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Local characterization of Jordan *-derivations on \(\mathcal{B}(H)\). (English) Zbl 1438.47071

Summary: Let \(H\) be an infinite-dimensional real Hilbert space and \(\mathcal{B}(H)\) the algebra of all bounded linear operators on \(H\). Assume that \(\delta:\mathcal{B}(H)\to\mathcal{B}(H)\) is a real linear map and \(P\in\mathcal{B}(H)\) is zero, or the unit element, or a nontrivial idempotent with infinite-dimensional range and infinite-dimensional kernel. It is shown that \(\delta\) satisfies \(\delta (A^2)=\delta (A) A^*+ A\delta (A)\) for all \(A\in\mathcal{B}(H)\) with \(A^2=P\) if and only if \(\delta\) is an inner Jordan \(^*\)-derivation. An example is also given to illustrate that this is not necessarily true when \(H\) is finite-dimensional.

MSC:

47B47 Commutators, derivations, elementary operators, etc.
47L30 Abstract operator algebras on Hilbert spaces
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