We study noncommutative ηand ρ-forms for homotopy equivalences. We prove a product formula for them and show that the ρ-forms are well defined on the structure set. We also define an index theoretic map from L-theory to C * -algebraic K-theory and show that it is compatible with the ρ-forms. Our approach, which is based on methods of Hilsum-Skandalis and Piazza-Schick, also yields a unified analytic proof of the homotopy invariance of the higher signature class and of the L 2 -signature for manifolds with boundary.

Abstract. We relate the spectral flow to the index for paths of selfadjoint Breuer-Fredholm operators affiliated to a semifinite von Neumann algebra, generalizing results of Robbin-Salamon and Pushnitski. Then we prove the vanishing of the von Neumann spectral flow for the tangential signature operator of a foliated manifold when the metric is varied. We conclude that the tangential signature of a foliated manifold with boundary does not depend on the metric. In the Appendix we reconsider integral formulas for the spectral flow of paths of bounded operators.

Abstract. We prove a higher Atiyah-Patodi-Singer index theorem for Dirac operators twisted by C * -vector bundles. We use it to derive a general product formula for η-forms and to define and study new ρ-invariants generalizing Lott's higher ρ-form. The higher Atiyah-Patodi-Singer index theorem of Leichtnam-Piazza can be recovered by applying the theorem to Dirac operators twisted by the Mishenko-Fomenko bundle associated to the reduced C * -algebra of the fundamental group.Key words. Atiyah-Patodi-Singer index theorem, higher index theory, Dirac operator, C * -vector bundle.AMS subject classifications. 58J22 (Primary); 58J28, 58J32 (Secondary).Introduction. In noncommutative geometry a compact space X is generalized by a unital C * -algebra A. By applying a noncommutative concept to the commutative C * -algebra A = C(X) one recovers its classical counterpart. An A-vector bundle on a Riemannian manifold M is a locally trivial bundle of projective A-modules. Its classical counterpart is a (complex) vector bundle on M × X. Thus the index theory of a Dirac operator on M twisted by an A-vector bundle is a variant of family index theory, where the base space, encoded by A, is noncommutative. If A = C(X), one obtains a Dirac operator on M twisted by a vector bundle on M × X, which we can consider as a vertical operator on the fiber bundle M ×X → X. In the realm of family index theory this situation is particularly simple since the fiber bundle is trivial and the metric on M does not depend on the parameter.Modelled on the family case, the superconnection formalism has been applied Dirac operators over C * -algebras in [Lo92b][Lo99b]. In the classical case, for the construction of a Bismut superconnection one needs that X is a smooth manifold. In the noncommutative case this is encoded in the choice of a projective system of Banach algebras (A i ) i∈IN0 with A 0 = A, and with injective structure maps whose images are closed under holomorphic functional calculus. By using the de Rham homology for algebras and the Chern character as defined in [K] from the K-theory of A to the de Rham homology of the projective limit A ∞ (which is assumed to be dense in A) one can formulate for closed M an index theorem for Dirac operators twisted by A-vector bundles in analogy to the Atiyah-Singer family index theorem. One gets numerical invariants from the index theorem by pairing the de Rham homology classes with reduced cyclic cocycles.By results of Lott [Lo99b] the Atiyah-Singer index theorem for Dirac operators over C * -algebras can be proven by adapting the superconnection proof of Bismut, as given in [BGV]. While the construction of the heat semigroup is completely analogous to the construction in [BGV], the main difficulty lies in the study of the large time limit of the heat semigroup. In the classical case, where the Dirac operator is a selfadjoint operator on a Hilbert space of L 2 -sections, the exponential decay of the heat semigroup on the complement of the kernel of the Dirac operator can be proven using the positivity of...

Abstract. We introduce a new topology, weaker than the gap topology, on the space of selfadjoint operators affiliated to a semifinite von Neumann algebra. We define the real-valued spectral flow for a continuous path of selfadjoint Breuer-Fredholm operators in terms of a generalization of the winding number. We compare our definition with Phillips' analytical definition and derive integral formulas for the spectral flow for certain paths of unbounded operators with common domain, generalizing those of Carey-Phillips. Furthermore we prove the homotopy invariance of the real-valued index. As an example we consider invariant symmetric elliptic differential operators on Galois coverings.

We define a new topology, weaker than the gap topology, on the space of selfadjoint unbounded operators on a separable Hilbert space. We show that the subspace of selfadjoint Fredholm operators represents the functor K 1 from the category of compact spaces to the category of abelian groups and prove a similar result for K 0 . We define the spectral flow of a continuous path of selfadjoint Fredholm operators generalizing the approach of Booss-Bavnek-Lesch-Phillips.