Actions of 2-groupoids on C*-algebras

点击翻译按钮获取中文摘要

Groupoids describe geometric spaces with singularities and are a central object of study in noncommutative geometry. Several problems in geometry and mathematical physics lead to singularities for which groupoids are inadequate; a typical example are foliations with non-Hausdorff holonomy groupoid. But these situations may be described in the framework of higher category theory using 2-groupoids. The goals of this project are to attach C*-algebras to such 2-groupoids; to study their actions on C*-algebras and crossed products for such actions, and the K-theory of the resulting C*-algebras. The natural notion of action for 2-groupoids is more flexible than the usual one and includes Green’s twisted group actions as a special case. Furthermore, it allows for proper actions on C*-algebras of holonomy 2-groupoids of foliations even if the holonomy groupoid is non-Hausdorff.