A reflexivity criterion for Hilbert C*-modules over commutative C*-algebras

A C*-algebra $A$ is C*-reflexive if any countably generated Hilbert C*-module $M$ over $A$ is C*-reflexive, i.e. the second dual module $M^{\mathrm{\prime }\mathrm{\prime }}$ coincides with $M$. We show that a commutative C*-algebra $A$ is C*-reflexive if and only if for any sequence $I_k$ of disjoint non-zero C*-subalgebras, the canonical inclusion ${\oplus }_k{I}_k\subset A$ doesn’t extend to an inclusion of $\prod_k I_k$.