Norms of generalized derivations with values in symmetric spaces

Abstract

Let $B(\mathcal{H})$ be the $\ast$-algebra of all bounded linear operators on a Hilbert space $\mathcal{H}$. A classical result due to Stamplfi shows that, for any $a\in B(\mathcal{H})$, the norm of the inner derivation ${\delta }_a=[a,\cdot ]$ on $B(\mathcal{H})$ is given by $\Vert {\delta }_a{\Vert }_{B(\mathcal{H})\to B(\mathcal{H})}=2\inf \{\Vert a-c1{\Vert }_{B(\mathcal{H})}:c\in \mathbb{C}\}$. In the present paper, we provide a formula for the norm of a generalized derivation implemented by self-adjoint operators from an ideal in $B(\mathcal{H})$, which partially answers a question posed by Fialkow and Loebl in the 1980s. We also consider this question in a more general setting of a symmetrically normed (not necessarily complete or separable) space affiliated with a properly infinite semifinite von Neumann algebra.

This version fixes an oversight in the proof of Theorem 1.3 in the previous version by referring to a lemma in [Ber-Sukochev, JFA, 2012]