Abstract
Let \(\,X\,\) be a completely regular Hausdorff space and \(\,\mathcal {B}o\,\) be the \(\sigma \)-algebra of Borel sets in X. Let \(C_b(X)\) (resp. \(B(\mathcal {B}o)\)) be the space of all bounded continuous (resp. bounded \(\mathcal {B}o\)-measurable) scalar functions on X, equipped with the natural strict topology \(\beta \). We develop a general integral representation theory of \((\beta ,\xi )\)-continuous operators from \(C_b(X)\) to a lcHs \((E,\xi )\) with respect to the representing Borel measure taking values in the bidual \(E''_\xi \) of \((E,\xi )\). It is shown that every \((\beta ,\xi )\)-continuous operator \(T:C_b(X)\rightarrow E\) possesses a \((\beta ,\xi _\mathcal {E})\)-continuous extension \({\hat{T}}:B(\mathcal {B}o)\rightarrow E''_\xi \), where \(\xi _\mathcal {E}\) stands for the natural topology on \(E''_\xi \). If, in particular, X is a k-space and \((E,\xi )\) is quasicomplete, we present equivalent conditions for a \((\beta ,\xi )\)-continuous operator \(T:C_b(X)\rightarrow E\) to be weakly compact. As an application, we have shown that if X is a k-space and a quasicomplete lcHs \((E,\xi )\) contains no isomorphic copy of \(c_0\), then every \((\beta ,\xi )\)-continuous operator \(T:C_b(X)\rightarrow E\) is weakly compact.
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Nowak, M. Integral Representation of Continuous Operators with Respect to Strict Topologies. Results Math 72, 843–863 (2017). https://doi.org/10.1007/s00025-017-0678-4
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DOI: https://doi.org/10.1007/s00025-017-0678-4