Let A be an algebra and let f be a nonconstant noncommutative polynomial. In the first part of the paper, we consider the relationship between A,A, the linear span of commutators in A, and spanf(A), ...the linear span of the image of f in A. In particular, we show that A,A=A implies spanf(A)=A. In the second part, we establish some Waring type results for images of polynomials. For example, we show that if C is a commutative unital algebra over a field F of characteristic 0, A is the matrix algebra Mn(C), and the polynomial f is neither an identity nor a central polynomial of Mn(F), then every commutator in A can be written as a difference of two elements, each of which is a sum of 7788 elements from f(A) (if C=F is an algebraically closed field, then 4 elements suffice). Similar results are obtained for some other algebras, in particular for the algebra B(H) of all bounded linear operators on a Hilbert space H.
Let A be a finite-dimensional algebra over a field F with char(F)≠2. We show that a linear map D:A→A satisfying xD(x)x∈A,A for every x∈A is the sum of an inner derivation and a linear map whose image ...lies in the radical of A. Assuming additionally that A is semisimple and char(F)≠3, we show that a linear map T:A→A satisfies T(x)3−x3∈A,A for every x∈A if and only if there exist a Jordan automorphism J of A lying in the multiplication algebra of A and a central element α satisfying α3=1 such that T(x)=αJ(x) for all x∈A. These two results are applied to the study of local derivations and local (Jordan) automorphisms. In particular, the second result is used to prove that every local Jordan automorphism of a finite-dimensional simple algebra A (over a field F with char(F)≠2,3) is a Jordan automorphism.
COMMUTING MAPS: A SURVEY Brešar, Matej
Taiwanese journal of mathematics,
09/2004, Volume:
8, Issue:
3
Journal Article
Peer reviewed
Open access
A map f on a ring A is said to be commuting if f(x) commutes with x for every x ∈ A. The paper surveys the development of the theory of commuting maps and their applications. The following topics are ...discussed: commuting derivations, commuting additive maps, commuting traces of multiadditive maps, various generalizations of the notion of a commuting map, and applications of results on commuting maps to different areas, in particular to Lie theory.
If a noncommutative polynomial
f
is neither an identity nor a central polynomial of
A
=
M
n
(
ℂ
)
, then every trace zero matrix in
A
can be written as a sum of two matrices from
f
(
A
)
−
f
(
A
)
. ...Moreover, “two” cannot be replaced by “one”.
Abstract
Three problems connecting functional identities to the recently introduced notion of a zero Lie product determined Banach algebra are discussed. The first one concerns commuting linear maps, ...the second one concerns derivations that preserve commutativity and the third one concerns bijective commutativity preserving linear maps.
Let R and S be nonassociative unital algebras. Assuming that either one of them is finite dimensional or both are finitely generated, we show that every derivation of R⊗S is the sum of derivations of ...the following three types: (a) adu where u belongs to the nucleus of R⊗S, (b) Lz⊗f where f is a derivation of S and z lies in the center of R, and (c) g⊗Lw where g is a derivation of R and w lies in the center of S.
In certain rings containing non-central idempotents we characterize homomorphisms, derivations, and multipliers by their actions on elements satisfying some special conditions. For example, we ...consider the condition that an additive map $h$ between rings $\mathcal{A}$ and $\mathcal{B}$ satisfies $h(x)h(y)h(z)=0$ whenever $x,y,z\in\mathcal{A}$ are such that $xy=yz=0$. As an application, we obtain some new results on local derivations and local multipliers. In particular, we prove that if $\mathcal{A}$ is a prime ring containing a non-trivial idempotent, then every local derivation from $\mathcal{A}$ into itself is a derivation.
Let A be a unital algebra over a field
with char
, and let
be linear maps. We say that f is a
-derivation if
for all
, and we say that f is a Jordan
-derivation if
for all
(here,
). We show that if ...the property that every Jordan
-derivation is a
-derivation holds in A, then so does in the algebra
for every commutative unital algebra S. We also show that every semiprime algebra A has this property. Combining these two results, it follows, in particular, that the classical Jordan derivations are derivations on the tensor product between a semiprime and a commutative algebra.
Let R and S be unital algebras. We show that if X is a d-free subset of R and S is finite dimensional, then the set X={x⊗s|x∈X,s∈S} is a d-free subset of the algebra R⊗S. The assumption that S is ...finite dimensional turns out to be necessary in general. However, we show that some important functional identities have only standard solutions on X even when S is infinite dimensional.
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