Local index theory for operators associated with Lie groupoid actions

We develop a local index theory for a class of operators associated with non-proper and non-isometric actions of Lie groupoids on smooth submersions. Such actions imply the existence of a short exact sequence of algebras, relating these operators to their non-commutative symbol. We then compute the connecting map induced by this extension on periodic cyclic cohomology. When cyclic cohomology is localized at appropriate isotropic submanifolds of the groupoid in question, we find that the connecting map is expressed in terms of an explicit Wodzicki-type residue formula, which involves the jets of non-commutative symbols at the fixed-point set of the action.

Smooth crossed product of minimal unique ergodic diffeomorphisms of a manifold and cyclic cohomology

Different diffeomorphisms can give the same [Formula: see text] crossed product algebra. Our main purpose is to show that we can still classify dynamical systems with some appropriate smooth crossed product algebras when their corresponding [Formula: see text] crossed product algebras are isomorphic. For this purpose, we construct two minimal unique ergodic diffeomorphisms [Formula: see text] and [Formula: see text] of [Formula: see text]. The [Formula: see text] algebras classification theory, smooth crossed product algebras considered by R. Nest and cyclic cohomology are used to show that [Formula: see text] and [Formula: see text] give the same [Formula: see text] algebra and induce different smooth crossed product algebras.

Monoidal Categories, 2-Traces, and Cyclic Cohomology

In this paper we show that to a unital associative algebra object (resp. co-unital co-associative co-algebra object) of any abelian monoidal category ( $\mathscr{C},\otimes$ ) endowed with a symmetric 2-trace, i.e., an $F\in \text{Fun}(\mathscr{C},\text{Vec})$ satisfying some natural trace-like conditions, one can attach a cyclic (resp. cocyclic) module, and therefore speak of the (co)cyclic homology of the (co)algebra “with coefficients in $F$ ”. Furthermore, we observe that if $\mathscr{M}$ is a $\mathscr{C}$ -bimodule category and $(F,M)$ is a stable central pair, i.e., $F\in \text{Fun}(\mathscr{M},\text{Vec})$ and $M\in \mathscr{M}$ satisfy certain conditions, then $\mathscr{C}$ acquires a symmetric 2-trace. The dual notions of symmetric 2-contratraces and stable central contrapairs are derived as well. As an application we can recover all Hopf cyclic type (co)homology theories.