Exotic cyclic cohomology classes and Lipschitz algebras

TL;DR

This paper introduces new cyclic cohomology classes for Lipschitz and Hölder algebras that reveal novel geometric invariants, distinguishable from classical smooth function invariants, and connect to higher algebraic K-theory.

Contribution

It develops a new class of Hochschild and cyclic cohomology that pair non-trivially with K-theory, extending noncommutative geometric invariants beyond smooth functions.

Findings

New cyclic cohomology classes for Lipschitz and Hölder algebras

These classes pair non-trivially with higher algebraic K-theory

They vanish on smooth function algebras, highlighting their non-classical nature

Abstract

We study the noncommutative geometry of algebras of Lipschitz continuous and H\"older continuous functions where non-classical and novel differential geometric invariants arise. Indeed, we introduce a new class of Hochschild and cyclic cohomology classes that pair non-trivially with higher algebraic $K$-theory yet vanish when restricted to the algebra of smooth functions. Said cohomology classes provide additional methods to extract numerical invariants from Connes-Karoubi's relative sequence in $K$-theory.