A note on the Akemann-Doner and Farah-Wofsey constructions

We remove the assumption of the continuum hypothesis from the Akemann-Doner construction of a non-separable $C^{\ast }$-algebra $A$ with only separable commutative $C^{\ast }$-subalgebras. We also extend a result of Farah and Wofsey’s, constructing ${\mathrm{\aleph }}_1$ commuting projections in the Calkin algebra with no commutative lifting. This removes the assumption of the continuum hypothesis from a version of a result of Anderson. Both results are based on Luzin’s almost disjoint family construction.