Hyperreflexivity of the derivation space of some group algebras. II (Q2880383)

Let \(X\) be a Banach space and let \(M\) be a closed linear subspace of \(B(X)\), the Banach algebra of all bounded linear operators on \(X\). For \(T\in B(X)\), one defines the distance of \(T\) to \(M\) as \(\operatorname{dist}(T,M)=\inf\{ \| T-S\| : S\in M\}\) and the Arveson distance as \(\alpha(T,M)=\sup\{ \inf\{ \| Tx-Sx\| : S\in M\}: x\in X,\;\| x\| \leq 1\}\). It is easily seen that \(\alpha(T,M) \leq \operatorname{dist}(T,M)\). If there exists a constant \(C\geq 1\) such that \(\operatorname{dist}(T,M) \leq C \alpha(T,M)\) for any \(T\in B(X)\), then \(M\) is said to be a hyperreflexive space of operators. In this paper, which is a continuation of [\textit{J. Alaminos}, \textit{J. Extremera} and \textit{A. R. Villena}, Math. Z. 266, 571--582 (2010; Zbl 1203.47019)], it is shown that the space of generalized derivations on the group algebra of a locally compact group is hyperreflexive and the same holds true for the space of derivations on the group algebra of a locally compact group which contains an open subgroup with polynomial growth.