Simple Proofs for Bochner-Schoenberg-Eberlein and the Bochner-Schoenberg-Eberlein Module Properties on <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.993901pt" id="M1" height="16.908pt" version="1.1" viewBox="-0.0657574 -13.9141 59.3156 16.908" width="59.3156pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-127" d="M442 569C442 628 407 660 362 660C328 660 292 648 255 616C200 568 165 508 123 334L106 264C80 242 52 222 23 202L30 176C54 191 76 205 100 222L87 166C73 103 79 43 95 19C108 0 139 -12 169 -12C242 -12 318 38 369 106L354 129C322 93 260 49 215 49C173 49 147 81 170 187L189 274C293 344 442 452 442 569ZM380 565C380 478 267 377 196 322L216 406C254 556 282 623 332 623C364 623 380 596 380 565Z"/></g><g transform="matrix(.012,0,0,-0.012,7.982,-7.578)"><path id="g50-113" d="M573 302C573 402 527 451 414 451C386 451 343 446 313 437L330 513L320 522C295 508 261 484 243 463L230 415C194 400 159 383 126 359L131 330C159 344 187 357 222 368L109 -147C96 -204 80 -214 18 -223L13 -255L256 -244L259 -212L236 -210C184 -205 180 -195 191 -141L219 -1C240 -10 268 -12 284 -12C352 4 431 48 484 104C543 166 573 240 573 302ZM481 290C481 165 381 37 305 37C280 37 249 56 235 71L302 395C328 399 353 402 372 402C427 402 481 378 481 290Z"/></g><g transform="matrix(.017,0,0,-0.017,15.837,0)"><path id="g113-41" d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"/></g><g transform="matrix(.017,0,0,-0.017,21.775,0)"><path id="g113-89" d="M748 650H522L515 622L546 617C580 611 587 604 565 575C518 513 469 451 419 393C376 474 349 534 330 580C318 609 325 612 361 618L383 622L392 650H151L144 622C214 616 224 612 257 543L360 327C270 218 187 124 159 95C106 40 92 34 26 28L17 0H252L259 28L236 31C189 37 188 47 209 78C249 136 308 210 377 294L478 79C494 44 487 37 449 32L418 28L409 0H673L680 28C596 34 591 39 554 116L436 361C526 469 574 521 604 553C659 612 669 614 739 622L748 650Z"/></g><g transform="matrix(.017,0,0,-0.017,34.214,0)"><path id="g113-45" d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z"/></g><g transform="matrix(.017,0,0,-0.017,41.003,0)"><path id="g113-66" d="M686 28C612 35 607 44 591 112C563 234 541 360 519 489L489 666L457 658L147 121C100 40 89 36 24 28L17 0H240L250 28C168 34 159 41 190 101L262 237H482C495 180 503 137 510 91C517 47 514 35 441 28L433 0H677L686 28ZM475 280H285L429 541H431L475 280Z"/></g><g transform="matrix(.017,0,0,-0.017,53.061,0)"><path id="g113-42" d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"/></g></svg>

Tavkoli, Shirin; Abazari, Rasoul; Jabbari, Ali

doi:https://doi.org/10.1155/2024/5893357

Journal of Function Spaces

On this page

Abstract Introduction Conclusion Data Availability Conflicts of Interest Authors’ Contributions References Copyright Related Articles

Research Article | Open Access

Volume 2024 | Article ID 5893357 | https://doi.org/10.1155/2024/5893357

Simple Proofs for Bochner-Schoenberg-Eberlein and the Bochner-Schoenberg-Eberlein Module Properties on

Shirin Tavkoli,¹Rasoul Abazari,¹and Ali Jabbari²

Academic Editor: Richard I. Avery

Received06 Nov 2023

Revised03 Apr 2024

Accepted05 Apr 2024

Published02 May 2024

Abstract

Let be a nonempty set, be a commutative Banach algebra, and . In this paper, we present a concise proof for the result concerning the BSE (Banach space extension) property of . Specifically, we establish that possesses the BSE property if and only if is finite and is BSE. Additionally, we investigate the BSE module property on Banach -modules and demonstrate that a Banach space serves as a BSE Banach -module if and only if is finite and represents a BSE Banach -module.

1. Introduction

The abbreviation BSE refers to the well-known Bochner-Schoenberg-Eberlein theorem, which provides a characterization of the Fourier-Stieltjes transforms of bounded Borel measures on locally compact abelian groups. In essence, this theorem describes the BSE property of the group algebra for a locally compact abelian group . For further information, we refer to [1–4]. Additionally, for integrable functions on the positive real line, denoted as , for the function space consisting of bounded complex-valued continuous functions on a locally compact Hausdorff space , and for characterizing the Fourier and Fourier-Stieltjes transforms on locally compact abelian groups, consult references [5–7].

Let be a commutative Banach algebra with the character space and be a Banach -module. We define a multiplier from into as an -module morphism from into , denoted by . For any , there exists a unique vector field on such that for all . When , we denote by , see [8], for more details.

A bounded continuous function on is called a BSE function if there exists a constant such that for any finite number of and complex numbers , the inequality holds, where is the first dual of . The BSE norm of , denoted as , is defined as the infimum of all such . The set of all BSE functions is denoted by .

A Banach algebra is called a BSE algebra if the BSE functions on are precisely the Gelfand transforms of the elements of , i.e., . This notion was introduced by Takahasi and Hatori in [9] and characterized by Kaniuth and Ülger in [10]. There are many literatures that contain interesting results on BSE algebras; see [10–20] for more details.

Takahasi in [21] generalized the BSE property to Banach modules. Let be a commutative Banach algebra with a bounded approximate identity and be a symmetric Banach -bimodule, i.e., , for all and . Let . Denote by . There exists such that . Now, define where is the closed linear span. Note that is independent of the choice of . Then, becomes a Banach -submodule of . Now, define and , for all . Hence, becomes a Banach -bimodule. Let be the class of all functions defined on such that . An element of is called a vector field on . The space is an -module by the following action:

Set

For each , define , for all . A vector field is called BSE if there exists such that for any finite number of and the same number , we have where denotes the dual space of the Banach space . Moreover, set

A vector field is called continuous if it is continuous at every . The class of all continuous vector fields in is denoted by and set . Let and . A Banach -module is called BSE if , for all . In [21], some examples of Banach algebras that have BSE module property such as group algebras on locally compact groups are given, and in [11], authors characterized module property of module extensions of Banach algebras.

Similar to , we define as the class of all functions defined on such that . By the similar module action that we have defined for , becomes an -module.

Let be a nonempty set and be a commutative Banach algebra. Suppose that and define

Then, with the pointwise product and the following norm becomes a commutative Banach algebra:

Some interesting results related to the maximal ideal space of and BSE property on it are given in [16]. Moreover, some results such as the regularity of and the existence of BED on are obtained in [22].

In the subsequent section, our initial focus is on examining the BSE property within the context of the -direct sum of Banach algebras. Through the insights gained from these investigations, we are able to provide a streamlined proof for the primary outcome established in [16]. This streamlined proof offers a clear and concise demonstration that can enhance comprehension of the BSE concept as it pertains to . Moving forward to Section 3, we delve into exploring the BSE module property concerning Banach modules over . Our key finding in this section establishes that for any -module, denoted as , to exhibit the BSE property, it is a necessary and sufficient condition that the module itself possesses the BSE attribute as an -module gate in the BSE module property on Banach modules over , and as a main result, we prove that for any -module, is BSE if and only if is BSE as an -module.

2. A Simple Proof for BSE Property on

Kamali and Abtahi proved the following result in [16].

Theorem 1. Let be a set and be a commutative semisimple Banach algebra. Then, is a BSE algebra if and only if is finite and is a BSE algebra.
In this section, we present an alternative and straightforward demonstration of the aforementioned outcome. Initially, we provide pertinent details pertaining to the direct sum of Banach algebras (modules). Let and be two commutative Banach algebras and such that . Consider the -direct sum algebra of and with the coordinate wise sum and product, i.e., for all , where is the Cartesian product of and . For , we equip by the following norm: Moreover, for , we equip by the following norm:

Throughout this section, the above notations are supposed to hold. Note that for , is the direct sum of and and we refer [17], for more details. To achieve our goal in this section, we need some results related to a direct sum of Banach algebras. We summarize some properties of the above structures as follows.

Proposition 2. (i), for , and are commutative Banach algebras(ii)The first dual of , for and , is ; this holds for any Banach spaces and (iii)Let and . Then, , for (iv) has a bounded approximate identity if and only if and have bounded approximate identity

Proof. (i) and (ii) are clear. (iii) is Lemma 2.1 in[17]. Let and be a bounded approximate identity for . Hence, there exists such that . This implies that . Moreover, for all . Thus, is a bounded approximate identity for . Similarly, becomes a bounded approximate identity for . For the case , similar to (12), we have for all . The rest of the proof is similar.
Conversely, suppose that and are bounded approximate identities for and , respectively. Then, there exist such that and . Now, we set and . Then, This means that is bounded net in . For the case , the proof is similar. For any , Thus, is bounded approximate identity for . For the case , we have This completes the proof.

The BSE property on direct sums of Banach algebras is investigated in [17], where the authors showed that for commutative semisimple Banach algebras and , is BSE if and only if and are BSE of Theorem 2.4 in [17]. Similarly, we have the following.

Theorem 3. Let and be two commutative semisimple Banach algebras and . Then, is BSE if and only if and are BSE.

Proof. The proof for is the proof of Theorem 2.4 in [17]. Now, let , , and be BSE. Similar discussion in the proof of Theorem 2.4 in [17] implies that , and moreover, for any , there exist and such that , , and . By , there exists such that for any finite number of and , we have Then, by (17) and letting , for all , we have This implies that Thus, and so is in . Similarly, by letting , for all in (17), we obtain that . These together imply that . This means that . Hence, is BSE. The converse is similar to the converse case of Theorem 2.4 in [17].

As an immediate result, we have the following result that plays an important role in the proof of the main result of this section.

Corollary 4. Let be commutative semisimple Banach algebras and . Then, is BSE if and only if each is BSE, for all .

Lemma 5. Let be a nonempty set and be a commutative Banach algebra. If is finite, then is a -direct sum of copies of , where is the cardinal number of .

Proof. If is finite, then there exists such that is isomorphic to . Hence, without loss of generality, we can see as follows: This implies that

If is a finite set and we show its cardinal number by , then we denote by . A bounded net is called a -weak approximate identity for , if , for all . Clearly, every bounded approximate identity for a Banach algebra is a -weak approximate identity. We now investigate the existence of bounded approximate identity for . We recall the following result.

Theorem 6 (see [16], Theorem 2.6). Let be a set, be a commutative Banach algebra, and . Then, has a -weak approximate identity if and only if is finite and has a -weak approximate identity.

Now, we are ready to give a simple proof for Theorem 1.

Proof of Theorem 1. Let be BSE. Then, by Corollary 5 in [9], it has a -weak approximate identity. Thus, Theorem 6 implies that is finite and has -weak approximate identity. Then, by employing Lemma 5, we see that is as . Then, by applying Corollary 4, we conclude that is BSE. The converse holds, clearly, because if is finite and is BSE, again by Corollary 4, is BSE and Lemma 5 implies that is BSE.

3. BSE Module Property of -Modules

The main aim of this section is the investigation of the existence of BSE module property on -modules, where is a commutative Banach algebra.

Lemma 7. Let be a nonempty set, be a commutative Banach algebra, and be in Banach -module. Then, is a Banach -module.

Proof. We define the left module action of on as follows: It is easy to verify that the above-defined left action is well defined and Thus, is a Banach -module.
Our main result in this section is the following result, and we give a short and simple proof for it.

Theorem 8. Let be a nonempty set, be a commutative Banach algebra, and be in Banach -module. Then, is a BSE Banach -module if and only if is finite and is a BSE Banach -module.

Before proving the above result, we give some results related to -direct sums of Banach algebra and their module properties. Let and be Banach - and -modules, respectively. For , we consider the Banach space with the norm . Then, by the following action, becomes a Banach -module:

From now on, we suppose that and have bounded approximate identities.

Proposition 9. if and only if there exist and such that

Proof. Let . Define by , by , by , and by , for all , , , and . These maps are linear and bounded. Finally, we define , , , and . Clearly, these maps are linear and bounded, and we have For any , (27) implies that Letting in (28) and (29) and possessing a bounded approximate identity for together imply that . Similarly, by letting , we obtain that . Moreover, we have and , for all and . These show that and . The converse can be verified easily. So the proof holds.

Proposition 10. Let , , be a Banach -module, and be Banach -module. Then, (i) and (ii) and as Banach spaces(iii), where such that and

Proof. (i)Let and . Then, . This means that . Hence, . Now, suppose that . Thus, . We claim that and . Assume towards a contradiction, . This implies that . These facts say that implies that . For any , similar to proof of Lemma 2.1 in [17], we haveOn the other hand, for any , So (31) and (32) imply that for all and . If , by letting and , we conclude that , for all . Hence, , for all . This means that , a contradiction. Thus, . Similarly, this holds for . Hence, if , then and . This shows that . (ii)Suppose that and such that and . Let and . Then, there exist , , , and such that, for every , we haveThen, for any and , (34) and (35) imply that These show that and . Now, let . Then, from (i), there exist and such that, for every , Thus, (37) implies that . This implies that , and so, we have . This implies that . Similarly, one can verify that . Thus, (ii) holds. (iii)Define by , for all . It is easy to verify that is a continuous homomorphism between Banach spaces. Moreover,This implies that Similarly, define by , for all . We have Then, (iv)Define , , and , for all and . Let and ; then, for any and , there exist , , and such thatLet and , where . Then, by Proposition 2 (iii), or . Thus, rearrange ’s as follows: By (iii) and Proposition 2 (ii), we have and , for any . Thus, there exist and such that First, we suppose that , so . Then, by employing (42) and (43) and the fact that , for all , we have where . Thus, Now, let ; then, similar to the above discussion, we have where . Hence, Moreover, from the continuity of and on and , we have .
Let . Then, there exists such that for any and , Then, ’s and ’s are similar to (44) and (45). Moreover, for any , and . Then, by employing (iii), there exist and such that and . If for any , we suppose that , and for , ; (50) implies that Thus, . Moreover, continuity of on implies that is continuous and this means that . Similarly, by letting , for and , for all , we conclude that .

Theorem 11. Let be a Banach -module and be Banach -module. Then, is a BSE Banach -module if and only if is a BSE Banach -module and is a BSE Banach -module.

Proof. Suppose that is a BSE Banach -module. Let and . Define by and , for all and . Since and are BSE, similar to the proof of Proposition 10 (iv), we can conclude that . On the other hand, is a BSE Banach -module, so there exists such that . Moreover, , for all . By Proposition 9, there exist and such that By letting , in the above equation, we have , for all . Then, Moreover, Thus, (53) and (54) imply that , for all . This implies that . This means that . Similarly, by letting in (52) and by the similar arguments as the above, we obtain that . Therefore, we have .
Now, let . Then, there exist and and (52) holds. Thus, there exists such that . Then, by Proposition 10 (iv), there exist and such that and . Choose an element such that . Then, This implies that . Thus, . Hence, we obtain that . This means that is a BSE Banach -module. Similarly, if satisfies , then we obtain . Thus, we have and so . This implies that is BSE a Banach -module.
Conversely, suppose that and are BSE Banach -module and -module, respectively. Let , where . By Proposition 10 (iv), there exist and such that , , and , for all and . Then, there exist and such that and . Define by , for all . Then, by Proposition 9, is in . Assume that and such that and . Then, On the other hand, from Proposition 10 (iv), is in such that and , for all and . Thus, (56) and (57) imply the , and consequently, . Now, we show that . Let . Again, by Proposition 9, there exist and satisfying (52). Thus, there exist and such that and . By similar argument in (56) and (57), we conclude that . Hence, . This completes the proof.

Example 1. (i)Let and be two commutative -algebras and and be closed ideals of and , respectively. From Theorem 3.1 in [21], and are BSE Banach - and -modules. Then, by Theorem 11, is a BSE Banach -module(ii)Let and be two compact abelian groups. Then, by Theorem 3.3 in [21], and , where , are BSE Banach - and -modules. Hence, Theorem 11 implies that is a BSE Banach -module

Proposition 12. has a bounded approximate identity if and only if has a bounded approximate identity and is finite.

Proof. Suppose that has a bounded approximate identity such that . This implies that it has a -weak approximate identity. Then, by Theorem 6, we conclude that is finite. For a fixed and any , define . Thus, and This shows that is a bounded net in . Moreover, Hence, is a bounded approximate identity for . Conversely, suppose that is finite and has a bounded approximate identity. Thus, by Lemma 5, is as . Then, Proposition 2 (iv) implies , and consequently, has a bounded approximate identity.

Proof of Theorem 4. Since has a bounded approximate identity, Proposition 12 implies that is finite and has a bounded approximate identity. Hence, by Lemma 5, one can see as and as . Now, by employing Theorem 11, the proof holds.

4. Conclusion

The BSE property and BSE module property in commutative Banach algebras and Banach modules are crucial for understanding the relationships between their multiplier spaces and maximal ideal spaces (character spaces). As mentioned earlier, the presence of these properties provides valuable insights into the structures of the spaces under investigation. In this study, we explore the BSE and BSE module properties of vector-valued functions belonging to . By examining the -direct sum of commutative Banach algebras, we present a concise and straightforward proof of the BSE property on , contrasting with the more complex proof presented in [16]. Furthermore, through the utilization of the -direct sum and the characterization of module multiplier spaces, we delve into the BSE module property of modules over . Our analysis reveals that is a BSE Banach -module if and only if is finite and is a BSE Banach -module.

Data Availability

The data supporting the findings of this study are available within the article.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

References

S. Bochner, “A theorem on Fourier-Stieltjes integrals,” Bulletin of the American Mathematical Society, vol. 40, no. 4, pp. 271–276, 2003.
View at: Publisher Site | Google Scholar
W. F. Eberlein, “Characterizations of Fourier-Stieltjes transforms,” Duke Mathematical Journal, vol. 22, no. 3, pp. 465–468, 1955.
View at: Publisher Site | Google Scholar
W. Rudin, Fourier Analysis on Groups, Interscience, Courier Dover Publications, New York, 1962.
I. J. Schoenberg, “A remark on the preceding note by Bochner,” Bulletin of the American Mathematical Society, vol. 40, pp. 277-278, 2003.
View at: Google Scholar
Z. Kamali and M. Lashkarizadeh, “The Bochner-Schoenberg-Eberlein property for ,” Journal of Fourier Analysis and Applications, vol. 20, no. 2, pp. 225–233, 2014.
View at: Publisher Site | Google Scholar
A. T.-M. Lau and H. Le Pham, “Characterisations of Fourier and Fourier–Stieltjes algebras on locally compact groups,” Advances in Mathematics, vol. 289, pp. 844–887, 2016.
View at: Publisher Site | Google Scholar
S.-E. Takahasi, K. Shirayanagi, and M. Tsukada, “Isomorphism problem in a special class of Banach function algebras and its application,” Constructive Mathematical Analysis, vol. 4, no. 3, pp. 305–317, 2021.
View at: Google Scholar
R. Larsen, An Introduction to the Theorey of Multipliers, Springer, New York, 1971.
View at: Publisher Site
S.-E. Takahasi and O. Hatori, “Commutative Banach algebras which satisfy a Bochner-Schoenberg-Eberlein-type theorem,” Proceedings of the American Mathematical Society, vol. 110, no. 1, pp. 149–158, 1990.
View at: Google Scholar
E. Kaniuth and A. Ülger, “The Bochner-Schoenberg-Eberlein property for commutative Banach algebras, especially Fourier and Fourier-Stieltjes algebras,” Transactions of the American Mathematical Society, vol. 362, no. 8, pp. 4331–4356, 2010.
View at: Publisher Site | Google Scholar
N. Alizadeh, S. Ostadbashi, A. Ebadian, and A. Jabbari, “The Bochner–Schoenberg-Eberlein property of extensions of Banach algebras and Banach modules,” Bulletin of the Australian Mathematical Society, vol. 105, no. 1, pp. 134–145, 2022.
View at: Publisher Site | Google Scholar
P. A. Dabhi, “Multipliers of perturbed Cartesian products with an application to BSE-property,” Acta Mathematica Hungarica, vol. 149, no. 1, pp. 58–66, 2016.
View at: Publisher Site | Google Scholar
P. A. Dabhi and R. S. Upadhyay, “The semigroup algebra is a Bochner–Schoenberg–Eberlein (BSE) algebra,” Mediterranean Journal of Mathematics, vol. 16, no. 1, 2019.
View at: Publisher Site | Google Scholar
A. Ebadian and A. Jabbari, “The Bochner-Schoenberg-Eberlein property for amalgamated duplication of Banach algebras,” Journal of Algebra and Its Applications, vol. 21, no. 8, article 2250155, 2022.
View at: Publisher Site | Google Scholar
M. Fozouni and M. Nemati, “BSE-property for some certain Segal and Banach algebras,” Mediterranean Journal of Mathematics, vol. 16, no. 2, 2019.
View at: Publisher Site | Google Scholar
Z. Kamali and F. Abtahi, “The Bochner-Schoenberg-Eberlein property for vector-valued -spaces,” Mediterranean Journal of Mathematics, vol. 17, no. 3, p. 94, 2020.
View at: Publisher Site | Google Scholar
Z. Kamali and M. Lashkarizadeh Bami, “The multiplier algebra and BSE property of the direct sum of Banach algebras,” Bulletin of the Australian Mathematical Society, vol. 88, no. 2, pp. 250–258, 2013.
View at: Publisher Site | Google Scholar
Z. Kamali and M. Lashkarizadeh, “The Bochner-Schoenberg-Eberlein property for totally ordered semigroup algebras,” Journal of Fourier Analysis and Applications, vol. 22, no. 6, pp. 1225–1234, 2016.
View at: Publisher Site | Google Scholar
E. Kaniuth, A. T. Lau, and A. Ülger, “Homomorphisms of commutative Banach algebras and extensions to multiplier algebras with applications to Fourier algebras,” Studia Mathematica, vol. 183, no. 1, pp. 35–62, 2007.
View at: Publisher Site | Google Scholar
S.-E. Takahasi and O. Hatori, “Commutative Banach algebras and BSE-inequalities,” Mathematica Japonica, vol. 37, pp. 47–52, 1992.
View at: Google Scholar
S.-E. Takahasi, “BSE Banach modules and multipliers,” Journal of Functional Analysis, vol. 125, no. 1, pp. 67–89, 1994.
View at: Publisher Site | Google Scholar
F. Abtahi and A. Pedaran, “On characterizations of the image of Gelfand transform and BED property for vector valued -algebras,” Journal of Mathematical Analysis and Applications, vol. 517, no. 1, article 126570, 2023.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2024 Shirin Tavkoli 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.

PDF Download Citation

Download other formats

Order printed copies

Views

56

Downloads

47

Citations