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 {, ., } and A-valued inner products on , 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.