Compact and ``compact'' operators on standard Hilbert modules over \(C^*\)-algebras (Q1790518)

\textit{D. J. Kečkić} and \textit{Z. Lazović} [Ann. Funct. Anal. 9, No. 2, 258--270 (2018; Zbl 1394.46047)] proved that, on a standard Hilbert module \(H_A\), where \(A\) is a unital \(W^*\)-algebra, there is a locally convex topology such that any ``compact'' operator (i.e., any operator in the norm closure of the linear span of the operators of the form \(\theta_{y, z}(x)=y \langle z, x \rangle\), \(x, y, z\in H_A\)) is compact (in the sense that it maps bounded sets into totally bounded sets). Let \(A\) be a unital \(C^*\)-algebra. In this paper, the author defines a topology on \(H_A\) and a topology on \({H_A}^\sharp\) (the extension of the module \(H_A\) by the algebra \(A^{**}\)) such that for them any ``compact'' operator is compact.