In the end of the seventies of the last century Alexander Miščenko and Anatolij Fomenko published a theorem on generalized Fredholm properties of elliptic operators acting on sections of certain bundles over compact manifolds, and derived an index theorem for them of a form which is similar to form of the Atiyah-Singer theorem. We explain certain parts of this generalization. The generalization allows to consider bundles which are not of finite rank, but which are finitely generated over a unital C*-algebra. We give examples on what happens when the algebra is the unitization of the algebra K(H) of compact operators on a separable Hilbert space.
On one hand, the generalized index and Fredholm theory seems to be of an interest for quantum theorists because the C*-algebra of compact operators has a significant role in measurements' concepts in the Quantum Theory. On the other hand, cohomology groups often serves as state spaces in the constrains' theories, .e.g., in the generalizations of the Batalin-Vilkoviskij theory, so it is desirarable to have a topology on them. Therefore it is convenient to know the topology of the images of differentials in order to assure at least the Hausdorff property of the cohomology groups. For finitely generated K(H)-bundles over compact manifolds and an elliptic operator or an elliptic complex the images of the operators are closed and the cohomology groups are Banach spaces. We explain this result and sketch its proof if the time permits.