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.

我们在Abelian上的Hilbert模块上定义可邻接算子的Schatten类 $C^{⁎}$-代数。希尔伯特太空舱延续了许多关键特征。特别是，Schatten类形成了紧凑型操作员的两个方面的理想，并配备了Banach规范和$C^{⁎}$具有预期属性的值跟踪。对于琐碎的希尔伯特束，我们表明可以使用Schatten-norm连续映射从基空间到希尔伯特空间光纤上的Schatten类中，通过纤维轨迹来双目地识别我们的Schatten类算子。作为应用程序，我们介绍$C^{⁎}$值的Fredholm行列式和算子zeta函数，并提出了交换条件下p可加无界Kasparov循环的概念。