Let D be an alternative division ring with characteristic different from two. The purpose of this paper is to characterize additive mappings f,g:D→D satisfying certain identities studied by Vukman, ...Brešar, and Catalano previously on an associative division ring. We also present some new identities concerning Lie and Jordan products, and provide a complete description of commuting maps on octonion algebras.
Consider a ring $R$, which is semiprime and also having $k$-torsion freeness. If $F, d : R\to R$ are two additive maps fulfilling the algebraic identity $$F(x^{n+m})=F(x^m) x^n+ x^m d(x^n)$$ for each ...$x$ in $R.$ Then $F$ will be a generalized derivation having $d$ as an associated derivation on $R$. On the other hand, in this article, it is also derived that $f$ is a generalized left derivation having a linked left derivation $\delta$ on $R$ if they satisfy the algebraic identity $$f(x^{n+m})=x^n f(x^m)+ x^m \delta(x^n)$$ for each $x$ in $R$ and $k\in \{2, m, n, (n+m-1)!\}$ and at last an application on Banach algebra is presented.
In this paper we prove that an additive mapping preserving ρ-orthogonality has to be a similarity (scalar multiple of a linear isometry). We apply this result to give full answer to question posed by ...Alsina, Sikorska and Tomás. Namely, unlike the recent result, now smoothness is not assumed. More precisely, we prove that a function f:X→Y from a normed space X into a normed space Y, satisfying the functional equation∀x,y∈Xf(y−ρ+′(x,y)‖x‖2x)=f(y)−ρ+′(f(x),f(y))‖f(x)‖2f(x), has to be a linear similarity. It is worth mentioning that a useful tool will be the concept of semi-smoothness. This notion will play a crucial role in the proof of the main result, and the proofs use methods of linear algebra.
We study orthogonality preserving operators between
C
∗
-algebras,
JB
∗
-algebras and
JB
∗
-triples. Let
T
:
A
→
E
be an orthogonality preserving bounded linear operator from a
C
∗
-algebra to a
JB
∗
...-triple satisfying that
T
∗
∗
(
1
)
=
d
is a von Neumann regular element. Then
T
(
A
)
⊆
E
2
∗
∗
(
r
(
d
)
)
, every element in
T
(
A
)
and
d operator commute in the
JB
∗
-algebra
E
2
∗
∗
(
r
(
d
)
)
, and there exists a triple homomorphism
S
:
A
→
E
2
∗
∗
(
r
(
d
)
)
, such that
T
=
L
(
d
,
r
(
d
)
)
S
, where
r
(
d
)
denotes the range tripotent of
d in
E
∗
∗
. An analogous result for
A being a
JB
∗
-algebra is also obtained. When
T
:
A
→
B
is an operator between two
C
∗
-algebras, we show that, denoting
h
=
T
∗
∗
(
1
)
then,
T orthogonality preserving if and only if there exists a triple homomorphism
S
:
A
→
B
∗
∗
satisfying
h
∗
S
(
z
)
=
S
(
z
∗
)
∗
h
,
h
S
(
z
∗
)
∗
=
S
(
z
)
h
∗
, and
T
(
z
)
=
L
(
h
,
r
(
h
)
)
(
S
(
z
)
)
=
1
2
(
h
r
(
h
)
∗
S
(
z
)
+
S
(
z
)
r
(
h
)
∗
h
)
.
This allows us to prove that a bounded linear operator between two
C
∗
-algebras is orthogonality preserving if and only if it preserves zero-triple-products.
Mappings preserving B-orthogonality Wójcik, Paweł
Indagationes mathematicae,
January 2019, 2019-01-00, 20190101, Volume:
30, Issue:
1
Journal Article
Peer reviewed
It is well known that a linear mapping T:X→Y preserving the Birkhoff orthogonality (i.e. ∀x,y∈Xx⊥By⇒Tx⊥BTy), has to be a similarity. For real spaces it has been proved by Koldobsky (1993); a proof ...including both real and complex spaces has been given by Blanco and Turnšek (2006). In the present paper the author would like to present a somewhat simpler proof of this nice theorem. Moreover, we extend the Koldobsky theorem; more precisely, we show that the linearity assumption may be replaced by additivity.
We investigate the class of operators preserving sesquilinear form. We show some properties similar to those characterizing mappings which preserve the inner product. Our considerations are carried ...out in spaces with inner product structure as well as in normed spaces. We also consider the class of additive mappings preserving orthogonality.
We investigate orthogonally additive mappings
f
:
E
→
G
from a real inner product space
(
E
,
|
|
·
|
|
)
of dimension at least 2 to a group
(
G
,
·
)
and show that there are additive functions
a
:
R
...→
G
and
b
:
E
→
G
such that
f
(
x
)
=
a
(
|
|
x
|
|
2
)
b
(
x
)
for all
x
∈
E
. Moreover the subgroup generated by the image of
f
is an abelian group, which is
n
-divisible for every positive integer
n
.
If
F
,
D
:
R
→
R
are additive mappings satisfying
F
(
x
n
y
n
)
=
α
(
x
n
)
F
(
y
n
)
+
β
(
y
n
)
D
(
x
n
)
for all
x
,
y
∈
R
, where
α
,
β
are automorphisms on
R
, then
F
becomes generalized left
(
...α
,
β
)
-derivation with associated Jordan left
(
α
,
β
)
-derivation
D
on
R
under suitable torsion restriction.
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