Biderivations and Commuting Linear Maps on Topologically Simple <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.3241005pt" id="M1" height="13.6595pt" version="1.1" viewBox="-0.0657574 -13.3354 24.4501 13.6595" width="24.4501pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g198-13" d="M608 643C551 682 486 687 443 687C299 687 188 608 188 504C188 361 341 312 465 312C483 312 501 313 518 315C454 195 366 93 314 59C255 82 201 109 141 109C73 109 33 83 33 52C33 13 76 -9 159 -9C218 -9 267 -1 314 15C354 0 402 -15 478 -15C575 -15 671 36 759 141L740 155C657 63 571 16 475 16C448 16 411 26 372 39C458 74 565 199 627 333C806 381 931 499 931 587C931 634 908 669 848 669C767 669 671 582 620 506C591 462 569 411 534 347C503 343 490 342 471 342C330 342 229 399 229 500C229 603 346 657 451 657C486 657 540 657 596 624L608 643ZM890 588C890 519 822 457 747 414C712 395 678 378 643 369C683 484 760 632 841 632C873 632 890 615 890 588ZM266 35C235 24 200 21 161 21C103 21 74 32 74 54C74 75 122 79 141 79C172 79 209 69 266 36V35Z"/></g><g transform="matrix(.012,0,0,-0.012,16.233,-7.578)"><path id="g50-43" d="M486 158C486 177 478 202 466 220C413 228 386 236 336 262C386 288 413 297 466 304C478 323 486 347 485 366C470 376 444 381 422 380C389 338 368 319 321 288C323 345 329 372 349 422C339 442 322 461 305 470C289 461 271 442 262 422C281 372 287 345 290 288C243 319 222 338 189 380C167 381 142 376 125 366C125 347 133 322 145 304C198 296 225 288 275 262C225 236 198 227 145 220C133 201 125 177 126 158C141 148 167 143 189 144C222 186 243 205 290 236C288 179 282 152 262 102C272 82 289 63 306 54C322 63 340 82 350 102C330 152 324 179 321 236C368 205 390 186 422 144C444 143 470 148 486 158Z"/></g></svg>-</span>Algebras

Ben Yahya, Abdelaziz

doi:https://doi.org/10.1155/2021/9920191

International Journal of Mathematics and Mathematical Sciences

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Abstract Introduction Preliminaries Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 9920191 | https://doi.org/10.1155/2021/9920191

Biderivations and Commuting Linear Maps on Topologically Simple -Algebras

Abdelaziz Ben Yahya¹

Academic Editor: Kaiming Zhao

Received22 Mar 2021

Revised31 May 2021

Accepted06 Jul 2021

Published17 Jul 2021

Abstract

Let be a topologically simple -algebra of arbitrary dimension. In this paper, we introduce the notion of semi-inner biderivation in order to prove that every continuous commuting linear mapping on is a scalar multiple of the identity mapping.

1. Introduction

Let be a Lie algebra over a field of a characteristic different from two. A linear map is called derivation on if it satisfies the following identity:for all .

A bilinear map is called biderivation on if it satisfies the following identities:

For all , which means that it is a derivation with respect to both components. In addition, if , will be called skew-symmetric biderivation. Let and be the bilinear map sending to ; it is straightforward to prove that is a biderivation of ; and the biderivations of this type are called inner biderivations of . A linear map is called a commuting linear map on if it satisfies the following identity: , for all . It is easy to show thatwhich implies that the bilinear map defined byis a skew-symmetric biderivation on .

Commuting maps and biderivations arose first in the associative ring theory [1, 2]. Since then, many authors have made considerable efforts to make their study very successful (see, for example, [3–10]). The way used in [8] requires the finiteness of the dimension of the simple Lie algebra. However, the purpose of this paper is to extend the results given in [8] concerning the commuting linear maps to topologically simple -algebras, which are of arbitrary dimension. To overcome the problem of the nonfiniteness of the dimension, we use some techniques related to these algebras. The -algebras are introduced by Schue in [11]. We recall that an -algebra over (the complex field) is a Lie algebra , which is also a complex Hilbert space with the inner product endowed with a (conjugate-linear) algebra involution such that , for all in .

Let be an -algebra; for subsets and of , we recall that denotes the closed subspace spanned by . is said to be semisimple as an -algebra if and only if . From [11], a finite-dimensional Lie algebra is semisimple as an -algebra if and only if it is semisimple in the usual sense. The -algebra will be called topologically simple if and only if there are no nontrivial closed ideals. In [11], the author shows that the -algebras are reductive, the semisimple ones are the Hilbert space direct sum of its closed topologically simple ideals, and the author also gives the classification of the topologically simple -algebras in the separable case. The classification of the topologically simple -algebras in the arbitrary dimensional case can be found in [12]. In [13], Schue shows that every semisimple -algebra has a Cartan decomposition relative to a Cartan subalgebra. We recall that a Cartan subalgebra of a semisimple -algebra is defined as a maximal self-adjoint abelian subalgebra.

The paper is organized as follows. In the second section, we give some definitions and basic results related to -algebras. In Section 3, we introduce the notion of semi-inner biderivation in order to show that every semi-inner biderivation on an arbitrary dimensional topologically simple -algebra is inner; using this result, we determine all continuous commuting linear maps on an arbitrary dimensional topologically simple -algebras.

2. Preliminaries

In this section, we summarize some basic results related to -algebras, collected from [14, 15]. First of all, we point out that the notation concerning Lie algebras follows principally from [8, 14]. Let be the complex numbers field, a semisimple -algebra, a fixed Cartan subalgebra of , and denotes the conjugation operator on , a root of relative to is a linear form commuting with the involution:

That is, for any , such that there exists , satisfying for any . The subspaceis called the root space associated to ; it follows from this that if is a root, then is also one and . The root space associated to the zero root is equal to the Cartan subalgebra , using the Jacobi identity; one proves that if is a root, then , and if is not a root, then . Let denote the set of nonzero roots of relative to , then we have the following Cartan decomposition , where is the usual Hilbert space direct sum.

Let be a root of relative to , then is a linear functional on ; this implies that there exists a unique vector such that where denotes the inner product of . Consequently, is self-adjoint, which means that and for any with . Then, we have the following result.

Lemma 1 (see [15]). The set is total in , i.e., for any , for all , implies .

Let be a Hilbert space, the orthogonal dimension of is denoted by , i.e., the cardinality of an orthonormal basis for . We will denote the cardinality of an arbitrary set by .

Now, we will define the root system relative to a Cartan subalgebra of the semisimple -algebra .

Definition 1. Let be a semisimple -algebra, a Cartan subalgebra of , and the set of nonzero roots of relative to . A subset of will be called a root system of if the following conditions are satisfied:(i)If , then (ii)If , such that , then We need some further notations; and will refer to the real and the rational fields, respectively. For a subset of , the set of all -linear combinations of elements of will be denoted by and the set of all -linear combinations of elements of by . If we write , then is obviously a root system. The following results will be useful in our main proofs.

Lemma 2. Let be a topologically simple -algebra and the set of its nonzero roots relative to some Cartan subalgebra . For any subset of , there exists a topologically simple -subalgebra of , with Cartan subalgebra , such that(i), where is the set of roots of (relative to ) and (ii) is finite-dimensional if is finite(iii) is infinite-dimensional and if is infinite

Proof. See Proposition 3 in [14].

Definition 2. Let be a topologically simple -algebra and the set of nonzero roots relative to a Cartan subalgebra . Let ; we say that is connected to if there are some such that .
Obviously, the connected relation is an equivalence relation on .

Lemma 3. Any two roots of a topologically simple -algebra are connected.

Proof. Let , then by Lemma 2 there exists a finite-dimensional simple -algebra of , with Cartan subalgebra , such that are roots of . Using Lemma 1.3 in [8], we obtain that and are connected in . Since , then and are connected in .

3. Semi-Inner Biderivations on -Algebras and Commuting Linear Maps on Topologically Simple -Algebras

In this section, we introduce the notion of semi-inner biderivation in order to show that every continuous commuting linear map on a topologically simple -algebra is a scalar multiple of the identity mapping. The aim of the first main theorem of this section is to prove that every semi-inner biderivation of a topologically simple -algebra is inner. To get this result, we have to show two lemmas.

Definition 3. Let be an -algebra and a biderivation of ; is said to be a semi-inner biderivation of if there exists two continuous linear maps and such thatwhere will be denoted by .

Remark 1. By using Lemma 2.1 in [8], any biderivation of a finite-dimensional simple complex Lie algebra is semi-inner (a linear map between two finite-dimensional vector spaces is continuous).
From now on, will represent an infinite-dimensional topologically simple -algebra of arbitrary dimension and where its Cartan decomposition is relative to a Cartan subalgebra ( is the set of nonzero roots relative to ).

Lemma 4. Let be a semi-inner biderivation of , then for any , we have .

Proof. For any , we select , such that , , and . Then, we haveLet , we denote by the set . LetFor some , the sums are orthogonal, , and , .
By equations (10) and (14), we haveIf we compare the two equations above and since , we obtain . In the same way, by considering with equations (10) and (14), . Similarly, considering the images and , we obtain , by equations (11)–(13).
If we put and , then equations (9) and (12) can be written as follows:Now, for any root is denoted by . By equations (16) and (17), we can writeThe two equations above imply thatBy comparing, one has . Due to the arbitrariness of , we obtain and . By Lemma 1, the set is total in and and are continuous, and we have and .

Lemma 5. Let be a semi-inner biderivation of , then there is a complex number such that

Proof. Let and such that , then we haveLetwhere and . Using equations (21) and (23), we obtainCombining equations (22) and (24), we obtain and for every .
For any , if we replace by in equation (24), we get ; this implies thatThe image of is computed. Similarly, we getwhere and .
For any , using equations (25) and (26), we haveBy combining equations (27) with (28), we first see that if . However, by taking . This means that for all , i.e., ker , which gives . Similarly, by taking , we have . Therefore, by equations (25) and (26), one can obtainUsing equation (29) and , it follows thatThen, comparing equations (27) with (28), we obtainLet , then by Lemma 2 there exists a finite-dimensional simple -algebra of , with Cartan subalgebra , such that is a root of . Now, let us prove that indeed; let , then .Then, is a biderivation of ; by Theorem 2.4 in [8], there exists such that for any . Then, for any and ; this implies thatEquations (30)–(33) imply that if such that . Additionally, for arbitrary connected roots , from Lemma 3, we conclude thatIf we pose in equation (34), we get our result.

Remark 2. In the above proof, there is another method to show that if : Indeed, using equation (23), we haveWe also have . Then, if .

Remark 3. In the case where is separable, Theorem 2 in [16] will facilitate some difficulties encountered in the above proof (the above proof will look like the proof of Lemma 2.2 in [8]).
Thanks to the above Lemmas, we can state our first main theorem.

Theorem 1. Let be a topologically simple -algebra. Then, is a semi-inner biderivation of if and only if it is inner, i.e., there is a complex number such that .

Proof. Let be a topologically simple -algebra, any inner biderivation of such that is a semi-inner biderivation withNow, let us prove the “only if direction.”
The first case: if is finite-dimensional, the result follows from Theorem 2.4 in [8]. The second case: if is infinite-dimensional, then, for a semi-inner biderivation of , by using Lemma 5, there is such that . For any and , we have . This implies that which means thatthis implies for all . For any , with , where , with except for a countable number. Therefore,In recent years, many authors have studied commuting linear maps of certain algebra structures; for some of these achievements, refer to [3, 4, 6, 8, 9, 17, 18]. It should be noted that this subject is not new since it was studied in 1957, exactly in Posner’s works [19]. As we said in the introduction, if is a commuting linear map on , then for any , and is a biderivation of . Using Theorem 1, the present theorem aims to determine all continuous commuting linear maps on topologically simple -algebras.

Theorem 2. Let be a topologically simple -algebra. Then, every continuous linear map on is a commuting linear map if and only if it is a scalar multiplication map on .

Proof. Let be a continuous commuting linear map on a topologically simple -algebra . Then, defined by is a semi-inner biderivation of . By Theorem 1, we have for some . Since is topologically simple -algebra and is arbitrary, therefore, we have .
The following Lemma is one of the interesting results given in (see p.7 in [3]).

Lemma 6. Let be a simple Lie algebra over an algebraically closed field of characteristic different from 2 such that ( is the cardinality of and is the dimension of ). For any skew-symmetric biderivation of , there exists such that

Remark 4. Let be a simple complex Lie algebra of countable dimension and a commuting linear map on . Then, is of the form where . Indeed, let be a commuting linear map on , then for all which is a skew-symmetric biderivation of . By Lemma 6, there exists , such that is of the form for all . Then, is of the form .
We mention here that this proof does not seem to work in our case when the underlying Hilbert space of the -algebra is infinite-dimensional. However, Corollary 2.4 in [3] may simplify some of the proofs in our paper.

4. Conclusions

This paper aimed to show that every continuous commuting linear map on a topologically simple -algebra is a scalar multiplication map on .

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that there are no conflicts of interest.

Acknowledgments

The author would like to thank Professor M. Ait Ben Haddou and Professor M. Raouyane.

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Copyright

Copyright © 2021 Abdelaziz Ben Yahya. 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.

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