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

We prove that any separable exact C*-algebra is isomorphic to a subalgebra of the Cuntz algebra ${\mathcal{O}}_2\mathrm{.}$ We further prove that if $A$ is a simple separable unital nuclear C*-algebra, then ${\cal O}2 \otimes A \cong {\cal O}2,$andif,inaddition,$A$ispurelyinfinite,then${\cal O}{\infty} \otimes A \cong A.$TheembeddingofexactC\ast -algebrasin$\OA{2}$ 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 ${\varphi }_x:A(x)\to {\mathcal{O}}_2$ such that for every continuous section $a$ of $A$ the function $x\mapsto {\varphi }_x(a(x))$ is continuous. Moreover, one can say something about the modulus of continuity of the functions $x\mapsto {\varphi }_x(a(x))$ in terms of the structure of the continuous field. In particular, we show that the continuous field $\theta \mapsto A{\theta}$ofrotationalgebrasposessesunitalembeddings$\phi_{\theta}$in${\cal O}_2$suchthatthestandardgenerators$u (\theta)$and$v (\theta)$aremappedto$\operatorname{Lip}^{1/2}$ functions.