Abstract
We prove that any derivation of the *-algebra \({LS({\mathcal{M}})}\) of all locally measurable operators affiliated with a properly infinite von Neumann algebra \({{\mathcal{M}}}\) is continuous with respect to the local measure topology \({t({\mathcal{M}})}\) . Building an extension of a derivation \({\delta:{\mathcal{M}}\rightarrow LS({\mathcal{M}})}\) up to a derivation from \({LS({\mathcal{M}})}\) into \({LS({\mathcal{M}})}\) , it is further established that any derivation from \({{\mathcal{M}}}\) into \({LS({\mathcal{M}})}\) is \({t({\mathcal{M}})}\) -continuous.
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Ber, A.F., Chilin, V.I. & Sukochev, F.A. Continuity of derivations of algebras of locally measurable operators. Integr. Equ. Oper. Theory 75, 527–557 (2013). https://doi.org/10.1007/s00020-013-2039-3
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DOI: https://doi.org/10.1007/s00020-013-2039-3