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 ℵ 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.

We prove a number of fundamental facts about the canonical order on projections in C * -algebras of real rank zero. Specifically, we show that this order is separative and that arbitrary countable collections have equivalent (in terms of their lower bounds) decreasing sequences. Under the further assumption that the order is countably downwards closed, we show how to characterize greatest lower bounds of finite collections of projections, and their existence, using the norm and spectrum of simple product expressions of the projections in question. We also characterize the points at which the canonical homomorphism to the Calkin algebra preserves least upper bounds of countable collections of projections, namely that this occurs precisely when the span of the corresponding subspaces is closed.2010 Mathematics Subject Classification. 47A46.

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We extend the classical Stone duality between zero dimensional compact Hausdorff spaces and Boolean algebras. Specifically, we simultaneously remove the zero dimensionality restriction and extend toétale groupoids, obtaining a duality with an elementary class of inverse semigroups.2010 Mathematics Subject Classification. 03C65, 06E15, 06E75, 06B35, 54D45, 54D70, 54D80.