Hilbert spaces over $C^*$-tensor categories, Fredholm modules and cyclic cohomology

TL;DR

This paper develops a framework for constructing Fredholm modules over algebras valued in generalized Hilbert spaces within rigid $C^*$-tensor categories, and introduces cyclic cohomology-based Chern characters that are homotopy-invariant.

Contribution

It extends the theory of Fredholm modules and Chern characters to the setting of $C^*$-tensor categories, providing new tools for noncommutative geometry.

Findings

Constructed Fredholm modules over algebras in $C^*$-tensor categories

Defined Chern characters with good periodicity properties

Proved homotopy invariance of the Chern characters

Abstract

We construct Fredholm modules over an algebra taking values in generalized Hilbert spaces over a rigid $C^*$-tensor category. Using methods of Connes, we obtain Chern characters taking values in cyclic cohomology. These Chern characters are well behaved with respect to the periodicity operator, and depend only on the homotopy class of the Fredholm module.