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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 235273, 30 pages
Quasimultipliers on -Algebras
1Department of Mathematics, Payamenoor University-Aligodarz Branch, Aligodarz, Iran
2Department of Mathematics and Computer Science, Amirkabir University of Technology, P.O. Box 15914, Tehran, Iran
3Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
Received 7 June 2010; Revised 24 October 2010; Accepted 25 January 2011
Academic Editor: Wolfgang Ruess
Copyright © 2011 Marjan Adib et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We investigate the extent to which the study of quasimultipliers can be made beyond Banach algebras. We will focus mainly on the class of -algebras, in particular on complete -normed algebras, , not necessarily locally convex. We include a few counterexamples to demonstrate that some of our results do not carry over to general -algebras. The bilinearity and joint continuity of quasimultipliers on an -algebra are obtained under the assumption of strong factorability. Further, we establish several properties of the strict and quasistrict topologies on the algebra of quasimultipliers of a complete -normed algebra having a minimal ultra-approximate identity.
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  in 1977. Further developments have been made, among others, by Vasudevan and Goel , Kassem and Rowlands , Lin [5, 6], Dearden , Argün and Rowlands , Grosser , Yılmaz and Rowlands , 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 -algebras or complete -normed algebras.
Definition 2.1. Let be a vector space over the field or . (1)A function is called an -seminorm on if it satisfies for all , if , for all and with , for all , if in , then for all .(2)An -seminorm on is called an -norm if, for any , implies that .(3)An -seminorm (or -norm) on is called -homogeneous ([13, page 160]; [14, pages 90, 95]), where , if it also satisfies for all and .(4)A -homogeneous -seminorm (resp., -norm) on is called, in short, a -seminorm (resp., -norm).
Definition 2.2. (1) A vector space with an -norm is called an -normed space and is denoted by ; if it is also complete, it is called an -space. Clearly, any -normed space is a metrizable TVS with metric given by , .
(2) An -seminorm (or -norm) on an algebra is called submultiplicative if An algebra with a submultiplicative -norm is called an -normed algebra; if it is also complete, it is called an -algebra. An algebra with a submultiplicative -norm is called an -normed algebra. A complete -normed algebra is also called a -Banach algebra in the literature.
(a) If is TVS, then its topology can be defined by a family of -seminorms (see [15, pages 48–51]; [16, pages 2-3]).
(b) If is a metrizable TVS, then may be defined by a single -norm (see [13, 15, 17]).
(c) If is a Hausdorff locally bounded TVS, then may be a single -norm for some , (see [13, 14]).
Note that if is an -normed space, then, for any , the set is a neighbourhood of 0 in , but it need not be a bounded set. In case, if is bounded for some , then becomes a Hausdorff locally bounded TVS and hence, by Theorem 2.3(c), a -normed space for some , .
Definition 2.4. Let be an algebra over or and a topology on such that is a TVS. Then the pair is called a topological algebra if it has a separately continuous multiplication. A topological algebra is said to be locally bounded if it has a bounded neighbourhood of 0 (see [18, page 39]). If is a complete Hausdorff locally bounded topological algebra, then its topology can be defined by a submultiplicative -norm , [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 -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 -normed and -normed spaces, and to [18, 21, 22] for various classes of topological algebras.
If and are topological vector spaces over the field or ), then the set of all continuous linear mappings is denoted by . Clearly, is a vector space over with the usual pointwise operations. Further, if , is an algebra under composition (i.e., , ) and has the identity given by .
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 be a complete metric space and a family of continuous real-valued functions on . If is pointwise bounded from above, then on a certain closed ball it is uniformly bounded above, that is, there exists a constant such that
Theorem 2.6 (see [14, page 39]; [19, page 465]). Let be an -space and any topological vector space. Let be a collection such that is pointwise bounded on . Then is equicontinuous; hence, for any bounded set in , is a bounded set in .
The following version is for bilinear mappings.
Theorem 2.7. Let and be -spaces and any TVS. (a)A collection of bilinear mappings from into is equicontinuous if and only if each is separately continuous and is pointwise bounded on . In particular, every separately continuous bilinear map is jointly continuous (see [13, page 172]; [19, page 489]).(b) Let be a sequence of separately continuous bilinear mappings such that exists for each . Then is equicontinuous and f is bilinear and jointly continuous (see [19,page 490]; [24, page 328]).
Definition 2.8. (1) A net in a topological algebra is called an approximate identity if
(2)An approximate identity in an -normed algebra is said to be minimal if for all .(3)An algebra is said to be left (resp., right) faithful if, for any , (resp., ) implies that ; is called faithful if it is both left and right faithful. One mentions that is faithful in each of the following cases:(i) is a topological algebra with an approximate identity (e.g., is a locally -algebra);(ii) is a topological algebra with an orthogonal basis .
Definition 2.9. A topological algebra is called(1)factorable if, for each , there exist such that ,(2)strongly factorable if, for any sequence in with , there exist and a sequence (resp., in with (resp., such that (resp., for all .
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  and its references). Using the terminology of , we state the following version in the nonlocally convex case.
Theorem 2.10 (see ). Let be a fundamental -algebra with a uniformly bounded left approximate identity. Then is strongly factorable.
Definition 2.11. Let be an -normed space (in particular, an -normed algebra). For any , let It is easy to see that if is a -norm, , (resp., a seminorm) on , then is a -norm (resp., a seminorm) on ; further, in these cases, we have alternate formulas for as for each (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 an -norm on , given by always exists for each . We are grateful to referee for pointing out that this assumption cannot be justified in view of the following counterexamples.
(1)First, need not be finite for a general -norm. For example, let , , . Then is an -norm on , but : for any , (2)Even when considering the subspace of those for which , then need not always be an -norm, since need not hold. For example, for a fixed sequence with , , consider the -algebra of sequences with and . Then for all multipliers of , but is not an -norm, that is, it makes the space 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 (cf. [14, Example , page 8]).
In view of the above remark, we will need to assume that is a -normed space (or a -normed algebra) whenever is considered for or .
Some useful properties of are summarized as follows.
Theorem 2.13 (see [14, pages 101-102]). Let be a -normed space (in particular, a -normed algebra) with . Then: (a)a linear mapping is continuous ;(b) is a -norm on ;(c) for all ;(d)for any , ; hence is a -normed algebra;(e)if is complete, then is a complete -normed algebra.
Remark 2.14. The referee has enquired if the present theory can be considered for the class of locally convex -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 , which need not be submultiplicative but satisfy the weaker condition ; however, for locally -convex -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 -convex algebra -algebras parallel to the one given by Phillips (see [29, pages 177–180]) for multipliers.
3. Multipliers on -Normed Algebras
In this section, we recall definitions and results on various notions of multipliers on an algebra (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 ). Let be an algebra over the field or .
(1)A mapping is called a(i)multiplier on if for all ,(ii)left multpilier on if for all ,(iii)right multiplier on if for all .(2)A pair of mappings is called a double multiplier on if for all .
Some authors use the term centralizer instead of multiplier (see, e.g., [30, 31]).
Let (resp., denote the set of all multipliers (resp., left multipliers, right multipliers) on and the set of all double multipliers on an algebra . For any , let be given by Clearly, , , and .
For convenience, we summarize some basic properties of multipliers in the following theorems for later references.
Theorem 3.2 (see ). Let be an algebra. Then, (a); (b)if is faithful, then and hence ;(c)if is commutative and faithful, then ;(d) and are algebras with composition as multiplication (i.e., ) and have the identity , ;(e) is a vector space; if, in addition, is faithful, then is a commutative algebra (without being commutative) with identity .
Theorem 3.3 (see ). Let be a faithful algebra. Then, (a)if , then (i) and are linear and (ii) and . In particular, every is linear;(b) is an algebra with identity under the operations (c)let . If , then ; if then ; (d)if is commutative, then is commutative and ; in fact, if , then .
Definition 3.4. One defines mappings , , and by
Theorem 3.5 (see ). Let be an algebra, and let , , and be the mappings as defined above. Then, (a), and are linear;(b) and are algebra homomorphisms, while is an algebra antihomomorphism;(c) (resp., is 1-1 is left (resp., right) faithful; is 1-1 is faithful;(d) (resp., is onto has left (resp., right) identity; is onto has an identity.
Theorem 3.6 (see ). Let be an algebra. (a)For any and , ; hence is a left ideal in .(b)For any and , ; hence is a left ideal in .(c)Suppose that is faithful. Then, for any and , hence is a two-sided ideal in .
Regarding the continuity of multipliers, we state the following.
(a) Suppose that is a strongly factorable -normed algebra. If (resp., ), then is linear and continuous (see [32, 33]).
(b) Suppose that is a faithful -algebra. If , then and are linear and continuous; in particular each is linear and continuous (see [31, 33]).
Convention. In the remaining part of this paper, unless stated otherwise, is a topological algebra and (resp., ) denotes the set of all continuous linear multipliers (resp., left multipliers, right multipliers) on and denotes the set of all double multipliers on with both and continuous and linear.
Definition 3.8 (see [31, 33]). Let be a topological algebra. The uniform operator topology (resp., the strong operator topology ) on is defined as the linear topology which has a base of neighborhoods of 0 consisting of all the sets of the form where is a bounded (resp., finite) subset of and is a neighborhood of 0 in . Clearly, . Note that the and topologies can also be defined on the multiplier algebras and in an analogous way. (The topology is also called the strict topology in the literature and denoted by .) There is an extensive literature on the and topologies (see, e.g., [30, 33–35, 37–39, 41–45]).
Theorem 3.9 (see ). Let be a faithful -algebra, and let denote any one of the algebras , , , and . Then, (a) and are topological algebras with separately continuous multiplication; (b) and are complete;(c) and have the same bounded sets; (d)if is metrizable, then on ; (e)if has a two-sided approximate identity, then is -dense in .
Remark 3.10. Let be an -normed algebra.(1)If is a -normed algebra, the -topology on , , and is given by the -norm the -topology on is given by the -norm (2)The -topology on , , and is given by the family of of -seminorms, where the -topology on is given by the family of -seminorms, where
Theorem 3.11. Let be a -normed algebra having a minimal approximate identity . Then, (a)for any , ; so each of the maps , , is an isometry and hence continuous; (b)for any , ;(c)if is complete, then is a -closed two-sided ideal in , under the identification .
Proof. (a) Let . Then
On the other hand,
Hence . Similarly, . Thus
(b) Let . Using (a), we have Similarly, Thus, .
(c) In view of Theorem 3.6(c), we only need to show that is -closed in . Let with . Choose such that . By part (a), is an isometry. Hence is a Cauchy net in . Then Thus is -closed in .
4. Quasimultipliers on -Algebras and -Normed Algebras
In this section, we consider the notion of quasimultipliers on -algebras and complete -normed algebra and extend several basic results of McKennon , Kassem and Rowlands , Argün and Rowlands , and Yılmaz and Rowlands  from Banach algebras to these classes of topological algebras.
Definition 4.1 (see [2, 4]). Let be an algebra. A mapping is said to be a quasimultiplier on if
for all .
The following Lemma shows in particular that every left multiplier, right multiplier, multiplier, and double multiplier on an algebra can be viewed as quasimultiplier on .
Lemma 4.2. Let be a faithful algebra. (a)For any , define by (b)For any , define an associated map by (c)For any , define an associated map by (d)For any , define an associated map by (e)For any , define an associated map by Then each of the maps defined above is a quasimultiplier on .
Proof. We only prove (e). Let , for any , In a similar way, .
Theorem 4.3. Suppose that is a strongly factorable -algebra. Then, (a)A map is a quasimultiplier on if and only if (b)every quasimultiplier on is bilinear; (c)every quasimultiplier on is jointly continuous.
Proof. (a) If is a quasimultiplier on , then clearly, for any ,
Conversely, let . Since is a strongly factorable and , there exist such that , . Then, using (4.8), Similarly, we obtain .
(b) Let and . Choose, as above, such that , . Then, Similarly, . Next, First, we show that is separately continuous. Let and with limit . Then converges to 0. By strong factorability, there exist a sequence and an element of such that , . Thus,
Now, the joint continuity of follows directly from Theorem 2.7(a).
Theorem 4.4. Let be a commutative algebra. Then, (a) for any quasimultiplier on and ;(b)if is also faithful, then a bilinear map is a quasimultiplier on if and only if
(a) By hypothesis,
(b) This is obvious.
For all , using (4.14), On the other hand, we have Comparing (4.16) and (4.17), we obtain
Now, for all , using (4.18) twice, By commutativity of and using (4.18) twice as above, Comparing (4.19) and (4.20), . Since this holds for all and is faithful, . Hence, for all , Since this holds for all and is faithful, .
A similar computation shows that . Hence is a quasimultiplier.
Definition 4.5. Let denote the set of all bilinear jointly continuous quasimultipliers on a topological algebra . Clearly, is a vector space under the usual pointwise operations. Further, becomes an -bimodule as follows. For any and , we can define the products and as mappings from into given by Then , so that is an -bimodule.
Definition 4.7. (1) A bounded approximate identity in a topological algebra is said to be ultra-approximate if, for all and , the nets and are Cauchy in (see ).
(2) A topological algebra is called -symmetric if, for each , there is a such that either or ).
Theorem 4.8. Let be an -algebra with a bounded approximate identity . Consider the following conditions. (a) is ultra-approximate.(b)For any , , and , the nets and are Cauchy in .(c) is -symmetric. Then (a) (b) (c). If is factorable, then (c) (a); hence (a), (b), and (c) are equivalent.
(a) (b) Suppose that in is ultra-approximate. Let and . Put and . Then and, for any , and which are Cauchy in by hypothesis.
(b) (c) Suppose that (b) holds, let , and suppose that . Since is complete, the map given by is well-defined. Since is continuous, for any , Since is a faithful -algebra, by Theorem 3.7(b), . Hence is -symmetric.
(c) (b) Suppose that (c) holds. Let and . By (c), there exist , such that ,. Then, for any , Thus, both and , being convergent, are Cauchy in .
Suppose that is factorable. Then (c) (a), as follows. Let and . By factorability, for some . Define the mappings by Then, for any , and so and . By (c), there exist , such that . Then Hence and are Cauchy in . So is ultra-approximate.
Theorem 4.9. Let be an -algebra having an ultra-approximate identity . Then, (a)each of the maps , , , and is a bijection;(b); (c).
Proof. (a) We give the proof only for . To show that is onto, let . Since is ultra-approximate, for each , the nets and are convergent. For each , define by
Then since, for any ,
Further, we have for any ,
that is, .
To show that, is one to one, let with . Then, for any , Since is faithful, . So . Consequently, by Theorem 3.3(c), also . Thus .
(b) Let and . Then, since and , Also
(c) For any and , Thus .
We obtain the following lemma for later use.
(a) If is a factorable -normed algebra having an approximate identity , then
(b) If is an -algebra having a minimal ultra-approximate identity , then, for any , Compare with [10, page 124].
Proof. (a) Let . Since is factorable, there exist such that . Then Since is complete and is ultra-approximate, for any , (say) exists. Since ,
Definition 4.11. Let be a -normed algebra. For any , we define Clearly,
Theorem 4.12. Let be a -normed algebra. Then, (a) is a -norm on ;(b)if is complete, then so is .
Proof. (a) For any ,
Also, for any and so it follows that .
Next, let , and let . Choose such that Then, Since is arbitrary, . Thus is a -norm on .
(b) Let be a -Cauchy sequence in . Then, for any , Therefore, for any in is a Cauchy sequence in . Since is complete, the map given by is welldefined. Clearly, is bilinear and, by Theorem 2.7(b), is jointly continuous. Further, for any , Hence, . Next, as follows. Let . Since is -Cauchy, there exists an integer such that that is, Let , . Fix any in (4.51); since in , letting in (4.51), Then, for any , using (4.51) and (4.52), Thus .
Theorem 4.13. Let be a factorable -normed algebra having a minimal approximate identity . Then (a)each of the maps , , , and defined above is a linear isometry;(b)for any and , .
Proof. (a) We give the proof only for . Clearly, is linear. Let . Then To prove the reverse inequality, let . There exists such that . For any , since