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

TL;DR

This paper removes the continuum hypothesis assumption from key constructions in $C^*$-algebra theory, extending results on non-separable algebras and projections in the Calkin algebra using Luzin's almost disjoint family.

Contribution

It eliminates the continuum hypothesis requirement from the Akemann-Doner and Farah-Wofsey constructions, broadening their applicability.

Findings

Constructed a non-separable $C^*$-algebra with only separable commutative subalgebras without continuum hypothesis.

Built $eth_1$ commuting projections in the Calkin algebra with no commutative lifting, removing continuum hypothesis constraints.

Extended Anderson's result under weaker set-theoretic assumptions.

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.