Abstract
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.
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