  An almost disjoint family $\mathcal A$ of subsets of $\mathbb N$ is said to
be $\mathbb R$-embeddable if there is a function $f:\mathbb N\rightarrow
\mathbb R$ such that the sets $f[A]$ are ranges of real sequences converging to
distinct reals for distinct $A\in \mathcal A$. It is well known that almost
disjoint families which have few separations, such as Luzin families, are not
$\mathbb R$-embeddable. We study extraction principles related to $\mathbb
R$-embeddability and separation properties of almost disjoint families of
$\mathbb N$ as well as their limitations. An extraction principle whose
consistency is our main result is: every almost disjoint family of size
continuum contains an $\mathbb R$-embeddable subfamily of size continuum. It is
true in the Sacks model. The Cohen model serves to show that the above
principle does not follow from the fact that every almost disjoint family of
size continuum has two separated subfamilies of size continuum. We also
construct in ZFC an almost disjoint family, where no two uncountable
subfamilies can be separated but always a countable subfamily can be separated
from any disjoint subfamily.
  Using a refinement of the $\mathbb R$-embeddability property called a
controlled $\mathbb R$-embedding property we obtain the following results
concerning Akemann-Doner C*-algebras which are induced by uncountable almost
disjoint families: a) In ZFC there are Akemann-Doner C*-algebras of density
$\mathfrak c$ with no commutative subalgebras of density $\mathfrak c$, b) It
is independent from ZFC whether there is an Akemann-Doner algebra of density
$\mathfrak c$ with no nonseparable commutative subalgebra. This completes an
earlier result that there is in ZFC an Akemann-Doner algebra of density
$\omega_1$ with no nonseparable commutative subalgebra.
