For an almost disjoint family of subsets of natural numbers, one can define the associated Ψ-space, which is a locally compact, separable and scattered space of the Cantor-Bendixson height

2. In 1977 Mrowka constructed a maximal almost disjoint family of size continuum for which the associated Ψ-space has the property that it's Cech-Stone compactification and one-point compactification coincide. In a joint work with Piotr Koszmider we constructed a non-commutative

analog of this example. This algebra is a non-stable and non-separable, separably represented AF-algebra which is an essential extension of the algebra K(l_2(c)) of compact operators on a

non-separable Hilbert space of density continuum by the algebra K(l_2) of compact operators on a separable Hilbert space. It also has the property that its multiplier algebra is *-isomorphic to its (minimal) unitization.

It follows from BDF-theory that for every extension of separable stable C*-algebra by K(l_2) has to be stable. Therefore our example shows that the similar statement is false, for non-separable

C*-algebras.