On R-embeddability of almost disjoint families and some problems concerning C*-algebras

An almost disjoint family A of subsets of N is said to be R-embeddable if there is a function f:N→R such that the sets f[A] are ranges of real sequences converging to distinct reals for distinct A∈A. It is well known that almost disjoint families which have few separations, such as Luzin families, are not R-embeddable. But Luzin families are small. Given a big almost disjoint family can we find its big subfamily which is R-embeddable? Can we control which reals are the limits along the elements of the almost disjoint subfamily? I will present our consistency and independence results answering such questions. One side-product of this research is a new result about real functions in the Sacks model. But our research was motivated by some set-theoretic questions concerning C*-subalgebras of B(ℓ_2) which is a noncommutative analogue of ℘(N). I will present the applications of our combinatorial results in this direction as well as further questions. The talk will be mainly based on a joint paper with Osvaldo Guzman and Michael Hrusak, the link to the preprint is https://arxiv.org/pdf/1901.00517.pdf.