Small masas of the Calkin algebra in the Cohen model

Abstract

We show that maximal abelian C*-subalgebras (masas) of the Calkin algebra (the algebra of all bounded operators on the separable Hilbert space modulo compact operators) may consistently have their densities strictly less than continuum and we describe many isomorphism types of such masas. Specifically, we prove that after adding any number of Cohen reals to a model of CH the algebra $C(K_{\mathcal A})$ of all complex-valued continuous functions on the Stone space $K_{\mathcal A}$ of a Boolean algebra $\mathcal A$ of cardinality $ω_1$ is $*$-isomorphic to a masa of the Calkin algebra if and only if $\mathcal A$ does not admit a countably generated ultrafilter. Moreover, for every such Boolean algebra we obtain $ω_2$ pairwise unitarily non-equivalent such masas, none of which has a commutative lift. We also show in ZFC that if a C*-algebra of the form $C(K)$ for any compact Hausdorff $K$ is $*$-isomorphic to a masa of the Calkin algebra, then no point of $K$ may have character smaller than $\mathfrak p$. Therefore, consistently, there may not be any masa of the Calkin algebra of density less than continuum.