The -pseudodifferential calculus on Galois coverings and a higher Atiyah-Patodi-Singer index theorem
The closed Friedman world model with the initial and final singularities as a non-commutative space
The most elegant definition of singularities in general relativity as b-boundary points, when applied to the closed Friedman world model, leads to the disastrous situation: both the initial and final singularities form the single point of the b-boundary which is not Hausdorff separated from the rest of space-time. We apply Alain Connes' method of non-commutative geometry, defined in terms of a C*-algebra, to this case. It turns out that both the initial and final singularities can be analysed as...
The cyclic homology of
The Frölicher-Nijenhuis bracket in non-commutative differential geometry.
The Local Index Formula in Noncommutative Geometry.
The quantum spheres and their embedding into quantum Minkowski space-time.
The variational approach to the Dirichlet problem in C*-algebras
The aim of this work is to develop the variational approach to the Dirichlet problem for generators of sub-Markovian semigroups on C*-algebras. KMS symmetry and the KMS condition allow the introduction of the notion of weak solution of the Dirichlet problem. We will then show that a unique weak solution always exists and that a generalized maximum principle holds true.
Toeplitz-Berezin quantization and non-commutative differential geometry
In this survey article we describe how the recent work in quantization in multi-variable complex geometry (domains of holomorphy, symmetric domains, tube domains, etc.) leads to interesting results and problems in C*-algebras which can be viewed as examples of the "non-commutative geometry" in the sense of A. Connes. At the same time, one obtains new functional calculi (of pseudodifferential type) with possible applications to partial differential equations and group representations.
Traces on the cone algebra with asymptotics
Trivial noncommutative principal torus bundles
A (smooth) dynamical system with transformation group ⁿ is a triple (A,ⁿ,α), consisting of a unital locally convex algebra A, the n-torus ⁿ and a group homomorphism α: ⁿ → Aut(A), which induces a (smooth) continuous action of ⁿ on A. In this paper we present a new, geometrically oriented approach to the noncommutative geometry of trivial principal ⁿ-bundles based on such dynamical systems, i.e., we call a dynamical system (A,ⁿ,α) a trivial noncommutative principal ⁿ-bundle if each isotypic component...
Twisted spectral triples and covariant differential calculi
Connes and Moscovici recently studied "twisted" spectral triples (A,H,D) in which the commutators [D,a] are replaced by D∘a - σ(a)∘D, where σ is a second representation of A on H. The aim of this note is to point out that this yields representations of arbitrary covariant differential calculi over Hopf algebras in the sense of Woronowicz. For compact quantum groups, H can be completed to a Hilbert space and the calculus is given by bounded operators. At the end, we discuss an explicit example of...