Hyperreflexivity of the derivation space of some group algebras

Let T be a continuous linear operator on a Banach algebra A. We address the question of whether the constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\rm sup}\{||aT(b)c||: a, b, c \in A, \, ab = bc = 0, ||a|| = ||b|| = ||c||=1\}}$$\end{document} being small implies that the distance from T to the space Der(A) of all continuous derivations on A is small. We show that this is the case for amenable group algebras. As a consequence, we deduce that Der(L1(G)) is hyperreflexive for each amenable group in [SIN].