Hyperreflexivity of the derivation space of some group algebras
Let \(A\) be a complex Banach algebra and \(X\) be a Banach \(A\)-bimodule. A linear map \(D:A\to X\) is called a derivation if \(D(ab)=D(a)\cdot b+a\cdot D(b)\) holds for arbitrary \(a, b\in A\). For instance, let \(x\in X\). Then \(\text{ad}_{x}(a)=a\cdot x - x\cdot a\) \((a\in A)\) defines a continuous derivation \(\text{ad}_{x}\); derivations of this type are called inner derivations. Denote by \(\text{Der}(A,X)\) the space of all continuous derivations from \(A\) to \(X\); in particular \(\text{Der}(A)\) is the space of all continuous derivations on \(A\). A Banach algebra \(A\) is said to be amenable if, for every dual \(A\)-bimodule \(X^*\), each element of \(\text{Der}(A,X^*)\) is an inner derivation. For a bounded linear operator \(T:A\to X\), let \(\text{der}(T)=\sup\{ \| T(ab)-T(a)\cdot b-a\cdot T(b)\|:a,b \in A, \| a\|=\| b\|=1\}\). If \(A\) is an amenable Banach algebra, then there exists a constant \(C>0\) such that, for every dual Banach \(A\)-bimodule \(X^*\) and for every bounded linear operator \(T:A\to X^*\), there exists \(\varphi \in X^*\) such that \(\| T- \text{ad}_{\varphi}\| \leq C \operatorname{der}(T)\). It follows from this that, under some additional conditions on \(A\), which are satisfied by a group algebra of an amenable group and by the algebra of compact operators on a separable Hilbert space, for instance, one has \(\text{dist}(T,\text{Der}(A))\leq C\operatorname{der}(T)\) for every bounded linear operator \(T\) on \(A\). Let \(X\) and \(Y\) be Banach spaces and let \(B(X,Y)\) be the Banach space of all bounded linear operators from \(X\) to \(Y\). The Arveson distance of \(T\in B(X,Y)\) to a closed linear subspace \({\mathcal S}\subseteq B(X,Y)\) is defined as \(\alpha(T,{\mathcal S})=\sup_{\| x\| \leq 1}\inf_{S\in {\mathcal S}}\| Tx-Sx\|\). One has \(\alpha(T,{\mathcal S}) \leq \text{dist}(T,{\mathcal S})\) for every space \({\mathcal S}\subseteq B(X,Y)\) and every \(T\in B(X,Y)\). If there exists a constant \(C\geq 1\) such that \(\text{dist}(T,{\mathcal S})\leq C \alpha(T,{\mathcal S})\), then \({\mathcal S}\) is said to be a hyperreflexive space of operators. It is proved that \(\text{Der} (L^{1}(G))\) is a hyperreflexive space whenever \(G\) is an amenable group in the class [SIN].
scientific article; zbMATH DE number 5797336
scientific article