Embedding of exact C*-algebras and continuous fields in the Cuntz algebra O_2

Abstract

We prove that any separable exact C*-algebra is isomorphic to a subalgebra of the Cuntz algebra ${\cal O}_2.$ We further prove that if $A$ is a simple separable unital nuclear C*-algebra, then ${\cal O}_2 \otimes A \cong {\cal O}_2,$ and if, in addition, $A$ is purely infinite, then ${\cal O}_{\infty} \otimes A \cong A.$ The embedding of exact C*-algebras in $\OA{2}$ is continuous in the following sense. If $A$ is a continuous field of C*-algebras over a compact manifold or finite CW complex $X$ with fiber $A (x)$ over $x \in X,$ such that the algebra of continuous sections of $A$ is separable and exact, then there is a family of injective homomorphisms $\phi_x : A (x) \to {\cal O}_2$ such that for every continuous section $a$ of $A$ the function $x \mapsto \phi_x (a (x))$ is continuous. Moreover, one can say something about the modulus of continuity of the functions $x \mapsto \phi_x (a (x))$ in terms of the structure of the continuous field. In particular, we show that the continuous field $\theta \mapsto A_{\theta}$ of rotation algebras posesses unital embeddings $\phi_{\theta}$ in ${\cal O}_2$ such that the standard generators $u (\theta)$ and $v (\theta)$ are mapped to $\operatorname{Lip}^{1/2}$ functions.