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In a previous paper [ 14 ], we have introduced the gauge-equivariant K -theory group $$ K_\mathcal{G}^0 (X)$$ of a bundle πX : X → B endowed with a continuous action of a bundle of compact Lie groups $$ p:\mathcal{G} \to B$$ . These groups are the natural range for the analytic index of a family of gauge-invariant elliptic operators ( i.e. , a family of elliptic operators invariant with respect to the action of a bundle of compact groups). In this paper, we continue our study of gauge-equivariant K -theory. In particular, we introduce and study products, which helps us establish the Thom isomorphism in gauge-equivariant K -theory. Then we construct push-forward maps and define the topological index of a gauge-invariant family.

The aim of this paper is to explain the notion of subspace defined by means of pseudodifferential projection and give its applications in elliptic theory. Such subspaces are indispensable in the theory of well-posed boundary value problems for an arbitrary elliptic operator, including the Dirac operator, which has no classical boundary value problems. Pseudodifferential subspaces can be used to compute the fractional part of the spectral Atiyah73-Patodi— Singer eta invariant, when it defines a homotopy invariant (Gilkey’s problem). Finally, we explain how pseudodifferential subspaces can be used to give an analytic realization of the topological K-group with finite coefficients in terms of elliptic operators. It turns out that all three applications are based on a theory of elliptic operators on closed manifolds acting in subspaces.

We study the homotopy invariants of free cochain complexes and Hilbert complex. This invariants are applied to calculation of exact values of Morse numbers of smooth manifolds.

An overview about C *-algebra bundles with a ℤ-grading is presented, with particular emphasis on classification questions. In particular, we discuss the role of the representable KK ( X ;−,−)-bifunctor introduced by Kasparov. As an application, we consider Cuntz-Pimsner algebras associated with vector bundles, and give a classification in terms of K -theoretical invariants in the case in which the base space is an n -sphere.