C-ASTERISK-ALGEBRAS GENERATED BY UNBOUNDED ELEMENTS

The main aim of this paper is to provide a proper mathematical framewo rk for the theory of topological non-compact quantum groups, where we have to deal with non-unital C--algebras. The basic concepts and resu lts related to the affiliation relation in the C--algebra theory are recalled. In particular natural topologies on the set of affiliated el ements and on the set of morphisms are considered. The notion of a C- -algebra generated by a finite sequence of unbounded elements is intro duced and investigated. It is generalized to include continuous quantu m families of generators. An essential part of the duality theory for C--algebras is presented including complete proofs of many theorems a nnounced in [17]. The results are used to develop a presentation metho d of introducing non-unital C--algebras. Numerous examples related ma inly to the quantum group theory are presented.