Operator \(K\)-theory and functor \(N_0\) some applications (Q2487400)

This paper gives a general introduction, firstly to main concepts of the abstract theory of operator algebras and Hilbert modules over them, including \(C^*\)-algebras, von Neumann algebras and Hilbert \(C^*\)-modules (Chapter I); secondly to \(K\)-theory of \(C^*\)-algebras (Chapter II). After these basic elements, a survey of previous papers of the author is given [Zap. Nauch Semin. POMI 266, 234--244 (2000; Zbl 1023.46076), English translation in J. Math. Sci., New York 113, No. 5, 675--682 (2003; Zbl 1044.46054), Vestn. Mosk. Univ., Ser. Mat., Mekh. 4, 55--58 (2000; Zbl 1074.46511), ``Generalized Lefschetz numbers of unitary endomorphisms of \(W^*\)-elliptic complexes'', in Int. Conf. Dedic. 80th Ann. V. A. Rokhlin, 56--58 (1999), Acta Appl. Math. 68, 137--157 (2001; Zbl 1013.46058) (Chapter III)]. So, the functor \(N_0\) from the category of von Neumann algebras to the category of Abelian groups is defined and the properties of this functor (in particular, its interaction with the functor \(K_0\)) are investigated. Moreover, the author points out some applications of the functor \(N_0\) in noncommutative geometry. Namely, the so-called normed semigroups possessing a ``sufficiently good'' topological structure and, in addition, a richer algebraic structure, which is similar to the module structure and their symmetrizations, are discussed. The Abelian semigroup \({\mathcal N}(A)\) as the most interesting example of a normed semigroup is studied.