Noncommutative Fr\'echet Spaces and Unbounded Bivariant K-Theory

TL;DR

This paper develops an abstract framework for higher order smooth systems on $C^*$-algebras within the Baaj-Julg $ ext{KK}$-theory context, advancing the understanding of unbounded bivariant $K$-theory.

Contribution

It introduces a novel abstract approach to higher order smooth systems on $C^*$-algebras in the setting of unbounded $ ext{KK}$-theory, expanding the theoretical foundation.

Findings

Provides a new framework for higher order smooth systems

Extends unbounded $ ext{KK}$-theory to broader contexts

Lays groundwork for future applications in noncommutative geometry

Abstract

In this paper we introduce an abstract approach to higher order smooth systems on $C^*$-algebras in contest of Baaj-Julg picture of $\KK$-theory.