  We remove the assumption of the continuum hypothesis from the Akemann-Doner
construction of a non-separable $C^*$-algebra $A$ with only separable
commutative $C^*$-subalgebras. We also extend a result of Farah and Wofsey's,
constructing $\aleph_1$ 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.
