Spectral Flow Argument Localizing an Odd Index Pairing

An odd Fredholm module for a given invertible operator on a Hilbert space is specified by an unbounded so-called Dirac operator with compact resolvent and bounded commutator with the given invertible. Associated with this is an index pairing in terms of a Fredholm operator with Noether index. Here it is shown by a spectral flow argument how this index can be calculated as the signature of a finite dimensional matrix called the spectral localizer.

FREDHOLM MODULES OVER GRAPH -ALGEBRAS

We present two applications of explicit formulas, due to Cuntz and Krieger, for computations in $K$-homology of graph $C^{\ast }$-algebras. We prove that every $K$-homology class for such an algebra is represented by a Fredholm module having finite-rank commutators, and we exhibit generating Fredholm modules for the $K$-homology of quantum lens spaces.

Compact metric spaces, Fredholm modules, and hyperfiniteness

We show that the existence of a finitely summable unbounded Fredholm module (h, D) on a C* algebra A implies the existence of a trace state on A and that no such module exists on the C* algebra of a non amenable discrete group. Both for the needs of non commutative differential geometry and of analysis in infinite dimension we are led to the better notion of the θ-summable Fredholm module.