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Commutants and derivation ranges

In this paper we obtain some results concerning the set ${\mathcal M} = \cup \bigl \lbrace \overline{R(\delta _A)}\cap \lbrace A\rbrace ^{\prime }\: A\in {\mathcal L(H)}\bigr \rbrace $, where $\overline{R(\delta _A)}$ is the closure in the norm topology of the range of the inner derivation $\delta _A$ defined by $\delta _A (X) = AX - XA.$ Here $\mathcal H$ stands for a Hilbert space and we prove t…