Nonlinear skew commuting maps on $\ast$-rings
UDC 512.5 Let $\mathcal{R}$ be a unital $\ast$-ring with the unit $I$. Assume that $\mathcal{R}$ contains a symmetric idempotent $P$ which satisfies $A{\mathcal{R}}P = 0$ implies $A=0$ and $A{\mathcal{R}}(I-P) = 0$ implies $A = 0$. In this paper, it is proved that if $\phi\colon\mathcal{R} \rightarr...
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| author | Kong, L. Zhang, J. Kong, L. Zhang, J. |
| author_facet | Kong, L. Zhang, J. Kong, L. Zhang, J. |
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Let $\mathcal{R}$ be a unital $\ast$-ring with the unit $I$. Assume that $\mathcal{R}$ contains a symmetric idempotent $P$ which satisfies $A{\mathcal{R}}P = 0$ implies $A=0$ and $A{\mathcal{R}}(I-P) = 0$ implies $A = 0$. In this paper, it is proved that if $\phi\colon\mathcal{R} \rightarrow \mathcal{R}$ is a nonlinear skew commuting map, then there exists an element $Z \in \mathcal{Z}_{S}(\mathcal{R})$ such that $\phi(X) = ZX$ for all $X \in \mathcal{R}$, where $\mathcal{Z}_{S}(\mathcal{R})$ is the symmetric center of $\mathcal{R}$.As an application, the form of nonlinear skew commuting maps on factors is obtained. |
| doi_str_mv | 10.37863/umzh.v74i6.801 |
| first_indexed | 2026-03-24T02:04:03Z |
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DOI: 10.37863/umzh.v74i6.801
UDC 512.5
L. Kong (School Math. and Statistics, Shaanxi Normal Univ., and Inst. Appl. Math., Shangluo Univ., China),
J. Zhang (School Math. and Statistics, Shaanxi Normal Univ., China)
NONLINEAR SKEW COMMUTING MAPS ON \ast -RINGS*
НЕЛIНIЙНI СКIСНI КОМУТУЮЧI ВIДОБРАЖЕННЯ НА \ast -КIЛЬЦЯХ
Let \scrR be a unital \ast -ring with the unit I . Assume that \scrR contains a symmetric idempotent P which satisfies A\scrR P = 0
implies A = 0 and A\scrR (I - P ) = 0 implies A = 0. In this paper, it is proved that if \phi : \scrR \rightarrow \scrR is a nonlinear skew
commuting map, then there exists an element Z \in \scrZ S(\scrR ) such that \phi (X) = ZX for all X \in \scrR , where \scrZ S(\scrR ) is the
symmetric center of \scrR . As an application, the form of nonlinear skew commuting maps on factors is obtained.
Нехай \scrR — унiтарне \ast -кiльце з одиницею I . Припустимо, що \scrR має симетричний iдемпотент P, для якого з
A\scrR P = 0 випливає A = 0, а з A\scrR (I - P ) = 0 — A = 0. У цiй статтi доведено, що якщо \phi : \scrR \rightarrow \scrR є нелiнiйним
скiсним комутуючим вiдображенням, то iснує елемент Z \in \scrZ S(\scrR ) такий, що \phi (X) = ZX для всiх X \in \scrR , де
\scrZ S(\scrR ) — симетричний центр \scrR . Як застосування отримано форму нелiнiйних скiсних комутуючих вiдображень на
факторах.
1. Introduction. Let \scrR be a ring. A map \phi : \scrR \rightarrow \scrR is called commuting if
\phi (X)X = X\phi (X) (1.1)
for all X \in \scrR . The usual goal when treating a commuting map is to describe its form. The
first important result on commuting maps is Posner’s theorem, which proved that the existence of a
nonzero commuting derivation on a prime ring \scrR implies that \scrR is commutative [12]. For X,Y \in \scrR ,
denote by [X,Y ] = XY - Y X the Lie product of X and Y. Accordingly, the commuting maps can
be written as [\phi (X), X] = 0 for all X \in \scrR . If \phi is additive, then for any X,Y \in \scrR , replacing X
by X + Y in Eq. (1.1) implies that
[\phi (X), Y ] = [X,\phi (Y )]
for all X,Y \in \scrR . Assuming that \phi is additive, Brešar [3] proved that additive commuting map \phi on
simple unital ring \scrR must be of the form
\phi (X) = ZX + f(X)
for some Z \in \scrZ (\scrR ) and additive f : \scrR \rightarrow \scrZ (\scrR ), where \scrZ (\scrR ) is the center of \scrR . The problem
of describing commuting maps is closely related with the theory of functional identities and many
results have been obtained on this subject. The reader is referred to the survey paper [5] and the
book [4]. Recently, Bounds [2] described commuting maps over the ring of strictly upper triangular
matrices. Brešar and Šemrl [6] gave the form of continuous commuting functions on matrix algebras.
Recall that a ring \scrR is called a \ast -ring if there is an additive map \ast : \scrR \rightarrow \scrR satisfying (XY )\ast =
= Y \ast X\ast and (X\ast )\ast = X for all X,Y \in \scrR . For X,Y \in \scrR , denote by [X,Y ]\ast = XY - Y X\ast the
skew Lie product of X and Y. The skew Lie product arose in the problem of representing quadratic
* This research was supported by National Natural Science Foundation of China (grant 11471199) and Scientific
Research Project of Shangluo University (21SKY104).
c\bigcirc L. KONG, J. ZHANG, 2022
826 ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 6
NONLINEAR SKEW COMMUTING MAPS ON \ast -RINGS 827
functionals with sesquilinear functionals and characterizing ideals [11, 14 – 16]. In the last decade,
skew Lie product has attracted attention of several authors [7 – 10, 13, 17, 18]. Motivated by the
above mentioned work, we introduce the concept of nonlinear skew commuting maps. A map \phi :
\scrR \rightarrow \scrR (without the additivity assumption) is called a nonlinear skew commuting maps if
[\phi (X), Y ]\ast = [X,\phi (Y )]\ast
for all X,Y \in \scrR .
Let \scrZ (\scrR ) be the centre of \scrR . An element X \in \scrR is called symmetric if X\ast = X, \scrZ S(\scrR ) =
= \{ X \in \scrZ (\scrR ) : X\ast = X\} is called symmetric center of \scrR . In this paper, we describe the form
of nonlinear skew commuting maps on \ast -rings. As an application, the form of nonlinear skew
commuting maps on factors is obtained.
2. Main result. In this section, we will prove the following theorem.
Theorem 2.1. Let \scrR be a unital \ast -ring with the unit I. Assume that \scrR contains a symmetric
idempotent P which satisfies: (\scrQ 1) A\scrR P = 0 implies A = 0, (\scrQ 2) A\scrR (I - P ) = 0 implies
A = 0. If a map \phi : \scrR \rightarrow \scrR (without the additivity assumption) satisfies
[\phi (X), Y ]\ast = [X,\phi (Y )]\ast
for all X,Y \in \scrR , then there exists an element Z \in \scrZ S(\scrR ) such that \phi (X) = ZX for all X \in \scrR .
It is clear that P \not = 0, P \not = I. Write P1 = P, P2 = I - P1. Put \scrR ij = Pi\scrR Pj , i, j = 1, 2. Then
\scrR = \scrR 11 +\scrR 12 +\scrR 21 +\scrR 22
and so for each A \in \scrR , A = A11 +A12 +A21 +A22, Aij \in \scrR ij , i, j = 1, 2.
We will complete the proof by several lemmas.
Lemma 2.1 ([1], Lemma 4). Let \scrR be a unital ring with the unit I. Assume that \scrR satisfies
A\scrR P1 = 0 implies A = 0 and A\scrR P2 = 0 implies A = 0. For Aii \in \scrR , i = 1, 2, if PiXAii =
= AiiXPi for all X \in \scrR , then there exists an element Z \in \scrZ (\scrR ) such that Aii = ZPi.
Lemma 2.2. \phi (I) \in \scrZ S(\scrR ).
Proof. For any Y \in \scrR , we have
[\phi (I), Y ]\ast = [I, \phi (Y )]\ast = 0,
which implies
\phi (I)Y = Y \phi (I)\ast (2.1)
for all Y \in \scrR . Taking Y = I in Eq. (2.1), we get \phi (I) = \phi (I)\ast . Hence, \phi (I) \in \scrZ S(\scrR ).
Lemma 2.3. For every X,Y \in \scrR , we have
\phi (X + Y ) - \phi (X) - \phi (Y ) \in \scrZ S(\scrR ).
Proof. For any X,Y, T \in \scrR , it follows that
[\phi (X + Y ) - \phi (X) - \phi (Y ), T ]\ast = [\phi (X + Y ), T ]\ast - [\phi (X), T ]\ast - [\phi (Y ), T ]\ast =
= [X + Y, \phi (T )]\ast - [X,\phi (T )]\ast - [Y, \phi (T )]\ast = 0.
Hence, \phi (X + Y ) - \phi (X) - \phi (Y ) \in \scrZ S(\scrR ).
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 6
828 L. KONG, J. ZHANG
Lemma 2.4. There exist elements Z1, Z2 \in \scrZ S(\scrR ) such that \phi (P1) = Z1P1 + Z2.
Proof. It follows from Lemma 2.2 that
0 = [P1, \phi (I)]\ast = [\phi (P1), I]\ast = \phi (P1) - \phi (P1)
\ast .
Hence, \phi (P1)
\ast = \phi (P1).
For any X \in \scrR , it is easy to check that
[P1, [P1, [P1, \phi (X)]\ast ]\ast ]\ast = [P1, \phi (X)]\ast .
Hence,
[P1, [P1, [\phi (P1), X]\ast ]\ast ]\ast = [\phi (P1), X]\ast .
Write \phi (P1) = S11 + S12 + S21 + S22. Since \phi (P1)
\ast = \phi (P1), the above equation becomes
(S11 + S12)XP1 - P1X(S11 + S21) + (S21 + S22)XP2 - P2X(S12 + S22) = 0. (2.2)
Taking X = X12 in Eq. (2.2) and multiplying by P2 from both sides, we get S21XP2 =
= S21X12 = 0 for all X \in \scrR . It follows from the condition (\scrQ 2) of Theorem 2.1 that S21 = 0.
Taking X = X21 in Eq. (2.2) and multiplying by P1 from both sides, we get S12XP1 = S12X21 = 0
for all X \in \scrR . It follows from the condition (\scrQ 1) of Theorem 2.1 that S12 = 0.
Taking X = X11 in Eq. (2.2), we get S11X11 = X11S11. It follows from Lemma 2.1 that
S11 = P1\phi (P1)P1 = ZP1 for some Z \in \scrZ (\scrR ). By \phi (P1)
\ast = \phi (P1), we have Z\ast P1 = ZP1,
and so Z\ast XP1 = ZXP1 for all X \in \scrR . It follows from the condition (\scrQ 1) of Theorem 2.1 that
Z\ast = Z, that is, Z \in \scrZ S(\scrR ). Similarly, taking X = X22 in Eq. (2.2), we get S22 = Z2P2 for some
Z2 \in \scrZ S(\scrR ). Hence,
\phi (P1) = S11 + S22 = ZP1 + Z2P2 = Z1P1 + Z2,
where Z1 = Z - Z2 \in \scrZ S(\scrR ).
Lemma 2.5. For every Xij \in \scrR ij , 1 \leq i \not = j \leq 2, we have
\phi (Xij) = Z1Xij .
Proof. Take any X12 \in \scrR 12 and let \phi (X12) = A11+A12+A21+A22. It follows from Lemma 2.4
that
P1\phi (X12) - \phi (X12)P1 = [P1, \phi (X12)]\ast = [\phi (P1), X12]\ast = Z1X12,
which implies that A12 = Z1X12 and A21 = 0.
Take any B \in \scrR and let \phi (B) = Y11 + Y12 + Y21 + Y22. Since
[B,\phi (X12)]\ast = [\phi (B), X12]\ast ,
we obtain
BA11 +BA12 +BA22 - A11B
\ast - A12B
\ast - A22B
\ast =
= Y11X12 + Y21X12 - X12Y
\ast
12 - X12Y
\ast
22. (2.3)
Multiplying Eq. (2.3) by P1 from the right, we get
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 6
NONLINEAR SKEW COMMUTING MAPS ON \ast -RINGS 829
BA11 - A11B
\ast P1 - A12B
\ast P1 - A22B
\ast P1 = - X12Y
\ast
12. (2.4)
Replacing B by P2BP1 in Eq. (2.4), we have
P2BA11 = - X12Y
\ast
12,
which implies P2BA11 = 0 for all B \in \scrR , and then A\ast
11B
\ast P2 = 0 for all B \in \scrR . It follows from
the condition (\scrQ 2) of Theorem 2.1 that A11 = 0. Similarly, multiplying Eq. (2.3) by P2 from the
left, and then replacing B by P1BP2, we can get A22 = 0. Hence,
\phi (X12) = A11 +A12 +A21 +A22 = Z1X12.
The proof of \phi (X21) = Z1X21 is similar.
Lemma 2.6. For every Xii \in \scrR ii, i = 1, 2, we have
\phi (Xii) = Z1Xii.
Proof. Take any X11 \in \scrR 11 and let \phi (X11) = S11 + S12 + S21 + S22. For any X12 \in \scrR 12, it
follows from Lemma 2.5 that
0 = [\phi (X12), X11]\ast = [X12, \phi (X11)]\ast ,
which implies that
X12S21 +X12S22 - S12X
\ast
12 - S22X
\ast
12 = 0. (2.5)
Multiplying Eq. (2.5) by P2 from the left and P1 from the right, we have S22X
\ast P1 = S22X
\ast
12 = 0
for all X \in \scrR , and so S22 = 0. Since
[X11, \phi (X12)]\ast = [\phi (X11), X12]\ast ,
it follows that
X11\phi (X12) = S11X12 + S21X12 - X12S
\ast
12. (2.6)
Multiplying Eq. (2.6) by P1 from both sides, we can get X12S
\ast
12 = 0, and so S12 = 0. Multiplying
Eq. (2.6) by P2 from both sides, we can get S21X12 = 0, and so S21 = 0. Multiplying Eq. (2.6) by
P1 from the left and P2 from the right, we have
X11\phi (X12)P2 = S11X12.
It follows from Lemma 2.5 that (S11 - Z1X11)X12 = 0 and so S11 = Z1X11. Hence,
\phi (X11) = S11 + S12 + S21 + S22 = Z1X11.
The proof of \phi (X22) = Z1X22 is similar.
Now we are in a position to prove the main theorem.
Proof of Theorem 2.1. It follows from Lemmas 2.5 and 2.6 that \phi (Xij) = Z1Xij , i, j = 1, 2.
For any X =
\sum 2
i,j=1
Xij \in \scrR , it follows from Lemma 2.3 that
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 6
830 L. KONG, J. ZHANG
\phi (X) - Z1X = \phi (X) - Z1
2\sum
i,j=1
Xij = \phi (X) -
2\sum
i,j=1
\phi (Xij) \in \scrZ S(\scrR ).
Define a map f : \scrR \rightarrow \scrZ S(\scrR ) by f(X) = \phi (X) - Z1X. Then we have
\phi (X) = Z1X + f(X)
for all X \in \scrR .
Since
[\phi (X), Y ]\ast = [X,\phi (Y )]\ast
for all X,Y \in \scrR , we obtain
[Z1X + f(X), Y ]\ast = [X,Z1Y + f(Y )]\ast .
Hence,
f(Y ) (X - X\ast ) = 0 (2.7)
for all X,Y \in \scrR .
For any X12 \in \scrR 12, replacing X by X12 in Eq. (2.7), we get
f(Y )X12 - f(Y )X\ast
12 = 0
for all Y \in \scrR . Multiplying the above equation by P2 from the right, we have f(Y )X12 = 0, and so
f(Y )P1 = 0.
For any X21 \in \scrR 21, replacing X by X21 in Eq. (2.7), we obtain
f(Y )X21 - f(Y )X\ast
21 = 0
for all Y \in \scrR . Multiplying the above equation by P1 from the right, we get f(Y )X21 = 0, and so
f(Y )P2 = 0. Hence,
f(Y ) = f(Y )P1 + f(Y )P2 = 0
for all Y \in \scrR , and, thus, \phi (X) = Z1X.
Let \BbbR be the real number field. We denote by H the complex Hilbert space and by B(H) the
algebra of all bounded linear operators on H. Let \scrA \subseteq B(H) be a von Neumann algebra. Recall
that \scrA is a factor if its center contains only the scalar operators.
Corollary 2.1. Let \scrA be a factor acting on a complex Hilbert space H. If a map \phi : \scrA \rightarrow \scrA
satisfies
[\phi (X), Y ]\ast = [X,\phi (Y )]\ast
for all X,Y \in \scrA , then \phi (X) = \alpha X for all X \in \scrA , where \alpha \in \BbbR .
References
1. Z. Bai, S. Du, Strong skew commutativity preserving maps on rings, Rocky Mountain J. Math., 44, № 3, 733 – 742
(2014).
2. J. Bounds, Commuting maps over the ring of strictly upper triangular matrices, Linear Algebra and Appl., 507,
132 – 136 (2016).
3. M. Brešar, Centralizing mappings and derivations in prime rings, J. Algebra, 156, № 2, 385 – 394 (1993).
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 6
NONLINEAR SKEW COMMUTING MAPS ON \ast -RINGS 831
4. M. Brešar, M. A. Chebotar, W. S. Martindale III, Functional identities, Birkhäuser-Verlag (2007).
5. M. Brešar, Commuting maps: a survey, Taiwanese J. Math., 8, № 3, 361 – 397 (2004).
6. M. Brešar, P. Šemrl, Continuous commuting functions on matrix algebras, Linear Algebra and Appl., 568, 29 – 38
(2019).
7. J. Cui, C. Li, Maps preserving product XY - Y X\ast on factor von Neumann algebras, Linear Algebra and Appl.,
431, № 5 – 7, 833 – 842 (2009).
8. J. Cui, C. Park, Maps preserving strong skew Lie product on factor von Neumann algebras, Acta Math. Sci. Ser. B.,
32, № 2, 531 – 538 (2012).
9. C. Li, Q. Chen, Strong skew commutativity preserving maps on rings with involution, Acta Math. Sin. (Engl. Ser.),
32, № 6, 745 – 752 (2016).
10. C. Li, F. Zhao, Q. Chen, Nonlinear skew Lie triple derivations between factors, Acta Math. Sin. (Engl. Ser.), 32,
№ 7, 821 – 830 (2016).
11. L. Molnár, A condition for a subspace of B(H) to be an ideal, Linear Algebra and Appl., 235, 229 – 234 (1996).
12. E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc., 8, № 6, 1093 – 1100 (1957).
13. X. Qi, J. Hou, Strong skew commutativity preserving maps on von Neumann algebras, J. Math. Anal. and Appl., 391,
№ 1, 362 – 370 (2013).
14. P. Šemrl, On Jordan \ast -derivations and an application, Colloq. Math., 59, № 2, 241 – 251 (1990).
15. P. Šemrl, Quadratic and quasi-quadratic functionals, Proc. Amer. Math. Soc., 119, № 4, 1105 – 1113 (1993).
16. P. Šemrl, Quadratic functionals and Jordan \ast -derivations, Stud. Math., 97, № 3, 157 – 165 (1991).
17. A. Taghavi, M. Nouri, V. Darvish, A note on nonlinear skew Lie triple derivations between prime \ast -algebras, Korean
J. Math., 26, № 3, 459 – 465 (2018).
18. W. Yu, J. Zhang, Nonlinear \ast -Lie derivations on factor von Neumann algebras, Linear Algebra and Appl., 437, № 8,
1979 – 1991 (2012).
Received 29.03.19
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 6
|
| id | umjimathkievua-article-801 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:04:03Z |
| publishDate | 2022 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/9a/d58d86575d557f056511f6fc0455989a.pdf |
| spelling | umjimathkievua-article-8012022-10-24T09:23:07Z Nonlinear skew commuting maps on $\ast$-rings Nonlinear skew commuting maps on $\ast$-rings Kong, L. Zhang, J. Kong, L. Zhang, J. Commuting maps; skew commuting maps; rings Commuting maps; skew commuting maps; rings UDC 512.5 Let $\mathcal{R}$ be a unital $\ast$-ring with the unit $I$. Assume that $\mathcal{R}$ contains a symmetric idempotent $P$ which satisfies $A{\mathcal{R}}P = 0$ implies $A=0$ and $A{\mathcal{R}}(I-P) = 0$ implies $A = 0$. In this paper, it is proved that if $\phi\colon\mathcal{R} \rightarrow \mathcal{R}$ is a nonlinear skew commuting map, then there exists an element $Z \in \mathcal{Z}_{S}(\mathcal{R})$ such that $\phi(X) = ZX$ for all $X \in \mathcal{R}$, where $\mathcal{Z}_{S}(\mathcal{R})$ is the symmetric center of $\mathcal{R}$.As an application, the form of nonlinear skew commuting maps on factors is obtained. УДК 512.5Нелiнiйнi скiснi комутуючi вiдображення на $\ast $-кiльцяхНехай $\mathcal{R}$ — унiтарне $\ast$ -кiльце з одиницею $I$. Припустимо, що $\mathcal{R}$ має симетричний iдемпотент $P$, для якого з $A{\mathcal{R}}P = 0$ випливає $A=0$, а з $A{\mathcal{R}}(I-P) = 0$ — $A = 0$. У цiй статтi доведено, що якщо $\phi\colon\mathcal{R} \rightarrow \mathcal{R}$ є нелiнiйним скiсним комутуючим вiдображенням, то iснує елемент $Z \in \mathcal{Z}_{S}(\mathcal{R})$ такий, що $\phi(X) = ZX$ для всiх $X \in \mathcal{R}$, де $\mathcal{Z}_{S}(\mathcal{R})$ — симетричний центр $\mathcal{R}$. Як застосування отримано форму нелiнiйних скiсних комутуючих вiдображень нафакторах. Institute of Mathematics, NAS of Ukraine 2022-07-07 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/801 10.37863/umzh.v74i6.801 Ukrains’kyi Matematychnyi Zhurnal; Vol. 74 No. 6 (2022); 826 - 831 Український математичний журнал; Том 74 № 6 (2022); 826 - 831 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/801/9254 Copyright (c) 2022 Liang Kong, Jianhua Zhang |
| spellingShingle | Kong, L. Zhang, J. Kong, L. Zhang, J. Nonlinear skew commuting maps on $\ast$-rings |
| title | Nonlinear skew commuting maps on $\ast$-rings |
| title_alt | Nonlinear skew commuting maps on $\ast$-rings |
| title_full | Nonlinear skew commuting maps on $\ast$-rings |
| title_fullStr | Nonlinear skew commuting maps on $\ast$-rings |
| title_full_unstemmed | Nonlinear skew commuting maps on $\ast$-rings |
| title_short | Nonlinear skew commuting maps on $\ast$-rings |
| title_sort | nonlinear skew commuting maps on $\ast$-rings |
| topic_facet | Commuting maps skew commuting maps rings Commuting maps skew commuting maps rings |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/801 |
| work_keys_str_mv | AT kongl nonlinearskewcommutingmapsonastrings AT zhangj nonlinearskewcommutingmapsonastrings AT kongl nonlinearskewcommutingmapsonastrings AT zhangj nonlinearskewcommutingmapsonastrings |