Schatten classes for Hilbert modules over commutative C⁎-algebras

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.