zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
On left derivations and related mappings. (English) Zbl 0703.16020

If X is a left R module, then an additive map D:RX is a left derivation if D(ab)=aD(b)+bD(a) for all a,bR, and is a Jordan left derivation if D(a 2 )=2aD(a) for all aR. The main theorem of the paper shows that if X is 6-torsion free and no nonzero submodule has an annihilator in R, then the existence of a nonzero Jordan left derivation forces R to be commutative. One corollary of this result shows that there are no nonzero Jordan left derivations of the algebra L(A) of continuous operators on A, a Hausdorff locally convex vector space, into either A or L(A). When D is a left derivation, the torsion assumption in the main theorem can be removed, and also, if D:RR, then D(R) is central when R is a semi-prime ring.

The authors apply their results to a Banach algebra A by showing that if D:AA is a continuous linear left derivation, then D(A)rad(A), and if D is a continuous linear Jordan derivation with D(x)x-xD(x)rad(A) for all xA, then again D(A)rad(A). A final application is to functional equations. Let X be a Banach space, B(X) the algebra of bounded linear operators on X, and f and g additive maps of B(X) into either X or B(X). If f(U)=U 2 g(U -1 ) for all invertible UB(X), then f=g and f(T)=Tf(I) for all TB(X).

Reviewer: C.Lanski

MSC:
16W25Derivations, actions of Lie algebras (associative rings and algebras)
16N60Prime and semiprime associative rings
16U70Center, normalizer (invariant elements) for associative rings
39B42Matrix and operator functional equations
46H99Topological algebras, normed rings and algebras, Banach algebras