On certain structural properties of Banach algebras associated with automorphisms (Q941962)

The author studies the Banach algebra \(B(A,\tau)\) generated by a Banach algebra \(A\) of bounded linear operators on a Banach space \(X\) and the operators \(\tau(G)\), corresponding to a representation of a group \(G\) by isometries of \(X\), with the property that \(\tau(t)a\tau(t)^{-1}\in A\) for all \(a\in A\) and \(t\in G\). There are two properties explored in this paper. Property \((\ast)\): For any finite sum \(b =\sum a_t\tau_t\), with \(a_t\in A \;(t\in G)\), the inequality \(\| b\|\geq\| a_e\|\) holds, from which it follows that for every \(t\in G\) the map \(\sum a_s\tau_s\mapsto a_t\) extends to \(N_t\colon B(A,\tau)\rightarrow A\). Property \((\ast\ast)\): \(b\in B(A,\tau)\) is zero exactly when \(N_t(b) = 0\) for each \(t\in G\). The author restricts his attention to the case where \(A\) is isomorphic to the algebra \(C(\Omega,\mathcal{B}(Y))\) of all continuous functions from a completely regular space \(\Omega\) into the bounded linear operators \(\mathcal{B}(Y)\) on a Banach space \(Y\). Then \(\tau\) induces an action of \(G\) on the centre of \(A\) which is assumed to have the form \([\tau_t z\tau_t^{-1}](\omega)= z(\varphi_t^{-1}(\omega))\) \((z\in\mathcal{Z}(A), \;\omega\in\Omega)\), where \(\varphi_t:\Omega\rightarrow\Omega\) is an automorphism. As a matter of fact, it is shown that \(B(A,\tau)\) has Property \((\ast)\) if the action of \(G\) on \(\Omega\) is topologically free.