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Abstract
Let be any unital *-algebra over the real or complex field
, and let
with
. Assume that
is a map. It is shown that,
satisfies
for all
if and only if
, the center of
,
and
for all
; if
, then
satisfies
for all
if and only if
,
and
for all
; if
and
is surjective, then
satisfies
for all
if and only if
,
, and
for all
.
Public Interest Statement
Preserver problem had attracted many mathematicians’ attentions for many years. In this paper, the authors discuss the general maps preserving strong (skew) -Lie commutativity on any algebras and give a complete characterization for such maps.
1. Introduction
Let be a ring. Then
is a Lie ring under the Lie product
Recall that a map
preserves commutativity if
whenever
for all
,
. The problem of characterizing linear (additive) maps preserving commutativity had been studied intensively on various rings and algebras (see Brešar, Citation1993; Brešar & Šemrl, Citation2005; Choi, Jafarian, & Radjavi, Citation1987) and the references therein).
In Bell and Daif (Citation1994), the authors gave the conception of strong commutativity preserving maps. Let be a subset of
. A map
is called strong commutativity preserving if
for all
. Note that a strong commutativity preserving map must be commutativity preserving, but the inverse is not true generally. Bell and Daif (Citation1994) proved that
must be commutative, if
is a prime ring and
admits a derivation or a non-identity endomorphism which is strong commutativity preserving on a right ideal of
. Brešar and Miers (Citation1994) proved that every strong commutativity preserving additive map
on a semiprime ring
is of the form
, where
, the extended centroid of
,
, and
is an additive map. Recently, Lin and Liu (Citation2008) obtained the similar result on a noncentral Lie ideal of a prime ring. Qi and Hou (Citation2010; Citation2012) gave a complete characterization of strong commutativity preserving surjective maps (without the assumption of additivity) on prime rings and triangular algebras, respectively.
Let be a *-ring. For any
,
denotes the skew Lie product of
and
. This kind of product is found playing a more and more important role in some research topics such as representing quadratic functionals with sesquilinear functionals, and its study has attracted many authors’ attention (see Brešar & Fosňer, Citation2000; Chebotar, Fong, & Lee, Citation2005; Cui & Hou, Citation2006) and the reference therein). Molnár (Citation1996) initiated the systematic study of this skew Lie product, and studied the relation between subspaces and ideals of
, the algebra of all bounded linear operators acting on a Hilbert space
.
Recall that a map is called zero skew Lie product preserving, if
whenever
for any
. Additive or linear maps preserving zero skew Lie products on various rings and algebras had been studied by many authors (see, Bell & Daif, Citation1994 and the references therein). More specially,
is strong skew commutativity preserving, if
for all
. It is obvious that strong skew commutativity preserving maps must be zero skew Lie product preserving. However, the inverse is not true generally. In Cui and Park (Citation2012), they proved that, if
is a factor von Neumann algebra, then every strong skew commutativity preserving map
on
has the form
for all
, where
is a linear bijective map satisfying
for all
and
is a real linear functional of
with
; particularly, if
is of type
, then
for each
, where
. Recently, Qi and Hou (Citation2013) generalized the above result to von Neumann algebras without central summand of type
.
Recall that commutes with
up to a factor
if
. Note that the concept of commutativity up to a factor for pairs of operators is important and has been studied in the context of operator algebras and quantum groups see (Brooke, Busch, & Pearson, Citation2002; Kassel, Citation1995). Motivated by this, a binary operation
, called
-Lie product of
and
, was introduced in Qi and Hou (Citation2009). Thus, we also can define the skew
-Lie product of
and
. Let
be a *-algebra over
, where
is a field with an involution
. For
and
, we call
the skew
-Lie product of
and
. It is obvious that the skew
-Lie product is the skew Lie product if
. Now, based on these concepts, we say that a map
is preserving strong
-Lie commutativity if
for all
; is preserving strong skew
-Lie commutativity if
for all
.
The purpose of this paper is to consider nonlinear strong (skew) -Lie commutativity preserving maps on general algebras with
. Let
be any unital algebra over any field
and
with
. Denote by
the center of
. Assume that
is a map. In Section 2, we prove that
preserves strong
-Lie commutativity if and only if
,
, and
for all
(Theorem 2.1). In Section 3, we furthermore assume that
is a *-algebra. It is shown that, if
, then
preserves strong skew
-Lie commutativity if and only if
,
and
for all
(Theorem 3.1); if
and
is surjective, then
preserves strong skew
-Lie commutativity if and only if
,
and
for all
(Theorem 3.2).
2. Maps preserving strong ![](//:0)
-Lie commutativity
In this section, we will give a characterization of nonlinear strong -Lie commutativity preserving maps on general algebras. The following is our main result.
Theorem 2.1
Let be any algebra with unit
over a field
, and let
with
. Assume that
is a map. Then
preserves strong
-Lie commutativity, that is,
satisfies
for all
, if and only if
,
and
for all
.
Proof
The “if” part is obvious. For the “only if” part, since , we have
. It follows that
as
.
In the sequel, we will complete the proof by considering two cases.
Case 1 .
Take any . Then
(2.1)
(2.1)
Multiplying from the left- and the right-hand side in Equation 2.1, respectively, one gets
and
Comparing the above two equations, we obtain for each
. It follows from the arbitrariness of
that
. This and Equation 2.1 imply
. Note that
. So
holds for all
, completing the proof of the theorem.
Case 2 .
Take any and note that
. Since
preserves strong
-Lie commutativity, we have
That is,
holds for all . Now, by Case 1, the theorem is true.
Combining Case 1 and Case 2, the proof of the theorem is complete.
3. Maps preserving strong skew ![](//:0)
-Lie commutativity
In this section, we will discuss the maps preserving strong skew -Lie commutativity on general algebras.
Theorem 3.1
Let be any *-algebra with unit
over the real or complex field
and let
with
. Assume that
is a map. If
, then
preserves strong skew
-Lie commutativity, that is,
satisfies
for all
, if and only if
,
and
for all
.
Proof
Still, one only needs to prove the “only if” part.
By the assumption, for any , we have
(3.1)
(3.1)
Taking in Equation 3.1, one gets
. Note that
and
. We obtain
(3.2)
(3.2)
Taking in Equation 3.1, one gets
, that is,
(3.3)
(3.3)
Taking in Equation 3.1, one has
(3.4)
(3.4)
This implies
Multiplying from both sides in the above equation, we get
(3.5)
(3.5)
Combining Equations 3.4 and 3.5, we have(3.6)
(3.6)
Note that . Equation 3.6 implies
(3.7)
(3.7)
In the following, we will prove the theorem by two cases.
Case 1 .
In this case, Equation 3.7 implies(3.8)
(3.8)
Multiplying from the right-hand side in Equation 3.8, by Equation 3.2, one gets
(3.9)
(3.9)
On the other hand, combining Equations 3.3 and 3.8, one has . Multiplying
from the left-hand side in this equation and noting that Equation 3.2, one gets
(3.10)
(3.10)
It follows from Equations 3.9 to 3.10 that . The proof is finished.
Case 2 .
Multiplying from the left- and the right-hand side in Equation 3.3, respectively, by Equation 3.2 again, one can obtain
and
Comparing the above two equations gets that is,
(3.11)
(3.11)
We claim , and so
. In fact, if
, Equation 3.11 implies
; if
, multiplying
from both sides in Equation 3.11, one has
(3.12)
(3.12)
as . On the other hand, Equation 3.4 implies
This and Equations 3.11–3.12 yield
Note that and Equation 3.2. The above equation can be reduced to
Multiplying from the right side in the above equation yields
(3.13)
(3.13)
Replacing by
in Equation 3.13, one can get
, that is,
(3.14)
(3.14)
Combining Equations 3.13 and 3.14, one achieves for all
. The claim holds.
Now, it follows from Equation 3.3 that , and so
holds for all
. The theorem holds.
We complete the proof of the theorem.
If is surjective and
, then the condition
in Theorem 3.1 can be deleted.
Theorem 3.2
Let be any *-algebra with unit
over the real or complex field
, and let
with
and
. Assume that
is a surjective map. Then
preserves strong skew
-Lie commutativity if and only if
,
and
for all
.
Proof
By Theorem 3.1, to complete the proof of the theorem, one only needs to prove .
Indeed, by checking the proof of Theorem 3.1, Equation 3.6 still holds, that is,
Since , the above equation reduces to
for all
. As
is surjective, there exists
such that
. So
. It follows from the fact
that
.
Acknowledgements
The authors wish to express their thanks to the referees for their helpful comments and suggestions.
Additional information
Funding
Notes on contributors
Xiaofei Qi
Xiaofei Qi is an associate professor in the Department of Mathematics at Shanxi University. She received her PhD degree in Mathematics from Shanxi University. Her research interests are functional analysis, operator algebras, and operator theory. She has published some research articles in reputed international journals.
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