Bijective linear maps preserving Lie products equal to rank-one nilpotents were recently shown to have nonstandard descriptions. In this paper, we show that bijective linear maps preserving Lie ...products equal to
, a rank-two matrix, are standard commutativity preserving maps.
The n-th Schrödinger algebra schn:=sl2⋉hn is the semi-direct product of the Lie algebra sl2 with the n-th Heisenberg Lie algebra hn. Let K be an algebraically closed field of characteristic zero. A ...Lie algebra L is zero product determined over K, if for every K-linear space V and every bilinear map ϕ:L×L→V with the property that ϕ(x,y)=0 whenever x,y=0, then there exists a linear map f:L,L→V such that ϕ(x,y)=f(x,y) for all x,y∈L. This article shows that the n-th Schrödinger algebra schn is zero product determined. Applying this result, product zero derivations and two-sided commutativity-preserving maps on schn are determined. Furthermore, quasi-derivations, linear anti-commuting maps, quasi-automorphisms and strong commutativity-preserving maps of schn are all obtained.
Let ϕ be a bijective linear map on the algebra of n×n complex matrices such that ϕ(e12)=e12 and ϕ(A),ϕ(B)=e12 whenever A,B=e12. The purpose of this paper is to describe ϕ. Surprisingly, ϕ has a ...different description from maps preserving zero Lie products.
A linear map ψ on a Lie algebra
over a field F with char
is called to be commuting (resp., skew-commuting) if
(resp.,
) for all
, and to be strong commutativity-preserving if
for all
. Let L be a ...finite-dimensional simple Lie algebra over an algebraically closed field F of characteristic 0, P a parabolic subalgebra of L. In this paper, firstly, we improve existing results about skew-symmetric biderivations on P by determining related linear commuting maps. Secondly, we classify the linear skew-commuting maps and the related symmetric biderivations on P, and so the biderivations of P are characterized. Finally, we classify the invertible linear strong commutativity-preserving maps of P.
Let Mn(D) be the ring of all n×n matrices over a division ring D, where n≥2 is an integer and let GLn(D) be the set of all invertible matrices in Mn(D). We describe maps f:GLn(D)→Mn(D) such that ...f(x),f(y)=x,y for all x,y∈GLn(D). The analogous result for singular matrices is also obtained.
Let
M
be a von Neumann algebra with no central summands of type
I
1
. Assume that
Φ
:
M
→
M
is a surjective map and
Φ
(
I
)
is an unitary operator. It is shown that
Φ
is strong bi-skew commutativity ...preserving (that is,
Φ
satisfies
Φ
(
A
)
Φ
(
B
)
∗
-
Φ
(
B
)
Φ
(
A
)
∗
=
A
B
∗
-
B
A
∗
for all
A
,
B
∈
M
) if and only if there exists a self-adjoint central operator
Z
∈
M
with
Z
2
=
I
such that
Φ
(
A
)
=
Z
A
Φ
(
I
)
for all
A
∈
M
. The strong bi-skew commutativity preserving maps on prime algebras with involution are also characterized.
Let H be an infinite dimensional separable complex Hilbert space and U be the group of all unitary operators on H. Motivated by the algebraic properties of surjective isometries of U that have ...recently been revealed, and also by some classical results related to automorphisms of the unitary groups of operator algebras, we determine the structures of bijective transformations of U that respect certain algebraic operations. These are, among others, the usual product of operators, the Jordan triple product, the inverted Jordan triple product, and the multiplicative commutator. Our basic approach to obtain these results is the use of commutativity preserving transformations on the unitary group.
Let
L be a Lie algebra over a field
F. We say that
L is zero product determined if, for every
F-linear space
V and every bilinear map
φ
:
L
×
L
→
V
, the following condition holds. If
φ
(
x
,
y
)
=
0
...whenever
x
,
y
=
0
, then there exists a linear map
f from
L
,
L
to
V such that
φ
(
x
,
y
)
=
f
(
x
,
y
)
for all
x
,
y
∈
L
. This article shows that every parabolic subalgebra
p
of a (finite-dimensional) simple Lie algebra defined over an algebraically closed field is always zero product determined. Applying this result, we present a method different from that of Wang et al. (2010)
9 to determine zero product derivations of
p
, and we obtain a definitive solution for the problem of describing two-sided commutativity-preserving maps on
p
.
Let be a unital prime ring containing a nontrivial idempotent P. Assume that Φ: → is a nonlinear surjective map. It is shown that Φ preserves strong commutativity if and only if Φ has the form ...Φ(A) = αA + f(A) for all A ∈ , where α ∈ {1, −1} and f is a map from into ( ). As an application, a characterization of nonlinear surjective strong commutativity preserving maps on factor von Neumann algebras is obtained.
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