Completely positive approximations and inductive systems

TL;DR

This paper introduces C*-encoding systems, a new framework for understanding when inductive limits of C*-algebras are nuclear and completely order isomorphic to C*-algebras, generalizing previous concepts and characterizations.

Contribution

It defines C*-encoding systems that characterize nuclear C*-algebras as inductive limits, extending prior models and simplifying the structural requirements.

Findings

C*-encoding systems are sufficient for limits to be C*-algebras.

For finite-dimensional systems, C*-encoding is also necessary.

Separable nuclear C*-algebras are exactly those that are limits of C*-encoding systems.

Abstract

We consider inductive systems of C*-algebras with completely positive contractive connecting maps. We define a condition, called C*-encoding, which is sufficient for the limit of the system to be completely order isomorphic to a C*-algebra and hence guarantees a unique C*-algebra associated to the limit. When the system consists of finite-dimensional C*-algebras, this condition is also necessary and thus characterizes when the limit is completely order isomorphic to a (nuclear) C*-algebra. C*-encoding systems generalize the NF systems of Blackadar and Kirchberg and the CPC*-systems of the author and Winter. Moreover, any system of completely positive approximations of a nuclear C*-algebra gives rise to a C*-encoding system. Consequently a separable C*-algebra is nuclear if and only if it is completely order isomorphic to the limit of a C*-encoding system. This gives an inductive limit description of all separable nuclear C*-algebras equivalent to the recent construction of the author and Winter but without the additional structure of order zero maps. Without these extra structural requirements, one can easily construct examples of our systems, which we demonstrate for all amenable group C*-algebras.