Geometrical aspects of Hilbert C*-modules

The aim of the present paper is to solve some major open problems of Hilbert C*-module theory by applying various aspects of multiplier C*-theory. The key result is the equivalence established between positive invertible quasi-multipliers of the C*-algebra of "compact'' operators on a Hilbert C*-module {M, <., .> } and A-valued inner products on M, inducing an equivalent norm to the given one. The problem of unitary isomorphism of C*-valued inner products on a Hilbert C*-module is considered and new criteria are formulated. Countably generated Hilbert C*-modules turn out to be unitarily isomorphic if they are isomorphic as Banach C*-modules. The property of bounded module operators on Hilbert C*-modules of being "compact'' and/or adjointable is unambiguously connected to operators with respect to any choice of the C*-valued inner product on a fixed Hilbert C*-module if every bounded module operator possesses an adjoint operator on the module. Every bounded module operator on a given full Hilbert C*-module turns out to be adjointable if the Hilbert C*-module is orthogonally complementary. Moreover, if the unit ball of the Hilbert C*-module is complete with respect to a certain locally convex topology, then these two properties are shown to be equivalent to self-duality.