The theory of multiplier modules of Hilbert C*-modules is reconsidered to obtain more properties of these special Hilbert C*-modules. The property of a Hilbert C*-module to be a multiplier C*-module is shown to be invariant with respect to consideration as a left or right Hilbert C*-module in the sense of imprimitivity bimodules in strong Morita equivalence theory. The interrelation of the C*-algebras of "compact" operators, the Banach algebras of bounded module operators and the Banach spaces of bounded module operators of a Hilbert C*-module to its C*-dual Banach C*-module, are characterized for pairs of Hilbert C*-modules and their respective multiplier modules. The structures on the latter are always isometrically embedded into the respective structures on the former. Examples are given for which continuation of these kinds of bounded module operators from the initial Hilbert C*-module to its multiplier module fails. However, existing continuations turn out to be always unique. Similarly, bounded modular functionals from both kinds of Hilbert C*-modules to their respective C*-algebras of coefficients are compared, and eventually existing continuations are shown to be unique.

I am grateful to David R. Larson for the years of fruitful collaboration during 1998-2002. David R. Larson from Texas A & M University made outstanding contributions to various mathematics subjects including operator theory, operator algebras and applied harmonic analysis, in particular wavelet and frame theory. As well, he was engaged in organizing the highly impacted annual Great Plains Operator Theory Symposium (GPOTS) during many years.

I thank Bartosz Kwasniewski, who attracted my attention to the double centralizer type approach to multiplier modules using only Banach C*-module constructions, and to problems about the equivalence of both these approaches in the case of Hilbert C*-modules, cf. [13, 14, 12, 17, 47].