Glasnik Matematicki, Vol. 46, No.1 (2011), 43-48.

SOME REMARKS ON DERIVATIONS IN SEMIPRIME RINGS AND STANDARD OPERATOR ALGEBRAS

Joso Vukman

Department of Mathematics and Computer Science, University of Maribor, FNM, Koroška 160, 2000 Maribor, Slovenia
e-mail: joso.vukman@uni-mb.si


Abstract.   In this paper identities related to derivations on semiprime rings and standard operator algebras are investigated. We prove the following result which generalizes a classical result of Chernoff. Let X be a real or complex Banach space, let L(X) be the algebra of all bounded linear operators of X into itself and let A(X) in L(X) be a standard operator algebra. Suppose there exists a linear mapping D:A(X)→ L(X) satisfying the relation 2D(A3)=D(A2)A+A2D(A)+D(A)A2+AD(A2) for all A in A(X). In this case D is of the form D(A)=AB-BA for all A in A(X) and some fixed B in L(X), which means that D is a linear derivation.

2000 Mathematics Subject Classification.   16W10, 46K15, 39B05.

Key words and phrases.   Prime ring, semiprime ring, Banach space, standard operator algebra, derivation, Jordan derivation, Jordan triple derivation.


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DOI: 10.3336/gm.46.1.07


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