On masas of the Calkin algebra generated by projections
Piotr Koszmider
TL;DR
This work investigates masas of the Calkin algebra $\mathcal{Q}(\ell_2)$ generated by projections, revealing a large diversity shaped by set theory. Under $CH$ it achieves complete $*$-isomorphic classifications and constructs $C(K)$-type masas for suitable $K$ with no commutative lifts, demonstrating masas with properties beyond the previously known types. In ZFC, it still yields a continuum-sized landscape of pairwise non-$*$-isomorphic masas built from projections via wide families and almost-masa frameworks, highlighting obstructions to liftings and the richness of the Calkin algebra's abelian substructures. Overall, the paper ties topology (Gelfand spaces), Boolean algebras, and lifting phenomena to show how set-theoretic assumptions shape the possible masa types in $\mathcal{Q}(\ell_2)$.
Abstract
Assuming the continuum hypothesis CH, we obtain complete $*$-isomorphic classification of maximal abelian self-adjoint subalgebras (masas) of the Calkin algebra $\mathcal Q(\ell_2)$ (bounded operators on a separable Hilbert space modulo compact operators) generated by projections. In particular, for any compact totally disconnected Hausdorff space $K$ of weight not exceeding the continuum and not admitting $G_δ$ points we construct under CH a masa of $\mathcal Q(\ell_2)$ which is $*$-isomorphic to the algebra $C(K)$ of complex-valued continuous functions on $K$. This, among others, shows that masas of the Calkin algebra could have rather unexpected properties compared to the previously known three $*$-isomorphic types of them generated by projections: $\ell_\infty/c_0$, $L_\infty$ and $\ell_\infty/c_0\oplus L_\infty$. It can be shown that some additional set-theoretic hypothesis, like CH, is necessary for such results. However, without making any additional set-theoretic assumptions we still construct a family of maximal possible cardinality (of the power set of $\mathbb R$) of pairwise non-$*$-isomorphic masas of $\mathcal Q(\ell_2)$ generated by projections and with properties unlike the three above examples.