On the Hahn-Banach theorem for Hilbert C*-modules

We show that for a given C*-algebra A and for any pair of Hilbert A-modules {{M, 〈., .〉},N ⊆ M} every bounded A-linear mapping r : N → A can be continued to a bounded A-linear mapping r : M → A so that (i) ‖r‖ = ‖r‖, (ii) r restricted to N equals r and (iii) the extended mappings of N ′ form a Banach A-submodule of M wherein the extensions {r n : n ∈ N} of the standardly embedded mappings {rn = 〈., n〉 : n ∈ N} ⊆ M ′ coincide with the latter, if and only if the multiplier C*algebra M(A) of A is monotone complete, if and only if the multiplier C*-algebra M(A) of A is additively complete. In this case the restrictions of the extended mappings r ∈ M to N ⊆ M equal zero automatically. This generalized Hahn-Banach theorem is valid for a particular pair {M,N ⊆ M} of Hilbert A-modules if and only if the bi-orthogonal complement N of the Hilbert A-submodule N taken with respect to the A-bidual Hilbert A-module {M, 〈., .〉} of M is a direct summand of M and the A-bidual Banach C*-modules N ′′ and (N) coincide. Classification numbers: 46L99, 46H25