Diagonalization of compact operators in Hilbert modules over C*-algebras of real rank zero

It is known that the classical Hilbert–Schmidt theorem can be generalized to the case of compact operators in Hilbert $A$-modules $H_A^{\ast }$ over a $W^{\ast }$-algebra of finite type, i.e. compact operators in $H_A^{\ast }$ under slight restrictions can be diagonalized over $A$. We show that if $B$ is a weakly dense $C^{\ast }$-subalgebra of real rank zero in $A$ with some additional property then the natural extension of a compact operator from $H_B$ to $H_A^\supset H_B$canbediagonalizedwithdiagonalentriesbeingfromthe$C^$-algebra$B$.