Decomposable approximations revisited

TL;DR

This paper revisits approximation properties of nuclear C*-algebras, demonstrating that outgoing maps can be asymptotically order-zero and multiplicative under certain conditions, refining understanding of their approximation behavior.

Contribution

It shows that outgoing maps in nuclear C*-algebras can be chosen to be asymptotically order-zero and multiplicative if the algebra and traces are quasidiagonal, extending previous results.

Findings

Outgoing maps can be asymptotically order-zero

Maps can be asymptotically multiplicative under quasidiagonality

Provides new insights into approximation properties of nuclear C*-algebras

Abstract

Nuclear C*-algebras enjoy a number of approximation properties, most famously the completely positive approximation property. This was recently sharpened to arrange for the incoming maps to be sums of order-zero maps. We show that, in addition, the outgoing maps can be chosen to be asymptotically order-zero. Further these maps can be chosen to be asymptotically multiplicative if and only if the C*-algebra and all its traces are quasidiagonal.