The Noncommutative Geometry of Graph C∗C^*-Algebras I: The Index Theorem

We investigate conditions on a graph $C^*$-algebra for the existence of a faithful semifinite trace. Using such a trace and the natural gauge action of the circle on the graph algebra, we construct a smooth $(1,\infty)$-summable semfinite spectral triple. The local index theorem allows us to compute the pairing with $K$-theory. This produces invariants in the $K$-theory of the fixed point algebra, and these are invariants for a finer structure than the isomorphism class of $C^*(E)$.Comment: 33 page

Non-Commutative Vector Bundles for Non-Unital Algebras

We revisit the characterisation of modules over non-unital $C^*$-algebras analogous to modules of sections of vector bundles. A fullness condition on the associated multiplier module characterises a class of modules which closely mirror the commutative case. We also investigate the multiplier-module construction in the context of bi-Hilbertian bimodules, particularly those of finite numerical index and finite Watatani index

Generalised time functions and finiteness of the Lorentzian distance

We show that finiteness of the Lorentzian distance is equivalent to the existence of generalised time functions with gradient uniformly bounded away from light cones. To derive this result we introduce new techniques to construct and manipulate achronal sets. As a consequence of these techniques we obtain a functional description of the Lorentzian distance extending the work of Franco and Moretti.Comment: 22 pages. Some imprecisions clarified compared to first versio

Reconstruction of manifolds in noncommutative geometry

We show that the algebra A of a commutative unital spectral triple (A,H,D) satisfying several additional conditions, slightly stronger than those proposed by Connes, is the algebra of smooth functions on a compact spin manifold.Comment: 67 pages, no figures, Latex; major changes, a new Appendix

Orbifolds are not commutative geometries

In this note we show that the crucial orientation condition for commutative geometries fails for the natural spectral triple of an orbifold M/G.Comment: 6 pages, Latex, no figure

Twisted Cyclic Cohomology and Modular Fredholm Modules

Connes and Cuntz showed in [Comm. Math. Phys. 114 (1988), 515-526] that suitable cyclic cocycles can be represented as Chern characters of finitely summable semifinite Fredholm modules. We show an analogous result in twisted cyclic cohomology using Chern characters of modular Fredholm modules. We present examples of modular Fredholm modules arising from Podle\'s spheres and from $SU_q(2)$