Representations on Rigged Spaces and Hilbert $$C^*$$-Modules

This chapter provides an introduction to \(*\)-representations of a \(*\)-algebra A on rigged spaces and Hilbert \(C^*\)-modules. A right (or left) rigged space is a right (or left) A-module equipped with an A-valued sesquilinear mapping which is compatible with the module action. First we explore \(*\)-representations on rigged spaces purely algebraically. Next we assume that the riggings are positive semi-definite; in particular, this holds for Hilbert \(C^*\)-modules. Then induced representations on “ordinary” Hilbert spaces can be defined and each imprimitivity bimodule of two \(*\)-algebras A and B yields an equivalence between \(*\)-representations of A and B. Finally, operators and \(*\)-representations on Hilbert \(C^*\)-modules are briefly investigated.