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

Bice, Tristan; Koszmider, Piotr

doi:10.1090/proc/13242

A note on the Akemann-Doner and Farah-Wofsey constructions
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by Tristan Bice and Piotr Koszmider PDF

Proc. Amer. Math. Soc. 145 (2017), 681-687

Abstract:

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.

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