Operator K-Theory and Functor N 0. Some Applications

In this paper, we present a general introduction to the K-theory of C ^*-algebras and survey of our previous papers, where the functor N ₀ from the category of von Neumann algebras to the category of Abelian groups was defined. We investigate the properties of this functor (in particular, its interrelation with the functor K ₀) and point out some applications of the functor N ₀ in noncommutative geometry. In addition, we recall facts of theory of C ^*-algebras, von Neumann algebras, and Hilbert C ^*-modules.