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Schatten classes for Hilbert modules over commutative C⁎-algebras
Journal of Functional Analysis ( IF 1.7 ) Pub Date : 2021-04-15 , DOI: 10.1016/j.jfa.2021.109042
Abel B. Stern , Walter D. van Suijlekom

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.



中文翻译:

交换C al-代数上希尔伯特模块的Schatten类

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。

更新日期:2021-04-21
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