A nonseparable scattered C*-algebra without nonseparable commutative subalgebras

The talk is based on the paper T. Bice, P. Koszmider, A note on the Akemann-Doner and Farah-Wofsey constructions, PAMS 145 (2017), no. 2, 681–687, where we remove the continuum hypothesis from the previous construction by Akemann and Doner of an algebra as in the title (A nonseparable C*-algebra with only separable abelian C*-subalgebras. Bull. London Math. Soc. 11 (1979), no. 3, 279-284). These constructions can be viewed as subalgebras of noncommutative versions of Psi-spaces, that is scattered, separable locally compact spaces of Cantor-Bendixson height 2. The main combinatorial "trick" which we use is to apply the Luzin almost disjoint family.