Schatten classes for Hilbert modules over commutative C*-algebras

TL;DR

This paper extends the concept of Schatten classes to adjointable operators on Hilbert modules over abelian C*-algebras, establishing their properties and applications in operator theory and noncommutative geometry.

Contribution

It introduces Schatten classes for Hilbert modules over commutative C*-algebras, generalizing classical results and applying them to Fredholm determinants, zeta functions, and Kasparov cycles.

Findings

Schatten classes form two-sided ideals of compact operators.

Identification of Schatten-class operators with fiberwise Schatten maps.

Introduction of C*-valued Fredholm determinants and zeta functions.

Abstract

We define Schatten classes of adjointable operators on Hilbert modules over abelian $C^*$-algebras. Many key features carry over from the Hilbert space case. In particular, the Schatten classes form two-sided ideals of compact operators and are equipped with a Banach norm and a $C^*$-valued trace with the expected properties. For trivial Hilbert bundles, we show that our Schatten-class operators can be identified bijectively with Schatten-norm-continuous maps from the base space into the Schatten classes on the Hilbert space fiber, with the fiberwise trace. As applications, we introduce the $C^*$-valued Fredholm determinant and operator zeta functions, and propose a notion of $p$-summable unbounded Kasparov cycles in the commutative setting.