On large submodules in Hilbert C*-modules

We consider several natural ways of expressing the idea that a one-sided ideal in a C & lowast;-algebra (or a submodule in a Hilbert C & lowast;-module) is large, and show that they differ, unlike the case of two-sided ideals in C & lowast;-algebras. We then show how these different notions, for ideals and for submodules, are related. We also study some permanence properties for these notions. Finally, we use essential right ideals to extend the inner product on a Hilbert C & lowast;-module to a part of the dual module. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.