C-ASTERISK-ALGEBRAS GENERATED BY UNBOUNDED ELEMENTS

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