Issue |
Wuhan Univ. J. Nat. Sci.
Volume 28, Number 5, October 2023
|
|
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Page(s) | 379 - 384 | |
DOI | https://doi.org/10.1051/wujns/2023285379 | |
Published online | 10 November 2023 |
Mathematics
CLC number: O177.1
Nonlinear A*B+B*A Type Derivations on *-Algebras
1
School of Science, Xi'an University of Posts and Telecommunications, Xi'an
710121, Shaanxi, China
2
Xi'an Modern Control Technology Institute, Xi'an
710065, Shaanxi, China
Received:
5
March
2023
Let be a unital *-algebra with the unit
and a nontrivial projection
. Suppose that
satisfies
implies
and
implies
In this paper, we prove that
is a nonlinear
type derivation on
if and only if
is an additive *-derivation. This is then applied to prime *-algebra, von Neumann algebras with no central summands of type
factor von Neumann algebras and standard operator algebras.
Key words: A*B+B*A type derivations / *-derivation / von Neumann algebras
Biography: ZHANG Fangjuan, female, Ph. D., Associate professor, research direction: operator algebras. E-mail: zhfj888@xupt.edu.cn; zhfj888@126.com
Fundation item: Supported by the National Natural Science Foundation of China (11601420) and the Natural Science Basic Research Plan in Shaanxi Province (2018JM1053)
© Wuhan University 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
0 Introduction
Let be a *-algebra over the complex field
For
we write
and
for the bi-skew Lie product, *-Jordan product and
product, respectively. These products have recently attracted the attention of many authors (see Refs. [1-14]).
Recall that a map is said to be an additive derivation if
and
for
Furthermore,
is an additive *-derivation if it is an additive derivation and
for
A map
(without the linearity assumption) is called a nonlinear
derivation if
for
where
Darvish et al[1] proved that every nonlinear
triple derivation on prime *-algebras is an additive *-derivation. A map
(without the linearity assumption) is called a nonlinear *-Jordan derivation if
for
The authors of Ref. [2] introduced the concept of *-Jordan-type derivation. Suppose that
is a fixed positive integer. Accordingly, a nonlinear *-Jordan-type derivation is a map
satisfying the condition
for all
where
Under some mild condition on a *-algebra
, they showed that
is a nonlinear *-Jordan-type derivation on
if and only if
is an additive *-derivation.
Motivated by the above results, we introduce the type derivations. Suppose that
is a fixed positive integer. A nonlinear
type derivation is a map
satisfying the condition
for all
where
In this paper, under some mild condition on a *-algebra
we prove that
is a nonlinear
type derivation on
if and only if
is an additive *-derivation.
1 The Main Result and Its Proof
Theorem 1 Let be a unital *-algebra with the unit
and a nontrivial projection
Suppose that
satisfies
(a) implies
and (b)
implies
.
If a map satisfies
for all
and
then
is an additive *-derivation.
Let and
Let
then
We can write every
as
where
denotes an arbitrary element of
Let
Then for all
where
Proof The proof is completed by the following several claims.
Claim 1
Claim 2
Using we obtain
which implies and then
Since
we obtain
From (1), we obtain
i.e.,
Using (1) and (2), we have
In the same manner, we obtain
Claim 3 For every we have
For all using
and Claim 2, we obtain
i.e.,
Using and Claim 2, we have
i.e.,
From (3) and (4), we obtain
Claim 4 For every we have
For all we have
which indicates
Claim 5 For every we have (i)
(ii)
Setting let us prove that
Based on Claim 4, we obtain
Since
it follows from Claim 1 and Claim 2 that
i.e., This together with
shows that
Using
Claim 1 and Claim 2, we obtain
i.e., This together with
shows that
And then
In the second case, we can similarly prove that the conclusion is valid.
Claim 6 For every we have
+
Setting since
applying Claim 1, Claim 2 and Claim 5(i), we obtain
i.e., This together with
shows that
In the same manner, by applying the above proof for
instead of
and Claim 5(ii) instead of Claim 5(i), we have
Claim 7 For every we have
Let we obtain
where
Since
we set
then
Using and Claim 6, we obtain
i.e.,
Claim 8 For each we have (i)
(ii)
Setting we obtain
i.e., This together with the fact
shows that
For all
take
Then
We get from Claim 7 that
i.e., Then
for all
Hence
for all
From (b), we obtain
and then
In the second case, we can similarly prove that the conclusion is valid.
Claim 9
is additive on
By Claims 6-8, is additive on
Claim 10
is additive on
and
for all
For all from Claim 2 we obtain
On the other hand, from Claim 2, we have
i.e.,
By adding (5) and (6), from Claim 3, we obtain
For all , we have
with
From (7), Claim 4 and Claim 9, we obtain
For all , we have
with
From (7) and Claim 9, we obtain
Claim 11
is an additive *-derivation on
For all from Claim 2 and Claim 4, we have
On the other hand, from (7), Claim 2 and Claim 4, we obtain
So we have
For all , we have
where
From Claim 10 and (7), we obtain
From this and Claim 10, we have proved that is an additive *-derivation. This completes the proof.
2 Corollaries
Now we give some applications of Theorem 1 to operator algebras. We say that is prime when for
, if
, then
or
It is easy to show that prime *-algebras satisfy (a) and (b), and the following corollary is immediate.
Corollary 1 Let be a prime *-algebra with unit
and a nontrivial projection. Then
is a nonlinear
type derivation on
if and only if
is an additive *-derivation.
Recall that a von Neumann algebra is weakly closed, self-adjoint algebra of operators on a Hilbert space
containing the identity operator
By Ref.[3], if a von Neumann algebra has no central summands of type
, then
satisfies (a) and (b). So the following corollary is obvious.
Corollary 2 Let be a von Neumann algebra with no central summands of type
. Then
is a nonlinear
type derivation on
if and only if
is an additive *-derivation.
is a factor von Neumann algebra means that its center only contains the scalar operators. Clearly,
is prime. So the following corollary is obvious from Corollary 1.
Corollary 3 Let be a factor von Neumann algebra. Then
is a nonlinear
type derivation on
if and only if
is an additive *-derivation.
is the algebra of all bounded linear operators on a complex Hilbert space
.
⊆
is all bounded finite rank operators.
⊆
is said to a standard operator algebra if it contains
. When
is a standard operator algebra, a more concrete form is achieved.
Corollary 4 Let be an infinite dimensional complex Hilbert space and
be a standard operator algebra on
containing the identity operator
Assume that
is closed under the adjoint operation. Let
be a nonlinear
type derivation. Then there exists
satisfying
such that
for all
.
Proof Since is prime, from Corollary 1, we obtain
is an additive *-derivation. According to the result of Ref.[4],
is linear, and then it is inner. Thus there exists
such that
. Hence
for all
. This indicates that
for certain
Take
then
and
for all
.
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