On masas of the Calkin algebra generated by projections

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.