# Decomposable Approximations Revisited

@article{Brown2016DecomposableAR, title={Decomposable Approximations Revisited}, author={Nathanial P. Brown and Jos'e R. Carri'on and Stuart A. White}, journal={arXiv: Operator Algebras}, year={2016}, pages={45-59} }

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.

#### 9 Citations

STRUCTURE OF NUCLEAR C*-ALGEBRAS: FROM QUASIDIAGONALITY TO CLASSIFICATION AND BACK AGAIN

- Mathematics
- Proceedings of the International Congress of Mathematicians (ICM 2018)
- 2019

I give an overview of recent developments in the structure and classification theory of separable, simple, nuclear C*-algebras. I will in particular focus on the role of quasidiagonality and… Expand

Nuclear dimension of simple stably
projectionless C∗-algebras

- Mathematics
- 2019

We prove that Z-stable, simple, separable, nuclear, non-unital C*-algebras have nuclear dimension at most 1. This completes the equivalence between finite nuclear dimension and Z-stability for… Expand

Quasidiagonal traces on exact $C^\ast$-algebras

- Mathematics
- 2015

Recently, it was proved by Tikuisis, White and Winter that any faithful trace on a separable, nuclear $C^\ast$-algebras in the UCT class is quasidiagonal. Building on their work, we generalise the… Expand

Decomposing nuclear maps

- Mathematics
- 2019

We show that the strengthened version of the completely positive approximation property of Brown, Carrion, and White---where the downward maps are asymptotically order zero and the upward maps are… Expand

O A ] 2 4 A ug 2 02 0 CLASSIFYING MAPS INTO UNIFORM TRACIAL SEQUENCE ALGEBRAS

- 2020

We classify -homomorphisms from nuclearC-algebras into uniform tracial sequence algebras of nuclear Z-stable Calgebras via tracial data. Introduction Over the last 10 years, the application of von… Expand

Almost finiteness, the small boundary property, and uniform property Gamma

- Mathematics
- 2020

We prove that for a free action $\alpha:G\curvearrowright X$ of a countable discrete amenable group on a compact metrizable space, the small boundary property is equivalent to the uniform property… Expand

Covering Dimension of C*-Algebras and
2-Coloured Classification

- Mathematics, Art
- Memoirs of the American Mathematical Society
- 2019

We introduce the concept of finitely coloured equivalence for unital *-homomorphisms between C*-algebras, for which unitary equivalence is the 1-coloured case. We use this notion to classify… Expand

Nuclear dimension of simple $$\mathrm {C}^*$$-algebras

- Mathematics
- Inventiones mathematicae
- 2020

<jats:p>We compute the nuclear dimension of separable, simple, unital, nuclear, <jats:inline-formula><jats:alternatives><jats:tex-math>$${\mathcal {Z}}$$</jats:tex-math><mml:math… Expand

Classifying maps into uniform tracial sequence algebras

- Mathematics
- 2020

We classify $^*$-homomorphisms from nuclear $C^*$-algebras into uniform tracial sequence algebras of nuclear $\mathcal Z$-stable $C^*$-algebras via tracial data.

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Recently, it was proved by Tikuisis, White and Winter that any faithful trace on a separable, nuclear $C^\ast$-algebras in the UCT class is quasidiagonal. Building on their work, we generalise the… Expand