Bochner algebras and their compact multipliers (Q2883998)

Let \(\Omega \) be a locally compact Hausdorff space and \(\lambda \) be a positive Radon measure on \(\Omega\) with \(\operatorname{supp}\lambda =\Omega\). Let \(\mathcal {U}\) be a nontrivial normed algebra and \(L^{1}\left (\Omega ,\mathcal {U}\right)\) be the space of all \(\lambda\)-Bochner integrable functions \(f\:\Omega \rightarrow \mathcal {U}\). For each two functions \(f\), \(g\) from \(\Omega \) into \(\mathcal {U}\), the pointwise multiplication of \(f\) and \(g\) is denoted by \(f\cdot g\). In the paper it is proved that \(L^{1}\left (\Omega, \mathcal {U}\right)\) is an algebra with pointwise multiplication if and only if \(\Omega \) is discrete and \(\inf \bigl \{\lambda \left (O\right) : O\subseteq \Omega {\text{ is an open nonempty set}}\bigr \} > 0\). Under this condition the authors characterize compact and weakly compact left multipliers on \(L^{1}\left (\Omega, \mathcal {U}\right)\).