Journal of Function Spaces

Volume 2014 (2014), Article ID 248925, 4 pages

http://dx.doi.org/10.1155/2014/248925

## The Arens Algebras of Vector-Valued Functions

^{1}Department of Science in Engineering, Faculty of Engineering, International Islamic University Malaysia, P.O. Box 10, 50728 Kuala Lumpur, Malaysia^{2}Department of Algebra and Functional Analysis, National University of Uzbekistan, Vuzgorodok, 100174 Tashkent, Uzbekistan

Received 4 September 2013; Accepted 5 December 2013; Published 23 February 2014

Academic Editor: Dragan Djordjevic

Copyright © 2014 I. G. Ganiev and O. I. Egamberdiev. 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.

#### Abstract

A class of Arens algebras of vector Banach algebra valued functions is considered. It is shown that Arens algebra of vector-valued functions is complete locally convex metrizable algebra.

#### 1. Introduction

Let be a measure space with finite measure and let be the Banach space of all measurable functions on such that , . In [1] the set is introduced and it is shown that is locally convex algebra in topology generated by the norms . Such algebras are called Arens algebras. In [2] it is shown that is algebra. Properties of algebra are considered in [3, 4]. Moreover, it is shown that dual of is and is reflexive. In [5] a class of noncommutative analogue of Arens algebra associated with semifinite von Neumann algebra and semifinite trace is introduced and in [6] it is shown that is algebra. In [6–8], properties of noncommutative Arens algebras are considered. Noncommutative Arens algebras associated with semifinite von Neumann algebra and center-valued trace are introduced in [9] and in [10] they are generalized for the case when is the Maharam trace.

Let be of Bochner maps from into a unital Banach algebra , integrable with degree , , that is, space of vector-valued functions, and In the present paper we prove that is complete locally convex metrizable algebra, which we call Arens algebra of vector-valued functions. It is shown that if dual of has Radon-Nikodym property, then dual space of is ; if is reflexive, then also is reflexive.

#### 2. Preliminaries

Let be a measure space with finite measure; let be the classes of all measurable functions on .

Throughout this paper, stands for a fixed initial Banach algebra with the norm . By we denote the classes of all measurable by Bochner mappings from to ; that is, It is known [11] that is algebra with respect to multiplication operation for any . By , , we denote the Banach space of valued measurable functions f on with the norm defined as that is, and with the norm defined as It is known [11] that the pair is a Banach algebra.

The following well known properties hold true.(a)For the Minkowski inequality is valid.(b)For and , where for and for , the Hölder inequality is valid.

Proposition 1. *
(i) If and then and
**(ii) If , , , , , and , then and .**
(iii) If a.e., , and , then and .*

*Proof. *(i) Let and . Since and we have that and

(ii) Let , , , , , and . As and from inequality we get that and = .

(iii) Let a.e., , and . Then . This means that and .

#### 3. The Arens Algebras of Vector-Valued Functions

In this section we introduce notion of the Arens algebras of vector-valued functions. Put that is, , and will consider in local convex topology is generated by system of norms . From (ii) Proposition 1 we get that , that is, the topology generated by countable system of norms . By Theorem III.2.2 [12] it means that topological vector space is metrizable space with respect to the metric

Theorem 2. * is a complete topological algebra.*

*Proof. *Let be a fundamental sequence in . Then is fundamental in for any . Since is complete there exist such that as , as
for all , implies that . This means that if . Therefore is limit of sequence and is an element of space . Since the inequality
is valid the multiplication is continuous on . Hence is a topological algebra.

Theorem 3. *
(i) If , , and a.e. then and .**
(ii) If and then and there exist *

*Proof. *(i) As we have that for any . Then from a.e. by Proposition 1(iii) we get that for any . This means that . Since for any we have

(ii) Let and ; then for all . By Proposition 1(i) for all . Therefore . We chose that . Then using Proposition 1(i) we get

*Theorem 4. is dense in and is dense in .*

*Proof. *Let . There exists sequence of idempotents from such that , where is unit element on . Then ; that is, . Let . Then and . As norm is order continuous in space for all we get that for all as . Therefore
as . The density in follows from density in .

Let and . Then we can define function by setting
for almost all .

We will denote . is subspace of , because for all .

*Theorem 5. is dense in .*

*Proof. *Let . Then for all .

Let be the sequence simple valued functions; that is, every has form , such that a.e. on . Then almost everywhere on .

Let . If we set we have that a.e. on and
By dominated convergence we have that
for all and of course .

*4. Dual of Arens Algebras *

*In this section we study dual of Arens algebras of vector-valued functions. It is shown that if dual of has Radon-Nikodym property, then dual space of is ; if is reflexive, then also is reflexive.*

*Definition 6 (see [13]). *Recall that a Banach space is said to possess the Radon-Nikodym property, with respect to , if for any -continuous vector measure of bounded total variation, there exists a Bochner integrable function such that
for all . It is known [13] that dual of Banach algebra , has the Radon-Nikodym property.

A class of Banach algebras whose duals have the Radon-Nikodym property is studied in [14].

*Theorem 7. If the dual space of Banach algebra has Radon-Nikodym property with respect to , then the dual of is the space
and for any there exists such that
*

*Proof. *Since is countably normed space by [15] dual of is the space , where . As has the Radon-Nikodym property with respect to by Theorem IV.1.1 [13] , where . Hence, the dual of is the space
where .

Let be a continuous linear functional on . Then by [15] there exists natural number and such that
It means . Using the general form of continuous linear functionals in (see [13], IV.1) we get that there exists such that

*Theorem 8. If is a reflexive Banach algebra then is reflexive.*

*Proof. *Let be a reflexive Banach algebra. Since every reflexive space has Radon-Nikodym property with respect to , by Theorem 7,
This means that is semireflexive space. By Theorem 2 is -space. Since by Corollary II.7.1 [16] every -space is barreled we have that is barreled. Hence by IV.5. [16] is reflexive.

*Conflict of Interests*

*Conflict of Interests*

*The authors declare that there is no conflict of interests regarding the publication of this paper.*

*Acknowledgment*

*Acknowledgment*

*The first author (I. G. Ganiev) acknowledges the MOHE Grant FRGS13-071-0312.*

*References*

*References*

- R. Arens, “The space ${L}^{\omega}$ and convex topological rings,”
*Bulletin of the American Mathematical Society*, vol. 52, pp. 931–935, 1946. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. R. Allan, “On a class of locally convex algebras,”
*Proceedings of the London Mathematical Society*, vol. 17, no. 3, pp. 91–114, 1967. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. J. Bhatt, “On Arens' algebra ${L}^{\omega}$,”
*Glasnik Matematički*, vol. 15(35), no. 2, pp. 305–314, 1980. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. J. Bhatt, “On Arens' algebra ${L}^{\omega}$. II,”
*Glasnik Matematički*, vol. 16(36), no. 2, pp. 297–306, 1981. View at Google Scholar · View at MathSciNet - A. Inoue, “On a class of unbounded operator algebras. II,”
*Pacific Journal of Mathematics*, vol. 66, no. 2, pp. 411–431, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. S. Zakirov, “Noncommutative Arens algebras,”
*Uzbek Mathematical Journal*, no. 1, pp. 17–24, 1997 (Russian). View at Google Scholar · View at MathSciNet - R. Z. Abdullaev, “Spaces dual to noncommutative Arens algebras,”
*Uzbek Mathematical Journal*, no. 2, pp. 3–7, 1997 (Russian). View at Google Scholar · View at MathSciNet - R. Z. Abdullaev and V. I. Chilin, “Arens algebras, associated with commutative von Neumann algebras,”
*Annales Mathématiques Blaise Pascal*, vol. 5, no. 1, pp. 1–12, 1998. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. A. Alimov, “Arens algebras associated with a center-valued trace,”
*Uzbek Mathematical Journal*, no. 1, pp. 12–18, 2000 (Russian). View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. A. Alimov, “Arens algebras of measurable operators for Maharam traces,” http://arxiv.org/abs/1307.2751v1.
- V. I. Chilin, I. G. Ganiev, and K. K. Kudaĭbergenov, “The Gelfand-Naĭmark theorem for
*C**-algebras-algebras over a ring of measurable functions,”*Russian Mathematics*, vol. 52, no. 2, pp. 58–66, 2008. View at Google Scholar · View at MathSciNet - L. V. Kantorovich and G. P. Akilov,
*Functional Analysis*, vol. 14, Pergamon Press, Oxford, UK, 2nd edition, 1982. View at MathSciNet - J. Diestel and J. J. Uhl Jr.,
*Vector Measures*, American Mathematical Society, Providence, RI, USA, 1977. View at MathSciNet - H. S. Mustafayev, “A class of Banach algebras whose duals have the Radon-Nikodym property,”
*Archiv der Mathematik*, vol. 87, no. 5, pp. 449–457, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. Z. Vulikh,
*Introduction to Functional Analysis*, Pergamon Press, New York, NY, USA, 1963. - H. H. Schaefer,
*Topological Vector Spaces*, Springer, New York, NY, USA, 1971. View at MathSciNet

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