[math/0302310] Hyperbolic group $C^*$-algebras and free-product $C^*$-algebras as compact quantum metric spaces

[math/0302310] Hyperbolic group $C^*$-algebras and free-product $C^*$-algebras as compact quantum metric spaces

An interesting paper on the metric geometry of the "dual absolute value of Dirac"

[math/0512040] A pairing between super Lie-Rinehart and periodic cyclic homology

[math/0512040] A pairing between super Lie-Rinehart and periodic cyclic homology

From the arXiv

Authors: Tomasz Maszczyk,
Comments: 11 pages,
Subj-class: K-Theory and Homology; Mathematical Physics,
MSC-class: Primary 16E40, 17B35, 19K56, Secondary 46L87

We consider a pairing producing various cyclic Hochschild cocycles, which led Alain Connes to cyclic cohomology. We are interested in geometrical meaning and homological properties of this pairing. We define a non-trivial pairing between the homology of a Lie-Rinehart (super-)algebra with coefficients in some partial traces and relative periodic cyclic homology. This pairing generalizes the index formula for summable Fredholm modules, the Connes-Kubo formula for the Hall conductivity and the formula computing the K0-group of a smooth noncommutative torus. It also produces new homological invariants of proper maps contracting each orbit contained in a closed invariant subset in a manifold acted on smoothly by a connected Lie group. Finally we compare it with the characteristic map for the Hopf-cyclic cohomology.