Quasimultipliers on 𝐹-Algebras

Abstract

We investigate the extent to which the study of quasimultipliers can be made beyond Banach algebras. We will focus mainly on the class of 𝐹-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.

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 of 𝐹-algebras or complete 𝑘-normed algebras.

2. Preliminaries

Definition 2.1. Let 𝐸 be a vector space over the field 𝕂(=ℝ or ℂ). (1)A function 𝑞∶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) if 𝛼_𝑛→0 in 𝕂, then 𝑞(𝛼_𝑛𝑥)→0 for all 𝑥∈𝐸.(2)An 𝐹-seminorm 𝑞 on 𝐸 is called an 𝐹-norm if, for any 𝑥∈𝐸, 𝑞(𝑥)=0 implies that 𝑥=0.(3)An 𝐹-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(F^′₃)𝑞(𝛼𝑥)=|𝛼|^𝑘𝑞(𝑥) for all 𝑥∈𝐸 and 𝛼∈𝕂.(4)A 𝑘-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 an 𝐹-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) An 𝐹-seminorm (or 𝐹-norm) 𝑞 on an algebra 𝐴 is called submultiplicative if 𝑞𝑥𝑦≤𝑞(𝑥)𝑞𝑦,∀𝑥,𝑦∈𝐴. (2.1) 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 𝜀>0, the set {𝑥∈𝐸∶𝑞(𝑥)≤𝜀} is a neighbourhood of 0 in 𝐸, but it need not be a bounded set. In case, if {𝑥∈𝐸∶𝑞(𝑥)≤𝜀} is bounded for some 𝜀>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. Let 𝐴 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 bounded if 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.
If 𝐸 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]). Let 𝑋 be a complete metric space and ℋ={𝑓_𝛼∶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 𝑓_𝛼(𝑥)≤𝐶,∀𝛼∈𝐽,𝑥∈𝐵. (2.2)

Theorem 2.6 (see [14, page 39]; [19, page 465]). Let 𝐸be 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. Let 𝐸 and 𝐹 be 𝐹-spaces and 𝐺 any TVS. (a)A collection ℋ  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) Let 𝑓_𝑛∶𝐸×𝐹→𝐺 be a sequence of separately continuous bilinear mappings such that lim_𝑛→∞𝑓_𝑛(𝑥,𝑦)=𝑓(𝑥,𝑦) exists for each (𝑥,𝑦)∈𝐸×𝐹. Then {𝑓_𝑛} is equicontinuous and f is bilinear and jointly continuous (see [19,page 490]; [24, page 328]).

Definition 2.8. (1) A net {𝑒_𝜆∶𝜆∈𝐼} in a topological algebra 𝐴 is called an approximate identity if lim_𝜆𝑒_𝜆𝑎=𝑎=lim_𝜆𝑎𝑒_𝜆,∀𝑎∈𝐴. (2.3)
(2)An approximate identity {𝑒_𝜆∶𝜆∈𝐼} in an 𝐹-normed algebra (𝐴,𝑞) is said to be minimal if 𝑞(𝑒_𝜆)≤1 for all 𝜆∈𝐼.(3)An algebra 𝐴 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 algebra 𝐴 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 {𝑎_𝑛} in 𝐴 with 𝑎_𝑛→0, there exist 𝑎∈𝐴 and a sequence {𝑏_𝑛} (resp., {𝑐_𝑛}) in 𝐴 with 𝑏_𝑛→0 (resp., 𝑐_𝑛→0) such that 𝑎_𝑛=𝑎𝑏_𝑛 (resp., 𝑎_𝑛=𝑐_𝑛𝑎) 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]). Let 𝐴 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‖_𝑞𝑞(𝑇(𝑥=sup))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 ‖𝑇‖_𝑞𝑞(𝑇(𝑥(𝑥)𝑞(𝑇(𝑥=sup))∶𝑥∈𝐴,𝑞=1=sup))∶𝑥∈A,𝑞(𝑥)≤1, (2.4) 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, for 𝑞 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, ‖𝑇‖_𝑞≥𝑞𝑇𝑛,𝑛²𝑞𝑛,𝑛²=𝑞𝑛²,𝑛𝑞𝑛,𝑛²=𝑛²+𝑛^1/22𝑛⟶∞. (2.5)(2)Even when considering the subspace of those 𝑇 for which ‖𝑇‖_𝑞<∞, then ‖⋅‖_𝑞 need not always be an 𝐹-norm, since (𝐹₅) need not hold. For example, for a fixed sequence (𝑝_𝑛) with 0<𝑝_𝑛≤1, 𝑝_𝑛→0, consider the 𝐹-algebra 𝐴 of sequences (𝑥_𝑛)⊆ℝ with |𝑥_𝑛|^𝑝𝑛→0 and 𝑞((𝑥_𝑛))=sup_𝑛≥1|𝑥_𝑛|^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 mapping 𝑇∶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 any 𝑆,𝑇∈CL(𝐴), ‖𝑆𝑇‖_𝑞≤‖𝑆‖_𝑞‖𝑇‖_𝑞; hence (CL(𝐴),‖⋅‖_𝑞) is a 𝑘-normed algebra;(e)if 𝐴 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 convex 𝐹-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 {𝑞_𝑛}, which need not be submultiplicative but satisfy the weaker condition 𝑞_𝑛(𝑥𝑦)≤𝑞_𝑛+1(𝑥)𝑞_𝑛+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 on 𝐹-Normed Algebras

In this section, we recall definitions and results on various notions of multipliers on an algebra 𝐴 (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]). Let 𝐴 be an algebra over the field 𝕂(ℝ or ℂ).
(1)A mapping 𝑇∶𝐴→𝐴 is called a(i)multiplier on 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 multiplier on 𝐴 if 𝑎𝑆(𝑏)=𝑇(𝑎)𝑏 for all a,𝑏∈𝐴.
Some authors use the term centralizer instead of multiplier (see, e.g., [30, 31]).
Let 𝑀(𝐴) (resp., 𝑀_ℓ(𝐴),𝑀_𝑟(𝐴)) denote the set of all multipliers (resp., left multipliers, right multipliers) on 𝐴 and 𝑀_𝑑(𝐴) the set of all double multipliers on an algebra 𝐴. For any 𝑎∈𝐴, let 𝐿_𝑎,𝑅_𝑎∶𝐴→𝐴 be given by𝐿_𝑎(𝑥)=𝑎𝑥,𝑅_𝑎(𝑥)=𝑥𝑎,𝑥∈𝐴. (3.1) Clearly, 𝐿_𝑎∈𝑀_ℓ(𝐴), 𝑅_𝑎∈𝑀_𝑟(𝐴), and (𝐿_𝑎,𝑅_𝑎)∈𝑀_𝑑(𝐴).
For convenience, we summarize some basic properties of multipliers in the following theorems for later references.

Theorem 3.2 (see [31]). Let 𝐴 be an algebra. Then, (a)𝑀_ℓ(𝐴)∩𝑀_𝑟(𝐴)⊆𝑀(𝐴); (b)if 𝐴 is faithful, then 𝑀(𝐴)⊆𝑀_ℓ(𝐴)∩𝑀_𝑟(𝐴) and hence 𝑀(𝐴)=𝑀_ℓ(𝐴)∩𝑀_𝑟(𝐴);(c)if 𝐴 is commutative and faithful, then 𝑀_ℓ(𝐴)=𝑀_𝑟(𝐴)=𝑀(𝐴);(d)𝑀_ℓ(𝐴) and 𝑀_𝑟(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]). Let 𝐴 be a faithful algebra. Then, (a)if (𝑆,𝑇)∈𝑀_𝑑(𝐴), then (i)  𝑆 and 𝑇 are linear and (ii)  𝑆∈𝑀_ℓ(𝐴) and 𝑇∈𝑀_𝑟(𝐴). In particular, every 𝑇∈𝑀(𝐴) is linear;(b)𝑀_𝑑(𝐴) is an algebra with identity (𝐼,𝐼) under the operations 𝑆₁,𝑇₁+𝑆₂,𝑇₂=𝑆₁+𝑆₂,𝑇₁+𝑇₂,𝜆𝑆₁,𝑇₁=𝜆𝑆₁,𝜆𝑇₁(),𝑆𝜆∈𝕂₁,𝑇₁𝑆₂,𝑇₂=𝑆₁𝑆₂,𝑇₂𝑇₁; (3.2)(c)let (𝑆₁,𝑇₁),(𝑆₂,𝑇₂)∈𝑀_𝑑(𝐴). If 𝑆₁=𝑆₂, then 𝑇₁=𝑇₂; if   𝑇₁=𝑇₂  then  𝑆₁=𝑆₂; (d)if 𝐴 is commutative, then 𝑀_𝑑(𝐴) is commutative and 𝑀_𝑑(𝐴)=𝑀(𝐴); in fact, if (𝑆,𝑇)∈𝑀_𝑑(𝐴), then 𝑆=𝑇.

Definition 3.4. One defines mappings 𝜇_ℓ∶𝐴→𝑀_ℓ(𝐴), 𝜇_𝑟∶𝐴→𝑀_𝑟(𝐴), and 𝜇_𝑑∶𝐴→𝑀_𝑑(𝐴) by 𝜇_ℓ(𝑎)=𝐿_𝑎,𝜇_𝑟(𝑎)=𝑅_𝑎,𝜇_𝑑(𝑎)=𝐿_𝑎,𝑅_𝑎,𝑎∈𝐴. (3.3)

Theorem 3.5 (see [31]). Let 𝐴 be an algebra, and let 𝜇_ℓ, 𝜇_𝑟, and 𝜇_𝑑 be the mappings as defined above. Then, (a)𝜇_ℓ,𝜇_𝑟,  and 𝜇_𝑑 are linear;(b)𝜇_ℓ and 𝜇_𝑑 are algebra homomorphisms, while 𝜇_𝑟 is an algebra antihomomorphism;(c)𝜇_ℓ (resp., 𝜇_𝑟) is 1-1 ⇔𝐴 is left (resp., right)   faithful; 𝜇_𝑑 is 1-1 ⇔𝐴 is faithful;(d)𝜇_ℓ (resp., 𝜇_𝑟) is onto ⇔𝐴 has left (resp., right) identity; 𝜇_𝑑 is onto ⇔𝐴 has an identity.

Theorem 3.6 (see [31]). Let 𝐴 be an algebra. (a)For any 𝑎∈𝐴 and 𝑇∈𝑀_ℓ(𝐴), 𝑇𝐿_𝑎=𝐿_𝑇(𝑎)∈𝜇_ℓ(𝐴); hence 𝜇_ℓ(𝐴) is a left ideal in 𝑀_ℓ(𝐴).(b)For any 𝑎∈𝐴 and 𝑇∈𝑀_𝑟(𝐴), 𝑇𝑅_𝑎=𝑅_𝑇(𝑎)∈𝜇_𝑟(𝐴);  hence 𝜇_𝑟(𝐴) is a left ideal in 𝑀_𝑟(𝐴).(c)Suppose that 𝐴 is faithful. Then, for any 𝑎∈𝐴 and (𝑆,𝑇)∈𝑀_𝑑(𝐴), 𝐿_𝑎,𝑅_𝑎(𝑆,𝑇)=𝐿_𝑇(𝑎),𝑅_𝑇(𝑎)∈𝜇_𝑑(𝐴),(𝑆,𝑇)𝐿_𝑎,𝑅_𝑎=𝐿_𝑆(𝑎),𝑅_𝑆(𝑎)∈𝜇_𝑑(𝐴); (3.4) hence 𝜇_𝑑(𝐴) is a two-sided ideal in 𝑀_𝑑(𝐴).

Regarding the continuity of multipliers, we state the following.

Theorem 3.7. (a) Suppose that 𝐴 is a strongly factorable 𝐹-normed algebra. If 𝑇∈𝑀_ℓ(𝐴) (resp., 𝑀_𝑟(𝐴)), then 𝑇 is linear and continuous (see [32, 33]).
(b) Suppose that 𝐴 is a faithful 𝐹-algebra. If (𝑆,𝑇)∈𝑀_𝑑(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., 𝑀_ℓ(𝐴),𝑀_𝑟(𝐴)) denotes the set of all continuous linear multipliers (resp., left multipliers, right multipliers) on 𝐴 and 𝑀_𝑑(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]). Let 𝐴 be a topological algebra. The uniform operator topology 𝑢 (resp., the strong operator topology 𝑠) on 𝑀_𝑑(𝐴) is defined as the linear topology which has a base of neighborhoods of 0 consisting of all the sets of the form 𝑁(𝐷,𝑊)=(𝑆,𝑇)∈𝑀_𝑑(𝐴)∶𝑆(𝐷)⊆𝑊,𝑇(𝐷)⊆𝑊, (3.5) 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),𝑀_ℓ(𝐴) and 𝑀_𝑟(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]). Let 𝐴 be a faithful 𝐹-algebra, and let 𝑀_𝑡(𝐴) denote any one of the algebras 𝑀_ℓ(𝐴), 𝑀_𝑟(𝐴), 𝑀(𝐴), and 𝑀_𝑑(𝐴). Then, (a)(𝑀_𝑡(𝐴),𝑢) and (𝑀_𝑡(𝐴),𝑠) are topological algebras with separately continuous multiplication; (b)(𝑀_𝑡(𝐴),𝑢) and (𝑀_𝑡(𝐴),𝑠) are complete;(c)𝑠 and 𝑢 have the same bounded sets; (d)if (𝑀_𝑡(𝐴),𝑠) is metrizable, then 𝑠=𝑢 on 𝑀_𝑡(𝐴); (e)if 𝐴 has a two-sided approximate identity, then 𝐴 is 𝑠-dense in 𝑀_𝑡(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 𝑀_ℓ(𝐴), 𝑀_𝑟(𝐴), and 𝑀(𝐴) is given by the 𝑘-norm ‖𝑇‖_𝑞𝑞(𝑇(𝑥=sup))𝑞(𝑥)∶𝑥∈𝐴,𝑥≠0; (3.6) the 𝑢-topology on 𝑀_𝑑(𝐴) is given by the 𝑘-norm ‖(𝑆,𝑇)‖_𝑞=max‖𝑆‖_𝑞,‖𝑇‖_𝑞,(𝑆,𝑇)∈𝑀_𝑑(𝐴). (3.7)(2)The 𝑠-topology on 𝑀_ℓ(𝐴), 𝑀_𝑟(𝐴), and 𝑀(𝐴) is given by the family of {𝑝_𝑎∶𝑎∈𝐴} of 𝐹-seminorms, where 𝑝_𝑎(𝑇)=𝑞(𝑇(𝑎)),𝑇∈𝑀_ℓ(𝐴),𝑀_𝑟(𝐴)or𝑀(𝐴); (3.8) the 𝑠-topology on 𝑀_𝑑(𝐴) is given by the family {𝑟_𝑎∶𝑎∈𝐴} of 𝐹-seminorms, where 𝑟_𝑎(𝑆,𝑇)=max𝑞(𝑆(𝑎)),𝑞(𝑇(𝑎)),(𝑆,𝑇)∈𝑀_𝑑(𝐴). (3.9)

Theorem 3.11. Let (𝐴,𝑞) be a 𝑘-normed algebra having a minimal approximate identity {𝑒_𝜆∶𝜆∈𝐼}. Then, (a)for any 𝑎∈𝐴, ‖𝐿_𝑎‖_𝑞=‖𝑅_𝑎‖_𝑞=𝑞(𝑎); so each of the maps 𝜇_ℓ∶𝐴→(𝑀_ℓ(𝐴),𝑢), 𝜇_𝑟∶𝐴→(𝑀_𝑟(𝐴),𝑢), 𝜇_𝑑∶𝐴→(𝑀_𝑑(𝐴),𝑢) is an isometry and hence continuous; (b)for any (𝑆,𝑇)∈𝑀_𝑑(𝐴), ‖𝑆‖_𝑞=‖𝑇‖_𝑞;(c)if (𝐴,𝑞) is complete, then 𝐴 is a 𝑢-closed two-sided ideal in 𝑀_𝑑(𝐴), under the identification 𝜇_𝑑∶𝑎→(𝐿_𝑎,𝑅_𝑎).

Proof. (a) Let 𝑎∈𝐴. Then𝐿_𝑎𝑞=sup_𝑏≠0𝑞𝐿_𝑎(𝑏)𝑞(𝑏)=sup_𝑏≠0𝑞(𝑎𝑏)𝑞(𝑏)≤sup_𝑏≠0𝑞(𝑎)⋅𝑞(𝑏)𝑞(𝑏)=𝑞(𝑎). (3.10) On the other hand, 𝐿_𝑎𝑞=sup_𝑏≠0𝑞(𝑎𝑏)𝑞(𝑏)≥𝑞𝑎𝑒_𝜆𝑞𝑒_𝜆≥𝑞𝑎𝑒_𝜆,∀𝜆∈𝐼; (3.11) so 𝐿_𝑎𝑞≥lim_𝜆𝑞𝑎𝑒_𝜆=𝑞lim_𝜆𝑎𝑒_𝜆=𝑞(𝑎). (3.12) Hence ‖𝜇_ℓ(𝑎)‖_𝑞=‖𝐿_𝑎‖_𝑞=𝑞(𝑎). Similarly, ‖𝜇_𝑟(𝑎)‖_𝑞=‖𝑅_𝑎‖_𝑞=𝑞(𝑎). Thus 𝜇_𝑑(𝑎)_𝑞=max𝐿_𝑎𝑞,𝑅_𝑎𝑞=𝑞(𝑎). (3.13)
(b) Let (𝑆,𝑇)∈𝑀_𝑑(𝐴). Using (a), we have ‖𝑆‖_𝑞=sup_𝑎≠0𝑞(𝑆(𝑎))𝑞(𝑎)=sup_𝑎≠0𝑅_{𝑆(𝑎)𝑞}𝑞(𝑎)=sup_𝑎≠0sup_𝑏≠01𝑞(𝑎)⋅𝑞𝑅_𝑆(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. (3.14) Similarly, ‖𝑇‖_𝑞=sup_𝑎≠0𝐿_{𝑇(𝑎)𝑞}𝑞(𝑎)=sup_𝑎≠0sup_𝑏≠0𝑞[𝑇(𝑎)𝑏]𝑞(𝑎)⋅𝑞(𝑏)=sup_𝑎≠0sup_𝑏≠0𝑞[𝑎⋅𝑆(𝑏)]𝑞(𝑎)(𝑏)≤‖𝑆‖⋅𝑞^𝑞. (3.15) Thus, ‖𝑆‖_𝑞=‖𝑇‖_𝑞.
(c) In view of Theorem 3.6(c), we only need to show that 𝜇_𝑑(𝐴) is 𝑢-closed in 𝑀_𝑑(𝐴). Let (𝑆,𝑇)∈𝑀_𝑑(𝐴) with (𝑆,𝑇)∈𝑢-cl(𝜇_𝑑(𝐴)). Choose {𝑎_𝛼∶𝛼∈𝐽}⊆𝐴 such that {𝐿_𝑎𝛼,𝑅_𝑎𝛼}→−^𝑢(𝑆,𝑇). By part (a), 𝜇_𝑑 is an isometry. Hence {𝑎_𝛼∶𝛼∈𝐽} is a Cauchy net in 𝐴. Then (𝑆,𝑇)=𝑢−lim_𝛼𝜇_𝑑𝑎_𝛼∈𝜇_𝑑(𝐴). (3.16) Thus 𝜇_𝑑(𝐴) is 𝑢-closed in 𝑀_𝑑(𝐴).

4. Quasimultipliers on 𝐹-Algebras and 𝑘-Normed Algebras

In this section, we consider the notion of quasimultipliers on 𝐹-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]). Let 𝐴 be an algebra. A mapping 𝑚∶𝐴×𝐴→𝐴 is said to be a quasimultiplier on 𝐴 if𝑚(𝑎𝑏,𝑐)=𝑎𝑚(𝑏,𝑐),𝑚(𝑎,𝑏𝑐)=𝑚(𝑎,𝑏)𝑐, (4.1) for all 𝑎,𝑏,𝑐∈𝐴.
The following Lemma shows in particular that every left multiplier, right multiplier, multiplier, and double multiplier on an algebra 𝐴 can be viewed as quasimultiplier on 𝐴.

Lemma 4.2. Let 𝐴 be a faithful algebra. (a)For any 𝑐∈𝐴, define 𝑚=𝑚_𝑐∶𝐴×𝐴→𝐴 by 𝑚_𝑐(𝑎,𝑏)=𝑎𝑐𝑏,∀(𝑎,𝑏)∈𝐴×𝐴. (4.2)(b)For any 𝑇∈𝑀_ℓ(𝐴), define an associated map 𝑚=𝑚_𝑇∶𝐴×𝐴→𝐴 by 𝑚_𝑇(𝑎,𝑏)=𝑎𝑇(𝑏),∀(𝑎,𝑏)∈𝐴×𝐴. (4.3)(c)For any 𝑇∈𝑀_𝑟(𝐴), define an associated map 𝑚=𝑚_𝑇∶𝐴×𝐴→𝐴 by 𝑚_𝑇(𝑎,𝑏)=𝑇(𝑎)𝑏,∀(𝑎,𝑏)∈𝐴×𝐴. (4.4)(d)For any 𝑇∈𝑀(𝐴), define an associated map 𝑚=𝑚_𝑇∶𝐴×𝐴→𝐴 by 𝑚_𝑇(𝑎,𝑏)=𝑎𝑇(𝑏),∀(𝑎,𝑏)∈𝐴×𝐴. (4.5)(e)For any (𝑆,𝑇)∈𝑀_𝑑(𝐴), define an associated map 𝑚=𝑚_(𝑆,𝑇)∶𝐴×𝐴→𝐴 by 𝑚_(𝑆,𝑇)(𝑎,𝑏)=𝑎𝑆(𝑏),∀(𝑎,𝑏)∈𝐴×𝐴. (4.6)Then each of the maps 𝑚∶𝐴×𝐴→𝐴 defined above is a quasimultiplier on 𝐴.

Proof. We only prove (e). Let  (𝑆,𝑇)∈𝑀_𝑑(𝐴), for any 𝑎,𝑏,𝑐,𝑑∈𝐴, 𝑚(𝑎𝑏,𝑐)=𝑚_(𝑆,𝑇)()=()𝑆(𝑐)[(𝑐𝑚𝑎𝑏,𝑐𝑎𝑏=𝑎𝑏𝑆)]=𝑎_(𝑆,𝑇)(𝑏,𝑐)=𝑎𝑚(𝑏,𝑐). (4.7) 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 map 𝑚∶𝐴×𝐴→𝐴 is a quasimultiplier on 𝐴  if and only if 𝑚(𝑎𝑏,𝑐𝑑)=𝑎𝑚(𝑏,𝑐)𝑑,∀𝑎,𝑏,𝑐,𝑑∈𝐴; (4.8)(b)every quasimultiplier 𝑚 on 𝐴 is bilinear; (c)every quasimultiplier 𝑚  on 𝐴 is jointly continuous.

Proof. (a) If 𝑚 is a quasimultiplier on 𝐴, then clearly, for any 𝑎,𝑏,𝑐,𝑑∈𝐴, 𝑚(𝑎𝑏,𝑐𝑑)=𝑎𝑚(𝑎,𝑐𝑑)=𝑎𝑚(𝑏,𝑐)𝑑. (4.9)
Conversely, let 𝑎,𝑏,𝑐∈𝐴. Since 𝐴  is a strongly factorable and 𝑎_𝑛={𝑏,𝑐,0,0,…}→0, there exist 𝑦,𝑧,𝑤∈𝐴 such that 𝑏=𝑤𝑦, 𝑐=𝑤𝑧. Then, using (4.8), 𝑚(𝑎𝑏,𝑐)=𝑚𝑎𝑤𝑦,𝑤𝑧=(𝑎𝑤)𝑚𝑦,𝑤𝑧=𝑎𝑚𝑤𝑦,𝑤𝑧=𝑎𝑚(𝑏,𝑐). (4.10) Similarly, we obtain 𝑚(𝑎,𝑏𝑐)=𝑚(𝑎,𝑏)𝑐.
(b) Let 𝑎,𝑏,𝑐∈𝐴 and 𝛼∈𝕂. Choose, as above, 𝑥,𝑦,𝑤∈𝐴 such that 𝑎=𝑤𝑥, 𝑏=𝑤𝑦. Then, 𝑚()=𝑚()()()()()().𝑎+𝑏,𝑐=𝑚𝑤𝑥+𝑤𝑦,𝑐𝑥+𝑦𝑤,𝑐=𝑥𝑚𝑤,𝑐+𝑦𝑚𝑤,𝑐=𝑚𝑤𝑥,𝑐+𝑚𝑤𝑦,𝑐=𝑚𝑎,𝑐+𝑚𝑏,𝑐 (4.11) Similarly, 𝑚(𝑐,𝑎+𝑏)=𝑚(𝑐,𝑎)+𝑚(𝑐,𝑏). Next, 𝑚(𝛼𝑎,𝑐)=𝑚(𝛼𝑤𝑥,𝑐)=(𝛼𝑤)𝑚(𝑥,𝑏𝑐)=𝛼𝑚(𝑤𝑥,𝑐)=𝛼𝑚(𝑎,𝑐). (4.12) First, we show that 𝑚 is separately continuous. Let 𝑎∈𝐴 and {𝑥_𝑛}⊆𝐴 with limit 𝑥. Then {𝑥_𝑛−𝑥} converges to 0. By strong factorability, there exist a sequence {𝑧_𝑛} and an element 𝑧 of 𝐴 such that z_𝑛→0, 𝑥_𝑛−𝑥=𝑧𝑧_𝑛. Thus, 𝑚𝑎,𝑥_𝑛−𝑚(𝑎,𝑥)=𝑚𝑎,𝑥_𝑛−𝑥=𝑚𝑎,𝑧𝑧_𝑛=𝑚(𝑎,𝑧)𝑧_𝑛⟶0. (4.13)
Now, the joint continuity of 𝑚 follows directly from Theorem 2.7(a).

Theorem 4.4. Let 𝐴 be a commutative algebra. Then, (a)𝑎𝑚(𝑏,𝑐)=𝑚(𝑏,𝑎)𝑐 for any quasimultiplier on 𝐴 and 𝑎,𝑏,𝑐∈𝐴;(b)if 𝐴 is also faithful, then a bilinear map 𝑚∶𝐴×𝐴→𝐴 is a quasimultiplier on 𝐴 if and only if𝑚𝑎²,𝑏=𝑎𝑚(𝑎,𝑏),𝑚𝑎,𝑏²=𝑚(𝑎,𝑏)𝑏∀𝑎,𝑏∈𝐴. (4.14)

Proof. (a) By hypothesis, 𝑎𝑚(𝑏,𝑐)=𝑚(𝑏,𝑐)𝑎=𝑚(𝑏,𝑐𝑎)=𝑚(𝑏,𝑎𝑐)=𝑚(𝑏,𝑎)𝑐. (4.15)
(b)(⇒) This is obvious.
(⇐) For all 𝑎,𝑏,𝑐∈𝐴, using (4.14), 𝑚(𝑎+𝑏)²=()𝑚()()()()().,𝑐𝑎+𝑏𝑎+𝑏,𝑐=𝑎𝑚𝑎,𝑐+𝑎𝑚𝑏,𝑐+𝑏𝑚𝑎,𝑐+𝑏𝑚𝑏,𝑐 (4.16) On the other hand, we have 𝑚(𝑎+𝑏)²,𝑐=𝑚𝑎²+𝑏²𝑎+2𝑎𝑏,𝑐=𝑚²,𝑐+𝑚𝑏²()()()().,𝑐+2𝑚𝑎𝑏,𝑐=𝑎𝑚𝑎,𝑐+𝑏𝑚𝑏,𝑐+2𝑚𝑎𝑏,𝑐 (4.17) Comparing (4.16) and (4.17), we obtain 2𝑚(𝑎𝑏,𝑐)=𝑎𝑚(𝑏,𝑐)+𝑏𝑚(𝑎,𝑐). (4.18)
Now, for all 𝑎,𝑏,𝑐,𝑑∈𝐴, using (4.18) twice,()()()()+12𝑚𝑎𝑏𝑑,𝑐=𝑎𝑏𝑚𝑑,𝑐+𝑑𝑚𝑎𝑏,𝑐=𝑎𝑏𝑚𝑑,𝑐2𝑑[(()+12𝑚𝑎𝑏,𝑐)]=𝑎𝑏𝑚𝑑,𝑐2()+1𝑑𝑎𝑚𝑏,𝑐2𝑑𝑏𝑚(𝑎,𝑐). (4.19) By commutativity of 𝐴 and using (4.18) twice as above, ()()()()()+12𝑚𝑎𝑏𝑑,𝑐=2𝑚𝑎𝑑𝑏,𝑐=𝑎𝑑𝑚𝑏,𝑐+𝑏𝑚𝑎𝑑,𝑐=𝑎𝑑𝑚𝑏,𝑐2[()(()+1𝑏𝑎𝑚𝑑,𝑐+𝑏𝑑𝑚𝑎,𝑐)]=𝑎𝑑𝑚𝑏,𝑐2()+1𝑎𝑏𝑚𝑑,𝑐2𝑑𝑏𝑚(𝑎,𝑐). (4.20) 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, 𝑑𝑚(𝑎𝑏,𝑐)=𝑎𝑏𝑚(𝑑,𝑐)=𝑎[𝑏𝑚(𝑑,𝑐)]=𝑎𝑑𝑚(𝑏,𝑐)=𝑑𝑎𝑚(𝑏,𝑐). (4.21) Since this holds for all 𝑑∈𝐴 and 𝐴 is faithful, 𝑚(𝑎𝑏,𝑐)=𝑎𝑚(𝑏,𝑐).
A similar computation shows that 𝑚(𝑎,𝑏𝑐)=𝑚(𝑎,𝑏)𝑐. Hence 𝑚 is a quasimultiplier.

Definition 4.5. Let 𝑄𝑀(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 (𝑎∘𝑚)𝑥,𝑦=𝑚𝑥𝑎,𝑦,(𝑚∘𝑎)𝑥,𝑦=𝑚𝑥,𝑎𝑦,𝑥,𝑦∈𝐴. (4.22) 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 𝜙_𝐴∶𝐴⟶𝑄𝑀(𝐴),𝜙_ℓ∶𝑀_ℓ(𝐴)(𝐴),𝜙⟶𝑄𝑀_𝑟∶𝑀_𝑟(𝐴)⟶𝑄𝑀(𝐴),𝜙_𝑑∶𝑀_𝑑(𝐴)⟶𝑄𝑀(𝐴), (4.23) by 𝜙_𝐴(𝑎)𝜙𝑥,𝑦=𝑥𝑎𝑦,𝑎∈𝐴,_ℓ(𝑇)𝑥,𝑦=𝑥𝑇𝑦,𝑇∈𝑀_ℓ(𝐴),𝜙_𝑟(𝑇)𝑥,𝑦=𝑇(𝑥)𝑦,𝑇∈𝑀_𝑟(𝐴),𝜙_𝑑(𝑆,𝑇)𝑥,𝑦=𝑥𝑆𝑦,(𝑆,𝑇)∈𝑀_𝑑(𝐴), (4.24) for all (𝑥,𝑦)∈𝐴×𝐴. By Lemma 4.2, these mappings are well defined.

Definition 4.7. (1) A bounded approximate identity {𝑒_𝜆∶𝜆∈𝐼} in a topological algebra 𝐴 is said to be ultra-approximate if, for all 𝑚∈𝑄𝑀(𝐴) and 𝑎∈𝐴, the nets {𝑚(𝑎,𝑒_𝜆)∶𝜆∈𝐼} and {𝑚(𝑒_𝜆,𝑎)∶𝜆∈𝐼} are Cauchy in 𝐴  (see [2]).
(2) A topological algebra 𝐴 is called 𝑚-symmetric if, for each 𝑆∈𝑀_ℓ(𝐴)∪𝑀_𝑟(𝐴), there is a 𝑇∈𝑀_ℓ(𝐴)∪𝑀_𝑟(𝐴) such that either (𝑆,𝑇)∈𝑀_𝑑(𝐴) or (𝑇,𝑆)∈𝑀_𝑑(𝐴)(see[4]).

Theorem 4.8. Let (𝐴,𝑞) be an 𝐹-algebra with a bounded approximate identity {𝑒_𝜆∶𝜆∈𝐼}. Consider the following conditions. (a){𝑒_𝜆∶𝜆∈𝐼} is ultra-approximate.(b)For any 𝑎∈𝐴, 𝑆∈𝑀_ℓ(𝐴), and 𝑇∈𝑀_𝑟(𝐴), the nets {𝑎𝑆(𝑒_𝜆)} and {𝑇(𝑒_𝜆)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 {𝑒_𝜆∶𝜆∈𝐼} in 𝐴 is ultra-approximate. Let 𝑆∈𝑀_ℓ(𝐴) and 𝑇∈𝑀_𝑟(𝐴). Put 𝑚₁=𝜙_ℓ(𝑆) and 𝑚₂=𝜙_𝑟(𝑇). Then 𝑚₁,𝑚₂∈𝑄𝑀(𝐴) and, for any 𝑎∈𝐴,  {𝑚₁(𝑎,𝑒_𝜆)}={𝑎𝑆(𝑒_𝜆)} and {𝑚₁(𝑒_𝜆,𝑎)}={𝑇(𝑒_𝜆)𝑎} which are Cauchy in 𝐴 by hypothesis.
(b) ⇒ (c) Suppose that (b) holds, let 𝑆∈𝑀_ℓ(𝐴)∪𝑀_𝑟(𝐴), and suppose that 𝑆∈𝑀_ℓ(𝐴). Since 𝐴 is complete, the map 𝑇∶𝐴→𝐴 given by 𝑇(𝑎)=lim_𝜆𝑎𝑆𝑒_𝜆,𝑎∈𝐴, (4.25) is well-defined. Since 𝑆 is continuous, for any 𝑎,𝑏∈𝐴, 𝑎𝑆(𝑏)=𝑎lim_𝜆𝑆𝑒_𝜆𝑏=𝑎lim_𝜆𝑆𝑒_𝜆𝑏=lim_𝜆𝑎𝑆𝑒_𝜆𝑏=𝑇(𝑎)𝑏. (4.26) Since 𝐴 is a faithful 𝐹-algebra, by Theorem 3.7(b), (𝑆,𝑇)∈𝑀_𝑑(𝐴). Hence 𝐴 is 𝑚-symmetric.
(c) ⇒(b) Suppose that (c) holds. Let 𝑆∈𝑀_ℓ(𝐴) and 𝑇∈𝑀_𝑟(𝐴). By (c), there exist 𝑇₁∈𝑀_𝑟(𝐴),  𝑆₁∈𝑀_ℓ(𝐴) such that (𝑆,𝑇₁),(𝑆₁,𝑇)∈𝑀_𝑑(𝐴). Then, for any 𝑎∈𝐴, 𝑎𝑆𝑒_𝜆=𝑇₁(𝑎)𝑒_𝜆⟶𝑇₁(𝑎),𝑇𝑒_𝜆𝑎=𝑒_𝜆𝑆₁(𝑎)⟶𝑆₁(𝑎). (4.27) Thus, both {𝑎𝑆(𝑒_𝜆)} and {𝑇(𝑒_𝜆)𝑎}, being convergent, are Cauchy in 𝐴.
Suppose that 𝐴 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 𝑆(𝑥)=𝑚(𝑐,𝑥),𝑇(𝑥)=𝑚(𝑥,𝑏),𝑥∈𝐴. (4.28) Then, for any 𝑥,𝑦∈𝐴, 𝑆()(𝑥)𝑇𝑦,𝑥𝑦=𝑚𝑐,𝑥𝑦=𝑚𝑐,𝑥𝑦=𝑆𝑦,𝑥𝑦=𝑚𝑥𝑦,𝑏=𝑥𝑚𝑦,𝑏=𝑥𝑇 (4.29) and so 𝑆∈𝑀_ℓ(𝐴) and 𝑇∈𝑀_𝑟(𝐴). By (c), there exist 𝑇₁∈𝑀_𝑟(𝐴), 𝑆₁∈𝑀_ℓ(𝐴) such that (𝑆,𝑇₁),(𝑆₁,𝑇)∈𝑀_𝑑(𝐴). Then 𝑚𝑎,𝑒_𝜆=𝑚𝑏𝑐,𝑒_𝜆=𝑏𝑚𝑐,𝑒_𝜆=𝑏𝑆𝑒_𝜆=𝑇₁(𝑏)𝑒_𝜆⟶𝑇₁(𝑏),𝑚𝑒_𝜆,𝑎=𝑚𝑒_𝜆,𝑏𝑐=𝑚𝑒_𝜆,𝑏𝑐=𝑇𝑒_𝜆𝑐=𝑒_𝜆𝑆₁(𝑐)⟶𝑆₁(𝑐). (4.30) Hence {𝑚(𝑎,𝑒_𝜆)∶𝜆∈𝐼} and {𝑚(𝑒_𝜆,𝑎)∶𝜆∈𝐼} are Cauchy in 𝐴. So {𝑒_𝜆∶𝜆∈𝐼} is ultra-approximate.

Theorem 4.9. Let (𝐴,𝑞) be an 𝐹-algebra having an ultra-approximate identity {𝑒_𝜆∶𝜆∈𝐼}. Then, (a)each of the maps 𝜙_𝐴, 𝜙_ℓ, 𝜙_𝑟, and 𝜙_𝑑 is a bijection;(b)𝜙_𝑑∣𝑀(𝐴)=𝜙_ℓ∣𝑀(𝐴)=𝜙_𝑟∣𝑀(𝐴); (c)𝜙_𝑑∣𝜇_𝑑(𝐴)=𝜙_𝐴.

Proof. (a) We give the proof only for 𝜙_𝑑∶𝑀_𝑑(𝐴)→𝑄𝑀(𝐴). To show that 𝜙_𝑑 is onto, let 𝑚∈𝑄𝑀(𝐴). Since {𝑒_𝜆} is ultra-approximate, for each 𝑥∈𝐴, the nets {𝑚(𝑥,𝑒_𝜆)∶𝜆∈𝐼} and {𝑚(𝑒_𝜆,𝑥)∶𝜆∈𝐼} are convergent. For each 𝑥∈𝐴, define 𝑆,𝑇∶𝐴→𝐴 by 𝑆(𝑥)=lim_𝜆𝑚𝑒_𝜆,𝑥,𝑇(𝑥)=lim_𝜆𝑚𝑥,𝑒_𝜆. (4.31) Then (𝑆,𝑇)∈𝑀_𝑑(𝐴) since, for any 𝑥,𝑦∈𝐴, 𝑥𝑆𝑦=𝑥lim_𝜆𝑚𝑒_𝜆,𝑦=lim_𝜆𝑚𝑥𝑒_𝜆,𝑦=𝑚𝑥,𝑦=lim_𝜆𝑚𝑥,𝑒_𝜆𝑦=lim_𝜆𝑚𝑥,𝑒_𝜆𝑦=𝑇(𝑥)𝑦. (4.32) Further, we have for any (𝑎,𝑏)∈𝐴×𝐴, 𝜙_𝑑(𝑆,𝑇)(𝑎,𝑏)=𝑎𝑆(𝑏)=𝑎lim_𝜆𝑚𝑒_𝜆,𝑏=lim_𝜆𝑚𝑎𝑒_𝜆,𝑏=𝑚(𝑎,𝑏); (4.33) that is, 𝜙_𝑑(𝑆,𝑇)=𝑚.
To show that, 𝜙_𝑑 is one to one, let (𝑆₁,𝑇₁),(𝑆₂,𝑇₂)∈𝑀_𝑑(𝐴) with 𝜙_𝑑(𝑆₁,𝑇₁)=𝜙_𝑑(𝑆₂,𝑇₂). Then, for any 𝑎,𝑏∈𝐴, 𝜙_𝑑𝑆₁,𝑇₁(𝑎,𝑏)=𝜙_𝑑𝑆₂,𝑇₂(𝑎,𝑏),or𝑎𝑆₁(𝑏)=𝑎𝑆₂(𝑏). (4.34) 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) Let 𝑇∈𝑀(𝐴) and 𝑥,𝑦∈𝐴. Then, since 𝑇∈𝑀_ℓ(𝐴)∩𝑀_𝑟(𝐴) and (𝑇,𝑇)∈𝑀_𝑑(𝐴), 𝜙_𝑑(𝑇)𝑥,𝑦=𝜙_𝑑(𝑇,𝑇)𝑥,𝑦=𝑥𝑇𝑦=𝑇𝑥𝑦. (4.35) Also 𝜙_ℓ(𝑇)𝑥,𝑦=𝑥𝑇𝑦=𝑇𝑥𝑦;𝜙_𝑟(𝑇)𝑥,𝑦=𝑇(𝑥)𝑦=𝑇𝑥𝑦. (4.36)
(c) For any 𝑎∈𝐴 and (𝑥,𝑦)∈𝐴×𝐴, 𝜙_𝑑𝜇_𝑑(𝑎)𝑥,𝑦=𝜙_𝑑𝐿_𝑎,𝑅_𝑎𝑥,𝑦=𝑥𝐿_𝑎𝑦=𝑥𝑎𝑦=𝜙_𝐴(𝑎)𝑥,𝑦. (4.37) Thus 𝜙_𝑑∣𝜇_𝑑(𝐴)=𝜙_𝐴.

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 {𝑒_𝜆∶𝜆∈𝐼}, then lim_𝜆𝑞𝑒_𝜆𝑎𝑒_𝜆−𝑎=0,∀𝑎∈𝐴. (4.38)
(b) If (𝐴,𝑞) is an 𝐹-algebra having a minimal ultra-approximate identity {𝑒_𝜆∶𝜆∈𝐼}, then, for any 𝑚∈𝑄𝑀(𝐴), lim_𝜆𝑒_𝜆𝑚𝑒_𝜆,𝑥=lim_𝜆𝑚𝑒_𝜆,𝑥(exists),∀𝑥∈𝐴. (4.39) Compare with [10, page 124].

Proof. (a) Let 𝑎∈𝐴. Since 𝐴 is factorable, there exist 𝑥,𝑦∈𝐴 such that 𝑎=𝑥𝑦. Then lim⁻⁻_𝜆𝑞𝑒_𝜆𝑎𝑒_𝜆−𝑎=lim⁻⁻_𝜆𝑞𝑒_𝜆𝑥𝑦𝑒_𝜆−𝑒_𝜆𝑥𝑦+𝑒_𝜆≤𝑥𝑦−𝑥𝑦₋₋lim_𝜆𝑞𝑒_𝜆𝑥⋅𝑞𝑦𝑒_𝜆−𝑦+lim⁻⁻_𝜆𝑞𝑒_𝜆(𝑥)(0)(0)𝑥𝑦−𝑥𝑦=𝑞⋅𝑞+𝑞=0. (4.40) Since 𝐴 is complete and {𝑒_𝜆} is ultra-approximate, for any 𝑥∈𝐴, lim_𝜆𝑚(𝑒_𝜆,𝑥)=𝑦 (say) exists. Since 𝑞(𝑒_𝜆)≤1, 𝑞𝑦−𝑒_𝜆𝑚𝑒_𝜆,𝑥≤𝑞𝑦−𝑒_𝜆𝑦+𝑞𝑒_𝜆𝑦−𝑒_𝜆𝑚𝑒_𝜆,𝑥≤𝑞𝑦−𝑒_𝜆𝑦+𝑞𝑒_𝜆𝑞𝑦−𝑚𝑒_𝜆,𝑥≤𝑞𝑦−𝑒_𝜆𝑦+𝑞𝑦−𝑚𝑒_𝜆,𝑥⟶0. (4.41)

Definition 4.11. Let (𝐴,𝑞) be a 𝑘-normed algebra. For any 𝑚∈𝑄𝑀(𝐴), we define ‖𝑚‖_𝑞𝑞𝑚=sup𝑥,𝑦𝑞(𝑥)𝑞𝑦∶𝑥,𝑦∈𝐴,𝑥,𝑦≠0. (4.42) Clearly, 𝑞𝑚𝑥,𝑦≤‖𝑚‖_𝑞𝑞(𝑥)𝑞𝑦,∀𝑥,𝑦∈𝐴. (4.43)

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 any 𝑚∈𝑄𝑀(𝐴), ‖𝑚‖_𝑞=0⟺sup_{𝑥≠0,𝑦≠0}𝑞𝑚𝑥,𝑦𝑞(𝑥)𝑞𝑦𝑚𝑚=0⟺𝑞𝑥,𝑦=0,∀𝑥,𝑦∈𝐴,𝑥≠0,𝑦≠0⟺𝑞𝑥,𝑦=0,∀𝑥,𝑦∈𝐴⟺𝑚=0. (4.44) Also, for any 𝛼∈𝕂,𝑞(𝛼𝑥)=|𝛼|^𝑘𝑞(𝑥) and so it follows that ‖𝛼𝑚‖_𝑞=|𝛼|^𝑘‖𝑚‖_𝑞.
Next, let 𝑚,𝑢∈𝑄𝑀(𝐴), and let 𝜀>0. Choose 𝑥,𝑦∈𝐴 such that ‖𝑚+𝑢‖_𝑞≤𝑞(𝑚+𝑢)𝑥,𝑦𝑞(𝑥)𝑞𝑦−𝜀. (4.45) Then, ‖𝑚+𝑢‖_𝑞≤𝑞𝑚𝑥,𝑦+𝑢𝑥,𝑦𝑞(𝑥)𝑞𝑦≤𝑞𝑚−𝜀𝑥,𝑦𝑞(𝑥)𝑞𝑦+𝑞𝑢𝑥,𝑦𝑞(𝑥)𝑞𝑦≤‖𝑚‖−𝜀_𝑞+‖𝑢‖_𝑞−𝜀. (4.46) Since 𝜀>0 is arbitrary, ‖𝑚+𝑢‖_𝑞≤‖𝑚‖_𝑞+‖𝑢‖_𝑞. Thus ‖⋅‖_𝑞 is a 𝑘-norm on 𝑄𝑀(𝐴).
(b) Let {𝑚_𝑖∶𝑖∈ℕ} be a ‖⋅‖_𝑞-Cauchy sequence in 𝑄𝑀(𝐴). Then, for any 𝑥,𝑦∈𝐴, lim⁻⁻⁻_𝑖,𝑗𝑞𝑚_𝑖𝑥,𝑦−𝑚_𝑗𝑥,𝑦=lim⁻⁻⁻_𝑖,𝑗𝑞𝑚_𝑖−𝑚_𝑗≤𝑥,𝑦₋₋₋lim_𝑖,𝑗𝑚_𝑖−𝑚_𝑗𝑞⋅𝑞(𝑥)𝑞𝑦=0. (4.47) Therefore, for any 𝑥,𝑦 in 𝐴,{𝑚_𝑖(𝑥,𝑦)} is a Cauchy sequence in 𝐴. Since 𝐴 is complete, the map 𝑚∶𝐴×𝐴→𝐴 given by 𝑚𝑥,𝑦=lim_𝑖𝑚_𝑖𝑥,𝑦,𝑥,𝑦∈𝐴, (4.48) is welldefined. Clearly, 𝑚 is bilinear and, by Theorem 2.7(b), 𝑚 is jointly continuous. Further, for any 𝑎,𝑏,𝑥,𝑦∈𝐴, 𝑚𝑎𝑥,𝑦𝑏=lim_𝑖𝑚_𝑖𝑎𝑥,𝑦𝑏=lim_𝑖𝑎𝑚_𝑖𝑥,𝑦𝑏=𝑎lim_𝑖𝑚_𝑖𝑥,𝑦𝑏=𝑎𝑚𝑥,𝑦𝑏. (4.49) Hence, 𝑚∈𝑄𝑀(𝐴). Next, ‖𝑚_𝑖−𝑚‖_𝑞→0 as follows. Let 𝜀>0. Since {𝑚_𝑖} is ‖⋅‖_𝑞-Cauchy, there exists an integer 𝑁≥1 such that 𝑚_𝑖−𝑚_𝑗𝑞<𝜀2,∀pairs𝑖,𝑗≥𝑁, (4.50) that is, 𝑞𝑚_𝑖−𝑚_𝑗𝑥,𝑦𝑞(𝑥)𝑞𝑦<𝜀2,∀pairs𝑖,𝑗≥𝑁,𝑥,𝑦∈𝐴,𝑥,𝑦≠0. (4.51) Let 𝑥,𝑦∈𝐴, 𝑥,𝑦≠0. Fix any 𝑖_𝑜≥𝑁 in (4.51); since 𝑚_𝑗(𝑥,𝑦)→𝑚(𝑥,𝑦) in 𝐴, letting 𝑗→∞ in (4.51), 𝑞𝑚_𝑖𝑜𝑥,𝑦−𝑚𝑥,𝑦𝑞(𝑥)𝑞𝑦≤𝜀2. (4.52) Then, for any 𝑖≥𝑁, using (4.51) and (4.52), 𝑞𝑚_𝑖𝑥,𝑦−𝑚𝑥,𝑦𝑞(𝑥)𝑞𝑦≤𝑞𝑚_𝑖𝑥,𝑦−𝑚_𝑖𝑜𝑥,𝑦𝑞(𝑥)𝑞𝑦+𝑞𝑚_𝑖𝑜𝑥,𝑦−𝑚𝑥,𝑦𝑞(𝑥)𝑞𝑦<𝜀2+𝜀2=𝜀. (4.53) Thus 𝑚−−−→_𝑖^{‖⋅‖𝑞}𝑚.

Theorem 4.13. Let (𝐴,𝑞) be a factorable 𝑘-normed algebra having a minimal approximate identity {𝑒_𝜆∶𝜆∈𝐼}. Then (a)each of the maps 𝜙_𝐴, 𝜙_ℓ, 𝜙_𝑟, and 𝜙_𝑑 defined above is a linear isometry;(b)for any 𝑎,𝑏∈𝐴 and 𝑚∈𝑄𝑀(𝐴), ‖𝑎∘𝑚∘𝑏‖_𝑞=𝑞(𝑚(𝑎,𝑏)).

Proof. (a) We give the proof only for 𝜙_𝑑∶𝑀_𝑑(𝐴)→𝑄𝑀(𝐴). Clearly, 𝜙_𝑑 is linear. Let (𝑆,𝑇)∈𝑀_𝑑(𝐴). Then 𝜙_𝑑(𝑆,𝑇)_𝑞=sup_{𝑥≠0,𝑦≠0}𝑞𝜙_𝑑(𝑆,𝑇)𝑥,𝑦𝑞(𝑥)𝑞𝑦=sup_{𝑥≠0,𝑦≠0}𝑞𝑥𝑆𝑦𝑞(𝑥)𝑞𝑦≤sup_{𝑥≠0,𝑦≠0}𝑞(𝑥)𝑞𝑆𝑦𝑞(𝑥)𝑞𝑦=sup_𝑦≠0𝑞𝑆𝑦𝑞𝑦=‖𝑆‖_𝑞. (4.54) To prove the reverse inequality, let 𝜀>0. There exists (𝑦≠0)∈𝐴 such that ‖𝑆‖_𝑞<𝑞(𝑆(𝑦))/𝑞(𝑦)+𝜀. For any 𝜆∈𝐼, since 0<𝑞(𝑦𝑒_𝜆)≤𝑞(𝑦)𝑞(𝑒_𝜆)≤𝑞(𝑦), 𝜙_𝑑(𝑆,𝑇)_𝑞≥𝑞𝜙_𝑑(𝑆,𝑇)𝑒_𝜆,𝑦𝑒_𝜆𝑞𝑒_𝜆𝑞𝑦𝑒_𝜆≥𝑞𝑒_𝜆𝑆𝑦𝑒_𝜆𝑞𝑦𝑒_𝜆≥𝑞𝑒_𝜆𝑆𝑦𝑒_𝜆𝑞𝑦; (4.55) hence, in view of factorability, using Lemma 4.10(a), 𝜙_𝑑(𝑆,𝑇)_𝑞≥lim_𝜆𝑞𝑒_𝜆𝑆𝑦𝑒_𝜆𝑞𝑦=𝑞𝑆𝑦𝑞𝑦>‖𝑆‖_𝑞−𝜀. (4.56) Since 𝜀>0 is arbitrary, we obtain ‖𝜙_𝑑(𝑆,𝑇)‖_𝑞=‖𝑆‖_𝑞=‖(𝑆,𝑇)‖_𝑞, and so 𝜙_𝑑 is an isometry.
(b) By (a), 𝜙_𝐴 is an isometry and so ‖𝑎∘𝑚∘𝑏‖_𝑞=sup_{𝑥≠0,𝑦≠0}𝑞(𝑎∘𝑚∘𝑏)𝑥,𝑦𝑞(𝑥)𝑞𝑦=sup_{𝑥≠0,𝑦≠0}𝑞𝑚𝑥𝑎,𝑏𝑦𝑞(𝑥)𝑞𝑦=sup_{𝑥≠0,𝑦≠0}𝑞𝑥𝑚(𝑎,𝑏)𝑦𝑞(𝑥)𝑞𝑦=sup_{𝑥≠0,𝑦≠0}𝑞𝜙_𝐴(𝑚(𝑎,𝑏))𝑥,𝑦𝑞(𝑥)𝑞𝑦=𝜙_𝐴(𝑚(𝑎,𝑏))_𝑞=𝑞(𝑚(𝑎,𝑏)). (4.57)

We next consider multiplication in 𝑄𝑀(𝐴) in various equivalent ways.

Definition 4.14 (see [2, 4]). Let 𝐴 be an 𝐹-algebra with an ultra-approximate identity {𝑒_𝜆∶𝜆∈𝐼} and 𝑚₁,𝑚₂∈𝑄𝑀(𝐴). Since 𝜙_𝑑 is onto, there exist (𝑆₁,𝑇₁),(𝑆₂,𝑇₂)∈𝑀_𝑑(A) such that 𝜙_𝑑𝑆₁,𝑇₁=𝑚₁,𝜙_𝑑𝑆₂,𝑇₂=𝑚₂. (4.58) By the definitions of 𝜙_ℓ and 𝜙_𝑟, 𝜙_ℓ𝑆₁=𝑚₁=𝜙_𝑟𝑇₁,𝜙_ℓ𝑆₂=𝑚₂=𝜙_𝑟𝑇₂. (4.59) Therefore, the product of 𝑚₁,𝑚₂ can be defined in any of the following ways:(i)𝑚₁∘_𝜙𝑑𝑚₂=𝜙_𝑑(𝑆₁,𝑇₁)∘_𝜙𝑑𝜙_𝑑(𝑆₂,𝑇₂)=𝜙_𝑑[(𝑆₁,𝑇₁)(𝑆₂,𝑇₂)]=𝜙_𝑑(𝑆₁𝑆₂,𝑇₂𝑇₁), (ii)𝑚₁∘_𝜙ℓ𝑚₂=𝜙_ℓ(𝑆₁)∘_𝜙ℓ𝜙_ℓ(𝑆₂)=𝜙_ℓ(𝑆₁𝑆₂), (iii)𝑚₁∘_𝜙𝑟𝑚₂=𝜙_𝑟(𝑇₁)∘_𝜙𝑟𝜙_𝑟(𝑇₂)=𝜙_𝑟(𝑇₂𝑇₁).
Note that, for any (𝑥,𝑦)∈𝐴×𝐴, 𝜙_𝑑𝑆₁𝑆₂,𝑇₂𝑇₁𝑥,𝑦=𝑥𝑆₁𝑆₂𝑦=𝜙_ℓ𝑆₁𝑆₂𝑥,𝑦, (4.60) also 𝑥𝑆₁𝑆₂𝑦=𝑇₂𝑇₁(𝑥)𝑦=𝜙_𝑟𝑇₂𝑇₁𝑥,𝑦. (4.61) Hence, 𝑚₁∘_𝜙𝑑𝑚₂=𝑚₁∘_𝜙ℓ𝑚₂=𝑚₁∘_𝜙𝑟𝑚₂.

Remark 4.15. (1) If 𝑚=𝜙_ℓ(𝑇) with 𝑇∈𝑀_ℓ(𝐴) and 𝑎∈𝐴, then, by Theorem 3.6(a), 𝑚∘_𝜙ℓ𝜙_𝐴(𝑎)=𝜙_ℓ(𝑇)∘_𝜙ℓ𝜙_ℓ𝐿_𝑎=𝜙_ℓ𝑇𝐿_𝑎=𝜙_ℓ𝐿_𝑇(𝑎)=𝜙_𝐴(𝑇(𝑎))∈𝜙_𝐴(𝐴). (4.62)
(2) If 𝑚=𝜙_𝑟(𝑇) with 𝑇∈𝑀_𝑟(𝐴) and 𝑎∈𝐴, then, by Theorem 3.6(b), 𝜙_𝐴(𝑎)∘_𝜙𝑟𝑚=𝜙_𝑟𝑅_𝑎∘_𝜙𝑟𝜙_𝑟(𝑇)=𝜙_𝑟𝑇𝑅_𝑎=𝜙_𝑟𝑅_𝑇(𝑎)=𝜙_𝐴(𝑇(𝑎))∈𝜙_𝐴(𝐴). (4.63)
(3) If 𝑚=𝜙_𝑑(𝑆,𝑇) and 𝑎∈𝐴, then, by Theorem 3.6(c), 𝑚∘_𝜙𝑑𝜙_𝐴(𝑎)=𝜙_𝑑(𝑆,𝑇)𝐿_𝑎,𝑅_𝑎=𝜙_𝑑𝑆𝐿_𝑎,𝑅_𝑎𝑇=𝜙_𝑑𝐿_𝑆(𝑎),𝑅_𝑆(𝑎)=𝜙_𝐴(𝑆(𝑎))∈𝜙_𝐴(𝐴). (4.64) 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. Let 𝐴 be a factorable complete 𝑘-normed algebra with a minimal ultra-approximate identity {𝑒_𝜆∶𝜆∈𝐼}. Then, (a)for any 𝑚₁,𝑚₂∈𝑄𝑀(𝐴), 𝑚₁⊙𝑚₂𝑥,𝑦=𝑚₁𝑥,lim_𝜆𝑚₂𝑒_𝜆,𝑦𝑥,𝑦∈𝐴×𝐴, (4.65) defines a product ⊙ on 𝑄𝑀(𝐴) so that (𝑄𝑀(𝐴),‖⋅‖_𝑞) is a complete 𝑘-normed algebra with identity 𝑚_𝑜=𝜙_𝑑(𝐼,𝐼);(b)for any 𝑚∈𝑄𝑀(𝐴) and 𝑎∈𝐴, 𝜙_𝐴(𝑎)⊙𝑚=𝑎∘𝑚,𝑚⊙𝜙_𝐴(𝑎)=𝑚∘𝑎; (4.66)(c)𝜙_𝐴(𝐴) is a two-sided ideal in 𝑄𝑀(𝐴); (d)If 𝐴 is factorable, then both 𝑀_ℓ(𝐴) and 𝑀_𝑑(𝐴) are isometrically algebraically isomorphic to 𝑄𝑀(𝐴), while 𝑀_𝑟(𝐴) is isometrically algebraically anti-isomorphic to 𝑄𝑀(𝐴);

Proof. (a) Let 𝑚₁,𝑚₂∈𝑄𝑀(𝐴) and (𝑥,𝑦)∈𝐴×𝐴. Choose (𝑆₁,𝑇₁),(𝑆₂,𝑇₂)∈𝑀_𝑑(𝐴) such that 𝜙_𝑑𝑆₁,𝑇₁=𝑚₁,𝜙_𝑑𝑆₂,𝑇₂=𝑚₂, (4.67) that is, 𝑥𝑆₁𝑦=𝑚₁𝑥,𝑦,𝑥𝑆₂𝑦=𝑚₂𝑥,𝑦. (4.68) Then, 𝑚₁⊙𝑚₂𝑥,𝑦=𝜙_𝑑𝑆₁𝑆₂,𝑇₂𝑇₁𝑥,𝑦=𝑥𝑆₁𝑆₂𝑦=𝑥𝑆₁𝑆₂𝑦=𝑥𝑆₁lim_𝜆𝑒_𝜆𝑆₂𝑦=𝑚₁𝑥,lim_𝜆𝑒_𝜆𝑆₂𝑦=𝑚₁𝑥,lim_𝜆𝑚₂𝑒_𝜆,𝑦. (4.69) 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) Let 𝑚∈𝑄𝑀(𝐴) and 𝑎∈𝐴. Using (a), for any (𝑥,𝑦)∈𝐴×𝐴, 𝑚⊙𝜙_𝐴(𝑎)𝑥,𝑦=𝑚𝑥,lim_𝜆𝜙_𝐴(𝑎)𝑒_𝜆,𝑦=𝑚𝑥,lim_𝜆𝑚𝑒_𝜆=(.𝑎𝑦=𝑚𝑥,𝑎𝑦𝑚∘𝑎)𝑥,𝑦 (4.70) Similarly, [𝜙_𝐴(𝑎)⊙𝑚](𝑥,𝑦)=(𝑎∘𝑚)(𝑥,𝑦).
(c) To show that 𝜙_𝐴(𝐴) is a two-sided ideal, let 𝑚∈𝑄𝑀(𝐴) and 𝑎∈𝐴. By Theorem 4.9(a), there exists (𝑆,𝑇)∈𝑀_𝑑(𝐴) such that 𝜙_𝑑(𝑆,𝑇)=𝑚. Using (b), for any (𝑥,𝑦)∈𝐴×𝐴, 𝑚⊙𝜙_𝐴(𝑎)𝑥,𝑦=(𝑚∘𝑎)𝑥,𝑦=𝑚𝑥,𝑎𝑦=𝜙_𝑑()(𝑎)𝑆,𝑇𝑥,𝑎𝑦=𝑥𝑆𝑎𝑦=𝑥𝑆𝑦=𝜙_𝐴(𝑆(𝑎))𝑥,𝑦. (4.71) Hence, 𝑚⊙𝜙_𝐴(𝑎)=𝜙_𝐴(𝑆(𝑎))∈𝜙_𝐴(𝐴). Similarly, 𝜙_𝐴(𝑎)⊙𝑚=𝜙_𝐴(𝑇(𝑎))∈𝜙_𝐴(𝐴).
(d) Suppose that 𝐴 is factorable. Then, by Theorem 4.13(a), each of the maps 𝜙_𝑑, 𝜙_ℓ, 𝜙_𝑟, and 𝜙_𝐴 is a linear isometry. If (𝑆₁,𝑇₁),(𝑆₂,𝑇₂)∈𝑀_𝑑(𝐴), then by definition 𝜙_𝑑𝑆₁,𝑇₁∘_𝜙𝑑𝜙_𝑑𝑆₂,𝑇₂=𝜙_𝑑𝑆₁,𝑇₁𝑆₂,𝑇₂. (4.72)
If 𝑆₁,𝑆₂∈𝑀_ℓ(𝐴), then by definition 𝜙_ℓ𝑆₁∘_𝜙ℓ𝜙_ℓ𝑆₂=𝜙_ℓ𝑆₁𝑆₂. (4.73)
If 𝑇₁,𝑇₂∈𝑀_𝑟(𝐴), then by definition 𝜙_𝑟𝑇₁∘_𝜙𝑟𝜙_𝑟𝑇₂=𝜙_𝑟𝑇₂𝑇₁. (4.74) Hence, 𝜙_𝑑∶𝑀_𝑑(𝐴)→𝑄𝑀(𝐴) and 𝜙_ℓ∶𝑀_ℓ(𝐴)→𝑄𝑀(𝐴) are algebraic isomorphisms and 𝜙_𝑟∶𝑀_𝑟(𝐴)→𝑄𝑀(𝐴) is algebraic anti-isomorphism.

Remark 4.17. If 𝐴 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, 𝜙_𝐴(𝑚(𝑒,𝑒))𝑥,𝑦=𝑥𝑚(𝑒,𝑒)𝑦=𝑚𝑥𝑒,𝑒𝑦=𝑚𝑥,𝑦. (4.75)

5. Quasistrict and Strict Topologies on 𝑄𝑀(𝐴)

In this section, we consider the quasistrict and strict topologies on 𝑄𝑀(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 {𝑒_𝜆∶𝜆∈𝐼}.

Definition 5.1. For any 𝑚∈𝑄𝑀(𝐴) and 𝑎,𝑏∈𝐴, we define mappings 𝑎∘𝑚,𝑚∘𝑎,𝑎∘𝑚∘𝑏∶𝐴×𝐴→𝐴 by (),(),(),𝑎∘𝑚𝑥,𝑦=𝑚𝑥𝑎,𝑦𝑚∘𝑎𝑥,𝑦=𝑚𝑥,𝑎𝑦𝑎∘𝑚∘𝑏𝑥,𝑦=𝑚𝑥𝑎,𝑏𝑦𝑥,𝑦∈𝐴×𝐴. (5.1)

Lemma 5.2. Let 𝑚∈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, ‖𝑎∘𝑚‖_𝑞=sup_{𝑥≠0,𝑦≠0}𝑞(𝑎∘𝑚)𝑥,𝑦𝑞(𝑥)𝑞𝑦=sup_{𝑥≠0,𝑦≠0}𝑞𝑥⋅𝑚𝑎,𝑦𝑞(𝑥)𝑞𝑦≤sup_{𝑥≠0,𝑦≠0}𝑞(𝑥)𝑞𝑚𝑎,𝑦𝑞(𝑥)𝑞𝑦=sup_𝑦≠0𝑞𝑚𝑎,𝑦𝑞𝑦=sup_𝑦≠0‖𝑚‖_𝑞𝑞(𝑎)𝑞𝑦𝑞𝑦=‖𝑚‖_𝑞𝑞(𝑎). (5.2) Similarly, ‖𝑚∘𝑎‖_𝑞≤‖𝑚‖_𝑞𝑞(𝑎).
(b) By Theorem 4.13(b), 𝑞(𝑚(𝑎,𝑏))=‖𝑎∘𝑚∘𝑏‖_𝑞. Further, using (a), ‖𝑎∘𝑚∘𝑏‖_𝑞≤‖𝑎∘𝑚‖_𝑞𝑞(𝑏)≤‖𝑚‖_𝑞𝑞(𝑎)𝑞(𝑏). (5.3)

Definition 5.3. (1) The quasistrict topology 𝛾 on 𝑄𝑀(𝐴) is determined by the family {𝜉_𝑎,𝑏(𝑚)∶𝑎,𝑏∈𝐴} of 𝑘-seminorms, where 𝜉_𝑎,𝑏(𝑚)=‖𝑎∘𝑚∘𝑏‖_𝑞=𝑞(𝑚(𝑎,𝑏)),𝑚∈𝑄𝑀(𝐴). (5.4) Compare with [2, page 109]; [4, page 558].
(2) The strict topology 𝛽 on 𝑄𝑀(𝐴) is determined by the family {𝜂_𝑎(𝑚)∶𝑎∈𝐴} of 𝑘-seminorms, where 𝜂_𝑎(𝑚)=max‖𝑎∘𝑚‖_𝑞,‖𝑚∘𝑎‖_𝑞,𝑚∈𝑄𝑀(𝐴). (5.5) Compare with [8, page 227].
Let 𝜏 denote the topology on 𝑄𝑀(𝐴) generated by the 𝑘-norm ‖⋅‖_𝑞.

Lemma 5.4. 𝛾⊆𝛽⊆𝜏on 𝑄𝑀(𝐴).

Proof. To show that 𝛾⊆𝛽, let 𝑎,𝑏∈𝐴. Then 𝜉_𝑎,𝑏(𝑚)=‖𝑎∘𝑚∘𝑏‖_𝑞≤‖𝑎∘𝑚‖_𝑞𝑞(𝑏)≤𝜂_𝑎(𝑚)𝑞(𝑏),𝑚∈𝑄𝑀(𝐴), (5.6) also 𝜉_𝑎,𝑏(𝑚)=‖𝑎∘𝑚∘𝑏‖_𝑞≤‖𝑚∘𝑏‖_𝑞𝑞(𝑎)≤𝜂_𝑏(𝑚)𝑞(𝑎),𝑚∈𝑄𝑀(𝐴). (5.7) Hence, 𝜉_𝑎,𝑏(𝑚)≤max𝜂_𝑎(𝑚),𝜂_𝑏(𝑚)𝑞(𝑎)𝑞(𝑏),𝑚∈𝑄𝑀(𝐴). (5.8) Let {𝑚_𝛼∶𝛼∈𝐽} be a net in 𝑄𝑀(𝐴) with 𝑚→−_𝛼^𝛽𝑚∈𝑄𝑀(𝐴). Then, for any 𝑎,𝑏∈𝐴, 𝜂_𝑎(𝑚_𝛼−𝑚)→0 and 𝜂_𝑏(𝑚_𝛼−𝑚)→0. Hence, 𝜉_𝑎,𝑏𝑚_𝛼−𝑚≤max𝜂_𝑎𝑚_𝛼−𝑚,𝜂_𝑏𝑚_𝛼−𝑚𝑞(𝑎)𝑞(𝑏)⟶0. (5.9) Thus 𝑚→−_𝛼^𝛾𝑚, and so 𝛾⊆𝛽.
To show that 𝛽⊆𝜏, Let 𝑎∈𝐴. Note that, for any 𝑥,𝑦∈𝐴, ()(𝐴),()()(𝐴).𝑎∘𝑚𝑥,𝑦=𝑚𝑥𝑎,𝑦=𝑥⋅𝑚𝑎,𝑦,𝑚∈𝑄𝑀𝑚∘𝑎𝑥,𝑦=𝑚𝑥,𝑎𝑦=𝑚𝑥,𝑎𝑦,𝑚∈𝑄𝑀 (5.10)
By Lemma 5.2(a), ‖𝑎∘𝑚‖_𝑞≤‖𝑚‖_𝑞𝑞(𝑎) and ‖𝑚∘𝑎‖_𝑞≤‖𝑚‖_𝑞𝑞(𝑎); hence, 𝜂_𝑎(𝑚)=max‖𝑎∘𝑚‖_𝑞,‖𝑚∘𝑎‖_𝑞≤‖𝑚‖_𝑞𝑞(𝑎),𝑚∈𝑄𝑀(𝐴). (5.11) Consequently, if {𝑚_𝛼} is a net in 𝑄𝑀(𝐴) with 𝑚→−_𝛼^𝜏𝑚∈𝑄𝑀(𝐴), then ‖𝑚_𝛼−𝑚‖_𝑞→0, and so, for any 𝑎,𝑏∈𝐴, 𝜂_𝑎𝑚_𝛼−𝑚≤𝑚_𝛼−𝑚_𝑞𝑞(𝑎)⟶0. (5.12) Thus 𝑚→−_𝛼^𝛽𝑚; that is, 𝛽⊆𝜏.

Theorem 5.5. 𝜙_𝐴(𝐴) is 𝛽-dense in 𝑄𝑀(𝐴) and hence 𝛾-dense in 𝑄𝑀(𝐴).

Proof. Let 𝑚∈𝑄𝑀(𝐴). Clearly {𝑚(𝑒_𝜆,𝑒_𝜆)}_𝜆∈𝐼⊆𝐴. We claim that 𝜙_𝐴(𝑚(𝑒_𝜆,𝑒_𝜆))→−^𝛽𝑚. Let 𝑎∈𝐴. We need to show that 𝜂_𝑎𝜙_𝐴𝑚𝑒_𝜆,𝑒_𝜆−𝑚⟶0. (5.13)
For any 𝑥,𝑦∈𝐴, by joint continuity of 𝑚,𝑞𝑎∘𝜙_𝐴𝑚𝑒_𝜆,𝑒_𝜆𝜙−𝑚𝑥,𝑦=𝑞_𝐴𝑚𝑒_𝜆,𝑒_𝜆−𝑚𝑥𝑎,𝑦=𝑞𝑥𝑎𝑚𝑒_𝜆,𝑒_𝜆𝑚𝑦−𝑚𝑥𝑎,𝑦=𝑞𝑥𝑎𝑒_𝜆,𝑒_𝜆𝑦−𝑚𝑥𝑎,𝑦⟶𝑞𝑚𝑥𝑎,𝑦−𝑚𝑥𝑎,𝑦=0. (5.14) Hence ‖𝑎∘{𝜙_𝐴(𝑚(𝑒_𝜆,𝑒_𝜆))−𝑚}‖_𝑞→0. Similarly, ‖{𝜙_𝐴(𝑚(𝑒_𝜆,𝑒_𝜆))−𝑚}∘𝑎‖_𝑞→0. Thus, 𝜂_𝑎𝜙_𝐴𝑚𝑒_𝜆,𝑒_𝜆−𝑚⟶0; (5.15) that is, 𝜙_𝐴(𝐴) is 𝛽-dense in 𝑄𝑀(𝐴). Since 𝛾⊆𝛽, it follows that 𝜙_𝐴(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 {𝑚_𝑖∶𝑖∈ℕ} be a 𝛾-Cauchy sequence in 𝑄𝑀(𝐴). For any 𝑥,𝑦∈𝐴, using Theorem 4.13(b), 𝑞𝑚_𝑖𝑥,𝑦−𝑚_𝑗𝑥,𝑦=𝑞𝑚_𝑖−𝑚_𝑗=𝑚𝑥,𝑦𝑥∘_𝑖−𝑚_𝑗∘𝑦_𝑞=𝜉_𝑥,𝑦𝑚_𝑖−𝑚_𝑗, (5.16) which implies that {𝑚_𝑖(𝑥,𝑦)} is a Cauchy sequence in 𝐴. Define 𝑚∶𝐴×𝐴→𝐴 by 𝑚(𝑥,𝑦)=lim_𝑖𝑚_𝑖(𝑥,𝑦). Clearly, 𝑚 is bilinear and, by Theorem 2.7(b), 𝑚 is jointly continuous. Further, for any 𝑎,𝑏,𝑥,𝑦∈𝐴, 𝑚𝑎𝑥,𝑦𝑏=lim_𝑖𝑚_𝑖𝑎𝑥,𝑦𝑏=𝑎lim_𝑖𝑚_𝑖𝑥,𝑦𝑏=𝑎𝑚𝑥,𝑦𝑏, (5.17) and so 𝑚∈𝑄𝑀(𝐴). Further, for any 𝑎,𝑏∈𝐴, 𝜉_𝑎,𝑏𝑚_𝑖−𝑚=𝑎∘𝑚_𝑖∘𝑏−𝑎∘𝑚∘𝑏_𝑞=𝑞𝑚_𝑖−𝑚(𝑎,𝑏)⟶0. (5.18) Hence 𝑚→−_𝑖^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 𝑆_𝑐,𝑇_𝑐∶𝐴→𝐴 given by𝑆_𝑐(𝑥)=𝑚(𝑐,𝑥),𝑇_𝑐(𝑥)=𝑚(𝑥,𝑐),𝑥∈𝐴, (5.19) define elements in 𝑀_ℓ(𝐴) and 𝑀_𝑟(𝐴), respectively, and it is easy to see that 𝜙_ℓ𝑆_𝑐=𝑐∘𝑚,𝜙_𝑟𝑇_𝑐=𝑚∘𝑐. (5.20)
Let {𝑚_𝑖∶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 {𝜙_ℓ(𝑆_𝑐)_𝑖} and {𝜙_𝑟(𝑇_𝑐)_𝑖}, where (𝑆_𝑐)_𝑖(𝑥)=𝑚_𝑖(𝑐,𝑥) and (𝑇_𝑐)_𝑖(𝑥)=𝑚_𝑖(𝑥,𝑐), are 𝜏-Cauchy in 𝑄𝑀(𝐴). Since 𝜙_ℓ and 𝜙_𝑟 are topological embeddings, the sequences {(𝑆_𝑐)_𝑖} and {(𝑇_𝑐)_𝑖} are ‖⋅‖_𝑞-Cauchy in 𝑀_ℓ(𝐴) and 𝑀_𝑟(𝐴), respectively. Both 𝑀_ℓ(𝐴) and 𝑀_𝑟(𝐴) are complete (Theorem 3.9) and so there exist 𝑆^(𝑐) in 𝑀_ℓ(𝐴) and 𝑇^(𝑐) in 𝑀_𝑟(𝐴) such that 𝑆_𝑐𝑖−𝑆^(𝑐)_𝑞⟶0,𝑇_𝑐𝑖−𝑇^(𝑐)_𝑞⟶0. (5.21)
Since 𝛾⊆𝛽, the sequence {𝑚_i} is 𝛾-Cauchy. As proved above, the space 𝑄𝑀(𝐴) is 𝛾-complete and so there exists an element 𝑚_𝑜 in 𝑄𝑀(𝐴) such thatlim_𝑖𝑚_𝑖𝑥,𝑦=𝑚_𝑜𝑥,𝑦,∀𝑥,𝑦∈𝐴. (5.22) For any 𝑎,𝑏∈𝐴, 𝜙_ℓ𝑆^(𝑐)(𝑎,𝑏)=lim_𝑖𝜙_ℓ𝑆^(𝑐)_𝑖(𝑎,𝑏)=lim_𝑖𝑎𝑚_𝑖(𝑐,𝑏)=𝑐∘𝑚_𝑜(𝑎,𝑏), (5.23) which implies that 𝜙_ℓ(𝑆^(𝑐))=𝑐∘𝑚_𝑜. Similarly, we can prove that 𝜙_𝑟(𝑇^(𝑐))=𝑚_𝑜∘𝑐. Thus, by (5.21), 𝑐∘𝑚_𝑖−𝑐∘𝑚_𝑜𝑞=𝜙_ℓ𝑆^(𝑐)_𝑖−𝜙_ℓ𝑆^(𝑐)_𝑞=𝑆^(𝑐)_𝑖−𝑆^(𝑐)_𝑞𝑚⟶0,_𝑖∘𝑐−𝑚_𝑜∘𝑐_𝑞=𝜙_𝑟𝑇^(𝑐)_𝑖−𝜙_𝑟𝑇^(𝑐)_𝑞=𝑇^(𝑐)_𝑖−𝑇^(𝑐)_𝑞⟶0, (5.24) which implies that 𝑚_𝑜 is the 𝛽-limit of the sequence {𝑚_𝑖} that is, 𝑄𝑀(𝐴) is 𝛽-complete.
(b) Suppose that 𝐴 is strongly factorable. Let {𝑚_𝛼∶𝛼∈𝐽} be a 𝛾-Cauchy net in 𝑄𝑀(𝐴). Replacing the sequence {𝑚_𝑖∶𝑖∈ℕ} by the net {𝑚_𝛼∶𝛼∈𝐽} in part (a), we obtain a map 𝑚∶𝐴×𝐴→𝐴 given by 𝑚(𝑥,𝑦)=lim_𝛼𝑚_𝛼(𝑥,𝑦). Then 𝑚 is bilinear; further, for any 𝑎,𝑏,𝑥,𝑦∈𝐴, 𝑚𝑎𝑥,𝑦𝑏=lim_𝛼𝑚_𝛼𝑎𝑥,𝑦𝑏=𝑎lim_𝛼𝑚_𝛼𝑥,𝑦𝑏=𝑎𝑚𝑥,𝑦𝑏. (5.25) 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 𝑚→−_𝛼^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 of 𝐴.

Theorem 5.8. (𝑄𝑀(𝐴),𝛾), (𝑄𝑀(𝐴),𝛽), and (𝑄𝑀(𝐴),𝜏) have the same bounded sets.

Proof. (a) Since 𝛾⊆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‖𝑎∘𝑚∘𝑏‖_𝑞≤𝑟,∀𝑚∈𝐻,or𝑞[𝑚(𝑎,𝑏)]≤𝑟,∀𝑚∈𝐻usingTheorem4.13(b). (5.26) For each 𝑎∈𝐴 and 𝑚∈𝐻, define 𝑀_𝑎∶𝐴→𝐴 by 𝑀_𝑎(𝑥)=𝑚(𝑎,𝑥),𝑥∈𝐴. (5.27) Then, ℱ={𝑀_𝑎∶𝑚∈𝐻}⊆CL(𝐴). By (5.26), for any 𝑥∈𝐴𝑞𝑀_𝑎(𝑥)=𝑞[𝑚(𝑎,𝑥)]≤𝑟(𝑎,𝑥),∀𝑚∈𝐻; (5.28) hence ℱ is pointwise bounded. Then, by the uniform boundedness principle (Theorem 2.6), there exists 𝑐=𝑐(𝑎)>0 such that 𝑀_𝑎𝑞≤𝑐,∀𝑚∈𝐻. (5.29) Consider now the family 𝑃={𝑝_𝑚∶𝑚∈𝐻} of 𝑘-seminorms on 𝐴 defined by 𝑝_𝑚(𝑎)=𝑀_𝑎𝑞=sup_𝑏≠0𝑞𝑀_𝑎(𝑏)𝑞(𝑏)=sup_𝑏≠0𝑞[𝑚(𝑎,𝑏)]𝑞(𝑏),𝑎∈𝐴. (5.30) For each 𝑚∈𝐻,𝑝_𝑚 is continuous on 𝐴 since, if {𝑎_𝑛}⊆𝐴 with 𝑎_𝑛→𝑎_𝑜 in 𝐴, then 𝑝_𝑚𝑎_𝑛−𝑝_𝑚𝑎_𝑜≤𝑝_𝑚𝑎_𝑛−𝑎_o=sup_𝑏≠0𝑞𝑀_{𝑎𝑛−𝑎𝑜}(𝑏)𝑞(𝑏)=sup_𝑏≠0𝑞𝑚𝑎_𝑛−𝑎_𝑜,𝑏𝑞(𝑏)≤sup_𝑏≠0‖𝑚‖_𝑞𝑞𝑎_𝑛−𝑎_𝑜𝑞(𝑏)𝑞(𝑏)=‖𝑚‖_𝑞𝑞𝑎_𝑛−𝑎_𝑜⟶0. (5.31) Then, by (5.29), the family 𝑃 is pointwise bounded. Applying Theorem 2.5, there exists a ball 𝐵=𝐵(𝑥_𝑜,𝑟)={𝑥∈𝐴∶𝑞(𝑥−𝑥_𝑜)≤𝑟} and a constant 𝐶>0 such that 𝑝_𝑚(𝑥)≤𝐶,∀𝑚∈𝐻,𝑥∈𝐵𝑥_𝑜,𝑟. (5.32) For any fixed 𝑎∈𝐴, we claim that 𝑝_𝑚(𝑎)≤2𝐶⋅𝑞(𝑎)𝑟. (5.33) 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+𝑥_𝑜,𝑥_𝑜∈𝐵(𝑥_𝑜,𝑟), as follows: 𝑞𝑡𝑎+𝑥_𝑜−𝑥_𝑜=𝑞(𝑡𝑎)=𝑡^𝑘(𝑎)≤𝑟⋅𝑞𝑞(𝑎)𝑞(𝑎)𝑞𝑥=𝑟,_𝑜−𝑥_𝑜=𝑞(0)=0<𝑟. (5.34) So, by (5.32), 𝑝_𝑚𝑡𝑎+𝑥_𝑜≤𝐶,𝑝_𝑚𝑥_𝑜≤𝐶. (5.35) Now, using (5.35) and the properties of 𝑘-norm again, 𝑝_𝑚(𝑎)=𝑝_𝑚1𝑡=1𝑡𝑎𝑡^𝑘𝑝_𝑚()≤𝑞(𝑎)𝑡𝑎𝑟𝑝_𝑚𝑡𝑎+𝑥_𝑜−𝑥_𝑜≤𝑞(𝑎)𝑟𝑝_𝑚𝑡𝑎+𝑥_𝑜+𝑝_𝑚𝑥_𝑜≤𝑞(𝑎)𝑟[]=(𝑎)𝐶+𝐶2𝐶⋅𝑞𝑟. (5.36) This proves our claim. Hence, using (5.33), for any 𝑚∈𝐻, ‖𝑚‖_𝑞=sup_{𝑎,𝑏≠0}𝑞[𝑚(𝑎,𝑏)]𝑞(𝑎)𝑞(𝑏)=sup_𝑎≠01𝑞(𝑎)sup_𝑏≠0𝑞[𝑚(𝑎,𝑏)]𝑞(𝑏)=sup_𝑎≠01𝑞(𝑎)𝑝_𝑚(𝑎)≤sup_𝑎≠01𝑞(𝑎).2𝐶⋅𝑞(𝑎)𝑟≤2𝐶𝑟. (5.37) Consequently, 𝐻 is 𝜏-bounded.
(b) This follows from (a) since 𝛾⊆𝛽⊆𝜏.

Acknowledgment

The authors are grateful to the referees for their several useful suggestions, including those mentioned in Remarks 2.12 and 2.14, which improved significantly the quality of this paper.