕd(S2,T2)=m2.
By the definitions of ϕℓ and ϕr,
(4.59)ϕl(S1)=m1=ϕr(T1),ϕl(S2)=m2=ϕr(T2).
Therefore, the product of m1,m2 can be defined in any of the following ways:(i)
m1∘ϕdm2=ϕd(S1,T1)∘ϕdϕd(S2,T2)=ϕd[(S1,T1)(S2,T2)]=ϕd(S1S2,T2T1),(ii)
m1∘ϕℓm2=ϕℓ(S1)∘ϕℓϕℓ(S2)=ϕℓ(S1S2),(iii)
m1∘ϕrm2=ϕr(T1)∘ϕrϕr(T2)=ϕr(T2T1).
Note that, for any(x,y)∈A×A,
(4.60)[ϕd(S1S2,T2T1)](x,y)=x(S1S2)(y)=[ϕl(S1S2)](x,y),
also
(4.61)x(S1S2)(y)=(T2T1)(x)y=[ϕr(T2T1)](x,y).
Hence, m1∘ϕdm2=m1∘ϕℓm2=m1∘ϕrm2.Remark 4.15.
(1) Ifm=ϕℓ(T) with T∈Mℓ(A) and a∈A, then, by Theorem 3.6(a),
(4.62)m∘ϕlϕA(a)=ϕl(T)∘ϕlϕl(La)=ϕl(TLa)=ϕl(LT(a))=ϕA(T(a))∈ϕA(A).
(2) Ifm=ϕr(T) with T∈Mr(A) and a∈A, then, by Theorem 3.6(b),
(4.63)ϕA(a)∘ϕrm=ϕr(Ra)∘ϕrϕr(T)=ϕr(TRa)=ϕr(RT(a))=ϕA(T(a))∈ϕA(A).
(3) Ifm=ϕd(S,T) and a∈A, then, by Theorem 3.6(c),
(4.64)m∘ϕdϕA(a)=ϕd[(S,T)(La,Ra)]=ϕd(SLa,RaT)=ϕd(LS(a),RS(a))=ϕA(S(a))∈ϕA(A).
In the sequel, we denote the product on QM(A) arising from (i), (ii), or (iii) by ⊙. Some properties of this product are given as follows.Theorem 4.16.
LetA be a factorable complete  k-normed algebra with a minimal ultra-approximate identity  {eλ:λ∈I}. Then, (a)
for anym1,m2∈QM(A),
(4.65)(m1⊙m2)(x,y)=m1(x,lim⁡λm2(eλ,y))((x,y)∈A×A),
defines a product  ⊙ on  QM(A) so that  (QM(A),∥·∥q) is a complete  k-normed algebra with identity  mo=ϕd(I,I);(b)
for anym∈QM(A) and  a∈A,
(4.66)ϕA(a)⊙m=a∘m,m⊙ϕA(a)=m∘a;(c)
ϕA(A) is a two-sided ideal in  QM(A);(d)
IfA is factorable, then both  Mℓ(A) and  Md(A) are isometrically algebraically isomorphic to  QM(A), while  Mr(A) is isometrically algebraically anti-isomorphic to  QM(A);Proof.
(a) Letm1,m2∈QM(A) and (x,y)∈A×A. Choose (S1,T1),(S2,T2)∈Md(A) such that
(4.67)ϕd(S1,T1)=m1,ϕd(S2,T2)=m2,
that is,
(4.68)xS1(y)=m1(x,y),xS2(y)=m2(x,y).
Then,
(4.69)(m1⊙m2)(x,y)=[ϕd(S1S2,T2T1)](x,y)=x(S1S2)(y)=x[S1(S2(y))]=x[S1(lim⁡λeλS2(y))]=m1(x,lim⁡λeλS2(y))=m1(x,lim⁡λm2(eλ,y)).
It is easy to verify that ∥m1⊙m2∥q≤∥m1∥q∥m2∥q, so that (QM(A),∥·∥q) is an F-normed algebra. Further, by Theorem 4.12, (QM(A),∥·∥q) is also complete.
(b) Letm∈QM(A) and a∈A. Using (a), for any (x,y)∈A×A,
(4.70)[m⊙ϕA(a)](x,y)=m(x,lim⁡λϕA(a)(eλ,y))=m(x,lim⁡λmeλay)=m(x,ay)=(m∘a)(x,y).
Similarly, [ϕA(a)⊙m](x,y)=(a∘m)(x,y).
(c) To show thatϕA(A) is a two-sided ideal, let m∈QM(A) and a∈A. By Theorem 4.9(a), there exists (S,T)∈Md(A) such that ϕd(S,T)=m. Using (b), for any (x,y)∈A×A,
(4.71)[m⊙ϕA(a)](x,y)=(m∘a)(x,y)=m(x,ay)=ϕd(S,T)(x,ay)=xS(ay)=xS(a)y=ϕA(S(a))(x,y).
Hence, m⊙ϕA(a)=ϕA(S(a))∈ϕA(A). Similarly, ϕA(a)⊙m=ϕA(T(a))∈ϕA(A).
(d) Suppose thatA is factorable. Then, by Theorem 4.13(a), each of the maps ϕd, ϕℓ, ϕr, and ϕA is a linear isometry. If (S1,T1),(S2,T2)∈Md(A), then by definition
(4.72)ϕd(S1,T1)∘ϕdϕd(S2,T2)=ϕd[(S1,T1)(S2,T2)].
IfS1,S2∈Mℓ(A), then by definition
(4.73)ϕl(S1)∘ϕlϕl(S2)=ϕl(S1S2).
IfT1,T2∈Mr(A), then by definition
(4.74)ϕr(T1)∘ϕrϕr(T2)=ϕr(T2T1).
Hence, ϕd:Md(A)→QM(A) and ϕℓ:Mℓ(A)→QM(A) are algebraic isomorphisms and ϕr:Mr(A)→QM(A) is algebraic anti-isomorphism.Remark 4.17.
IfA has an identity e, then A may be identified with QM(A) as follows. Let m∈QM(A). Then m(e,e)∈A and, for any x,y∈A,
(4.75)ϕA(m(e,e))(x,y)=xm(e,e)y=m(xe,ey)=m(x,y).


## 5. Quasistrict and Strict Topologies onQM(A)

In this section, we consider the quasistrict and strict topologies onQM(A) and extend several results from [2, 4, 8]. Throughout we will assume, unless stated otherwise, that (A,q) is a factorable complete k-normed algebra having a minimal ultra-approximate identity {eλ:λ∈I}.Definition 5.1.
For anym∈QM(A) and a,b∈A, we define mappings a∘m,m∘a,a∘m∘b:A×A→A by
(5.1)(a∘m)(x,y)=m(xa,y),(m∘a)(x,y)=m(x,ay),(a∘m∘b)(x,y)=m(xa,by),(x,y)∈A×A.Lemma 5.2.
Letm∈QM(A) and  a,b∈A. Then (a)
∥a∘m∥q≤∥m∥qq(a),∥m∘a∥q≤∥m∥qq(a),(b)
∥a∘m∘b∥q=q(m(a,b))≤∥m∥qq(a)q(b).Proof.
(a) By definition,(5.2)‖a∘m‖q=sup⁡x≠0,y≠0q[(a∘m)(x,y)]q(x)q(y)=sup⁡x≠0,y≠0q[x⋅m(a,y)]q(x)q(y)≤sup⁡x≠0,y≠0q(x)q(m(a,y))q(x)q(y)=sup⁡y≠0q(m(a,y))q(y)=sup⁡y≠0‖m‖qq(a)q(y)q(y)=‖m‖qq(a).
Similarly, ∥m∘a∥q≤∥m∥qq(a).
(b) By Theorem4.13(b), q(m(a,b))=∥a∘m∘b∥q. Further, using (a),
(5.3)‖a∘m∘b‖q≤‖a∘m‖qq(b)≤‖m‖qq(a)q(b).Definition 5.3.
(1) Thequasistrict topologyγ on QM(A) is determined by the family {ξa,b(m):a,b∈A} of k-seminorms, where
(5.4)ξa,b(m)=‖a∘m∘b‖q=q(m(a,b)),m∈QM(A).
Compare with [2, page 109]; [4, page 558].
(2) Thestrict topologyβ on QM(A) is determined by the family {ηa(m):a∈A} of k-seminorms, where
(5.5)ηa(m)=max⁡⁡{‖a∘m‖q,‖m∘a‖q},m∈QM(A).
Compare with [8, page 227].
Letτ denote the topology on QM(A) generated by the k-norm ∥·∥q.Lemma 5.4.
γ⊆β⊆τon QM(A).Proof.
To show thatγ⊆β, let a,b∈A. Then
(5.6)ξa,b(m)=‖a∘m∘b‖q≤‖a∘m‖qq(b)≤ηa(m)q(b),m∈QM(A),
also
(5.7)ξa,b(m)=‖a∘m∘b‖q≤‖m∘b‖qq(a)≤ηb(m)q(a),m∈QM(A).
Hence,
(5.8)ξa,b(m)≤max⁡⁡{ηa(m),ηb(m)}q(a)q(b),m∈QM(A).
Let {mα:α∈J} be a net in QM(A) with mα→βm∈QM(A). Then, for any a,b∈A, ηa(mα-m)→0 and ηb(mα-m)→0. Hence,
(5.9)ξa,b(mα-m)≤max⁡⁡{ηa(mα-m),ηb(mα-m)}q(a)q(b)⟶0.
Thus mα→γm, and so γ⊆β.
To show thatβ⊆τ, Let a∈A. Note that, for any x,y∈A,
(5.10)(a∘m)(x,y)=m(xa,y)=x⋅m(a,y),m∈QM(A),(m∘a)(x,y)=m(x,ay)=m(x,a)y,m∈QM(A).
By Lemma5.2(a), ∥a∘m∥q≤∥m∥qq(a) and ∥m∘a∥q≤∥m∥qq(a); hence,
(5.11)ηa(m)=max⁡⁡{‖a∘m‖q,‖m∘a‖q}≤‖m‖qq(a),m∈QM(A).
Consequently, if {mα} is a net in QM(A) with mα→τm∈QM(A), then ∥mα-m∥q→0, and so, for any a,b∈A,
(5.12)ηa(mα-m)≤‖mα-m‖qq(a)⟶0.
Thus mα→βm; that is, β⊆τ.Theorem 5.5.
ϕA(A) is  β-dense in  QM(A) and hence  γ-dense in  QM(A).Proof.
Letm∈QM(A). Clearly {m(eλ,eλ)}λ∈I⊆A. We claim that ϕA(m(eλ,eλ))→βm. Let a∈A. We need to show that
(5.13)ηa[ϕA(m(eλ,eλ))-m]⟶0.
For anyx,y∈A, by joint continuity of m,(5.14)q[a∘{ϕA(m(eλ,eλ))-m}(x,y)]=q[{ϕA(m(eλ,eλ))-m}(xa,y)]=q[xam(eλ,eλ)y-m(xa,y)]=q[m(xaeλ,eλy)-m(xa,y)]⟶q[m(xa,y)-m(xa,y)]=0.
Hence ∥a∘{ϕA(m(eλ,eλ))-m}∥q→0. Similarly, ∥{ϕA(m(eλ,eλ))-m}∘a∥q→0. Thus,
(5.15)ηa[ϕA(m(eλ,eλ))-m]⟶0;
that is, ϕA(A) is β-dense in QM(A). Since γ⊆β, it follows that ϕA(A) is γ-dense in QM(A).Theorem 5.6.
(a)(QM(A),γ) and  (QM(A),β) are sequentially complete.
(b)  If, in addition,(A,q) is strongly factorable, then  (QM(A),γ) and  (QM(A),β) are complete.Proof.
(a) Let{mi:i∈ℕ} be a γ-Cauchy sequence in QM(A). For any x,y∈A, using Theorem 4.13(b),
(5.16)q[mi(x,y)-mj(x,y)]=q[(mi-mj)(x,y)]=‖x∘(mi-mj)∘y‖q=ξx,y(mi-mj),
which implies that {mi(x,y)} is a Cauchy sequence in A. Define m:A×A→A by m(x,y)=limimi(x,y). Clearly, m is bilinear and, by Theorem 2.7(b), m is jointly continuous. Further, for any a,b,x,y∈A,
(5.17)m(ax,yb)=lim⁡imi(ax,yb)=a[lim⁡imi(x,y)]b=am(x,y)b,
and so m∈QM(A). Further, for any a,b∈A,
(5.18)ξa,b(mi-m)=‖a∘mi∘b-a∘m∘b‖q=q[(mi-m)(a,b)]⟶0.
Hence mi→γm. So (QM(A),γ)  is sequentially complete.
Next we show that(QM(A),β) is sequentially complete. We first note that, if m∈QM(A), then, for each c∈A, the mappings Sc,Tc:A→A given by(5.19)Sc(x)=m(c,x),Tc(x)=m(x,c),x∈A,
define elements in Mℓ(A) and Mr(A), respectively, and it is easy to see that
(5.20)ϕl(Sc)=c∘m,ϕr(Tc)=m∘c.
Let{mi:i∈ℕ} be a β-Cauchy sequence in QM(A), and let c∈A. It follows from the definition of the β-topology that the sequences {ϕℓ(Sc)i} and {ϕr(Tc)i}, where (Sc)i(x)=mi(c,x) and (Tc)i(x)=mi(x,c), are τ-Cauchy in QM(A). Since ϕℓ and ϕr are topological embeddings, the sequences {(Sc)i} and {(Tc)i} are ∥·∥q-Cauchy in Mℓ(A) and Mr(A), respectively. Both Mℓ(A) and Mr(A) are complete (Theorem 3.9) and so there exist S(c) in Mℓ(A) and T(c) in Mr(A) such that
(5.21)‖(Sc)i-S(c)‖q⟶0,‖(Tc)i-T(c)‖q⟶0.
Sinceγ⊆β, the sequence {mi} is γ-Cauchy. As proved above, the space QM(A) is γ-complete and so there exists an element mo in QM(A) such that(5.22)lim⁡imi(x,y)=mo(x,y),∀x,y∈A.
For any a,b∈A,
(5.23)[ϕl(S(c))](a,b)=lim⁡i[ϕl(S(c))i](a,b)=lim⁡iami(c,b)=(c∘mo)(a,b),
which implies that ϕℓ(S(c))=c∘mo. Similarly, we can prove that ϕr(T(c))=mo∘c. Thus, by (5.21),
(5.24)‖c∘mi-c∘mo‖q=‖ϕl(S(c))i-ϕl(S(c))‖q=‖(S(c))i-S(c)‖q⟶0,‖mi∘c-mo∘c‖q=‖ϕr(T(c))i-ϕr(T(c))‖q=‖(T(c))i-T(c)‖q⟶0,
which implies that mo is the β-limit of the sequence {mi} that is, QM(A) is β-complete.
(b) Suppose thatA is strongly factorable. Let {mα:α∈J} be a γ-Cauchy net in QM(A). Replacing the sequence {mi:i∈ℕ} by the net {mα:α∈J} in part (a), we obtain a map m:A×A→A given by m(x,y)=limαmα(x,y). Then m is bilinear; further, for any a,b,x,y∈A,
(5.25)m(ax,yb)=lim⁡αmα(ax,yb)=a[lim⁡αmα(x,y)]b=am(x,y)b.
Hence, using strong factorability as in Theorem 4.3(c), it follows that m is jointly continuous and so m∈QM(A). Again, as in part (a), it follows that mα→γm and consequently (QM(A),γ)  is complete. That QM(A) is β-complete also follows by the argument similar to the above one.Remark 5.7.
The authors do not know whether part (b) of the above theorem can be proved without the assumption of the strong factorability ofA.Theorem 5.8.
(QM(A),γ),  (QM(A),β), and  (QM(A),τ) have the same bounded sets.Proof.
(a) Sinceγ⊆τ, every τ-bounded set is γ-bounded. Let H be any γ-bounded set in QM(A). Then, for each a,b∈A, there exists a constant r=r(a,b)>0 such that(5.26)‖a∘m∘b‖q≤r,∀m∈H,orq[m(a,b)]≤r,∀m∈H(usingTheorem4.13(b)).
For each a∈A and m∈H, define Ma:A→A by
(5.27)Ma(x)=m(a,x),x∈A.
Then, ℱ={Ma:m∈H}⊆CL(A). By (5.26), for any x∈A(5.28)q[Ma(x)]=q[m(a,x)]≤r(a,x),∀m∈H;
hence ℱ is pointwise bounded. Then, by the uniform boundedness principle (Theorem 2.6), there exists c=c(a)>0 such that
(5.29)‖Ma‖q≤c,∀m∈H.
Consider now the family P={pm:m∈H} of k-seminorms on A defined by
(5.30)pm(a)=‖Ma‖q=sup⁡b≠0q[Ma(b)]q(b)=sup⁡b≠0q[m(a,b)]q(b),a∈A.
For each m∈H,pm is continuous on A since, if {an}⊆A with an→ao in A, then
(5.31)|pm(an)-pm(ao)|≤pm(an-ao)=sup⁡b≠0q[Man-ao(b)]q(b)=sup⁡b≠0q[m(an-ao,b)]q(b)≤sup⁡b≠0‖m‖qq(an-ao)q(b)q(b)=‖m‖qq(an-ao)⟶0.
Then, by (5.29), the family P is pointwise bounded. Applying Theorem 2.5, there exists a ball B=B(xo,r)={x∈A:q(x-xo)≤r} and a constant C>0 such that
(5.32)pm(x)≤C,∀m∈H,x∈B(xo,r).
For any fixed a∈A, we claim that
(5.33)pm(a)≤2C⋅q(a)r.
If a=0, this is obvious. Suppose that a≠0. For simplification, put t=(r/q(a))1/k.  Then, q is k-homogeneous, and we have ta+xo,xo∈B(xo,r), as follows:
(5.34)q(ta+xo-xo)=q(ta)=tk⋅q(a)≤rq(a)q(a)=r,q(xo-xo)=q(0)=0<r.
So, by (5.32),
(5.35)pm(ta+xo)≤C,pm(xo)≤C.
Now, using (5.35) and the properties of k-norm again,
(5.36)pm(a)=pm(1tta)=(1t)kpm(ta)≤q(a)rpm[ta+xo-xo]≤q(a)r[pm(ta+xo)+pm(xo)]≤q(a)r[C+C]=2C⋅q(a)r.
This proves our claim. Hence, using (5.33), for any m∈H,
(5.37)‖m‖q=sup⁡a,b≠0q[m(a,b)]q(a)q(b)=sup⁡a≠01q(a)sup⁡b≠0q[m(a,b)]q(b)=sup⁡a≠01q(a)pm(a)≤sup⁡a≠01q(a).2C⋅q(a)r≤2Cr.
Consequently, H is τ-bounded.
(b) This follows from (a) sinceγ⊆β⊆τ.



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*Source: 235273-2011-03-29.xml*