On $\mathbb R$-embeddability of almost disjoint families and Akemann–Doner C$^*$-algebras

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: 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.