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The cb-norm approximation of generalized skew derivations by elementary operators. (English) Zbl 1478.16040

Summary: Let \(A\) be a ring and \(\sigma:A\to A\) a ring endomorphism. A generalized skew (or \(\sigma\)-)derivation of \(A\) is an additive map \(d:A\to A\) for which there exists a map \(\delta:A\to A\) such that \(d(xy)=\delta(x)y+\sigma(x)d(y)\) for all \(x,y\in A\). If \(A\) is a prime \(C^*\)-algebra and \(\sigma\) is surjective, we determine the structure of generalized \(\sigma\)-derivations of \(A\) that belong to the cb-norm closure of elementary operators \(\mathcal{E}\ell(A)\) on \(A\); all such maps are of the form \(d(x)=bx+axc\) for suitable elements \(a,b,c\) of the multiplier algebra \(M(A)\). As a consequence, if an epimorphism \(\sigma:A\to A\) lies in the cb-norm closure of \(\mathcal{E}\ell(A)\), then \(\sigma\) must be an inner automorphism. We also show that these results cannot be extended even to relatively well-behaved non-prime \(C^*\)-algebras like \(C(X,\mathbb{M}_2)\).

MSC:

16W25 Derivations, actions of Lie algebras
47B47 Commutators, derivations, elementary operators, etc.
46L07 Operator spaces and completely bounded maps
16N60 Prime and semiprime associative rings
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