Nuclearity and CPC*-systems

TL;DR

This paper demonstrates how separable nuclear C*-algebras can be constructed as limits of finite-dimensional systems with increasingly orthogonality-preserving maps, enabling a multiplication structure to be defined on the limit.

Contribution

It introduces CPC*-systems, generalizing NF systems beyond quasidiagonality, and shows how to equip the limit with a C*-algebra structure.

Findings

Limits of CPC*-systems form C*-algebras

Maps become more orthogonality preserving

Generalization of NF systems beyond quasidiagonality

Abstract

We write arbitrary separable nuclear C*-algebras as limits of inductive systems of finite-dimensional C*-algebras with completely positive connecting maps. The characteristic feature of such CPC*-systems is that the maps become more and more orthogonality preserving. This condition makes it possible to equip the limit, a priori only an operator space, with a multiplication turning it into a C*-algebra. Our concept generalizes the NF systems of Blackadar and Kirchberg beyond the quasidiagonal case.