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Abstract

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:

$\bullet$ every almost disjoint family of size $\mathfrak c$ contains an $\mathbb R$-embeddable subfamily of size $\mathfrak c$.

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 $\mathsf{ZFC}$ an almost disjoint family where no two uncountable subfamilies can be separated but every countable subfamily can be separated from any disjoint subfamily.

Using a refinement of the $\mathbb R$-embeddability property called the controlled $\mathbb R$-embedding property we obtain the following results concerning Akemann–Doner C$^*$-algebras which are induced by uncountable almost disjoint families:

$\bullet$ In $\mathsf{ZFC}$ there are Akemann–Doner C$^*$-algebras of density $\mathfrak c$ with no commutative subalgebras of density $\mathfrak c$.

$\bullet$ It is independent from $\mathsf{ZFC}$ whether there is an Akemann–Doner algebra of density $\mathfrak c$ with no non-separable commutative subalgebra.

This completes an earlier result that there is in $\mathsf{ZFC}$ an Akemann–Doner algebra of density $\omega _1$ with no non-separable commutative subalgebra.