Norms of elementary operators (Q2760471)

Let \(E\) be a Banach space that contains an infinite dimensional complemented subspace with a Schauder basis, and \(A\) an algebra of bounded operators on \(E\) that contains the finite rank operators and endowed with the operator norm. For a given elementary operator \(Ta=\sum_{i=1}^\ell a_iab_i\in \mathcal{E}\ell\) on \(A\), a new one is defined by \(T^fa=\sum_{i=1}^\ell b_iaa_i\). The main result of the paper states that the map \(T\mapsto T^f\) is not continuous on \(\mathcal{E}\ell\), and it is claimed that this a variant of an unpublished result of Runde (to appear in J. Operator Th.), with an elementary proof. An editorial note says that an argument similar to the main theorem for the case of Hilbert space was provided by A. W. Wickstead.