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Extending derivations. (English) Zbl 0544.46039

Let (\({\mathcal A},G,\tau)\) be a \(C^*\)-dynamical system, where G is a compact abelian group, \({\mathcal A}^{\tau}\) be the fixed point algebra, and \(\delta_ 0:{\mathcal D}_ 0\to {\mathcal A}^{\tau}\) be a \({}^*\)- derivation defined on a \({}^*\)-subalgebra \({\mathcal D}_ 0\) of \({\mathcal A}^{\tau}\). Suppose that there are unitaries \(u(\gamma)\) in each spectral subspace \({\mathcal A}^{\tau}(\gamma)\) (\(\gamma \in \hat G)\), such that \(u(0)=1\), \(u(\gamma_ 1)u(\gamma_ 2)u(\gamma_ 1+\gamma_ 2)^*\in {\mathcal D}_ 0\) and \(u(\gamma){\mathcal D}_ 0u(\gamma)^*\subseteq {\mathcal D}_ 0,\) and that there is a family of traces of \({\mathcal A}\) which separates the centre of \({\mathcal A}^{\tau}\). Then \(\delta_ 0\) extends to a \({}^*\)-derivation, commuting with G, on a domain including \(\{\) \(u(\gamma)\}\) if and only if
(i) \(u(\gamma)\delta_ 0(u(\gamma)^*.u(\gamma))u(\gamma)^*-\delta_ 0(.)\) is a bounded inner derivation on \({\mathcal A}^{\tau}\), for all \(\gamma\) in \(\hat G\), and
(ii) \(\phi [\delta_ 0(u(\gamma_ 1)^*u(\gamma_ 2)^*u(\gamma_ 1)u(\gamma_ 2))u(\gamma_ 2)^*u(\gamma_ 1)^*u(\gamma_ 2)u(\gamma_ 1)]=0\) for any trace \(\phi\) on \({\mathcal A}\), \(\gamma_ 1,\gamma_ 2\in \hat G\).

MSC:

46L55 Noncommutative dynamical systems
47B47 Commutators, derivations, elementary operators, etc.
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References:

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