Nuclearity and generalized inductive limits

Abstract: One of Alain Connes' seminal results establishes that any semi-discrete (or injective or amenable) von Neumann algebra can be written as a direct limit of dimensional von Neumann algebras. In the C*-setting however, such a concise characterization is not possible: the direct C*-analogue of semi-discreteness is nuclearity, and most nuclear C*-algebras do not arise as the direct limits of finite dimensional C*-algebras. Nonetheless, by generalizing the notion of inductive limits of C*-algebras, Blackadar and Kirchberg were able to characterize quasidiagonal nuclear C*-algebras as those arising as (generalized) inductive limits of finite dimensional C*-algebras. In joint work with Wilhelm Winter, we give a further generalization of this construction, which gives us a complete characterization of separable nuclear C*-algebras as those arising from a (generalized) inductive limit of finite dimensional C*-algebras.

operator algebras

Audience: researchers in the topic

Conference on operator algebras and related topics in Istanbul, 2021