Diagonalization of compact operators on Hilbert modules over $C^*$-algebras of real rank zero

The classical Hilbert–Schmidt theorem can be extended to compact operators on Hilbert $\mathscr A$-modules over $W^*$-algebras of finite type; i.e., with minor restrictions, compact operators on $\mathscr H_\mathscr A^*$ can be diagonalized over $\mathscr A$. We show that if $B$ is a weakly dense $C^*$-subalgebra of $\mathscr A$ with real rank zero and if some additional condition holds, then the natural extension from $\mathscr H_\mathscr B$ to $\mathscr H_\mathscr A^*\supset\mathscr H_\mathscr B$ of a compact operator can be diagonalized so that the diagonal elements belong to the original $C^*$-algebra $\mathscr B$.