\(B_ r\)-complete spaces which are not B-complete (Q800601)

Let F be a closed subspace of \(\ell^{\infty}\). Let \(F^*\) be the closure of F in \((\ell^{\infty})''[\sigma ((\ell^{\infty})'',(\ell^{\infty})')].\) Let \({\mathcal F}_ F\) be the topology of Mackey \(\mu ((\ell^{\infty})',\ell^{\infty}+F^*).\) In this paper the following result is proved: If the strong dual of F is separable and F is of infinite codimension in \(F^*\) there is a topology \({\mathcal F}\) on \((\ell^{\infty})'\) which has the following properties: 1. \({\mathcal F}\) is coarser than \({\mathcal F}_ F\). 2. \((\ell^{\infty})'[{\mathcal F}]\) is \(B_ r\)-complete. 3. \((\ell^{\infty})'[{\mathcal F}]\) is not B-complete. A precise example of a \(B_ r\)-complete space which is not B-complete can be constructed by taking \(F=c_ 0\). An example of a \(B_ r\)-complete space with a non- complete quotient is given also.