On Derivations of Operator Algebras with Involution

  • 1 DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE, FNM UNIVERSITY OF MARIBOR Koroška Cesta 160 2000 MARIBOR, SLOVENIA
  • 2 INSTITUTE OF MATHEMATICS, PHYSICS AND MECHANICS DEPARTMENT IN MARIBOR Gosposvetska 84 2000 MARIBOR, SLOVENIA

Abstract

The purpose of this paper is to prove the following result. Let X be a complex Hilbert space, let L(X) be an algebra of all bounded linear operators on X and let A(X) ⊂ L(X) be a standard operator algebra, which is closed under the adjoint operation. Suppose there exists a linear mapping D : A(X) → L(X) satisfying the relation 2D(AA*A) = D(AA*)A + AA*D(A) + D(A)A*A + AD(A*A) for all A ∈ A(X). In this case, D is of the form D(A) = [A,B] for all A ∈ A(X) and some fixed B ∈ L(X), which means that D is a derivation.

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