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Zbl 0751.46041
Local derivations.
(English)
[J] J. Algebra 130, No.2, 494-509 (1990). ISSN 0021-8693

The main result of this paper is:\par Theorem A. If $\delta$ is a norm-continuous linear map of a von Neumann algebra ${\cal R}$ into a dual ${\cal R}$- bimodule ${\cal M}$ with the property that for each $A$ in ${\cal R}$ there is a (norm-continuous) derivation $\delta\sb A$ such that $\delta(A)=\delta\sb A(A)$, then $\delta$ is itself a derivation.\par In the above theorem, $\delta$ is an example of a local derivation. It follows from the above that:\par Theorem B. If $\delta$ is a norm-continuous linear mapping of a von Neumann algebra ${\cal R}$ into itself such that, for each $A$ in ${\cal R}$, there is a $T\sb A$ in ${\cal R}$ for which $\delta(A)=AT\sb A-T\sb A A$, then there is a $T$ in ${\cal R}$ such that $\delta(A)=AT-TA$ for all $A$ in ${\cal R}$.\par The proof is interesting and the result is new even in finite dimensions.\par Examples are given of local derivations which are not derivations (both finite and infinite dimensional examples).\par We note in closing that the author proves that every local derivation of the polynomial in $n$ variables over $\bbfC$ into the polynomials in $m$ variables over $\bbfC$ is a derivation if $n\leq m$. (The list of $m$ variables contains the list of $n$ variables.).
[M.E.Walter (Boulder)]
MSC 2000:
*46L57 Derivations etc. in $C^*$-algebras
46L10 General theory of von Neumann algebras
16W20 Morphisms of associative rings
46L40 Automorphisms of C*-algebras
16D10 General module theory (assoc. rings and algebras)
13D03 (Co)homology of commutative rings and algebras

Keywords: cohomology; norm-continuous linear map; von Neumann algebra; bimodule; local derivation

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